Physics and Combinatorics, Procs of the Nagoya 2000 Intl Workshop

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Proceedings of the Nagoya 2000 International Workshop

Physics and

Combinatorics 2000 !£"*»

Editors

Anatol N. Kirillov and Nadejda Liskova

Physics and

Combinatorics 2000

Proceedings of the Nagoya 2000 International Workshop

Physics — and — Combinatorics 2000 Graduate School of Mathematics, Nagoya University 21-26 August, 2000

Editors

Anatol N. Kirillov Professor of Graduate School of Mathematics, Nagoya University

Nadejda Liskova PhD in Physics and Mathematics

V f e World Scientific wb

Singapore • New Jersey • London • Hong Kong L

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Cover: Illustration by Nadya Kirillova

PHYSICS AND COMBINATORICS Proceedings of the Nagoya 2000 International Workshop Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4642-0

Printed in Singapore by World Scientific Printers

PREFACE This volume contains the Proceedings of the Workshop "Physics and Combinatorics" held at the Graduate School of Mathematics, Nagoya University, Japan, during August 21-26, 2000. The workshop organizing committee consisted of Kazuhiko Aomoto, Fumiyasu Hirashita, Anatol Kirillov, Ryoichi Kobayashi, Akihiro Tsuchiya, and Hiroshi Umemura. This is the second Workshop in a series of workshops with common title "Physics and Combinatorics" which have been held at the Graduate School of Mathematics, Nagoya University. The first one was held in 1999 and happened to be successful. In the preface to the Proceedings of the Workshop "Physics and Combinatorics, Nagoya 1999" (World Scientific Publisher, 2000) we had explained the purpose and ideas behind the Workshop. Here we would like to repeat: "The purpose of the Workshop in Nagoya was to get together a group of scientists actively working in Combinatorics, Representation Theory, Special Functions, Number Theory, and Mathematical Physics to acquaint the participants with some basic results in their fields and discuss existing and possible interactions between the mentioned subjects." The present volume contains contributions on • algebra-geometric approach to the representation theory; • algebraic and tropical combinatorics; • birational representations of affine symmetric group and integrable systems; • Grothendieck polynomials; • (t, g)-analogue of characters of finite dimensional representations of quantum affine algebras; • quantum Teichmiiller theory; • complex reflections groups; • Integrable systems: quantum Cologero-Moser models; • Statistical Physics: Bethe ansatz, exclusion statistics, ... The Workshop "Physics and Combinatorics, Nagoya 2000" appeared to be successful, and we hope both respective researchers and graduate students can find many interesting and useful facts and results in this volume of Proceedings. Organizers would like to take an opportunity and to thank all participants of the Workshop, all contributors to this volume, and anonymous referees for their speedy work. Anatol Kirillov Nadejda Liskova v

CONTENTS

Preface

v

Vanishing Theorems and Character Formulas for the Hilbert Scheme of Points in the Plane M. Haiman

1

Exclusion Statistics and Chiral Partition Function K. Hikami

22

Forrester's Constant Term Conjecture and its ^-analogue J. Kaneko

49

On the Spectrum of Dehn Twists in Quantum Teichmiiller Theory R. Kashaev

63

Introduction to Tropical Combinatorics A. Kirillov

82

Bethe' s States for Generalized XXX and XXI models A. Kirillov andN. Liskova

151

Transition on Grothendieck Polynomials A. Lascoux

164

Tableau Representation for Macdonald's Ninth Variation of Schur Functions J. Nakagawa, M. Noumi, M. Shirakawa and Y. Yamada

180

T-analogue of the ^-characters of Finite Dimensional Representations of Quantum Affine Algebras H. Nakajima

196

VII

viii Generalized Holder's Theorem for Multiple Gamma Function M. Nishizawa

220

Quantum Calogero-Moser Models: Complete Integrability for All the Root Systems R. Sasaki

233

Green Functions Associated to Complex Reflection Groups G(e, 1, n) T. Shoji

281

Simplification of Thermodynamic Bethe-Ansatz Equations M. Takahashi

299

A Birational Representation of Weyl Group, Combinatorial .R-matrix and Discrete Toda Equation Y. Yamada

305

VANISHING T H E O R E M S A N D C H A R A C T E R F O R M U L A S FOR T H E HILBERT SCHEME OF P O I N T S IN T H E P L A N E ( A B B R E V I A T E D VERSION) MARK HAIMAN Dept. of Mathematics, U. G. San Diego La Jolla, CA, 92093-0112 E-mail: [email protected] In an earlier paper, 13 we showed that the Hilbert scheme of points in the plane Hn = Hilb n (t?) can be identified with the Hilbert scheme of regular orbits C2™ //Sn. Using our earlier result and a recent result of Bridgeland, King and Reid,4 we prove vanishing theorems for tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish the conjectured character formula for diagonal harmonics of Garsia and the author.8 In particular we prove that the dimension of the space of diagonal harmonics is (n + 1)"- 1 . This is a preliminary report. We state the main results and outline the proofs. Detailed proofs, a more systematic study of the applications, and a fuller exposition will be given in a future publication.

1

Introduction

In an earlier paper, 13 we showed that the Hilbert scheme of points in the plane Hn = Hilb"(C?) can be identified with the Hilbert scheme of regular orbits C 2 " I/Sn for the action of 5 n permuting the factors in the Cartesian product C2™ = (C2)™. This identification gives rise to two different tautological vector bundles on the Hilbert scheme. From the universal family F C Hn x C2

(1)

"i

Hn we get the usual tautological bundle B of rank n, whose sheaf of sections is 7T*0F, the sheaf of regular functions on F, pushed down to Hn. The Hilbert scheme of S„-orbits also has a universal family Xn

C

(C?n//S„)xC?n

"|

(2)

n

Hn = O //Sn 1

2

giving rise to a bundle P of rank n\ with sections p*Ox„- This bundle P carries an Sn action affording the regular representation on every fiber. Here we give a vanishing theorem for the higher sheaf cohomology of all bundles of the form P ® B®1 on Hn. We also identify the space of global sections of P®B®1 with the coordinate ring R(n, I) of a subspace arrangement Z(n,l) in C 2 " 4 " 2 ', called a polygraph. The polygraph was introduced in ( 13 ), where we used a freeness property of its coordinate ring as the main technical tool to derive other results. Here we shall see more clearly the nature of the link between the polygraph and the Hilbert scheme, and the reason why the former carries geometric information about the latter. The trivial bundle OH„ occurs as a direct summand of P, and the natural ample line bundle 0ff n (l) is the highest exterior power of B. As special cases of our vanishing theorem we therefore recover the previously known vanishing theorems of Danila 9 for the tautological bundle B and of Kumar and Thomsen 14 for the ample line bundles OH„ (k), k > 0. Prom our first vanishing theorem we derive a second, giving higher cohomology vanishing for the bundles P®B®1 over the zero fiber H° in Hn. Again we explicitly identify the space of global sections. For Z = 0 in particular, we find that the space of global sections H°(H°, P) is isomorphic to the ring Rn of coinvariants for the "diagonal" action of Sn on C 2 ", or equivalently to the space of harmonics for that action. The coinvariant ring Rn was the subject of a series of conjectures, presented in ( 10 ), relating its character as a doubly graded 5 n -module to enumerations of various well-known combinatorial objects, such as trees and parking functions. In the spring of 1992, Garsia and the author discussed these conjectures with C. Procesi. Procesi suggested that it might be possible to determine the dimension and character of Rn by identifying it as the space of global sections of a vector bundle on H®. Following this suggestion, the author obtained a formula expressing the doubly graded character of Rn in terms of Macdonald polynomials, assuming the validity of suitable geometric hypotheses. Subsequently, Garsia and the author 8 showed that the combinatorial conjectures would follow from this master formula. Using the results in ( 13 ) together with the second vanishing theorem here, we can now prove the geometric hypotheses needed to justify the formula. As a particular consequence, we obtain the dimension formula dimi? n = (n + l ) n - 1 .

(3)

Another consequence is that the q, ^-Catalan polynomial Cn(q, t) studied in (8) and ( u ) is the Hilbert series of the 5 n -alternating part of Rn, and therefore has non-negative coefficients. This has also been proven very recently by Garsia

3

and Haglund, 7 who gave a remarkably simple combinatorial interpretation of Cn(q,t). Our methods can be applied to show that other expressions related to the character formula for Rn are character formulas for suitable doubly-graded ,Sn-modules, and hence have non-negative coefficients. This establishes some positivity conjectures made in ( 2 ), as will be explained in the expanded version of this report. Now we state our primary vanishing theorem, whose proof is discussed in Section 4. Theorem 1 For all I we have Hi(Hn,P®B®l)

= Q fori>0,

(4)

and H°(Hn,P®B®l)

= R{n,l),

(5)

where R(n, I) is the coordinate ring of the polygraph Z(n, I) C C ? n + 2 / . To properly explain the meaning of the identity in (5) we must review the definition of the polygraph. It was denned in (13) as a certain union of linear subspaces in C 2 n + 2 ' ! but it is better here first to describe it from a Hilbert scheme point of view. Let Z = Xn x Fl/Hn

(6)

be the fiber product over Hn of Xn and I copies of F. Since Xn is a closed subscheme of Hn x C271 and F is a closed subscheme of Hn x C 2 , we have that Z is a closed subscheme of Hn x C 2 " x (C 2 )' = Hn x C 2ri + 2i . The polygraph Z(n, I) is the image of the projection of Z on the factor C 2 " 4 " 2 '. Next we describe Z(n, I) in elementary terms. To each point / £ Hn of the Hilbert scheme there corresponds a subscheme V(I) C C2 of the plane. Counting the points of V{I) with multiplicities we get a 0-cycle a(I) = ^ mjPj of total weight ^ m « = n- We may identify o(I) with a multiset, or unordered n-tuple with possible repetitions, of points in the plane
=C2n/Sn,

(7)

called the Chow morphism. A point of Xn is just a tuple (I, Pi,... , Pn), such that a(I) is equal to the unordered n-tuple [ P i , . . . , P n ] . A point of F is a pair (/, Q) such that Q € V(I). Hence a point of Z is a tuple (I,Pi,...,Pn,Qi,...,Ql)

(8)

4

such that a {I) = [ P i , . . . , P n ] and Qi € { P i , . . . ,Pn} for all 1 < i < I. 2

(9)

2

Projecting on C ™+ ') w e see that Z(n,l) is the set of points ( P i , . . . , P n , Q i , . . . , Qi) € C 2 ^ 2 ' satisfying (9). Given a global regular function on Z(n,l), we may compose it with the projection Z -» Z(n,l) to get a global regular function on Z, which is the same thing as a global section of P ® B®1. Hence we have a natural injective ring homomorphism R(n,l)^H0(Hn,P®B®l)

= O(Z),

(10)

where R(n,l) = 0(Z(n,l)) is the coordinate ring of the polygraph. The meaning of the equal sign in (5) is that this homomorphism is an isomorphism. For the character formula application we will need a vanishing theorem along the lines of Theorem 1, but for the restriction of the tautological bundles to the zero fiber H° =
= 0 fori>0,

(11)

and H°(H°n,P®B®l)=R(n,l)/I,

(12)

where R(n, I) is the polygraph coordinate ring and I — mR(n, I) is the ideal generated by the homogeneous maximal ideal m in the subring C[x, y]Sn C R(n,I). Here C[x,y] = C[xx,yi,... ,xn,yn] is the coordinate ring of<[?n. Again let us make a few clarifying remarks. As explained above, there is a geometrically natural homomorphism R(n, I) -> H°(Hn,P®B®1). Restricting global sections of P ® B®1 from Hn to the zero fiber, we get a homomorphism R(n,l)^H°(H°,P®B®1),

(13)

which a priori might not be surjective. The content of (12) is that this homomorphism is surjective and its kernel is the ideal I. We remark that it is obvious that the ideal I is contained in the kernel of the homomorphism in

5

(13), since it is the ideal of the scheme-theoretic fiber over 0 of the projection Z(n, I) - • S^C 2 . What is not obvious, but is so according to the theorem, is that the kernel is not larger than this. 2

The Bridgeland—King—Reid theorem

We derive our main results using our earlier results on Hilbert schemes and polygraphs, combined with a recent general theorem of Bridgeland, King and Reid concerning Hilbert schemes of orbits M//G such that M//G is what is known as a crepant resolution of singularities of M/G. It is well-known that Hn is a crepant resolution of C 2 " /Sn via the Chow morphism. Therefore, as a consequence of our identification of C 2 " //Sn with Hn, the Bridgeland-KingReid theorem applies in the case M = C2™, G = Sn. Let us describe the relevant set-up and state their theorem. In general, M is a non-singular complex projective variety and G is a finite group of automorphisms of M such that the induced action on the canonical sheaf UJM is locally trivial. This implies that M/G has Gorenstein singularities, with canonical sheaf u) descended from % • In our situation G will act linearly on a complex vector space M, so the condition means that G is a subgroup of SL(M). More specifically, we are interested in the action of Sn on C 2 ". This is a subgroup of SL(2n), since every element w € Sn acts with determinant e(w) = ± 1 on C™, and therefore with determinant e(w)2 = 1 on C? n = C" © C " . For a generically chosen point x 6 M, the stabilizer of x is trivial, so the orbit Gx has \G\ elements. Such an orbit is said to be regular. Each regular orbit corresponds to a point of Hilb' G ' (M), and the Hilbert scheme of G-orbits M//G is defined to be the closure of the set of these points in Hilb' G '(M), with the induced reduced subscheme structure. It is a component of the locus of all points / G Hilb' ' (M) having the property that I is G-invariant and 0(M)/I affords the regular representation of G. We ignore other components of this locus, if any. There is a Chow morphism a:M//G-^M/G

(14)

which restricts to the obvious isomorphism on the open subset of M//G consisting of points corresponding to regular orbits. The Chow morphism is projective, so in the event that M//G is non-singular, it is a resolution of singularities. In that case, if we also have CJM//G — G*WMIG, the resolution is said to be crepant. If M is a vector space this just means that wM//G — O is trivial.

6

Let X C (M//G) x M be the universal family, inherited from the universal family on Hilb | G | (M). We have a G-equivariant commutative diagram X

—£-•

M

{

P[ M//G

(15)

—?—>• MjG.

Bridgeland, King and Reid define a functor $:D(M//G)-*DG(M)

(16)

from the bounded derived category of coherent sheaves on M//G to the bounded derived category of coherent G-equivariant sheaves on M, by the formula $ = Rf*op*.

(17)

Note that p is flat, so we can write p* instead of Lp* here. The Bridgeland-King-Reid theorem has two parts: a criterion for M//G to be a crepant resolution of M/G, and, more importantly for us, a generalized McKay correspondence, expressed in a strong form as an equivalence of derived categories, whenever their criterion holds. Theorem 3 ( 4 ) Suppose that the Chow morphism M//G —• M/G satisfies the following smallness condition: for every d, the locus of points x £ M/G such that dimcr _1 (x) > d has codimension at least 2d— 1. Then M//G is a crepant resolution of singularities of M/G, and the functor $ is an equivalence of categories. Let us apply this to our case, M = C 2 " and G = Sn. Identifying
"1

1

(18)

Hn —2—> S H ? . The Hilbert scheme Hn is non-singular by a theorem of Fogarty,6 and it is known 13 ' 15 that UJH„ = Onn (more generally, Hilb n (M) possesses a nondegenerate symplectic form whenever M does). This shows directly that C2™ //Sn is a crepant resolution of C ? n / 5 n ; we don't need the BridgelandKing-Reid criterion for this. The Bridgeland-King-Reid criterion does hold, however. This follows either from the description of the fibers of the Chow morphism due to Briangon,3 or from the observation in (4) that, conversely

7

to Theorem 3, the criterion holds whenever G preserves a symplectic form on M and M//G is a crepant resolution. Hence we have the following result. Corollary 2.1 The Bridgeland-King-Reid functor $ = Rf* ° p* for the diagram (18) is an equivalence of categories $ : D(Hn)^Ds"(C2n).

(19) 2

Let x , y = xi,yi,... ,xn,yn be the coordinates on C ™. Since C2™ is Sn affine, we may identify D (C2™) with the bounded derived category of Snequivariant finitely generated C[x, y]-modules. Then the functor Rf* is identified with RTxn • Since p is finite and therefore affine, RTxn ° p* is naturally isomorphic to RTHn(P <£>—). This gives us an alternate description of the Bridgeland-King-Reid functor and a corresponding reformulation of Corollary 2.1. Corollary 2.2 The functor $ = RT(P ® —) is an equivalence of categories $:D(Hn) -+Ds»{Cn). Using this we can also reformulate our main theorem. Proposition 2.3 Theorem 1 is equivalent to the identity in D s " ( C 2 n ) $B9,*R{n,l),

(20)

where the isomorphism is given by the map R(n, I) —>• $_B®' obtained by composing the canonical natural transformation T —• RT with the homomorphism R(n,l)^T(P®B®1) in (10). We prove identity (20), and thus Theorem 1, by using the inverse Bridgeland-King-Reid functor * : DSn (C 2 ") -> D(Hn), which also has a simple description in our case. In general, as observed in ( 4 ), the inverse functor $ can be calculated using Grothendieck duality as the right adjoint of $ , given by the formula V = (p*(ux®Lf*-)f.

(21)

We can simplify this using the following result from ( 13 ). Proposition 2.4 The line bundle 0(1) = AnB is the twisting sheaf induced by a natural embedding of Hn as a scheme projective over Sn€? . Writing 0(1) also for its pullback to Xn, we have that Xn is Gorenstein with canonical sheaf "xn = O ( - l ) . We need an extra bit of information not contained in the proposition. There are two possible equivariant Sn actions on 0 x „ ( l ) - One is the trivial action coming from the definition of Oxn(l) as P*OHU{V). The other is the twist of the trivial action by the sign character of Sn. The latter action is the correct one, in the sense that identification wxn — 0{-\) is an 5„-equivariant

8

isomorphism for this action. This can be seen by a careful examination of the proof of Proposition 2.4 given in ( 13 ). Taking this into account, and using the fact that Ox„(—l) is pulled back from Hn, we have the following description of the inverse functor. Proposition 2.5 The inverse of the functor $ in Corollary 2.2 is given by ¥ = 0(-l)®(p„L/'-)e. e

Here — denotes the functor of Sn-alternants, e is the sign representation. 3

(22) e

i.e., A = B.omsn(e,A),

where

Prior results on Hilbert schemes and polygraphs

To derive our vanishing theorem from the Bridgeland-King-Reid isomorphism, we need some results from ( 13 ). First, of course, we need the identification of Hn with C?™//5„, in order to have Theorem 3 apply at all. More importantly, we need the theorem on polygraphs that was the main technical tool in ( 13 ), in order to calculate the inverse isomorphism $ applied to the polygraph coordinate ring R(n,l). For this calculation we also need a proposition describing Xn locally where V(I) is not concentrated at a point. Finally, for the application to character formulas, we need the characters of the fibers of the tautological bundle at certain distinguished points of Hn. We now briefly review all these results. In ( 13 ), we defined the isospectral Hilbert scheme to be the set-theoretic fiber product Xn in the diagram (18), with its induced reduced structure as a subscheme of Hn x C? n . The identification of Hn with C2™ //Sn and of Xn with the universal family over C2™ //Sn then follows once it is shown that the morphism p: Xn -»• Hn in (18) is flat. Since the Hilbert scheme Hn is nonsingular, this is equivalent to Xn being Cohen-Macaulay. What we actually proved in ( 13 ) is the stronger result cited above as Proposition 2.4. Via the projection / : Xn -> C 2 ", the coordinates x , y on C 2 " can be regarded as global functions on Xn. For the geometric argument in ( 13 ), we needed to know that the sheaf of regular functions Oxn is flat as a sheaf of C[y]-modules, a fact which we obtained using a theorem on the coordinate rings of polygraphs. We will restate this theorem for use again here. The polygraph Z(n, I) and its coordinate ring R(n, I) have already been defined in the introduction. Let us rephrase the definition in the form given in ( 13 ). We write x , y , a , b = xi,yi,...

,xn,yn,ai,h,...

,ahbi

(23)

9

for the coordinates on C2n+21. To each function / : { 1 , . . . ,1} ->• { 1 , . . . ,n} there corresponds a linear subspace Wf = V(If)

C C2"^',

where

If = (a* - xf{i),

bt - ym

:l
The polygraph Z(n, I) is the union of these subspaces. Hence its coordinate ring is R(n,l) = C[x,y, a, b]/I(n,l),

where

I(n,l) = f]lf. (25) / The main technical theorem on polygraphs is as follows. Proposition 3.1 The polygraph coordinate ring R(n, I) is a free C[y]-module. For present purposes, we need to strengthen this slightly in two ways. Any automorphism of C2 induces an automorphism of C 2 n + 2 ' ; and the corresponding automorphism of C[x,y,a, b] leaves invariant the defining ideal I(n,l) of Z{n,l). In particular, this is the case for translations in the x-direction, which also leave invariant the ideal (y) and hence I(n,l) + (y). This implies that any of the coordinates #,, a, is a non-zero-divisor in R(n, l)/(y), yielding the following two corollaries. Corollary 3.2 The coordinate ring R(n,l) is a free C[xi,y]-module. Corollary 3.3 The coordinate ring R(n, I) has a free resolution of length n—1 as a C[x,y]-modw/e. The polygraph ring R(n,l) can be defined with any ground ring 5 in place of C, following the same recipe as in (24)-(25). In (13) we showed that Proposition 3.1 is valid in this more general setting. Here we will only need the case when S is a polynomial ring over C. Any automorphism of S[x, y] as an S'-algebra extends to an automorphism of R(n, I), giving the next corollary. Corollary 3.4 Let S be a C-algebra and let y' denote the image of y under some automorphism of S[x,y] as an S-algebra. Then S <8>c R(n,l) is a free % ! , • • • ,y'n]-module. In addition to the results on polygraphs we need a local structure theorem for Xn, which enables us to assume by induction on n that desired geometric results hold locally over the open locus in Hn consisting of points / such that V(I) is not concentrated at a single point. Proposition 3.5 Let Uk Q Xn be the open set consisting of points (I, Pi,... , Pn) for which {Pi,... , P*} and {Pk+i, • • • , Pn) are disjoint. Then Uk is isomorphic to an open set in Xk x Xn~k> in a manner compatible with the projection on C 2 ". The pullback F' = F x Xn/Hn of the universal family to Xn decomposes over Uk as the disjoint union of the pullbacks of the universal families from Hk and J?n-fe-

10

In Section 5, we will make reference to the fixed points of a natural torus action on Hn. The two-dimensional torus group T 2 = (C*) 2

(26)

acts on C2 as the group of 2 x 2 invertible diagonal matrices. This action induces an equivariant action on all schemes and bundles under discussion. The T 2 -action on the Hilbert scheme Hn has finitely many fixed points, namely, the points corresponding to monomial ideals I C C[ar,y]. For such an / , the exponents (r, s) of monomials xrys not in J form the diagram of a partition \i of n. We denote the corresponding fixed point J by 7M. The motivating application for the geometric results in (13) was the proof of the positivity conjecture for Macdonald polynomials via the identification of the Macdonald polynomial H^(z;<7, t) with the character of the fiber of the tautological bundle P at the distinguished point 7M. Proposition 3.6 The character as an Sn x T2-module of the fiber P(Ip) of P at Ip is given by FP(IJ

= Hll(z;q,t),

(27)

in the notation of Section 5 (specifically, T denotes the Frobenius series as defined by (44), and Hft(z;q,t) the transformed Macdonald polynomial in (46)/ 4

Main results

In this section we outline the proofs of Theorems 1 and 2. We begin with Theorem 1. We have a map fl(n,0-+$£®'

(28)

in the derived category DSn{£?n), and by Proposition 2.3, it suffices to show that this map is an isomorphism. Applying the inverse functor * yields a map VR(n,l)-*B®1

(29)

in D(Hn), and we can equally well show that this is an isomorphism. Let C be the third vertex of a distinguished triangle C [ - l ] -»• *R(n, I) -* B®1 -J- C We are to show that C = 0.

(30)

11

We may compute ^R(n,l) as follows. By Corollary 3.3, the C[x,y]algebra R(n,l) has a free resolution of length n — 1. In derived category terminology this means that there is complex of free C[x, y]-modules A. =

>• 0 -+ A n _i -+

> A x - + A 0 -> 0 - > • • •

(31)

quasi-isomorphic to R(n,l). Using the formula for * from Proposition 2.5, we have "9R(n,l) = 0{—1) ® (/>*/* A.) e . Moreover, since p is flat, and since the functor -e is a natural direct summand of the identity functor, this formulation provides us with a locally free resolution of ^R(n,l). As B®1 is a 1 sheaf, the map ^R(n, I) -> B® in (29) is represented by an honest homomorphism of complexes. The object C is represented by the mapping cone of this homomorphism, namely, the complex of locally free sheaves • C 2 -+ Ci -> B®1 -¥ 0,

0 -> Cn -¥ e

(32)

where C» = 0(—1) ® (p»/*Aj_i) . We are to prove that this complex is exact. For this we will make use of the following fundamental result of homological algebra. Proposition 4.1 (Intersection T h e o r e m 1 6 1 7 1 8 ) Let 0 -> Cn -> • • • ->• Ci -> Co —>• 0 be a bounded complex of locally free coherent sheaves on a Noetherian scheme X. Denote by Supp(C) the union of the supports of the homology sheaves Hi(C). Then every component o/Supp(C.) has codimension at most n in X, where n is the length of the complex. In particular, if C. is exact on an open set U C X whose complement has codimension exceeding n, then C. is exact. Let U C Hn be the open set of points / such that V(I) contains at least two distinct points, that is, such that a{I) is not a single point with multiplicity n. Using Proposition 3.5, we can show by induction on n that the map R(n,l) —t $B®1 in (28) restricts to an isomorphism on the open set corresponding to U in C? n . Although the derived category is not a local object, the sheaf operations used to define the functors $ and \£ do localize, so we can conclude that the map ^R(n, I) ->• B®1 in (29) is an isomorphism on U, and hence the mapping cone complex in (32) is exact on U. In other words, the support Supp(C) of the object C is disjoint from U. The complement of U in Hn is isomorphic to C2 x iJ°, so it has dimension ra+1 and codimension n — 1. We are not yet ready to apply Proposition 4.1; we first need to enlarge U to an open set whose complement has codimension n + 1. Let Ux C Hn be the open set consisting of ideals / such that x generates the tautological fiber B(I) = C[x,y]/I as a C-algebra, or equivalently, such that {1, x,... , xn~1} is a basis of B(I). Let Uy be defined similarly, with y

12

in place of x. Our desired open set will be U U Ux U Uy. Its complement is isomorphic to C2 x (H° \ (Ux U [/„)), which has codimension n + 1 by the following lemma. Lemma 4.2 77ie complement H° \ (Ux U C/y) of of Ux U C/y in the zero fiber has dimension n — 3. Proof. Let V denote the complement. We interpret the statement that a locus has negative dimension to mean that it is empty. Then the lemma holds trivially for n = 1, so we can assume n > 2. We consider the decomposition of H® into affine cells as in ( 3 ' 5 ), and show that each cell intersects V in a locus of dimension at most n — 3. There is one open cell, of dimension n — 1. This cell is actually UxnH°, so it is disjoint from V. There is also one cell of dimension n — 2. It has non-empty intersection with Uy, so its intersection with V also has dimension at most n — 3. In fact this intersection has dimension exactly n — 3, since the complement of Uy is the zero locus of a section of the line bundle AnB = Oil). All remaining cells have dimension less than or equal to

n-3.

•

We digress briefly to point out the geometric meaning of this lemma. For / in the zero fiber, the fiber B{I) is an Artin local C-algebra with maximal ideal (x,y). The point / belongs to Ux U Uy if and only if the maximal ideal is principal, that is, B(I) has embedding dimension one, or equivalently, the corresponding closed subscheme V(I) is a subscheme of some smooth curve through the origin in C 2 . In this case I is said to be curvilinear. Lemma 4.2 says that the non-curvilinear locus has codimension two in the zero fiber. All that we now require to complete the proof of Theorem 1 is the following lemma, together with a corresponding version with Uy in place of Ux that clearly follows from it by symmetry. Lemma 4.3 The map ^R(n, I) -> B®1 restricts to an isomorphism on Ux. Outline of proof. The lemma is proven by a calculation in local coordinates on the open set Ux and its preimage U'x in Xn. The calculation has two ingredients. First, using Corollary 3.4, we can show that Lf*R(n, I) reduces to the sheaf f*R(n,l) on U'x. In local coordinates, this sheaf is associated to the algebra 0(U'X) ®c[x,y] R(n,l). The desired result takes the form of an isomorphism between the S^-alternating part of this algebra and another algebra representing the sheaf B®1 on Ux. The isomorphism in question and its inverse can be written down explicitly. • Next we turn to the proof of Theorem 2. As we will see, it follows as a corollary to Theorem 1, once we have an appropriate resolution of the structure sheaf of the zero fiber. Such a resolution was found in ( u ) , and the

13

demonstration that Theorem 1 implies Theorem 2 in the case I = 0 was given in ( 12 ). Here we treat the case where I is arbitrary, and give a somewhat simpler proof using the functorial interpretation. To begin with, we describe the relevant resolution. The tautological sheaf B is a sheaf of OH„ -algebras, the quotient of O ® C[x, y] by the ideal sheaf of the universal family. As such it comes with a canonical homomorphism i: O —> B. Since B is locally free, there is a trace homomorphism r : B —• O sending a section / of B to the trace of multiplication by / . More explicitly, for a section represented by a polynomial f(x,y), we have n

Tf = Y,f{*uVi)-

(33)

»=i

The right hand side here is a symmetric polynomial, an element of C[x, y ] S n , and thus makes sense as a regular function on Hn, via the Chow morphism. Since ^ T ( I ) = 1, we see that ^ T splits the canonical map i, giving a direct sum decomposition B = O © B',

(34)

where B' is the kernel of ^ r . Let Q be the rank-2 free 0Hn-module sheaf Q = 0®c (C 2 )*, where (C 2 )* is the dual vector space of C 2 . We may identify (C 2 )* with the homogeneous component of degree 1 in the polynomial ring C[x, y]. Then the realization of B as a quotient algebra of O ® C[x, y] yields a homomorphism of sheaves of C-modules s: Q —»• B. We have the following characterization of the structure sheaf of the zero fiber. Proposition 4.4 ( u ' 1 2 ) Let J be the sheaf of ideals in B generated by the subsheaf B' and the image of the homomorphism s: Q —» B. Then B/J is isomorphic to OHO as a sheaf of Onn -algebras. The content of this can be rephrased as follows. Let j : B' <-» B be the inclusion homomorphism. We can compose the homomorphism (j © s) 1B : (B' © Q) B -> B ® B with multiplication in B to get a homomorphism A*s ° (0' ® s) ® 1B) : {B' © Q) ® B -* B whose image is the sheaf of ideals J in the proposition. Then we have an exact sequence of sheaves of Onn modules (B'®Q)®B->B-+OHO-*0.

Now SpecB dimensional, fiber H® has equal to the

(35)

is the universal family F, which is Cohen-Macaulay and 2nsince it is flat and finite over the smooth variety Hn. The zero dimension n — 1. The sheaf B' ® Q is locally free of rank n + 1, codimension of the zero fiber in F. Hence the ideal sheaf J is

14

locally a complete intersection ideal, minimally generated by any local basis of B' © Q. It follows that the sequence in (35) extends to a Koszul complex 0->An+1(5'©<9)®£-> • A 2 ( 5 ' © Q) ® B -> ( £ ' © Q) ® B ->• B ->• C»Ho -> 0, (36) which is a locally free resolution of OH° • Let V. be the deleted resolution, that is, the complex in (36), but with the final term (9#o omitted. For every I, the complex of locally free sheaves V.&B®1 is isomorphic in the derived category to 0#o ®B®1. Note that every term in V. ® B®1 is a direct summand of a sum of tensor powers of B. The functor $ is the right derived functor of r ^ „ ° p*, and Theorem 1 implies that the terms in the complex V. B®1 are acyclic for the latter functor. Hence we can compute $(OHO B®1) as TXnp*(V. B®1). This last complex has non-zero terms only in degrees i < 0, while the homology modules H1$(OH° ® B®1) are non-zero only in degrees i > 0. Together these facts imply that $(OHO ® B®1) reduces to a complex concentrated in degree zero, or in other words, to a module, and that the complex Txnp*(V. $(0Ho

® B®1) -> 0. (37)

To complete the calculation of $ ( 0 ^ o ® B®'), we need to make the map explicit. This can be done, and the result is that the image of is the the ideal J C R(n, I + 1) generated by x, y and all expressions of the form 1 "

f(x,y)--ytf(xi,yi),

(38)

for / 6 C[x,y]. Here we have written x,y instead of a, b for the coordinates in R{n,l + 1) = <J>(B ® 5®') corresponding to the generators of the first tensor factor B. We write a, b — a±, b\,... , ai, bi as usual for the coordinates corresponding to generators of B®1. Let I C R(n, I) be the ideal generated by the homogeneous maximal ideal m in the subring C[x,y] S n , as in the statement of Theorem 2. To complete the proof, we show that the inclusion of R(n, I) as the subalgebra of R(n, 1 +1) generated by the variables x, y, a, b induces an isomorphism

£:R(n,l)/I->R(n,l

+ l)/J.

(39)

15

This suffices, as Theorem 2 amounts to the identity $(£>Ho B®1) S R(n, l)/I.

(40)

19

By a theorem of Weyl, the ideal I is generated by the power-sums Ph,k — 127=1 xiVi > for ft + A > 0. If f(x,y) is any polynomial, then the expression in (38) is congruent modulo (x,y) to /(0,0) — £ $^£=i f{xi,Vi)This last quantity is symmetric and vanishes at x = y = 0, so it belongs to 7. By Weyl's theorem, I is generated by such expressions, so we have J = IR(n,l + 1) 4- (x,y). In particular, the inclusion R(n,l) <-»• jR(n,Z + 1) makes I a subset of J, so the homomorphism £ in (39) is well-defined. The variables x, y generate R(n, I +1) as an i?(n, Z)-algebra, and since they vanish modulo J, we see that £ is surjective. To show that £ is injective, we construct its left inverse. The inclusion R(n,l) <-> R(n,l + 1) is split by a trace map ^ T : R(n,l + 1) -t R(n,l), which is a homomorphism of R(n, Z)-modules satisfying ^rf(x,y) = ^ Y%=i f(xt,yi). Since - r is an R(n, Z)-module homomorphism, it carries IR(n, I + 1) into I. The ideal (x, y) C R(n, I + 1) is generated as an R(n, Z)-module by the monomials xhyk with ft + fc > 0. We have ^r{xhyk) = Ph,k, so the trace map carries (x, y) and hence also J into I. Therefore it induces a well-defined homomorphism ^ r : R(n, I +1)/ J —> R(n, l)/I, and the fact that it is a homomorphism of R(n, £)-modules implies that i-r o f = 1. 5

Application to character formulas

In ( 10 ), we presented a series of combinatorial conjectures by the author and others concerning the ring of coinvariants Rn = C[x,y}/I

(41)

for the action of Sn on C 2 ". Here I = mC[x,y] is the ideal generated by the homogeneous maximal ideal m of the subring of invariants C[x, y ] S n . As a doubly-graded 5 n -module, Rn is isomorphic to the space DHn of harmonics for the Sn action on C 2 ", that is, the space of polynomials / ( x , y) annihilated by all 5 n -invariant differential operators p(9x, dy) without constant term. We call DHn the space of diagonal harmonics, because the Sn action on C2™ is the diagonal action on the direct sum of two copies of the natural permutation representation on C". It is convenient to keep track of the character of Rn or any doubly graded SVi-module using its Frobenius series, a notion which combines the notions of Hilbert series and Frobenius characteristic. Let A be the algebra of symmetric functions in variables z = zi,z2,..., and let A n be its homogeneous

16

component of degree n. Recall that the Probenius characteristic is the linear map F: (Sn characters) -» A„ X

sending the irreducible character \ explicit formula f

to the Schur function s\(z).

X - ^ E x(«0Pr(»)(*), wes„

(42) It has the

(43)

wherepT(w)(z) is the power-sum symmetric function indexed by the partition of n corresponding to the decomposition of w into disjoint cycles. Using the Probenius characteristic, one can extract the characters and other information about representations of the symmetric groups Sn from calculations in the algebra of symmetric functions. We define the Frobenius series of a doubly-graded 5„-module A — 0 r s Ar>s to be the symmetric function with coefficients in the ring of formal Laurent series Z[[g, t]][g - 1 ,t - 1 ] rA(z;q,t)

= ^2trq'Fchax(Arit).

(44)

r,s

By construction, the coefficients of TA(z; q, t) are actually in N[[q, £]][ k, for all 1 < k < n. The idea behind the name is this: picture a one-way street with n parking spaces numbered 1 through n. Suppose that n cars arrive in succession, each with a preferred parking space given by f(i) for the i-th car. Each driver proceeds directly to his or her preferred space and parks there, or in the next available space, if the desired space is already taken. The necessary and sufficient condition for everyone to park without being forced to the end of the street is that / is a parking function. The weight of / is the quantity w(f) = J27=i /(*) — *• ** measures the total frustration experienced by the drivers in having to pass up occupied parking spaces.

17

The symmetric group acts on the set P of parking functions by permuting the cars (that is, the domain of / ) and this action preserves the weight. Let A be the graded permutation representation A = © d C P d , where Pd = {/ € P ; w(f) = d}, and let e be the sign representation. We can now state the two conjectured character specializations for the diagonal harmonics. Conjecture 5.1 The Probenius series of the diagonal harmonics DHn, or equivalently of the ring of coinvariants Rn, satisfies (i) FRn&q, 1) = !F(e <8> A)(z;q), as above, and

(ii) q^TRn{z;q,q-')

where A is the parking function

module,

= £|*|=„ '#£-£>«*(*)•

Part (i) of the conjecture describes the singly graded character of Rn with respect to degree in the y variables, ignoring the x-degree. Part (ii) describes the character with respect to the difference between the x and y-degrees, or what is the same, the character of the SL(2)-action on Rn. Ignoring the grading entirely, the character of Rn as an 5„-module is given by the Frobenius characteristic JrRn{z; 1,1). Setting q = 1 in part (i) of Conjecture 5.1 we have the following corollary. Corollary 5.2 The representation of Sn on DHn and Rn is equivalent to e <E> A, and its dimension is therefore the number of parking functions, dim J R n = (n + l ) " - 1 .

(45)

We remark that the permutation action of Sn on parking functions is known to be equivalent to the action of Sn on the finite Abelian group Q/(n + 1)Q, where Q = Z n / Z , Sn acts on Z n by permuting the factors, and the subgroup Z under the fraction bar is the diagonal subgroup generated by ( 1 , 1 , . . . ,1). Since Q is free of rank n—1, Q/(n+l)Q has order ( n + l ) n _ 1 . One can show directly that this description agrees with the q = 1 specialization of part (ii) of the conjecture. It is also interesting to note that Q is the root lattice of type An-\ and Sn is the Weyl group. The reader may consult ( 10 ) for discussion of the extension (or lack of it) to other Weyl groups. In ( 8 ), Garsia and the author conjectured an exact formula for TRn(z; q, t) in terms of Macdonald polynomials and proved that it implies the two specializations in Conjecture 5.1. To state the conjectured formula we use the so-called transformed integral form Macdonald polynomials H„,(z;q,t), 8

12

13

(46)

as defined in ( ), ( ), or ( ). We work in the algebra AQ( 9I< ) of symmetric functions with coefficients in Q(q,t), and define as in ( 2 ' 8 ) a linear operator

18

V on AQ( 9]4 ) whose eigenfunctions are the Macdonald polynomials, by the formula VH„ = pWq^Hp.

(47)

Here fi' denotes the partition conjugate to /i, and n(n) is the statistic «(/*) = £ ( * ~ 1)W-

(48)

i

Our conjectured character formula for the diagonal harmonics is as follows. Conjecture 5.3 We have FRn(z;q,t)

= Ven(z),

(49)

where en{z) is the n-th elementary symmetric function. The motivation for this conjecture was exactly the geometric picture by means of which it will be proven here. At the time, we had to take the geometric facts as unproved assumptions, although the ability of these assumptions to explain Conjecture 5.1 struck us as strong evidence that they must be true. The results of this paper and (13) confirm their validity. Theorem 4 Conjecture 5.3 is true, and consequently Conjecture 5.1 and Corollary 5.2 are true also. Using our preceding results, we can reduce the proof of this theorem to a calculation. The Bridgeland-King-Reid isomorphism is equivariant with respect to the T 2 action, so it induces an isomorphism KT (Hn) —> (£2,1) from t h e Grothendieck group of torus-equivariant coherent Ksnx7 sheaves on H„ to the Grothendieck group of finitely-generated doubly-graded 5 n -equivariant C[x, y]-modules. (Note that T2-equivariance is the same thing as double grading for C[x, y]-modules.) By Theorem 2 in the case I = 0, this isomorphism carries OH" to i?„. If A is any T 2 -equivariant coherent sheaf on Hn, then its cohomology modules Mi = Hl(Hn, A) are finitely-generated doubly-graded C[x, y ] 5 n -modules. As such, each has a Hilbert series %Mi, and we write XA(q,t)=

^(-lYHMiiq,

t)

(50)

i

for the Hilbert series Euler characteristic. The Euler characteristic is additive on exact sequences, so it is well-defined as a functional on objects A in the Grothendieck group KT (Hn). To carry out our calculation, we need the following fundamental result, known as the Atiyah-Bott Lefschetz formula.1 For simplicity we state it in the notation of our particular case.

19

Proposition 5.4 Let H% = {JM : \fi\ = n} be the set of T2-fixed points of Hn. For every T2-equivariant algebraic vector bundle A on Hn, we have

where Cll — AlT*Hn is the i-th exterior power of the cotangent bundle, that is, the bundle whose sections are the regular differential i-forms. The Probenius series FM(z; q, t) is additive on exact sequences and hence well-defined for M e Ks"x7 (C 2 "). Of course TM{z; q, t) is merely a compact notation for the simultaneous Hilbert series of all the doubly-graded modules Homs n (V A , M), as Vx ranges through irreducible representations of Sn- Thus the Atiyah-Bott formula immediately implies the corresponding result with TA in place of HA (in which case we write XTA in place of X-A)By Theorem 2, we have TRn = XT{P®OH°), and w e c a n use the resolution (36) to replace P ® Ofjo with the alternating sum of the vector bundles P®B® Al(B' 0 Q). The denominator in (51) is the product J ] ( l -
Y[ (1 - q-a(xh1+l^)(l

- q1+a^h-l^x)).

(52)

Here x ranges over the cells in the diagram of the partition /x, and the arm a{x) and leg l(x) denote the number of cells strictly east and north of x, respectively, with the diagram drawn in the French manner with the origin at the southwest corner. At I,i we have TiB(I^) = B^(q, t), where

( 53 )

**.(«,*) = J2 ^

Here r and s are the row and column coordinates of a cell in /x, indexed from (0,0) for the cell at the origin. Since B =* B' ® O, we have HB'(I^) = Bn(q, t) — 1, and hence

J2(-l)iU(AiB'(I„))=Il^q,t)=

JJ

(1-tV).

(54)

(r,«)?i(0,0)

We have %Q{I^) = q + t at every point and hence E ( - 1 ) ^ ( A * Q ( / M ) ) = (1 - ?)(1 - *)•

(55)

20

Combining these yields X ) ( - 1 ) ^ ( A « ( B ' 0 $)(!„)) = (1 - q)(l - t)n M («, t).

(56)

i

Putting all this together with Proposition 3.6, we get the explicit formula for TRn in terms of Macdonald polynomials TT>

<7.n rt _ v " (i-g)(i-Wg.*)g,.(g.*)g>.(*;g.*) "^ ' 9 ' ^ " ^ n x € M ( l - g-«(»)tl+'(-))(l - gl+a(*) t -!(x))-

.__. (57>

In ( 8 ), we found the expansion of e n (z) in terms of the Macdonald polynomials Hp(z; q, t) and thereby showed that the expression in (57) is just Ve„(z). This proves Theorem 4. Acknowledgment This research was supported in part by N.S.F. Mathematical Sciences grants DMS-9701218 and DMS-0070772. References 1. M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic differential operators, Bull. Amer. Math. Soc. 72 (1966), 245-250. 2. F. Bergeron, A. M. Garsia, M. Haiman, and G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods and applications of analysis 6 (1999), no. 3, 363-420. 3. Joel Briangon, Description de BilbnC{x,y}, Invent. Math. 41 (1977), no. 1, 45-89. 4. Tom Bridgeland, Alastair King, and Miles Reid, Mukai implies McKay: the McKay correspondence as an equivalence of derived categories, Electronic preprint, arXiv:math.AG/9908027, 1999. 5. Geir Ellingsrud and Stein Arild Str0mme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987), no. 2, 343-352. 6. John Fogarty, Algebraic families on an algebraic surface, Amer. J. Math 90 (1968), 511-521. 7. A. M. Garsia and J. Haglund, A proof of the q,t-Catalan positivity conjecture, Submitted to Discrete Mathematics (September, 2000). 8. A. M. Garsia and M. Haiman, A remarkable q,t-Catalan sequence and q-Lagrange inversion, J. Algebraic Combin. 5 (1996), no. 3, 191-244.

21

9. Danila Gentiana, Sur la cohomologie d'un fibre tautologique sur le schema de Hilbert d'une surface, Electronic preprint, arXiv:math.AG/9904004, 1999. 10. Mark Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17-76. 11. , t,q-Catalan numbers and the Hilbert scheme, Discrete Math. 193 (1998), no. 1-3, 201-224, Selected papers in honor of Adriano Garsia (Taormina, 1994). 12. , Macdonald polynomials and geometry, New perspectives in geometric combinatorics (Billera, Bjorner, Greene, Simion, and Stanley, eds.), MSRI Publications, vol. 38, Cambridge University Press, 1999, pp. 207-254. 13. , Hilbert schemes, polygraphs, and the Macdonald positivity conjecture., Submitted to Journal of the A.M.S. (October, 2000). 14. Shrawan Kumar and Jesper Funch Thomsen, Probenius splitting of Hilbert schemes of points on surfaces, Electronic preprint, arXiv:math.AG/9911181, 1999. 15. Hiraku Nakajima, Lectures on Hilbert schemes of points on surfaces, American Mathematical Society, Providence, RI, 1999. 16. Christian Peskine and Lucien Szpiro, Syzygies et multiplicity, C. R. Acad. Sci. Paris Ser. A 278 (1974), 1421-1424. 17. Paul Roberts, Le theoreme d'intersection, C. R. Acad. Sci. Paris Ser. I Math. 304 (1987), no. 7, 177-180. 18. , Intersection theorems, Commutative algebra (Berkeley, CA, 1987), Springer, New York, 1989, pp. 417-436. 19. Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939.

EXCLUSION STATISTICS A N D CHIRAL PARTITION FUNCTION KAZUHIRO HIKAMI Department of Physics, Graduate School of Science University of Tokyo Hongo 7-3-1, Bunkyo, Tokyo 113-0033, Japan E-mail: [email protected] We consider the statistical mechanics of quasi-particles which obey the exclusion statistics. Based on the recursion relation of the restricted partition function, we study the thermodynamic properties. We also apply this method to the SIM,N supersymmetric system based on the super Yangian symmetry and the path configuration sum.

1

Introduction

In 1991 Haldane introduced the exclusion statistics as quantum fractional statistics which interpolates the boson and fermion *. The exclusion statistics is a generalization of the Pauli exclusion principle, and is defined by SGa = -Y,ga0SNp,

(1.1)

P

where Np is the number of /3-particle, and Ga corresponds to the dimension of the Hilbert space of a-particle. Parameters gap is called the statistical interaction; ga$ = 0 denotes the free bosonic system and gap = 5ap is the free fermionic system. We see that the exclusion statistics differs from a notion of "anyon" which is related to the braid group (see, e.g., Ref. 2 ) . Since Haldane applied the exclusion statistics to the quasi-particles of the fractional quantum Hall effect and of the spin chains, thermodynamics of the exclusion gas has been widely studied 3 ' 4 ' 5 . The connection with the Bethe ansatz was also discussed 6 ' 7 . In the case of the one-component exclusion gas, the partition function was shown to coincide with the partition function of the Calogero model confined in the harmonic potential 8 . This result is consistent with the fact that the idea of the exclusion statistics originates from the eigenfunction of the Calogero-Sutherland model, which is mathematically called the Jack polynomial. On the other hand, based on studies of the character of the conformal field theory, the universal chiral partition function 22

23

(UCPF) is proposed [9];

(u,y\q)=

J2

9- m'

Bm-iA'

rn=0 restrictions Q

na = l y*m"

((l.-B)m+|u)Q 9

1.2)

where (
(*;«)„ = ( l - t ) ( l - * ? ) - - - ( l - t « n - 1 ) ,

(t\q)o = l.

It is conjectured that the UCPF with (1.3) becomes the partition function of the exclusion gas satisfying Eq. (1.1). Our purpose below is to reveal the relationship between the exclusion statistics and the restricted partition function, and to study the thermodynamics of the quasi-particles. This paper is organized as follows. In Section 2 we study the ideal g-on gas in detail. We introduce the restricted partition function, and we shall show that the recursion relation of the restricted partition function gives the thermodynamics of the system without the explicit form of the partition function. In Section 3 we consider a generalization to the multi-component case. We first construct the super Rogers-Szego polynomial in terms of the path configuration sum. As seen from the correspondence with the character of affine Lie algebra for non-super case [10], we can expect that the SRS polynomial gives the character of the affine super Lie algebra. In fact this SRS polynomial coincides with the restricted partition function of the super Yangian invariant spin chains, and this supports our observation. We then apply the recursion relation method to this (super) Yangian invariant system, and study the thermodynamic properties of the quasi-particles. We shall especially give the distribution function and the central charge. The last section is devoted to the concluding remarks. 2 2.1

Ideal
We consider the one-component exclusion gas satisfying 6G

(2.1)

24

where g = 0 and g = 1 respectively denotes the free boson and fermion. We call the particle obeying Eq. (2.1) "g-on", and study the partition function for the ideal g-on gas. For a meanwhile, we suppose that g is a positive integer. We set for brevity Vk = e-M^-ri,

(2.2)

A = e/3w""<\

(2.3)

where (3 is an inverse of the temperature p = -j^-f, and ek is the energy of the fc-th level. Parameters \x and /ihole denote the chemical potentials for a particle and a hole, respectively. We further introduce the restricted partition function ipk', "restricted" means that ifk denotes the partition function when the particle can occupy the energy level only up to the fc-th level from the ground state. We then obtain the partition function by a large fc limit; E = lim
(2.4)

k—>oo

Due to the definition of the exclusion statistics (2.1), we have a constraint for the restricted partition function. To help an understanding, we introduce a kind of "motif". The restricted partition function
v ( r 2 ) = (o)+(i) = x + v0, v4 s=2) = (oo) + (io) + (oi) = A2 + A (V0 + VL), <4 9 = 2 ) = (000) + (100) + (010) + (001) + (101) = A3 + A2 (V0 + Vx + V2) + W0 V2. One easily finds that positions rij of ' 1 ' in motif satisfy a constraint, nj+i -rij>g,

(2.5)

which is a translation of the definition of the exclusion statistics (2.1). See also that the number of states of N particles occupying G states is given by [3] (N-l)(l-g))\ {G + N\(G-gN-(l-g))l

(

'

25

From a condition (2.5), we see that the restricted partition function
1

^ ^ .

(2.7)

Initial conditions for these recursion relation are given by ifo = 1, if i = A + V0,
+ V0 + Vi),

(2.8)

The grand partition function E' 9 ' is thus computed by use of the recursion relation (2.7) in a large k limit (2.4). It should be noted that the recursion relation (2.7) is solved easily for g = 0 and g = 1 cases. We get oo

•.

oo

^^nt-A-i-^).

^

n=0

= Il(A + e-^->), (2.9) n=0

which respectively denote the well-known grand partition functions for the free (chiral) boson and fermion. We thus see that a parameter g indeed intertwines the quantum statistics of the boson and the fermion. We should note that above computation of the partition function of g-on is essentially same with that of the monomer-polymer problem [11]. 2.2

Character for g-on Gas

We shall solve the recursion relation (2.7) for arbitrary g. Hereafter we suppose that the particle has a linear dispersion relation, e{k) = k. Here the momentum k is written as 2?r kj = -y-rij, tor rij G Z>o, Li

where L denotes the size of the system, and the particle is assumed to be rightmoving. By this reason character such as Eq. (1.2) is called chiral partition function. With these assumptions we substitute in Eq. (2.7) Vn=qnx,

(2.10)

where q and the fugacity x are defined by q = exp(-27r / 9/L),

x = exp(/3/i).

(2.11)

26

By setting /^hoie = 0, the recursion relation (2.7) reduces to ^ 1 = ^9)+X9n^92g+1.

(2-12)

To solve this recursion relation with an initial condition (2.8), we introduce the generating function F^9\t) = F^9\t;x,q) by oo

F ^ (*) = $ > < ? > (*,«)*"•

(2-13)

71 = 0

We see that the generating function satisfies the g-difference equation as 9-1

(1 - t) F^{t)

= 1 + x^2qk-1tk

+xq9-H9

F{9)(qt).

(2.14)

k-l

This difference equation is a linear combination of the simplified qdifference equation, (l-t)K(t)

xt2aqbK(qt),

=l +

where a and b are arbitrary parameters. We find that a solution of this equation is given by °°

n 2a

an2 + (b-a)n

K(t) = Y,x t n=0

^

^

Proof is rather straightforward; z °° + (b-a) nan *~~* „ r u 2 a n y

11

K

w = rh + £ ^ 2

(t;q)n+i ,a n 2 + (b+a) n-+-6

~ \ - t

+

q n ST^ „n2an/„J.\2 J^Ly ^ x (qt)

1

XI

1

z£2V

\-t

((1 1

^n =L0

-t)(qt;q)n+i an <J

r

+-r-ftf(«t). 1-i

By use of this result, we find that the g-difference equation (2.14) is solved by S - 1 °°

JT(9)(t) = ^ y " x n + l - * M . t 9 » + f c « fc=0n=0

2 n 2 + ( | _ l + fc)n+fc_l + (Sfci0

—

.

(2.15)

27

As a result we obtain the restricted partition function as [8] 9-1

9)

^ =E

N-(g-l)n-k n

E

fc=0 0 n?7—x-n=0

(2.17)

l 9 , 9 j n

Though g may take only positive integer in the recursion relation (2.7) of the restricted partition function, the character (2.17) is naturally extended to arbitrary non-negative g. This formula exactly coincides with the grand partition function for the Calogero model confined in an external harmonic potential [12], N

d2

wc = E(-s?+«V)+^to-i) E ( ^ ^ j=\

1

l<j
v

•>

K/

(™)

See that Eq. (2.17) supports an equality (1.3). We note that character (2.17) is factorized for several g's. As was pointed out in eqs. (2.9), the (chiral) boson and fermion cases are rewritten respectively as oo

n=0

..

oo

X q

n=0

Due to the Rogers-Ramanujan identity (a = 0,1),

h

fe?)n

~ T }J L (i-? 5 n - i -°)(i-9 5 ' i - 4 + o ) 1 =

V^

fqn(10n+l+2a)

_q(5n+2-a)(2n+l)\

(2.19)

we can interpret that that the g = 2-on gas acts like a fractional boson (only every 2 per 5 energy levels can be occupied). It should be noted that the recursion relation (2.12) with g = 2 was used to prove these RogersRamanujan identities [13]. The character (2.17) as a grand partition function for the Calogero model (2.18) can be constructed based on the Bethe ansatz type equation

28

without using the recursion relation of the restricted partition function. We introduce the Bethe ansatz type equation [14] as N

nj=*i + (fl-l)5Ifl(ni-n').

(2-20)

where 8(x) is a step function; fl,

x>0,

[0,

x<0.

Quantum number Ij takes a positive integer, and tij corresponds to the momentum of the chiral particles. The ground state is given by Ij = j — 1. The total energy is a summation of the momenta, JV

£({/}) = 5 > ,

(2.21)

i=i

and the partition function is given by oo

N <

&\x,q)=Ytx Z g\x,q),

Z%\x,q) =

N=0

£

^ ^ ' •

(2-22)

config. Ij \I\=N

It is easy to see that, by fixing ni < ri2 < • • • < njv, the momenta rij fulfill a condition (2.5) of the exclusion statistics. To compute the partition function from the Bethe ansatz equation (2.20), we rewrite the energy as

£({/}) = Y,"J = £ 4 + & - l ) I > ( n i -"*) = E J i + j

j

hk

l

^N(N-l).

j

As a consequence, we get the partition function for TV particles as 2-N (N-l) J7>0

ijlr>ij which coincides with Eq. (2.17).

{q;q)N

29 2.3

Thermodynamics I

Once we have the explicit form of the partition function (2.17) in a q-series, we can study the physical properties in a thermodynamic limit q —> 1 (or L —> oo in Eq. (2.11)) (see, e.g., Refs. 9,15). This method was originally used to derive the nontrivial identity of the dilogarithm function from the asymptotic behavior of the Andrews-Gordon identity [16]. When we set x = 1 in Eq. (2.17), we have in a limit q —> 1 H^9' ~ / dn exp [ n log x + - n2 log q -

J *

\

2

/ dt

logq J1

1 .

t

J

The saddle point of the integral is fixed by w9 = 1 - to,

(2.24)

where we have set w = qn. With a solution of Eq. (2.24), we have ~(s) ~ exp ( -—*— L(l - w)) , V logq) which gives the effective central charge as c(g) = \L{1-

W).

7T2

(2.25)

Here the function L(z) is the Rogers dilogarithm function defined by

n=l =

_ r iog(i-*)df f Jo

+

1 *

logzlog(1

_ z)>

(2 26)

and satisfies the following identities (see, e.g., Ref. 17 for review); 7T 2

L(x) + L(l - x) = L(l) = —-,

(2.27a)

L(x) + L(y) = L(xy) + L{^—^-) + L(^—^-). (2.27b) 1 — xy 1 — xy We plot the central charge (2.25) as a function of the statistical interaction g in Figure 1. From the identities of the dilogarithm function, we find a duality, cig-1) = 1 - c(g),

(2.28)

30 i 1 1 1

1 1 1

i i 1 1 1 1 1 1 1 1 1 1

0.8

0

0

1 5

I 10

I < 15

20

9 Figure 1. Effective central charge c(g) is plotted for the statistical interaction g.

which represents a particle-hole duality for an ideal g-on gas [5,18]. We also have c(2)

2

2)=5' which supports a result of the Rogers-Ramanujan identity (2.19). 2.4

C (

5'

Thermodynamics II

In the previous section, we have computed the thermodynamics of the ideal g-on gas from the explicit form of the grand partition function E^(x.q). In this section we apply another method; based on the recursion relation of the restricted partition function, we shall derive the central charge without using the explicit form of the partition function. We suppose that the restricted partition function ipk is asymptotically given by
(2.29)

in k —> 00. In a terminology of the inverse scattering method, this function corresponds to the Jost function and the recursion relation can be viewed as the Lax formalism. By substituting this function into the recursion relation (2.7), we see that w satisfies VX9'1

w9 = 1 - Xw,

(2.30)

31

where we have set V = e - ^ - ^ with the energy e. See that the same equation has already appeared in Eq. (2.24) as a saddle point equation. As the restricted partition function tpk is approximately regarded as the partition function H in a large k limit (2.4), we see that an average occupation number (n av ) is given by [19] 1

Pi

"raoMlog^

(2 31)

-

By use of the asymptotic form (2.29) and the algebraic equation (2.30), we get

Figure 2 denotes an average number of particle (n av ) (2.32) for several g. We have (n a v )| _ = 1/g. We can check that the well-known results for the boson (g = 0) and the fermion (g = 1) are recovered from Eq. (2.32); (flav/Bose =

\n&v)Fermi

, y-1 _ i '

— *

, y - \•

We also have a particle-hole duality,
(2-33)

where the average number of hole is computed by ("hole) ^ T ' -K77.

k

7 log^fc-

9(/3/i ho i e )

Based on a distribution function (2.31) we can get the central charge in a simple way. We set A = 1 hereafter. As the energy is given by E = /d£e(nav), the specific heat is r1 i / dV-logw(V).

Cv = -2k^T

(2.34)

*

JO

Recall that w(V) is a solution of the algebraic equation (2.30) with A = 1 which follows from the recursion relation (2.7). As the specific heat Cy is proportional to the central charge, we finally obtain the effective central charge as [20]

c(<7) = - 4 [ dV ± logw(V). 7T Jo

V

(2.35)

32

Figure 2. Average number
This formula gives the same result with Eq. (2.25). We stress that the central charge can be computed only from the recursion relation of the restricted partition function without an explicit form of the grand partition function. 3 3.1

Generalization: Yangian Symmetry Background: Physical Spin Chains

In our previous section we have demonstrated in a case of the ideal <7-on gas that the thermodynamic properties can be derived only from the recursion relation for the restricted partition function. Hereafter we shall try to generalize this strategy for multicomponent cases. A crucial key in the following is the Yangian symmetry [21] and its intimate relationship with the level-1 WZNW theory [22-24]. This relationship has revealed from studies of the onedimensional spin chains with inverse-square interactions whose Hamiltonians are given by WHS

(Zj ~ Zk){zk

E

(Xj -

l<3
Zj Zfz

E

l<j
Xk)2

~ Zj)

'jk-

jki

(3.1a)

(3.1b)

33

Here L denotes the number of spins, and Pjfe is the permutation operator acting on the j - t h and fe-th spins. The former is called the Haldane-Shastry (HS) model [25,26], and a position Zj of the j - t h spin is on a unit circle; Zj = exp(2 7rij/L). The latter is the Polychronakos-Frahm (PF) model [27,28], and a position Xj of spin is the zero point of the i - t h order Hermite polynomial satisfying

Both two Hamiltonians are integrable and Yangian invariant for arbitrary L; explicit form of the Yangian currents are constructed in Refs. 22,29. Remarkable is that the simultaneous eigenstates of these systems can be constructed combinatorially in terms of "motif", a sequence of '0' and ' 1 ' which respectively represents absence or presence of quasi-particles at each energy level [22]. The dispersion relation of the quasi-particle excitations of the HS spin chain (3.1a) is square while that of the PF chain (3.1b) is linear. This fact is consistent with the representation of the Yangian conserved operators in terms of the level-1 current algebra [23,24]. Therein the first conserved operator of the Yangian invariant systems is identified with the Virasoro generator Lo, and consequently we can conclude that the Hamiltonian of the PF spin chain is proportional to Lo in a large L limit, Lo oc limi_ >00 WpF [29]. Furthermore thanks to the linear dispersion relation of the PF chain, the energy spectrum is equidistant, and we can exactly compute the partition function Zi(q) of the PF chain for any L spins. This partition function Z^q) is shown to be related to a generalization of the Rogers-Szego (RS) polynomial, and we have the recursion relation for this restricted partition function Z^q) [29]. This property holds also in the supersymmetric case, i.e., P in the Hamiltonians (3.1) is the supersymmetric permutation operator (see below), and the partition function for the supersymmetric S@M,N PF chain is given as [30] Z?
= T r ^ - « ™ = HlM>N)(x,y;q)

,

(3.2)

X=J/=1

where E0 is a constant and WL ' (x,y; q) is the super Rogers-Szego polynomial (see Appendix for definition). As the PF spin chain (3.1b) preserves the Yangian symmetry for arbitrary L spins, it is natural to conclude that the recursion relation of the SRS polynomial is responsible for the thermodynamics of the quasi-particle excitations. In the following, we shall consider the supersymmetric case based on a result of Ref. 30. We consider the super Lie algebra g(-M%N (see e.g., Ref. 31).

34

We suppose that M > 1. For our convention, we set B = B + U B _ where B+ = {ei,- ••,£*/}, ab

We use E

B _ = {€M+I,£M+2,-

• • ,CM+N}-

as a set of generators of the supersymmetric Lie algebra g(-M,N\

[Eaf> , E c d } = SbcEad — (—i)(p(°)+p(6)Hp(c)+p(d))

sadlEcb

where we assign the parity function p(a) as 0,

for ea € B + ,

1,

foreaeB_.

p(a)

With these generators, the permutation operator P is written as M+N

^(-lfWE'^E1".

P=

o,6=l

The generators E

a6

and the permutation operator P act on bases ea as follows; Eabec = 8bcea, Pea®eb

(-l)p^p^eb^ea.

=

We note that the permutation operator is also represented as [32] *ij=

M+N 2-^i Ci,aCj,bCh<>cj,a,

(3-3)

a,6=1

where Cji0 is a bosonic (resp. fermionic) annihilation operator for e0 € B+ (resp. ea € B_), and we have a constraint for each i; M+N / „ a=l

C

i,a C i,<*

=

*•

The representation (3.3) makes it clear a relationship between the Hamiltonians (3.1) and the supersymmetric t-J model with long-range interactions [33]. We comment on a definition of the super Yangian symmetry. It is defined from the fundamental commutation relation [34],

R12(Zi - 6 ) T ( 6 ) T ( 6 ) = T ( 6 ) T ( 6 ) R12(Zi - 6 ) ,

(3.4)

where R(£) is the rational solution of the graded Yang-Baxter equation [35], R(t)=Z

+ hP.

(3.5)

35

Here h is a parameter. The super Yangian algebra is then constructed from a set of generators Tgb, which are defined by M+N

T ( 0 = J2 (-l)pib)Tab(£)®Eba,

(3.6)

a,6=1

r a 6 (o = sab(-i)p^ 3.2

-ft^r f e a 6 r f c _ 1 -

(3.7)

Path Configuration Sum

We shall show that the SRS polynomial, which plays a role of the restricted partition function (3.2), can be constructed from the path configuration sum. This correspondence was pointed out for the S£M case [36] based on a result of Ref. 29 where a representation of "motif" was given recursively. We define the energy function H : B x B -» {0,1} as [37]

{

0,

if e„ X €;,, or ea = eb € B _ ,

1,

if ea >- £6, or ea = eb e B + .

(3.8)

This is related to the order of the spectral parameter of the supersymmetric solution of the YBE [35]. It is noted that the energy function H has a symmetry, HStM'N (C„, Cfc) = 1 - HUN-" (eM+N-b, tM+N-a)We define a path configuration with finite length; s= (si,S2,...,sL+i),

(3.9)

where S j S B , and we denote the length of s as £(s) — L + 1. The energy and the weight of the path configuration s are defined respectively as t(s)-l

E(s)=

J2 3-H(8j,8j+1),

wt(s) = ^

S j

.

(3.10)

(3.11)

With these notations, we define the character of the path configuration by

ch«SL=

J2

9S(l)ewt(l).

s:e{s)=L+l

(3.12)

36

This path configuration sum first appeared in calculation of the one-point function of the integrable lattice model by the Baxter corner transfer matrix method (see e.g. Ref. 38), and its importance was recognized by the correspondence with the character of the affine Lie algebra [10]. We now give the explicit form of these characters. The character chiS^, can be naturally decomposed by a boundary condition as M+N

chSL=

chS(*\

\_\ a=l

where ch4a)=

Y.

(3-13)

s_-.e(s)=L+i

When

we

introduce

a

set

of

the

^-polynomials

±L

V?\...tf™,b<£\...,b™)by ^ ' • ^ ( i - y i f

(a) ch«SL

,
L(L+1)

we see by definition that x *z,+i

=

a

1

) ,

M

foreaGB+, 1

-/i - W;
foreaGB.

satisfies the recursion relation,

K L '

M

'

dia

g ( x i ' • • •' XM, Vi, • • •>

VN),

where each element of (M + N) x (M + N) matrix M is given by This recursion relation is solved, and we have a generating function for xL follows;

y

bL

L=0

(xy,q)

\^ti>L'

tL =

*a+l

l

I

\j=o+l

\

/

.G(M,N){tl

as

37

Here the function G^M,N^(t) is defined in Eq. (A.14). This set of identities can be derived by identifying the recursion relation of the character b"L\x,y;q) with Eq. (A.17). From these equalities, we can conclude that the character is written in terms of the super Rogers-Szego (SRS) polynomial (A.15), and especially we have

chSJ,1)=giL-.ff

(M,N)

(x, y, q - 1 \

(3.14)

The motif, eigenstates of the super Yangian invariant spin system, can be constructed from this identification of the character with the SRS polynomial which is known to be connected with the partition function of the PF spin chain. We define the motif m = ( m i , . . . , TTIL) from the path s with £(s) = L + l and SL+I = ei

by

rrii = H(susi+i)

€ {0,1}.

(3.15)

The energy (3.10) of the path is rewritten in terms of the motif m as E(m) = '^2 j -rrij.

(3.16)

As the character for this decomposed path is identified with the SRS polynomial (3.14) and the SRS polynomial is represented by Eq. (A.19), we find that the degeneracy of the motif m is given by the skew super Schur function for the border strip (fci + 1, k
k2

We give examples in the following tables. Compare them expression of the SRS polynomial Hi,(x,y;q) in Appendix. border strip m E{m) m E(m) 0 (1,1) 000 0 1 <2> 100 1 010 2 001 3 m E(m) border strip 110 3 00 0 (1,1,1) 101 4 01 2 (1,2) 011 5 10 1 (2,1) 111 6 11 3 (3)

with the explicit border strip (1,1,1,1) (2,1,1) (1,2,1) (1,1,2) (3,1) (2,2) (1,3) (4)

38

3.3

Thermodynamics

We study the thermodynamics of quasi-particles for the super Yangian invariant system. From the fact that the Yangian symmetry preserves for spin chains with arbitrary L spins, the partition function ZL ' \q) of the PF spin chain with finite L spins can be regarded as the restricted partition function
= zM,

z = l-(l-V)w,

(3.17)

where V — exp(—/? (e — /i)) and e is the energy. As was shown in the case of the ideal g-on gas, we can compute the central charge as 6 Z-1 1 = -~2 / d ^ - l o g ^ O 7T Jo

V

= M - l + y.

(3.18)

The average occupation number of the quasi-particle is also given in terms of the solution of Eq. (3.17) by M

=

_

-JW)logw V

" ~iTv

z +

1

T^~z ' M + ^ J V '

(3 19)

'

See that we have the particle-hole duality between S£M,N and S^JV,M cases; {*£»•» (V)) + Ki"-"

(V-1))

= 1.

(3.20)

We list some interesting properties below. 1.

s£lX, We have c = 1/2, and from Eq. (3.19) we get

^ = dhhg{1+n

(3 21)

-

39

We thus see that the theory coincides with the free (chiral) fermion theory. This can also be supported from the character; oo

c h ^ 1 ^ ) = lim H£'l)(x,y;q)

= TT (x + qn y).

L—>oo

(3.22)

•*•-*-

n=0

2.

slM; We have c — M — 1, which exactly coincides with the central charge of the level-1 S£M WZW theory. In fact we recover the character of the level-1 WZW model in the large L limit as [29] ch Af (g) =

lim L=l

q^L2H[Mfi\x;q-').

(3.23)

L—• oo

mod M

The average occupation number (3.19) of the quasi-particle is given by
log ( l + V-h + • • • + V^

.

(3.24)

This distribution function denotes that of the Gentile statistics [39], in which at most M — 1 quasi-particles can occupy the same energy levels though the energy is scaled by 1/M. We have (n a v )| e _ 4 _ 0 0 = (M— 1)/M. 3. s£i. N,

The central charge is computed as c = N/2, which indicates that the theory can be described by the JV sets of the chiral fermions. Indeed it can be checked from the character, N

ch{1>N\q)=

limoHiL1'N)(x,y]q)

oo

= l[(j[(x i=\

+ yiqn)).

(3.25)

n=0

Although, the distribution function of quasi-particles differs from that of the fermion case, and we have, e.g., in N = 2 case,

In**) = ^ - l o g f1 { av ; 4.

3GM H

+ 2

^+ ^/ITW

2

S(.M,M'I

We have the central charge c = \M — 1, and the distribution function of the quasi-particle is given by K v ) = g^~

log ((1 + V*)

(1 + W r + . • • + V^))

,

(3.26)

40

which is a new statistics as a supersymmetric extension of the Gentile statistics (3.24). This result also supports a validity of our results as a supersymmetric algebra. We have (n av )| £ _ > _ 00 = 1. In M = 1 case we recover the free fermion theory (3.21), and in M = 2 case we realize 2 sets of fermions, (n a v 22 ) = 2 (n^ 2 ). We plot an average occupation of states at a temperature T > 0 for cases (M,N < 2) and the free fermion case in Fig. 3. See Ref. 40 for application of these symmetry to the edge states of the quantum Hall states. S#M,N

e-|x Figure 3. Average occupation number of quasi-particles is plotted for some of the super S^M,N cases. A bold line is the distribution function for the free fermion, (ra av ) = V / ( l + V ) , which corresponds to the stiti case. Two solid lines denote those for sli and s^2,2- A dotted line shows for st.2,1 case. A gray solid and dotted line are respectively for the 8^3,3 and st.3 cases. The energy is given in units / 3 _ 1 .

4

Concluding Remarks

We have studied the chiral partition function and the thermodynamics of quasi-particle which obeys the exclusion statistics. In a one-component case, we have computed explicitly the chiral partition function, and established the relationship between the statistical interaction and the chiral partition function. Based on this simple example, we have reviewed how to study the thermodynamic properties only from the recursion relation of the restricted

41

partition function without an explicit form of the partition function. As an application to the multi-component case, we have adopted the super Yangian invariant system; for example we have a spin model such as the supersymmetric extension of the Haldane-Shastry model and the Heisenberg model with infinite spins. The restricted partition function of this system is given in terms of the super Rogers-Szego polynomial, and we have reconstructed the SRS polynomial in terms of the path configuration sum. This representation of the SRS polynomial is shown to give the degeneracy of the motif, which is the eigenstates of the Yangian invariant system. As we have the correspondence between the SIM Rogers-Szego polynomial and the character of the S£M affine Lie algebra, we can expect that our formula is useful in studying the character of the affine super Lie algebra (see Ref. 41 for recent work). Using the recursion relation of the SRS polynomial, we have revealed the thermodynamic properties of quasi-particles in the super Yangian invariant system. We have computed the central charge and the distribution function explicitly. These properties can be applicable for future studies of the physical problems such as the edge states of the fractional quantum Hall states and the disordered system [40,42-44]. Acknowledgement The author would like to thank B. Basu-Mallick for collaboration in Ref. 30. He also thanks A. N. Kirillov and F. Hirashita for kind hospitality during Workshop. Appendix A A .1

Super Polynomial Super Schur Polynomial

We consider the super Schur polynomial associated with the supersymmetric S#M,N Lie algebra. We suppose M > 1. We set B = B + U B _ , where B+ = { c i , . . . , CM},

B _ = {CM+I, £M+2, • • •, CM+AT}.

Two sets B+ and B _ respectively denote the even and odd set. To get the S£M case, we should treat B _ as an empty set; B _ = . We fix an ordering in B as ei

-< £2 -< • • • -< C M + J V -

(A.1)

42

The super Schur function appears in the character for the S£M,N super Lie algebra [31], and it is defined combinatorially as follows [45]. For a given Young diagram A, the Young tableaux T is given by filling a number 1, 2 , . . . , M + N in A by the following rules; • The entries in each row are increasing, allowing the repetition of elements in {i\ti e B+}, but not permitting the repetition of elements in {i|e; e B-}, • The entries in each column are increasing, allowing the repetition of elements in {i\ti € B _ } , but not permitting the repetition of elements in {iheB+}. To define the super Schur polynomial, we set the weight for the Young tableaux T as M+N

wt(T)= Y,

m e

(A-2)

" ">

a=l

where ma is the number of a's in T. In terms of the weight of T, the super Schur polynomial S\(x,y) is defined as Sx(x,y)

=

ewt
£

(A.3)

tableaux T of s h a p e A

where we use e £i , £M

e +*,

for ct £ B + , foreM+iGB_.

Due to the rule of filling a number, the Young diagram A must be contained by the (M,N) hook (see Fig. 4). By this reason the super Schur function S\(x,y) is often called as the hook Schur function. We note that the super Schur polynomial can be rewritten as Sx(x,y)

=det(cAi+i-i)1
where c„ = cn(x, y) is defined by

njLi(i+«yj)

v\

leu-**) «=on

e

(A.4)

43

M

N Figure 4. (M, N) hook Young diagram.

Hereafter we denote sx(x) as the usual Schur function. For our convention we use the elementary polynomial e ^ x ) , complete symmetric polynomial hi(x), and elementary super polynomial EL{X) defined respectively by eL{x) = s{llj]{x),

hL{x) = s[L](x),

EL(x,y)

= S[lL](x,y).

(A.5)

Note that M

M

H(l+tXi)

^ek(x)tk,

=

=1

(A-6)

k=0

M

oo

1

(A.7) fe=0

»=i

IIj=i(l+*!/j)

fc=o

The skew super Schur function Sx/^x,]/) can be also defined in a same way by filling numbers in the skew Young diagram \/fi following the rules before. It is easy to see that we have [46] S\/Ii(x,y)=

]T

sx/„(x)s„,/fi,(y),

(A.9)

where /x' and v' denote partitions conjugate to fi and v respectively. For the super skew Schur function, we have a Jacobi-Trudi type formula, Sx/^(x,y)

=

det(EX'.-^.-i+j{x,y))1^.J^r.

(A.10)

Throughout this paper, we denote ( m i , m 2 , . . . ,mr) as the border strip (or, rim hook, or ribbon), which is the connected skew Young diagram with no

44

2 x 2 square; T mi I (7711,7712,..., m r )

T

(A.11)

=

1

T m3

I

fP For instance, (2,1,4,1,1)

nB. Due to Eq. (A. 10), the skew super Schur

function for ( m i , . . . , m r ) is given by E,m1-\—\-m\ '(mi,.,.,m >(z,2/)

0

1

Em.._,

0

vl.2

•••

(A.12)

:

0

£?„

Super Rogers-Szego Polynomial

We define the super Rogers-Szego polynomial HL '

(x,y;q) as [30]

HiM'N\x,y,q)

£

(q;q)i I\fLl(l^)ai

M

N

y±\bj q

n^r-n^-n(-f) _1

_1

X

• nJLl(9 ;9 )6,- » =^

7= 1

a,i>0,bj>0

(A.13) (q;q)i 7 ; T-rM+i\ , . % •yM + N a.—L l l t = l W9,)a;

The SRS polynomial has a generating function; when we set G^N\t;x,y;q) by G ( M l i V ) (') = ^ A T Ttx: > ~l . . Ui=i{ ^q)oo-Uj=i(-tyjq~1>q

^-r—. *)

G^M'N\t)

(A- 14 )

45

we have oo

G{M,N),t)

y

=

TT(M,N),

H

L

\

(WD . tL

( A 1 5 )

It is straightforward to see that the function G^M,N^(t) satisfies the qdifference equation, I N

\

/M

\

• &M
n ( l +tVj)

CSM'N\t),

(A.16)

which gives the recursion relation of the SRS polynomial as

HiM>N\x,y;q) max(M,N)

=

.

.

(-l)k-l{^^-(ek(x)-(-l)kqL-kek(y)).HiMJkN\x,y^

E fe=l

KHiHjL-k

(A.17) Here we have used the generating function of the elementary polynomials (A.6). Recalling Eq. (A.8), we can rewrite the g-difference equation (A.16) as

G^N\qt)

(Y,{-l)LEL{x,y)A.GlM>N\t),

= \L=0

/

and we obtain another type of recursion relation of the SRS polynomial; HiM'N\x,y-,q)^J2(-ir^^^.Ek(x,y).H^f\x,y;q). fc=l

(A.18)

{QiQjL-k

As was proved in Ref. 30 the SRS polynomial can be rewritten as H^N\x,y;q)

=E

?* L(L+1) - Er -* (mi + - m ' , -S(m l .....m P >(»,»),

E

(A.19)

mi>l

where S(mil...,„»,.) (a:, y) is the skew super Schur polynomial (A.12). The essential point of the proof of above equality is that the skew super Schur polynomial (A. 12) fulfills r S(mi,...,mr)(x,y)

= E(

_ 1

)

t + 1

E

rnr+--+m,-i+l(x,y)

' 5 ^ ,

m ,._,)(2:,t/).

46

Then we see that the expression (A. 19) also satisfies the recursion relation (A. 17). We give some explicit form of the SRS polynomials below; H0(x,y;q)

= 1,

Hi(x,y;q)

=

Sa(x,y),

#2(z, y;q) =q- Sr\(x, y) + SUJ(X, y),

H3(x,y;q)

= q3 • Sr\(x,y) + q2 • Srp(x,y)+q-

Hi{x, y; q)=q6-

Sn{x, y) + q5 • Srp(x,y) + Q3 • Srrn(x,y)

Sr\(x,y)

+

Srm(x,y),

+ q4 • Sn(x, y) + q3 • S n(z, y)

+ q2 • Srp(x,y)+q-

S

yjx,y)

+ Srrm(x,y). A.3

(q,t)-Deformation

of the Super Schur Polynomial

In the S£M case, the Rogers-Szego polynomial H(L (x;q) coincides with a limit t —> 0 of the Macdonald polynomial with one-row Young diagram P[L](x;q,t). It is therefore natural to define the super Macdonald polynomial for the one-row Young diagram by a ^-deformation of the SRS polynomial TT(M,N),

•,

n (**,;„)„ -11 (-zyjq-WU itzx^q)^

YJ (-tzyjq-1^-1)^

i=1

_ ^

(t;q)L

~ h ^ ^

p(M,N), {L]

,

(X V q,t) Z

'' '

L

'

(A.20) This polynomial has a following property; P^'N)(x,y;q,t) = ] T U [ L ] , A • Sx(x,y) <—> P[L]{x;q,t) A

= ^ U [ L ] , A • sx(x). A

(A.21) Further studies should be required on these polynomials. References 1. F. D. M. Haldane: Phys. Rev. Lett. 67, 937 (1991). 2. A. Shapere and F. Wilczek, eds.: Geometric Phases in Physics, Advanced Series in Mathematical Physics 5 (World Scientific, Singapore, 1989).

47

3. 4. 5. 6.

7. 8. 9.

10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

Y.-S. Wu: Phys. Rev. Lett. 73, 922 (1994). S. B. Isakov: Phys. Rev. Lett. 73, 2150 (1994). C. Nayak and F. Wilczek: Phys. Rev. Lett. 73, 2740 (1994). D. Bernard and Y.-S. Wu: In New Developments of Integrable Systems and Long-Ranged Interaction Models (Edited by M.-L. Ge and Y.-S. Wu) (World Scientific, Singapore, 1995) pp. 10-20. T. Fukui and N. Kawakami: Phys. Rev. B 51, 5239 (1995). K. Hikami: Phys. Lett. A 205, 364 (1995). A. Berkovich and B. M. McCoy: In Statistical Physics on the Eve of the 21st Century (Edited by M. T. Batchelor and L. T. Wille) (World Scientific, Singapore, 1999) pp. 240-256. E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado: Lett. Math. Phys. 17, 69 (1987). O. J. Heilmann and E. H. Lieb: Phys. Rev. Lett. 24, 1412 (1970). F. Calogero: J. Math. Phys. 12, 419 (1971). G. E. Andrews: q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra (AMS, Providence, 1986). N. Kawakami: J. Phys. Soc. Jpn 62, 4163 (1993). W. Nahm, A. Recknagel, and M. Terhoeven: Mod. Phys. Lett. A 8, 1835 (1993). B. Richmond and G. Szekeres: J. Austral. Math. Soc. (Sereis A) 31, 362 (1981). A. N. Kirillov: Prog. Theor. Phys. Suppl. 118, 61 (1995). A. K. Rajagopal: Phys. Rev. Lett. 74, 1048 (1995). K. Hikami: Phys. Rev. Lett. 80, 4374 (1998). P. Bouwknegt and K. Schoutens: Nucl. Phys. B 547, 501 (1999). V. G. Drinfeld: Sov. Math. Dokl. 36, 212 (1988). F. D. M. Haldane, Z. N. C. Ha, J. C. Talstra, D. Bernard, and V. Pasquier: Phys. Rev. Lett. 69, 2021 (1992). D. Bernard, V. Pasquier, and D. Serban: Nucl. Phys. B 428, 612 (1994). K. Schoutens: Phys. Lett. B 331, 335 (1994). F. D. M. Haldane: Phys. Rev. Lett. 60, 635 (1988). B. S. Shastry: Phys. Rev. Lett. 60, 639 (1988). A. P. Polychronakos: Phys. Rev. Lett. 70, 2329 (1993). H. Frahm: J. Phys. A: Math. Gen. 26, L473 (1993). K. Hikami: Nucl. Phys. B 441, 530 (1995); J. Phys. Soc. Jpn. 64, 1047 (1995). K. Hikami and B. Basu-Mallick: Nucl. Phys. B 566, 511 (2000). V. G. Kac: Commun. Math. Phys. 53, 31 (1977).

48

32. F. D. M. Haldane: In Correlation Effects in Low Dimensional Electron Systems (Edited by A. Okiji and N. Kawakami) (Springer, Berlin, 1994) pp. 3-20. 33. Y. Kuramoto and H. Yokoyama: Phys. Rev. Lett. 67, 1338 (1991). 34. M. L. Nazarov: Lett. Math. Phys. 21, 123 (1991). 35. P. P. Kulish and E. K. Sklyanin: J. Sov. Math. 19, 1596 (1982). 36. A. N. Kirillov, A. Kuniba, and T. Nakanishi: Commun. Math. Phys. 185, 441 (1997). 37. K. Hikami and R. Inoue: J. Phys. A: Math. Gen. 33, 4081 (2000). 38. R. J. Baxter: Exactly Solved Models in Statistical Mechanics (Academic Press, London, 1982). 39. G. Gentile: Nuovo Cim. 17, 493 (1940). 40. K. Hikami: Phys. Rev. B 6 1 , 4473 (1999). 41. V. G. Kac and M. Wakimoto: math-ph/0006007 . 42. R. A. J. van Elburg and K. Schoutens: Phys. Rev. B 58, 15704 (1998). 43. S. B. Isakov, T. Martin, and S. Ouvry: Phys. Rev. Lett. 80, 580 (1999). 44. L. G. C. Rego and G. Kirczenow: Phys. Rev. B 59, 13080 (1999). 45. A. Berele and A. Regev: Adv. Math. 64, 118 (1987). 46. A. Molev: In Kirillov's Seminar on Representation Theory (Edited by G. I. Olshanski) (Amer. Math. Soc, Providence, 1998), AMS Translations Series 2 181, pp. 109-137.

FORRESTER'S C O N S T A N T T E R M C O N J E C T U R E A N D ITS g-ANALOGUE JYOICHI KANEKO Graduate School of Mathematics, Kyushu University Ropponmatsu Fukuoka 810-8560, Japan E-mail: kaneko @math. kyushu-u. ac.jp We describe our approach to the Forrester's conjectured constant term identity which is equivalent to a new kind of generalization of the Selberg integral. We also deal with the ^-analogue of the conjecture in a parallel way and give a simple proof in the case Ni = 1

1

Introduction

Suppose a, b, k 6 Z>o and let N

H(x1,...,xN;a,b,k)

N

„ a Xl) (l

:= J ] ( l 1 - ,

- - ) 6 f [ ( l - -)*'. •

The Morris constant term identity

13

CT{x}H(x1,...,xN;a,b,k)

£

/

.

.

.

(1.1)

Xj

states that =MN(a,b,k)

(1.2)

where CT{X} denotes the constant term in the Laurent polynomial and TT1 r(o + 6 + l + fcpr(l+ &(/ + !))

** ( uu\ MNia b k) :

> > = 2 r(a + i +fcor(6+ i + *or(i + *)-

(L3)

In 4 , after some numerical computations, Forrester suggested the following remarkable generalization of this identity and proved the special cases a = 6 = 0 (general k, No,Ni) and k = 1 (general a, b, N0,N±). Conjecture N

CT{x}

JJ i,j=N0+l,i^j

(l-^;)H(xi,...,xN;a,b,k)

=

MNo{a,b,k)

J

(1.4) 1

"ij X

M

(j + l)r((fc + l)j + a + b + kN0 + l)r((fc + l)(j + 1) + kN0) T k

(

+ 1 ) r ( ( f c + 1)j

' + a + kNo + l)r((fc + l ) j + b + kN0 + 1)' 49

50

where N = N0 + Ni. Subsequently a (/-analogue of the conjecture was formulated by Baker and Forrester * where several cases including the N± = 2 case were proved. All these proofs are strongly of combinatorial nature. It should be noted here that Forrester formulated a more general constant term conjecture associated to a partition N = N0 + Nx H h Nm, on which not much has been known. In this paper we shall briefly describe our approach to the conjecture invoking the integration formula of Jack polynomials in 9 ' 1 0 . We also deal with the (/-analogue in a parallel way and give a simple proof in the case JVi =2. 2

Forrester's conjecture

It is known evaluation.

4

that the conjecture (1.4) is equivalent to the following integral N

f

1

N

n*?"^ -'*)*- n *jNi+i x

1

II

tti-ti?

N0+l
=

X

(ti-tj)2kdh---dtN

II l
^ f r 1 T(v + kl)T(x + HN-1l))T(l + k(l + 1)) AA T(x + y + k(N - 1 + / ) ) r ( l + k) "yr1 (j + l)r(y + (fc + 1) j + kN0))T(x + k(N-l~N0H V(x + y + k(N-l + N0+j)+j) y

j) - j)

r((fc + l)(j + l) + fcJVo)

AA

r(i + *)

•

(2J)

This is clearly equivalent to the symmetrized form:

/ n frNi{i-u)y-ifN,N0(tu-.-,tN) r

J

N

[0A]N f=l

n (u-tj)2kdt1---dtN

l
ATI

NolNil

(RHSof(2.1)),

(2.2)

51

where fN,N0(h,-..,tN)

••=

(*«i - - ' ^A,,, )

22

1_1

l
(2.3)

x

n

{th-thf-

ji<J2,{ji,J2}C{l,...,JV}\{ii,...,Jjv 0 }

We rewrite (2.3) by invoking on the integration formula of Jack polynomials. Let A = ( A i , . . . , XN) be a partition of length < N and denote by |A| its weight: |A| = Ai + • • • + AJV- For a parameter a, define

(a)la)=n(°-^i-1)^

(2-4)

j=l

where (a) n = a(a + 1) • • • (a + n — 1) and (a)o = 1. Let J^* (t) be the Jack polynomial corresponding to the partition A with the parameter a 12>14. They form a basis o f symmetric polynomials in t\,..., £/v with coefficients in Q(a), the rational function field in a. Moreover we have 6 ' 7 :

r

/

N

^ w l l f ^-'O"-1

IT

(u-tjfkdt1---dtN (2.5)

= . H , ! ^ ^ , , , , ) ^ ) ^ ^ - ^ ^

+ y)[i:)

(2k(N-l)+x where N

fj

r ( l + fa-)r(a; +

fc(t-l))r(y

+

fc(t-l))

i=l

Suppose the symmetric polynomial polynomials:

/JV,JV0 (i)

is expressed in terms of Jack

/W*) = £ ^ J ? W -

(2-7)

x Here we notice that the same kind of expansion, namely, the square of the difference product in terms of Schur polynomials, was extensively investigated in 3 . Substituting (2.7) into (2.2), we see that the integral formula (2.2) is

52

equivalent to

y

CA

(kN)[i](k(N -i) + x-Nl + 1)P

X

x

(2k(N - 1) + x + y - JVx + l)[i]

= kN(Nl-i)N}_

JVo_1

h

N0\ JA

' - Ar + ' 1Hfc^-i-OK-i (X-NX + (a: + i, - JVx + 1 + ^ - 1 + 0 ) ^ - 1 (2.8)

x

" j y 1 (x - Nt + 1 +fe(JVi- 1 - j ) ) M - i - i ( l / + *M> + j))> j=o (a; + y - iVi + 1 +fc(iV+ AT0 + j - 1 ) ) J V 1 + J - I JVi-l

x J J ( n - f c ^ o + J + l)),-. Now define partitions K, /i, ^ by No+l

K = {2N! - 2,2JVi - 3 , . . . , JVi, ^ - 1 , . . . ,Ni - 1), JVo+l

(2.9)

/x = ( J V i - l , . ? . , J V i - l , JVi - 2 , . . . , 1), i/ = ( J V 1 - l , ^ 1 - 2 , . . . , l ) . and write Cx = Cx{kN)[i\x

= x-N1 + l + k{N-l),Y

=x+y - ^ + 1 +

21*^-1).

Then it is straightforward to rewrite (2.8) to get

£^{X)\\{Y)<1 = C W ^ -

rf\

(2-10)

where CN,No = fe^^-i) _w [ ^ ( i +fc(JV0+ J + 1)),. We verified (2.10) in the cases Ni = 2 , 3 by calculating C\'s 9 , and the case N\ = N - 1 by using the Chu-Vandermonde formula for the generalized binomial coefficients 10 , but it still seems difficult to give a full proof and we content ourselves with showing a partial factorization associated to the first column of v (Theorem 2.1 below).

53

One needs to prove the LHS of (2.10) is actually a polynomial and is divisible by (Y — X)vk , i.e., the LHS vanishes when one substitutes X = Y-(il)fc + j - l for all (i,j) with 1 < i < Nx - 1,1 < j < Vi = iVi - i:

Ec -(Idizlrf = 0.

(2.n)

The case i = j = 1 of (2.11) is clear because substituting ti — • • • = t^ — 1 in (2.7) yields £ ) A ^ A = 0 in view of (2.13) below. We can rewrite the equality (2.11) by using the Chu-Vandermonde formula for the generalized binomial coefficients which we now recall. The generalized binomial coefficient (*) is defined by

j < " > ( i + t l t . . . , 14- tn) = Ji°\i») E (*) 4a) £;;; n : tn) „ W where J A a ) ( l n ) := J J rem 5.4]:

o )

J^ '{ln)

(

^

( W ) has the following explicit formula [14, Theo-

4 0 ) d B ) = « |A| (S)l a) -

(2-13)

It holds that (A) is independent of the dimension n in the definition (2.12) and ( ) = 0 unless /x C A [7, Theorem 3], where (j, C X means //, < Xi for every i. The Chu-Vandermonde formula is ([14, Theorem 12.1]):

E ( _ 1 ) M < # ^ = !t#

(2.14)

This follows from change of variables t, with 1 — i i ; 1 < i < N in (2.5) and the definition of the generalized binomial coefficients. We apply this formula to the left-hand side of (2.11):

(r-(t-i)fc + j - i ) y _ ^ ,

Miii-yk-j

+ ijUfX

so that (2.11) is equivalent to

s-^^(g*e))-* «,15,

54

(-) where, since ((i - l)fc — j + l)Tk = 0 for a partition r of r» > JVi - i, the summation is taken over r with Ti < Ni — i. We observe that the sum — — (1) r bein fixed is t h e 1 J T,x->T C\ it) ( S ) coefficient of fc-^W" ) ^ (*i--M i n the expansion of /JV,JV0(1 + * i , • . . , 1 + i/v) in terms of Jack polynomials. By this rewriting, it is now easy to show Theorem 2.1 The equality (2.11) holds in the cases 1 < i < Ni — l,j = 1. Proof. Let fm(h,... ,tu) be the homogeneous part of degree m of the inhomogeneous polynomial /JV,.IV0(1 + *i, • • •, 1 + tN), so that fm(h,. • •, tN) is identically zero if m < Ni(Nx - 1). Let fm(tl,...,tN)

= Y^TdTJr

(ti,S,tN)

( 2 - 16 )

= E ^ft-m^W".'") For the proof of (2.15) with j = 1 it suffices to show that

E

«i^5

= 0.

(2.17,

(1)

Since ((i — l)&)rfc = 0 when Tj > 0, the summation is over the partitions with n = 0. Substituting U = • • • = £/v = 0 in (2.16) gives fm(tl, • • • ,ti-i,0,

Since nAr0+i
_

. . . ,0) = 2_^dTJTk (*!,...

,U-i).

^i)2 *s translation invariant, we see

/TO(ti,...,«<_!,0,...,0)=0, provided i < iVi — 1, and this implies that dT = 0 if Tj = 0 . We have thus proved the theorem. • We note that our proof of the conjecture in the case Ni = N — 1 10 was carried out by comparison of expansion of fm (t\,..., £JV) in terms of monomial symmetric polynomials and in terms of Jack polynomials.

55

3

q—analogue

Fix q with 0 < q < 1 and set (x)oo = (x;q)oo = IliSo( 1 ~ XQ1)' (x)" a

{x;q)a = (x)00/(xq )00.

=

Define

r,i - 1 - g a _ r « ( a ; + 1 ) [J *'~ 1 - q Tq(x) • We shall denote by QAJV(^) the product Hi 0 and let

- Qtj) and by Ajv(t)

N

Hq(x1,...,xn;a,b,k):=1[[(xl;q)a(^;q)b j=i

JJ x

'

(
i
Xl

(3.1)

x

i

The g-Morris constant term identity 13>6>5 states that CT^Hgix!,...

,xN;a,b,k)

= MN(a,b,k;q)

(3.2)

where ^

i

u u ^

TT

Tq(a + b+l + kl)Tq(l + k(l + l))

In 1 Baker and Forrester have given a conjecture extending this identity to read: Conjecture 3.1 For a,b,k € Z> 0 we have

CT M

JJ N0+l
-M

No{(

(l-qkXi-)(l-qk+lX^)Hq(x1,...,xN;a,b,k) J

h h „\ IfT1 r ? ( ( f c + 1)j + a + b + kN° + 1 ) r ( ( f c + 1 ) ( j ' + 1 } + kNo) ' ' 'q> f}0 Tq((k + l)j +a + kN0 + l)Tq((k + l)j + b + kN0 + l)

:;n'[(Hi)(j

/i

r fc+2+ i)],

*( ) ' (3.4)

w/jere N = N0 + Nx.

56

They proved this conjecture in the special cases a = X and a = b = 0 by using a theorem of Bressoud and Goulden 2 . Also the special case Ni = 2 was proved by adapting the method of Zeilberger 15 . In this section we formulate the conjecture in an analogous way as in the case of Forrester's conjecture by employing the ^-integration formula of Macdonald polynomial P\(h,...,tN;q,t). Define the (/-integral of a function f(t\,..., t^r) over [0,1]N by f

= (l-q)N

f(tu...,tN)dqh---dqtN

f(qs\...,qs»)qs>---qs».

T

It is known 1 that the conjectured constant term identity (3.4) is equivalent to the following g-Selberg type integral evaluation N

if^jr^)) wjLv'

j^z^At)dqH-dqtN

((q^Y(RHS

=

(3.5)

of (3.4))

^ (?)a+6 '

where b = —x, a = x + y — 1 and 3

iV 0 +l<J<7<JV

l

(3.6)

l
l

°

Let us first recall the ^-integration formula of Macdonald polynomials [8, Proposition 5.2]. Let A = ( A i , . . . , XN) be a partition of length < N and A' be the conjugate partition. For each square s = (i,j) in the diagram of A, let a(s) = Xi - j ,

a'(s)

l(s) = X'j-i,

=j-l l'(s)=i-l

and put

Y[(l-qa{s)tl^+1),

hx(q,t) = sex sex

»>i

57

where n(A) := £ ( t - l)Ai. For a Macdonald polynomial P\(ti,...,

ijv; 9, £) we have

AT

= 0 ta(?)+2*

X

a

(?)(O£^

^ r g (ifc + l ) r g ( » + (JV - t)fc + Xj)Tq(y + {N- i)k) y rq(k + l)r,(ar + y + (2iV - i - l)fc + Af)

s

<37>

=' »^ht^n(%»-^

-

where gSN(x,y;k):=

=

f

JJ^-S^S-

J[o,i]» V[

JJ

(9"*i)oo ! < £ } < „

1*k(q1-ktj/Uhkdgt1.-dgtN

„^(^)+2fc 2 (?) TT r9(ifc + l ) r , ( x + (N - i)k)Tq(y + (N - i)k) ii

r,(A + l)r,(a: + i/ + ( 2 J V - t - l ) J k )

'

{

J

Note that ni
n «?*(«i-*tJ-/*i)2*=,*AAr(t) n *?*~v_v*«)2*-i.

l
l
Let A denotes the antisymmetrizer: A / ( t i , . . . , t j v ) = ]T) e(cO/(i
58

Hence symmetrization of the integrand of LHS of (3.7) gives

f

Px(t-,q,qk)f[txf1ytp^AN(t)t2ik-\q'-Hj/ti)2k_1dqt1--.dqtN

N

J[o,i] - L"! •J

~[j=i

J

{Q j)o0

SN(x,y;k) h^qk)

(qh;qk)N9

[[{qX+y+{N_i)k^.

Since N t-k(N-i)^

j=l

YJ

#k(„i-l
l
2k'

the integral evaluation (3.5) is equivalent to

J[o,i]" f=\

(<7^)oo

1 S

^

N

No

X

EL^ 1 " 1 i=l

=

(U-q-ktj)(ti-qk+1tj)dqt1---dgtN

II N0+l
N N Ajyf^) (^) (RKS ™\T {x + y) J V (q) q

J

a+b

v

of (3.4)), "'

where ^

_

<^\kN{N-\)l2+Nx(Nx-\)l2q-k(k-V)N{N-\)l±-kN1{Ni-\)l2_

Therefore if we write FN,No(tl,---,tN)=&N(t)-1 N0

1

xA^A^Jlt^-

i=l

J! N0+l
(U-q-ktj)(ti-qk+1tj)),

(3.9)

59 the symmetrized form of this evaluation is f

rTj-MJV-il-iVi (g*j)oo

x

II

A

,.,

t2ik~1(Q1~ktj/ti)2k-iFN,No{tu...,tN)dqt1---dqtN

(3.10)

l
= NIAN(T-^M)N(^Sky(RHS

V r ^ x + 2/) / V (q)a+b J

of (3.4) ; y ).

v

Let us expand the symmetric homogeneous polynomial -F/v,iVo (*i > • • • i t/v) in terms of Macdonald polynomials P\(ti,..., ijv; 9,<7fc) Fiv,iVo(ii,...,^) = ^ C A F A ( t ; ^ g ' ! ) ,

(3.11)

A

and substitute this into (3.10). Then by straightforward (but tedious) computation we can rewrite (3.10) as

Mg,« fc ) ,ti (9* +y+ * (<_1) ~ JVl+1 )^ ~

Y ;;

(g*i 9 % " f f P + 1)0" + 1)], Nyr1 (gM J+ N 0+ i) +1) ,

x n (^+':(jvo+i))i n (^-fc'-jvi+i)Wl_1 n ( ^ - ^ - ^ - ^ v - j - i JVi-l

No-l

j=0

Ni-1

(=0

j=0

n (?x+y+':(jvo+j)-jvi+i)iv:-i+1!

JVo-1

JVi-1

J=0

j=0

x n ^+V+U-NI+I)N\-1

(3.12) where Siv = k(^N(N! - 1)(N - Nt - 1) + J^fJV! - 1)(M - 2)) - ^ ( J V i - l ) ( J V i + l).

(3.13)

60

Put __

{q

(nNk\(q,qk)

Cx = cx , \

, x = «f-JV»+1, r = q*+y-Ni+i+«N-i)

(314)

h\{q,q") so that (3.12) becomes - ( X ^ Y ) ^

(^^W-D/^^-i)^)

= c

(3.15)

where

(3.16) and C w = ±k(N! - 1)((N(N - 1) - TV!) + ^(Ni

- 1)(M - 2)

(3.17)

Once the conjecture is formulated in this way, it is rather easy to prove the case Ni = 2. In fact, one can readily show that the partitions appearing in the left-hand side of (3.15) are A(1) = (2,1 JV - 2 ),A( 2 ) = (17V), so that the left-hand side of (3.15) is a polynomial in X and Y of degree at most one in each variable. Since

X^^-VI^YX-1)^

=X-Y,

it suffices to show that the left-hand side of (3.15) is divisible by X — Y and the coefficient of X is CN,N0(Q) — CJV,JV-2(
£61=0.

(3.18)

A

This is implied by Proposition 3.2 We have FN,No(l,qk,...,q^N-^)

= 0.

(3.19)

Proof. The term corresponding to the unit element of the symmetric group in the alternating sum is zero because of the factor £;v-i — q~ktjv. The vanishing of the remaining terms is a consequence of the factor qk A^r(t 0 .( 1 ),..., £CT(;v)) in view of the following lemma.

61

Lemma 3.3 For any element a ^ the unit in the symmetric group SN, there exists a pair (i,j) such that 1 < i < j < N and a(i) — a(j) — 1 Proof. This is easily verified by induction on N. To obtain the formula of Cxw, recall that the Macdonald polynomial P\(xi,..., XM ;q,t) is an eigenfunction of the operator D\ = D\(q,t) with the eigenvalue eA = ex(q,t) = J^Zi * A r _ i 9 A i [12, Chapter 6, (4.15)]: DiP\(x1}.

..,xN;

q, t) = e\P\(xi,.

..,xN; q, t),

where N

_

D1=YtMx;t)Tq<Xi,

Ai(x;t) = Yl

i=i

&i

Xi

_

Xi

Xi

x

i

and TQtXi is the shift operator: {Tq,Xif)(xi,...,xN)

=

f(xi,...,qxi,...,xN)-

k

Apply Di with t = q to both sides of (3.11) (N0 = N-2) and then substitute ti = g fc ( l_1 ), i = 1 , N. By the lemma above we have

(DxFw-aXl,^...,^*- 1 *) AN(l,qk,...,qk(N-V;qk)(TqitNFNtN-2)(l,qk,...,qkiN-1))

=

Q

>(W+i)(iv-3) (1 - g)(l - g 2fc + 2 )(l qkN+1)(qk;qk)N k l ' {1 - q + )(l - qk)N

The right-hand side is e A ( i , C ^ + e A ( 2 ) C ^ = (eA(i) - exm)C^)

qk{N~1)+1)C^),

= (1 - q)(l ~

so that KAr+i)(N-a) (1 - qkN+1)(l

_ A(1)

~

q

*

+ qk+1)(qk\

(1 - qHN-i)+i)(i

_ qk)N

qk)N •

Now the coefficient of X in the left-hand side of (3.15) is —q

^\m

- 0A(2> = (1 - q

q

~

K

'^

)0 A (i)

•KAT+ixw-a) (1 - g fcJV+1 )(l + qk+1)(qk;qk)N ' (1-
This coincides with CJV,;V-2(
62

References 1. Baker T. H. and Forrester P. J., Generalizations of the q-Morris constant term identity, J. Comb. Theory Ser. A. 81 (1998), 69-87. 2. Bressoud D. M. and Goulden I. P., Constant term identities extending the q-Dyson theorem, Trans. Amer. Math. Soc. 291(1985), 203-228. 3. Di Francesco P., Gaudin M., Itzykson C. and Lesage F., Laughlin's wave functions, Coulomb gases and expansions of the discriminant, Internat. J. Modern Phys. A9(1994), 4257-4351. 4. Forrester P. J., Normalization of the wavefunction for the CalogeroSutherland model with internal degree of freedom, Internat. J. Modern Phys. B9(1995), 1243-1261. 5. Habsieger L., line q-integrale de Selberg et Askey, SIAM J. Math. Anal. 19(1988), 969-986. 6. Kadell K., The Selberg-Jack symmetric functions, Adv. Math. 130(1997), 33-102. 7. Kaneko J., Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal. 24(1993), 1086-1110. 8. Kaneko J., q-Selberg integrals and Macdonald polynomials, Ann. Sci. Ecole. Norm. Sup., 4eserie, 29(1996), 583-637. 9. Kaneko J., On Forrester's Generalization of Morris constant term conjecture, Contemp. Math. 254(2000), 271-282. 10. Kaneko J., Forrester's conjectured constant term identity II, preprint, 2000. 11. Lassalle M., Some combinatorial conjectures for shifted Jack polynomials, Ann. Combin. 2(1998), 145-163. 12. Macdonald I. G., Symmetric Functions and Hall Polynomials, Second Edition, Clarendon Press, Oxford, 1995. 13. Morris W. G., Constant term identities for finite and affine root system, Ph. D. thesis, University of Wisconsin, Madison, 1982. 14. Stanley R. P., Some combinatorial properties of Jack symmetric functions, Adv. Math. 77(1989), 76-115. 15. Zeilberger D., A Stembridge-Stanton style elementary proof of the Habsieger-Kadell q-Morris identity, Discrete Math. 79(1989), 313-322.

O N T H E S P E C T R U M OF D E H N TWISTS IN Q U A N T U M TEICHMULLER THEORY R. M. KASHAEV Steklov Mathematical Institute at St. Petersburg 27 Fontanka, St. Petersburg 191011, Russia and Helsinki Institute of Physics, P. 0. Box 9, (Siltavuorenpenger FIN-00014 University of Helsinki, Finland E-mail: [email protected]

20C)

The operator realizing a Dehn twist in quantum Teichmiiller theory is diagonalized and continuous spectrum is obtained. This result is in agreement with the expected spectrum of conformal weights in quantum Liouville theory at c > 1. The completeness condition of the eigenvectors includes the integration measure which appeared in the representation theoretic approach to quantum Liouville theory by Ponsot and Teschner. The underlying quantum group structure is also revealed.

1

Introduction

The quantization problem of Teichmiiller spaces of punctured surfaces is connected to quantum Chern-Simons theory with non-compact gauge groups (and thereby to 2+1-dimensional quantum gravity 1,2:2) and non-rational quantum conformal field theory in two dimensions. In particular, quantum Teichmiiller theory is expected to describe the space of conformal blocks in quantum Liouville theory 21 . Prom the mathematical point of view this can be useful in three-dimensional topology, in particular for construction of new link and three-manifold invariants. The Teichmiiller spaces of punctured surfaces have been quantized in two different but essentially equivalent ways: as (degenerate) Poisson manifolds 8,3 , and as simplectic manifolds with Weil-Petersson simplectic structure 13 . The both approaches employ the real analytic coordinates closely related to the Penner parameterization of the decorated Teichmiiller spaces 17 . The main outcome of the theory is the construction of a particular projective (infinite dimensional) representation of the mapping class groups of punctured surfaces. The projective factor has been shown to be related to the Virasoro central charge in quantum Liouville theory 14 . In this paper we describe the spectrum of Dehn twists in the quantum Teichmiiller theory, derive an explicit formula for the associated with braidings i?-matrix (square of which is a Dehn twist), and uncover the underlying infinite dimensional representations of the quantum group Uq(sl(2)). Those 63

64

representations were considered in papers 20-18>19. The paper is organized as follows. In section 2 we briefly remind the main properties of the non-compact quantum dilogarithm (closely related to the double sine function, see for example 9 ) . This is the main technical object entering practically all the constructions of the paper. In section 3 we describe the algebraic system underlying the quantum Teichmuller theory and its realization in terms of the quantum dilogarithm. Section 4 is a survey of the part of papers 13 ' 14 which describes the mapping class group representation in terms of the basic algebraic system of section 3. Solution of the Dehn twist spectral problem is described in section 5 (proof of Theorem 1 is to appear in a separate publication). Section 6 contains derivation of the i?-matrix and revealing the associated quantum group structure. 2

Quantum dilogarithm

Let complex b have a nonzero real part 5R6 ^ 0. The non-compact quantum dilogarithm, eb(z), is defined by the integral formula 5 eb{z) = exp

- /

/„„^„ ,/„ . / ^ .. sinh(w6) :smh(w/b)w

(!)

in the strip |3z| < |3c(,|, where

b-1)^.

c^Mb +

In particular, when 96 2 > 0, we have the following product formula: eb(z) = (e2^+^b;

< 7 2 W ( e 2 ^ - C t ) 6 ~ X ; fU,

(2)

where q = e,7rb\

q=

e-i7,b~2.

Using the symmetry properties eb(z) = e-b(z) =

e1/b(z),

we assume in what follows $tb > 0,

96 > 0.

Function eb(z) can be analytically continued in variable z to the entire complex plane as a meromorphic function with essential singularity at infinity, and with the following characteristic properties:

65

poles and zeros ( e 6 ( z ) ) ± 1 = 0 <S> z = T{cb + m\b + n\b-1),

m,n£Z>0;

(3)

behavior at infinity depending on the direction along which the limit is taken, 1

\axgz\ > § + argfc 2

emz

eb(z)

-m(l+2c2b)/6

\&Tgz\

< | _

l "arg * 6 ~z

0(i6- 1 z;-6" : 2 ) 0(\bz;b2)

ar

g fr

- 11 2 < arg b

(4)

arg z + | | < arg ft,

^ (? 2 ;? 2 )«,

where

0(«; r) = J2 e'WTn2+i2nzn,

3 r > 0;

inversion relation e 6 (z)e ( ,(-z) =

i 2 i

- - ^1+2^)/6;

e

(5)

functional equations eb(z - \b±1/2) = (1 + e 2 ^ ± l z ) e6(* + i6 ±1 /2);

(6)

unitarity when b is either real or on unit circle, ( 1 - | 6 | ) 9 * = 0 => e ^ I ) = l/e 6 (z);

(7)

e&(p) e6(q) = e6(q) e6(p + q) e 6 (p),

(8)

pentagon equation 2

if self-adjoint operators p and q in L (E) satisfy the Heisenberg commutation relation [p,q] = (27ri) -1 ; integral analogue of Ramanujan's summation formula (x ++ uu) f eebb(x JR eb(x + v)'

2wiwx

dx

e

b(u -V-Cb)

=

eb(w

+ Cb)

c^2v\w(v+cb)+\n(l-4cl)/12

eb(u -v + w - cb) e =

where

b(v

-U-W

+ Cb)

c-2iy\w(u-ch)-\n(l-4ci)/12

/g)

eb(v - u + cb) eb(-w - cb) Q(v + cb) > 0,

S ( - u + cb) > 0,

3(i> - u) < 9to < 0.

(10)

66

Restrictions (10) can actually be relaxed by deforming the integration path in the complex x plane, keeping the asymptotic directions of the two ends within the sectors ± ( | arga;| — n/2) > argfr. The enlarged in this way domain for the variables u, v, w in eqn (9) has the form: |axg(iz)| < 7r - argfo,

z e {w,v - u - w,u - v - 2c;,}.

(11)

As the matter of fact the pentagon equation (8) is equivalent to the integral Ramanujan formula (9), see 7 for the proof. 3

The basic algebraic s y s t e m

Introduce an important notation. Let V be a vector space. For any natural 1 a* = id ® • • • ® id ®a ® id ® • • • id G E n d y ® m , i—1 times

ie a stands in the i-th position. If b £ EndF® fc for some 1 < k < m and {*i,*2, - - • ,'fe} C { 1 , 2 , . . . , m } , we write bi,i 2 ...i 2 = 4*! 8 U j 2 ® • • • ® tj fc (b).

Note also that the permutation group § m is naturally represented in y ® m : Pff(a;i ® • •• ®Xj ® • ••) = xCT-i(i) ® • •• i&Xv-ni) ® . . . ,

cr £ S m .

(12)

The projective representation of the mapping class groups of punctured surfaces, associated with the quantum Teichmuller theory, is based on the concrete realization of the following algebraic system of equations on two invertible elements A € End V and T e EndV® 2 , A3 = id,

(13)

T12T13T23 = T23T12,

(14)

AiT 12 A 2 = A 2 T 21 Ai,

(15)

T 12 A!T 21 =CAiA 2 P ( i2),

(16)

where C is a (nonzero) complex number, P(12) being denned in eqn (12) with S 2 9cr = (12): 1 i-+2 >-)• 1.

67

3.1

Useful notation

In practical calculations it is very convenient to use the following notation which permits to avoid writing explicitly operator A. For any a G End V we denote a^AfcafcA"1,

z-k = A^a f c A f c .

(17)

Obviously

where the last two equations follow from eqn (13). Besides, it is useful to denote P

(kl...mk)

P(kl...mk)-Ak

= hkP(kl...m),

P

(kl...m),

(18)

where (kl... m) is cyclic permutation, (kl... m): k i->- I H-» . . . H->- m i-» k. Eqns (15), (16) in this notation take very compact form

3.2

Ti2 = T 5 i ,

(19)

Ti2T2i=CP(12i).

(20)

Realization through the quantum dilogarithm

Take V = L 2 (E). Let self-adjoint operators p, q satisfy the Heisenberg commutation relation [p,q] = (27ri)-1. Then operators A =

2 2 e-iT/3ei37rq e,7T(p+q)

i2

T = e *e^(e6(qi +

p2

_

£

£ 2 ^

q2))-i G L

(21)

2(R2))

(22)

do satisfy equations (13) - (16) with ( = emc'>/3. Note that operator A here is unitary and is characterized by the equations (up to a normalization factor) AqA-1=p-q,

ApA" 1 = - q ,

while operator T is unitary if (1 — \b\)Qb = 0.

68

4

The mapping class group representation

Here we recall how the mapping class groups of punctured surfaces are represented in terms of system (13) - (16). 4-1

Decorated ideal triangulations

We call a two-cell in a cell complex (or CW-complex, see 4 ) triangle if exactly three boundary points of the corresponding two-disk are mapped to the zero-skeleton. We shall also call zero-cells and one-cells vertices and edges, respectively. Let E be a compact oriented (possibly with boundary) surface with finite non-empty set Vs of marked points called also punctures. Definition 1 A cell complex decomposition of E is called ideal triangulation if • the zero-skeleton coincides with Vz; • all two-cells are triangles. Two ideal triangulations are considered equivalent if there exists an isotopy of E fixed on <9E U Vs deforming one into another. We suppose that E admits ideal triangulations (for example, each boundary component must contain at least one marked point). Any ideal triangulation of E has one and the same number of triangles ns- Denote T(T) the set of triangles in ideal triangulation r. Definition 2 Ideal triangulation r is called decorated if • each triangle is provided by a marked corner; • all triangles are numbered, ie a bijective numbering mapping f:{l,...,n£}->T(r) is fixed. Graphically (see below) in a triangle we put the corresponding integer inside of it, and asterisk at the marked corner. The set of all decorated ideal triangulations of E will be denoted A s . For terminological convenience and if no confusion is possible, in what follows we sometimes will use the term triangulation as a substitute for decorated ideal triangulation. Action of the permutation group. The permutation group § „ s naturally acts in A s from the right, A s x S „ s 9 (r,a)

H-> TCT

e As,

69 by changing the numbering mapping: 777 = f O a.

Changing of a marked corner. If r € A s and 1 < i < n s , then triangulation pi(r) is obtained from r by changing the marked corner of triangle f{i) as is shown in figure 1.

Figure 1. Transformation pi changes the marked corner of triangle f(i).

The flip transformation. Let two distinct triangles f (i), f(j) have a common edge and their marked corners be as in the lhs of figure 2, so the common edge is a diagonal of a quadrilateral combined of the two triangles. Then tri-

O * # Figure 2. The flip transformation Uy clockwise "rotates" one diagonal of the quadrilateral until it matches another diagonal.

angulation Uij(r) is obtained from T by replacing the common edge by the opposite diagonal of the quadrilateral, assigning the numbers and marked corners to new triangles as is shown in the rhs of figure 2. Note that this flip transformation Wy implicitly depends on the triangulation it transforms as at fixed i and j it is not defined on all triangulations. The properties. We have the following properties: Pi0 Pi ° Pi = id, Wjfc o ujik o Wij = uiij o ojjk,

(23) (24)

(Pr 1 x Pi) ° "a = un o (p~l x Pj), Uji o Pi o unj — (ij) o fa x pj).

(25) (26)

The first equation is evident since there are only three possibilities to mark a corner in a triangle. The other three equations are proved pictorially in figures 3 - 5 . Any two (decorated ideal) triangulations can be transformed to each other by a (finite) composition of elementary transformations, ie pi, ui^ and permutations. This follows from the known fact that any two ideal triangulations

70

Figure 3. Proof of the pentagon equation (24).

p{

w

X

pj^.

ii

Figure 4. Proof of the symmetry relation (25).

(without decoration) can be transformed one into another by a composition of flips 15 , and that all possible decorations of a fixed ideal triangulation are transitively acted upon by compositions of permutations and pi transformations. 4-2

Quantum theory

Suppose we are given a solution to eqns (13) - (16). For each r G A s and 1 < i < n-£ assign F(T,pi(T))=AieEndV®n*.

(27)

71

-H"i) ° Pi x

PJ

Figure 5. Proof of the inversion relation (26).

Let i / j be such that triangles f(i) and f (j) are as in the lhs of figure 2. Then we put F(r, Uij (r)) = Ty € End V®n*.

(28)

Finally, for any permutation a 6 S „ s set F(r,rcr) = Pff £ E n d V ® " s , where operator Pa is defined by eqn (12). Ensured by the consistency of eqns (13) - (16) with eqns (23) - (26), mapping F can be extended to an operator valued function F(r, r') on As x A s such that for any r, r', r " £ A s F(T,r) = l,

F(r,r')F(r',r")F(r",r)eC-{0}.

(29)

In particular (when r " = r ) , F(r,r')F(r',r)6C-{0}.

(30)

As an example, deduce operator F(r,w^ (r)). Denoting r' = w^ (r) and employing eqn (30) as well as definition (28), we have F(r,o;yi(r)) = F( W i j (r'),r') ~ ( F ^ o ^ r ' ) ) ) - 1 = Ty 1 ,

(31)

where we denote by ~ an equality up to a numerical factor. The mapping class or modular group" ME of S naturally acts in As- By construction we have the following invariance property of function F: F ( / ( r ) , / ( r ' ) ) = F(r,r'), a

V/G A4 E .

T h e group of homeomorphisms permuting the marked points, factored wrt the connected component of the identical homeomorphism.

72

This enables us to construct a projective representation of MY,'Mz3

/ H> F ( r , / ( r ) ) € E n d F ® n s .

Indeed, F(T,/(T))F(T,MT))

= F(r,/(r))F(/(r),/(/l(r))) ~ F(r,/h(r)).

Any Dehn twist, at least if it is along a non-separating contour, is equivalent to the Dehn twist of an annulus (with two marked points on the two boundary components) along its only non-contractible loop denoted a in figure 6. As an operator it is given in terms of operator T. Indeed, from figure 6 it follows that W12 o Da{r) = r. So, using eqn (31), we obtain F(r, £ Q ( r ) ) = F f o w f / M ) ^ Tr 2 \

(32)

where the normalization is to be fixed.

Figure 6. The Dehn twist along contour a followed by the flip transformation uj\2 does not change the initial triangulation.

5

Diagonalizing the Dehn twist

In this section we work within the quantum Teichmiiller theory, ie we consider solution (21), (22) of system (13) - (16). We assume that (1 - |6|)3* = 0, so the range of mapping F in this case is given by unitary operators. Properly normalized quantum Dehn twist of the annulus has the form: D„ = r e T r 2 1 e i 2 , r z - ,

za = ( p 1 + q 2 ) / 2 ,

Q= e^'\

where the exponential factor removes the spurious degree of freedom, see 13>14. Using formula (22) we rewrite it equivalently DQ=ei2'r(c'2»-c')e6(pa+qa), where the self-adjoint operators pa = qi + p2 - (q2 + Pi)/2,

qa = ( q 2 - p i ) / 2

satisfy the Heisenberg commutation relation [Pa,q«] = (27ri)_1.

73 To diagonalize Da, we use the fact that mutually commuting (and conjugate to each other in the case when |6| = 1) unbounded operators L± = 2cosh(2 7 r6 ±1 q a ) +

2 b±lp e

*

°

also commute with Da: [L-,L+] = [D a ,L±] = 0. Geometrically operator L+ is nothing else but the quantized geodesic length (hyperbolic cosine thereof) of contour a, see 3 , s . Let us consider the "coordinate" basis (a;|, x £ M, where q a is diagonal and Pc is a differentiation operator, {x\qa = x{x\,

(x\pa =

2^-g^(x\,

and define one parameter family of vectors: (i|

} =

e ^ + x + C5-i0^ e _ i 2 w ( x + C b ) , eb(s -x - cb + IO)

=

s e K

Theorem 1 Vectors \as) are eigenvectors of operators L j , D a : L±|a s ) = |a s )2cosh(27r6 ±1 s) )

D Q |a s ) = | a s ) e i 2 ^ 2 - c ' ) .

(34)

They are orthogonal to each other: (ar\as)

= v(s)~16(\r\-\s\),

v(s) = 4sinh(27r6s) sinh(27r6 _1 s),

(35)

and complete in L 2 (R); \as)v(s) ds{as\ = 1.

(36)

Jo

The continuous spectrum of Dehn twists in quantum Teichmiiller theory has also been observed by V. Fock6. It is worth noting that measure v(s)ds in eqn (36) also appears in the representation theory of the non-compact quantum group Uq(sl(2,R)) 18 ' 19 . In the next section we reveal that this is not accidental. Private communication.

74

6

Braiding and .R-matrix

Braiding of a disk with two vertices is naturally associated with the contour (isotopy class thereof) surrounding the vertices. Moreover, square of the braiding is Dehn twist along the associated contour. Figure 7 shows how the braiding of triangulation r of a disk with two interior and two boundary marked points can be removed by a sequence of elementary transformations: r = (13)(24) o (p3 x pi1) 0W14 o ( Wl3 x uj2i) ° (pi x W23) ° P31

OB~1(T).

Using the construction of subsection 4.2, the corresponding quantum braiding

"(13)(24)o(p3 x

1 Pl

)

4- Pi X "23

Figure 7. Braiding along contour a followed by the indicated sequence of transformations brings back to the initial triangulation r .

operator has the form (up to a normalization factor) B a = F ( T , Ba(r))

~ P(13)(24) R 1234,

with R = R1234 = A j

A3T41T31T42T32A1A3

= T14Ti3T42T32,

(37)

where in the second equation we have used notation (17) and eqn (19). As a consequence of relations (13) - (16) operator R G End V4 solves the following

75

Yang-Baxter equation R1234R1256R3456 = R3456R1256R12341

(38)

and in the case corresponding to realization through the quantum dilogarithm it is in fact (as will be shown below) Uq(sl(2)) (more precisely, the modular double thereof 6 ) universal i?-matrix evaluated at a reducible infinite dimensional representation acting in L 2 (E 2 ). Note that eqn (37) is a particular case of the realization of universal i?-matrices in the Drinfeld doubles of Hopf algebras in terms of solutions of the pentagon equation associated with the Heisenberg doubles 12 . 6.1

Relation to Hopf algebras

To identify the generators of the quantum group, we first rewrite the i?-matrix in the form R = Ad(T1-21T43)T24 = Ad(A 2 Tr 2 1 A^ 1 T 43 )T 24 ,

(39)

where Ad(a)x = a x a - 1 ,

and we have used the pentagon equation (14). Let now operator T be written as a sum T = ^ea®ea, a a

where two operator sets {e a } and {e } are realizations of mutually dual linear bases of two (mutually dual) Hopf sub-algebras Aj and AT in the associated to T Heisenberg double H(Aj). Their co-products are given by the formulae A(e 0 ) = Ad(T- 1 )(l ® e„),

A(e a ) = Ad(T)(e a ® 1).

From eqn (39) we conclude that the i?-matrix is also decomposed into a sum R = Y, EQ ® E° a

with Ea = Ad(A 2 T- 1 )(l ® e a ) = Ad(A 2 )A(e a ),

(40)

a

(41)

1

1

E = Ad(A 2 " T 21 )(l ® e°) = Ad(A 2 - )A'(e°),

where A' is the opposite co-product. These are also realizations of the Hopf algebras Aj and Ay, but this time within the Drinfeld double D(Aj).

76

6.2

Quantum Teichmuller theory and liq(sl(2))

Here we will see what quantum group and in which representation does correspond to our solution (21), (22). From eqns (22), (2) we see that all elements {e a } can be thought to be generated by operators p and e27r6 q with the co-product A(p) = pi + p 2 ,

A(e 27r6±lq ) = e ^ ^ i i + P s ) + e 2 ^ ± l q \

Similarly dual elements {e a } are generated by q and e27r6 co-product A(e27r6±1(p_q)) =

A(q) = q i + q2,

2 e

( p_q ) with the

* " 6 ± 1 ( p i - q i - q 2 ) + e2'r!>±1(P2-q2)]

From eqn (40) we deduce that operators g12 = p i - q2,

f± =

2 e

^±l(qi-q2)

+ e2x6±

1

(P2-q2)

( 4 2 )

generate the set {E a }, while according to eqn (41) the generators of the dual set {E a } are the operators q i

-

P 2

= -g21,

ga^'^-qi)

+ e2,b^(Pl-qi)

= f±

( 4 3 )

In what follows we restrict our attention to the sub-algebra corresponding to the positive exponent of parameter b. Proposition 1 Operators fm„ = f+ n ; g m n (mn = 12 or 21) with the relations [Sl2i§21J — U,

[§mn>'mnj 1

==

2

[fi2,f2i] = (q - q- )^ ""^

'0'mnj 2

- e ^

[Snmi'mnJ — \o*mm 21

),

q = e'*b*;

and the twisted co-product Av = A d ^ ^ 2 1 ® 8 1 2 " 8 1 2 ^ 2 1 ) ) o A,

ipeR,

A(g m „) = gmn ® 1 + 1 ® gmn, 2

A(f 12 ) = fi 2 ® e ^ 1 2 + 1 ® f12,

A(f21) = e 2 7 ^ 2 1 ® f21 + f21 ® 1.

generate a Hopf algebra Qv. Algebra Q^/2 is closely related with Uq{s\{2)). Definition 3 c Quantum group Uq(sl(2)) is a Hopf algebra with • generators c

EJFJKJK-1

;

T h i s definition is that of 19 without concretizing the star-structure, since the latter depends on a solution to the equation (1 — |6|)06 = 0.

77

• relations KE = qEK,

KF = q~lFK,

[E, F] = -{K2 - K~2)/(q

-

q'1);

• co-product A(K) = K®K,

A{X) = X®K

+ K~1®X,

X = E or F.

Proposition 2 There exists a faithful Hopf algebra homomorphism »j:W g (Bl(2))^ff w/2 such that f

f 1

q-i'1

q-Q' '

The generators of algebra (^ depend on only three combinations of the Heisenberg operators Pi,qt (i = 1,2) z/3 = (Pi - qi + P2 - q2)/2, P/3 = (Pi + qi - P2 - q2)/2,

q/3 = (-Pi + qi + p2 - q2)/2

which satisfy the commutation relations Iz/3,P/3] = [z/3,q/3] = 0,

[P/3,q/3] = (271-i)-1.

Subscript /? here refers to the associated to our operators contour around the interior vertex of a triangulated disk (with one interior vertex) in figure 8. In particular, in quantum Teichmiiller theory operator ZQ describes the spurious degree of freedom associated with contour /?. Substituting these definitions

g Figure 8. Triangulated disk with one interior vertex. into eqns (42), (43) we obtain gl2 = Z/3 + P/3,

g21 = Z/3 - P/3,

b

sinhfrbfo - p0 + cb)),

nb +p

3 c

fi2 = 2e^ ^e ^ < - ^ nb

+c

f2i = -2e ^-^ ^e~

2lTb

^

sinh(7r6(z0 + p0 + cb)),

78

while i?-matrix (37) takes the form R

= ^ - ( P ^ U P * - * , ) Ad hf*:*01])

Mq* + zft - q& -

P & ))-i.

\e6(,P/32 +z/3 2 J/ (44)

On the eigenvectors (£,x| of operators 2/3 and q^, "1

(£,x\z0=£(£,x\,

(Z,x\q0=x{Z,x\,

£)

((i,x\p(, = — — {Z,x\, Z7TI OX

algebra Qv is irreducibly represented for each fixed £ € K, and these are exactly the representations from Schmiidgen's classification list 20 considered by Ponsot and Teschner in 18>19. Theorem 2 / / 96 = 0, the following equation holds {£,x\r)(a) =-JT-l{i-Cb)(a){^,x\,

Va e Uq{s\{2)),

where representations na, a G i(M — C&), are defined in eqn (12) of 19. Now it becomes clear the relevance of the result of Ponsot and Teschner on the decomposition of the tensor products of the representations under consideration to the spectral problem of Dehn twists in quantum Teichmiiller theory considered in section 5: the decomposition of tensor products of representations into irreducibles is essentially equivalent to the spectral problem of the i?-matrix, while square of the latter is nothing else but a Dehn twist. Elaboration of this connection is to be published elsewhere. 7

Conclusion

In this paper we have described the spectrum of Dehn twists (Theorem 1) in the quantum Teichmiiller theory and demonstrated appearance of certain infinite dimensional representations of the quantum group Uq(sl(2)) (Theorem 2) studied in 20>18>19. The Dehn twist spectrum coincides with that of the operator e'2nL°, where LQ is the Virasoro (infinitesimal light-cone translation) operator in quantum Liouville theory. The quantum group structure indicates a relationship between the two approaches to quantum Liouville theory: one through quantization of the Teichmiiller spaces and another through representation theory of a non-compact quantum group. The result of 2 on the representation theory of the quantum Lorentz group is also likely to be relevant here. Quantum Teichmiiller theory should be connected to quantum Liouville theory on a space-time lattice 7 where the non-compact quantum dilogarithm plays the central role too. The first step in this direction could be the following

79 observation. For any integers j and t with odd sum, operators Xj,t defined through the relations Xj,t = D - ' e - ^ P + ^ D * ,

j + t=l

(mod 2),

satisfy the quantum lattice Liouville equation Xj,t+iXj,t-i

= (1 + qxj+i,t)(l + q3Xj-i,t),

q = emb\

with periodicity property \j,t = Xj+2,t- Here pair of self-adjoint operators p and q satisfy the Heisenberg commutation relation [p,q] = (i27r)_1 and D = el27rq e;,(p-(-q) is the quantum Dehn twist operator (up to normalization) considered in section 5. Clearly, the quantum Dehn twist in this case is the "light-cone" traslation operator D_1

Xj,tD =

Xj-i,t+i-

This again is in agreement with formula D ~ e'2lrL°. It is thus possible that such simple quantum evolution model describes the zero modes (responsible for the spectrum of conformal weights) in quantum lattice Liouville model of arbitrary (even) length. We have derived the i?-matrix (associated with braidings in the mapping class groups) in terms of the non-compact quantum dilogarithm, formula (44), which first has been suggested by Faddeev in 6 as the universal Uq(sl(2)) i?-matrix for the corresponding modular double. Note that more general formula (37) directly follows from the canonical embedding of the Drinfeld doubles of Hopf algebras into tensor product of two Heisenberg doubles 12 . Thus the explanation of section 6 can be considered as a geometrical view on the pure algebraic construction of 12 . It is also worth mentioning that this i?-matrix is the direct non-compact analogue of the finite dimensional .R-matrix introduced in 10 which leads to a specialization of the colored Jones link invariants (polynomials) 16 with the particularly remarkable asymptotic behavior n . In this light it would be interesting to study the corresponding non-compact analogue of the Jones invariants. Acknowledgments I wish to thank L. D. Faddeev for valuable discussions and encouragement. I am grateful to the organizers of the International Workshop on Physics and Combinatorics, Nagoya, August 21-26, 2000 for giving me the opportunity to report about these results. This work is supported by RFBR grant 99-0100101, INTAS grant 99-01705, and Finnish Academy.

1. A. Achucarro, P. K. Townsend: Chern-Simons actions for three dimensional anti-De Sitter supergravity theories. Phys. Lett. B 180 (1986), 89-92. 2. E. Buffenoir, Ph. Roche: Tensor products of principal unitary representations of quantum Lorentz group and Askey-Wilson polynomials. Preprint math.QA/9910147. 3. L. Chekhov, V. V. Fock: Quantum Teichmuller space. Theor. Math. Phys. 120 (1999) 1245-1259, math.QA/9908165. 4. B. A. Dubrovin, A. T. Fomenko, S. P. Novikov: Modern Geometry — Methods and Applications. Part III. Introduction to Homology Theory. Springer-Verlag 1990. 5. L. D. Faddeev: Discrete Heisenberg-Weyl group and modular group. Lett. Math. Phys. 34 (1995), 249-254, hep-th/9504111. 6. : Modular double of quantum group. Preprint math.QA/9912078. 7. L. D. Faddeev, R. M. Kashaev, A. Yu. Volkov: Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality. Preprint hep-th/0006156. 8. V. V. Fock: Dual Teichmuller spaces. Preprint dg-ga/9702018. 9. M. Jimbo, T. Miwa: Quantum KZ equation with \q\ = 1 and correlation functions of the XXZ model in the gapless regime. J. Phys. A: Math. Gen. 29 (1996), 2923-2958. 10. R. M. Kashaev: A link invariant from quantum dilogarithm. Mod. Phys. Lett. A, Vol. 10, No. 19 (1995), 1409-1418, q-alg/9504020. 11. : The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39 (1997), 269-275, q-alg/9601025. 12. : The Heisenberg double and the Pentagon relation. St. Petersburg Math. J. Vol. 8, No. 4 (1997), 585-592, q-alg/9504020. 13. : Quantization of Teichmuller spaces and the quantum dilogarithm. Lett. Math. Phys. 43 (1998), 105-115, q-alg/9705021. 14. : Liouville central charge in quantum Teichmuller theory. Proc. of the Steklov Inst, of Math. Vol. 226 (1999), 63-71, hep-th/9811203. 15. L. Mosher: Mapping class groups are automatic. Annals of Math., Vol. 142 (1995), 303-384. 16. H. Murakami, J. Murakami: The colored Jones polynomials and the simplicial volume of a knot. Preprint math.GT/9905075. 17. R. C. Penner: The decorated Teichmuller space of punctured surfaces. Commun. Math. Phys. 113 (1987), 299-339. 18. P. Ponsot, J. Teschner: Liouville bootstrap via harmonic analysis on a

81

19.

20. 21.

22.

noncompact quantum group. Preprints DIAS-STR-99-14, ESI-791, LPM99/46, hep-th/9911110. : Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of Uq(sl(2, ffi)). Preprint DIAS-STP-00-15, LPM00/21, math.QA/0007097. K. Schmiidgen: Operator representations of ZY?(s((2,E)). Lett. Math. Phys. 37 (1996), 211-222. H. Verlinde: Conformal field theory, two-dimensional quantum gravity and quantization of Teichmuller space. Nuclear Phys. B 337 (1990), 652-680. E. Witten: 2 + 1-dimensional gravity as an exactly soluble system. Nuclear Phys. B 311 (1988/89), 46-78.

INTRODUCTION TO TROPICAL

COMBINATORICS

A N A T O L N. K I R I L L O V Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya 486-8602, Japan and Steklov Mathematical Institute at St. Petersburg Fontanka 27, St.Petersburg 191011, Russia E-mail: kirillovQmath. nagoya-u. ac.jp We construct birational representations of the symmetric, affine symmetric and extended affine symmetric groups on the affine spaces A™ and A n ( n + 1 > / 2 which are tropical versions of the action of these groups on the set of transportation matrices and Lascoux-Schiitzenberger's action of the symmetric group on the set of semistandard Young tableaux. We show that tropical version of the RobinsonSchensted-Knuth correspondence (RSK for short) is a birational automorphism of the affine space A"

2

which intertwines two birational actions of the symmetric

2

group on the space A n . We show that tropical version of RSK has many properties which are similar to those of the classical RSK. We study tropical version of statistics cocharge c„ and proof, in particular, that it is invariant with respect to the birational action of the symmetric group on the space A " ( " + 1 ' ' 2 constructed in this paper. We prove that tropical version of Schiitzenberger's involution satisfies the discrete Hirota-Miwa type equations (the Pliicker relations for the 9-th variation of Schur functions). This result allows to find explicit formula for the tropical version of Schiitzenberger's involution as a generating function for certain weighted non-crossing lattice paths.

CONTENTS

1 2

Introduction Basic Definitions 2.1 Young tableaux 2.2 Triangles and Gelfand-Tsetlin patterns 2.3 Gelfand-Tsetlin patterns and Young tableaux 2.4 Exponents and Littlewood-Richardson numbers 2.5 Bender-Knuth transformations 2.6 Schutzenberger involution 2.7 Action of the symmetric group on the set of Young tableaux 2.8 Plane partitions 2.9 Charge and cocharge 2.10 Transportation matrices and transportation polytope 2.11 Action of the extended affine symmetric group W{A{n_l) on the space of triangles Xn and on the set of GT-patterns 82

83

3

4

2.12 Action of the symmetric group Sn on the set of matrices 2.13 Robinson-Schensted-Knuth correspondence Explicit formulae: piece-wise linear versions 3.1 Piece—wise linear action of involutions Sk on the space of triangles 3.2 Schutzenberger involution qn-\ 3.3 Continuous analog of the Robinson-Schensted-Knuth correspondence Explicit formulae: tropification 4.1 Tropification and ultradiscretization 4.2 Tropical images of involutions Sj and sf 4.3 Tropical image of involution aa 4.4 Tropical image of Schutzenberger involution 4.5 Tropical image of Robinson-Schensted-Knuth correspondence 4.6 Tropical image of the inverse Robinson-Schensted-Knuth correspondence 4.7 Tropical image of cocharge 4.8 Tropical image of charge 4.9 Energy function and charge

Appendix: Explicit formulae for small n 1

Introduction

This notes presents Introduction to the subject which we call by Tropical Combinatorics. In a few words, by Tropical Combinatorics we mean a branch of Mathematics dealing with the properties and applications of tropical image (tropification) of the Piece—wise Linear Combinatorics. By tropical image of a piece—wise linear function we mean a rational function constructed from the former by certain rules, see Section 4. It happens that tropification of certain piece—wise linear functions which have appeared in Combinatorics and Representation Theory 4>5.6>7>14>i5,2i,24,25,3i,32^ have many striking and unexpected properties, and closely connected to some famous discrete integrable equations of Statistical Physics 17>22.35.40. F o r example, the tropical images of Schiitzenberger's involution qn~i and Robinson-Schensted-Knuth's correspondence RSK are closely connected to the 9-th variation of Schur functions introduced by I.G. Macdonald, see e.g. 34 , Chapter I, Section 3, Exercise 21, and discrete Hirota-Miwa's type equations. Such a connection not only allows to obtain explicit piece—wise linear formulae for the latter, but also throw

84

new light on their combinatorial, algebraic and algebra-geometric properties. For example, the Robinson-Schensted-Knuth correspondence is a remarkable birational automorphism of the n 2 -dimensional affine space which intertwines two birational representations of the symmetric group Sn on the affine space A" 2 . The Piece—wise Linear Combinatorics takes its origin in a series of papers by I.M. Gelfand and A.V. Zelevinsky 14 ' 15 about good bases in irreducible finite dimensional representations of the general linear algebra gl (n), and has been further developed by A.D. Berenstein and A.V. Zelevinsky 6 ' 7 . A.D. Berenstein and A.N. Kirillov 4.23>24, H. Knight and A.V. Zelevinsky 25 , M. Kashiwara 20 , M. Kashiwara and T. Nakashima 21 , G. Lusztig 3 1 ' 3 2 ' 3 3 and many others. One of the main subject of the Piece-wise Linear Combinatorics is to study relationships between different parameterizations of bases in an irreducible finite dimensional representation of a Lie algebra. Recall that even if the canonical basis of an irreducible finite dimensional representation of a Lie algebra g is defined uniquely, its parameterizations depend on a choice of reduced decomposition of the longest element of the Weyl group W(g). It is one of the most challenging open problem of Piece—wise Linear Combinatorics to find for any semisimple finite dimensional Lie algebra g an explicit formula for the transition map that relates parameterizations of canonical basis corresponding two reduced decompositions of the longest element of the Weyl group W(g). Essential progress to the solution of this problem for the Lie algebra of type A was made in 3 . As far as I am aware, the tropification map, i.e. a map which transfers the maximum max(a, 6) to the sum a + b and the sum a + b to the product a -b, was invented and widely used since the early sixties by the "French School" led by Marcel-Paul Schutzenberger, in connection with the theory of formal languages, finite automata, operations research and theoretical computer science. We refer the reader to 1>9>18 for further details about these subjects. According to J.-E. Pin 37 , the name "tropical semiring" for a commonly known (max, +)-semiring was invented by Dominique Perrin in honor of the pioneering works in this field by brazilian mathematician Imre Simon. The (max, +)-semirings were widely used for algebraic description of Discrete Event Dynamical Systems, see e.g. *. Recently, the tropical semirings were used for investigation of Soliton Cellular Automata, see e.g. 17,23,39,40 The name " Tropical Combinatorics" was invented by the author in honor of Marcel-Paul Schutzenberger (1920-1996), one of the most creative and influential combinatorialists of 20th century. In this notes we will consider only the case of the Lie algebra of type A. There exist several bases in irreducible finite dimensional representation of

85

gl(n). For example, Schur-Weyl-Young basis, Gelfand-Tsetlin bases, Canonical basis, Crystal bases, PBW-bases, String basis, Verma basis and many others. The main purpose of this notes is to study combinatorial relationships between the Schur-Weyl-Young basis and the Gelfand-Tsetlin one. There exists a simple and well-known bijection between the set of (integral) Gelfand-Tsetlin patterns (GT-patterns for short) Kx n Zn(-n+1^2 and that of semistandard Young tableaux STY(\, < n), see Section 2.3. This bijection allows to transfer certain combinatorial structures given on the set of semistandard Young tableaux to those given on the set of integral G T patterns. It turns out that in many cases combinatorial structures on the set of integral GT-patterns which came from the set of semistandard Young tableaux, have a natural extension to the whole Gelfand-Tsetlin polytope Kx((3) and even to the whole space of triangles Xn. This is a basic idea of the present paper. Among such combinatorial structures on the set of semistandard Young tableaux, we are going to study in the present paper, are the following: • exponents of Young tableau; • Bender-Knuth transformations; • Lascoux-Schutzenberger's action of the symmetric group on the set of Young tableaux STY{\, < n); • Schutzenberger transformation; • Lascoux-Schtzenberger's statistics cocharge and charge. Let us say a few words about the content of our paper. In Section 2 we recall some basic notation, definitions and results which we are going to use throughout the paper, and tropical images of which we are going to study in Section 4. This material is standard except, probably, Section 2.12 and Theorem 2.28. In Section 2.12 we define the action of the symmetric group on the space of matrices which is compatible with the Robinson-SchenstedKnuth correspondence, its inverse and the upper action of the symmetric group on the "space of plane partitions" introduced by A. Lascoux and M.P. Schiitzenberger 30 . In order to illustrate the power of tropical approach, let us give some historical remarks. In the paper by the author and A.D. Berenstein 24 a certain piece-wise linear action of the affine symmetric group W ( J 4 „ _ ! ) on the space of triangles Xn was introduced and studied. This action had originated from the familiar action of the symmetric group on the set of Young tableaux introduced by A. Lascoux and M.-P. Schutzenberger 30 . Recall 24 that the generators Sj, 1 < j < n — 1, of the symmetric group Sn are defined as a

86

product of Bender-Knuth's involutions: Sj = (*1*2 • • • * n - l ) i * l ( * l * 2 • • ' * n - l ) ~ J ' . 2

(1.1)

24

It is easy to check that s = 1. One can show, see , that the Coxeter relation {sjSj+if = 1, 1 < j < n - 2, is a corollary of the following relation ( i i ^ ) 6 = 1 between Bender-Knuth's involutions t\ and <2- The main problem is to show that s» and Sj commute if \i — j \ > 2. In 24 we had proved more precise statement, namely, that the action of involution Sj changes only the entries of the j - t h row of triangle x, and depends only on the entries of the (j — l)-th, j - t h and (j + l)-th rows of the latter. The fact that s$ and Sj commute if \i — j \ > 2 follows from the above statement immediately. The proof of this more precise statement given in 24 is based on yet another product formula for involution Sj-. SJ = Rj-ijRj-2,j

• • • Ri,j@jRi,j ''' Rj—2,jRj—i,ji

1 S 3' ^ n — 1-

(1-2)

Precise definitions of piece-wise linear involutions Rjk and (linear) transformation 9j may be found in 24 , p.92, or in Section 2, Exercises 6 and 7. Essential point is that the action of involutions Rij and that of transformation 9j change only the j - t h row of a triangle x, and depends only on the entries of the (j — l)-th, j - t h and (j + l)-th rows of the latter. This observation and formula (1.2) imply the more precise statement about involution Sj mentioned above, and, therefore, imply that involutions Si and Sj commute if \i — j \ > 2. Both formulae (1.1) and (1.2) for the involutions Sj were used by the author to obtain an explicit (if complicated) expression for the piece-wise linear action of Sj on the space of triangles. In the max-min form the latter expression appeared to be messy and untractable even for j = 3, see e.g. 24 , (3.14). As a big surprise for the author was an observation that after tropification the messy max-min expression for the action of S3 on the space of triangles X4 has drastically simplified. We give a simple explicit formula for the action of tropical version of the involution Sj in Section 4.3, together with a sketch of the proof that transformations Sj, 1 < j < n — 1, defined by (4.11), indeed satisfy the defining relations of the symmetric group on. A similar situation had happened with the author's attempts to find explicit formulae for the action of the symmetric group Sn on the space of matrices Mn. Our construction of the symmetric group action on the space of matrices had appeared first as the transform by means of the RobinsonSchensted-Knuth correspondence and its inverse of the upper action of the symmetric group on the set of plane partitions:

aj = {RSK)-1osf

oRSK.

87

Namely, let d € Mnxn(Z > 0) be a matrix with non-negative integer entries, and (P, Q) be a pair of semistandard Young tableaux which corresponds to d according to the Robinson-Schensted-Knuth correspondence dR-^(P,Q), then aj(d) =

(RSK)-1(P,sj(Q)).

Note that explicit piece-wise linear formulae for the transformations RSK, RSK~l and Sj had been known for some time 4 ' 24 . To compute the action of <jj explicitly, it remained only to plug one explicit formula to another. The formula for OJ obtained in this way was messy and untractable. However after tropification of the corresponding formulae for RSK, (RSK)^1 and Sj some unexpected cancelations had happened, and the final formulae for CTJ has drastically simplified, see Section 2.12. In my opinion, the main advantage of the tropical approach to the piece-wise linear combinatorics is a possibility to use the " advanced determinantal calculus" including applications of the noncrossing lattice paths theory, see e.g. 16 - 27 ) and Plucker's relations for the 9-th variation of Schur functions introduced by I.G. Macdonald, see e.g. 34 . In Section 2.11 we recall 24 the definition of the affine symmetric group action on the set of Gelfand-Tsetlin patterns and that on the space of triangles. We also construct on the space of triangles an action of the extended affine symmetric group. Having in mind many recent applications of piecewise linear and birational representations of extended affine Weyl groups to the theory of discrete dynamical systems 36 , Cellular Automata and Ultradiscrete systems, see e.g. 17 ' 40 and the literature quoted therein, birational and piece-wise linear representations of the (extended) affine symmetric group constructed in the present paper are hopefully of some use. In Section 3 we review several piece-wise linear formulae: • for the action of involutions Sk on the space of triangles 24 . Recall that involutions s^ being restricted to the set of Gelfand-Tsetlin patterns and after that transformed to the set of Young tableaux, correspond to the symmetric group action on the former set, introduced by A. Lascoux and M.-P. Schiitzenberger 29 ; • for the Schiitzenberger involution 4 ' 25 . • for the Robinson-Schensted-Knuth correspondence 4 . In the same paper 4 the piece-wise linear formulae for the inverse Robinson-Schensted-Knuth correspondence and for the Hillman-Grassl bijection which is a generalization of the RSK, have been obtained and proved.

We give tropical version for the inverse RSK in Section 4. The piece-wise linear formula for the latter can be obtained by means of the ultradiscretizaton map; tropical versions of the Hillman-Grassl bijection and its inverse will be considered in a separate publication. The main body of our paper is Section 4 which is devoted to the description and study of the tropical version for • Schiitzenberger involution; • Lascoux-Schutzenberger's action of the symmetric group on the set of Young tableaux; • Robinson-Schensted-Knuth correspondence and its inverse; • action of the symmetric group on the space of matrices; • cocharge and charge. The main purpose of Section 4 is to show that after passing to tropical version, the items mentioned above still have "nice combinatorial" properties which are similar to those they had before. For example, see Theorems 4.27 and 4.29, let RSK

be the tropical version of the Robinson-Schensted-Knuth correspondence {RSK), then

0 if

dkj

=

A \

ii) if

dij

—

/c

D/

,

then

B \

,

then

c/

RSK

RSK

where P = SP and Q = §Q, and S = qn-i k n ( n + l ) / 2 _ , A n ( n + l ) / 2 d e n o t e s the tropical image of Schiitzenberger's involution; Hi) RSK(hf\d)) = Qjj(y(Q))> if 1 < J < " - 1, where h, ' (d) and Qjj (y) stands for the tropical versions of the row exponents h® (d) of a matrix d G Mnxn(R), see Section 2.10, and the exponents Kj(x) of a triangle x 6 Xn, see Section 2.4, correspondingly; iv) s]p(RSK{d)) = RSKio-jid)) if 1 < j < n - 1, see Theorem 4.27 for explanation of notation we have used.

89 Another example which shows fruitfulness of the tropical approach to Piece-wise Linear Combinatorics is familiar statistics charge and cocharge, introduced by A. Lascoux and M.-P. Schiitzenberger 2 9 . In 24 we had suggested conjectural piece—wise linear formula for the statistics cocharge cn on the space of triangle Xn. In particular, we made a conjecture that statistics cocharge cn is invariant with respect to the action of the affine symmetric group W(A„_ 1 ) on the space of triangles Xn. Here we prove this conjecture. Let us briefly explain the main steps of the proof. First of all we compute the tropical image cn(y), called by tropical cocharge, of the formula for statistics cocharge, suggested in 24 . The next step is to study the ratio cn(y) = c„(y)/c„_i(y). It is easy to see that ci(y) = c2(y) = 1, and cn(y) = n " = i Cjiv)- The main tool of our proof is two inductive formulae for the Laurent polynomial cn(y), namely, (4.26) and (4.27). We state here only their corollary, namely, the following inductive formula: -

/ \

cn+i{y)=

Cn{y)cn(sn(y))

, Cn-i(y)

j

1

yn+l,n+lQn,n{y)f^n n_Pn-i{y)P M A n+i{y)

UMv))k,n\fe=i

(1.3)

It is not difficult to see from (4.26) and (4.27) that if n > 2 be an integer, then Laurent's polynomial cn+i(y)cn(y) is a s n -invariant, and furthermore, the function cn+i(y) is a Sj-invariant for all j , 1 < j < n — 1. These two statements imply that statistics tropical cocharge cn(y) is a 5„-invariant. Because it is clear from the very definition of statistics tropical cocharge cn(y) that the latter is invariant with respect to the action of affine shift operator Te, see Section 4.2, we conclude that Laurent's polynomial c(y) is a W ^ ^ ^ ) invariant. We also introduce the tropical version of statistics charge cn{y). It follows from 8 ' 2 8 that the ultradiscretization of the statistics charge cn(y) being restricted to the set dominant weight semistandard Young tableaux STY+(< n), coincides with statistics charge c„(«) on the set STY+(< n), introduced by A. Lascoux and M.-P. Schiitzenberger. Combining this result with the one about relationships between (tropical) statistics cn{y) and Cn(y), one can deduce that the ultradiscritization of statistics cocharge cn(y) being restricted to the set of semistandard Young tableaux, coincides with statistics cocharge c„(»), introduced by A. Lascoux and M.-P. Schiitzenberger. The main tool to obtain inductive formulae for statistics tropical cocharge Cn+i(y) is to study the tropical versions for the exponents Kj(x)oi a triangle x G Xn, and that for the row exponents M '(d) of matrix d 6 M „ x n ( R ) . In this notes we consider mainly the algebraic and combinatorial aspects of Tropical Combinatorics. Very interesting connections of the latter with

90 Crystal bases, Canoniacal basis, Toric varieties, Birational Geometry, Discrete and Ultradiscrete dynamical systems will be considered in subsequent publications. 2

Basic Definitions

In this section we collect some basic notation, definitions, and results dealing with combinatorics of Young tableaux and Gelfand-Tsetlin patterns. For more details and proofs see n. 14 . 24 - 34 . 2.1

Young tableaux

Let A be a partition and (3 be a composition, denote by STY (A, (3) the set of semistandard Young tableaux of shape A and weight (or content) p. We refer the reader to n , p.1-4, for a condensed explanation of the above notions. We also will use notation STY(\, < n) = ] J STY(X,0) to denote the set of all \0\=n semistandard Young tableaux of shape A filled by the numbers from 1 to n; notation STY+(X,< n) = TT STY(X,fi) to denote the set of semistandard Young tableaux which have a dominant weight n — (MI > A*2 > • • • > Mn); notation STY(< n) = TT STY(X, < n) to denote the set of all semistandard Young tableaux filled by the numbers from 1 to n, and that STY+(< n) to denote the set of dominant weight semistandard Young tableaux filled by the numbers from 1 to n. 2.2

Triangles and Gelfand-Tsetlin

patterns

Let n be a positive integer. By definition, the space of triangles Xn consists of all sequences x = (x^, x^n~l\ ...,x^), where x^ = (x\j,..., Xjj) € W for j = 1 , . . . , n. As a vector space, Xn ~ K s . In the sequel we will display a triangle x — (x^n\... ,x^) € K " either follow to a traditional notation for Gelfand-Tsetlin patterns: %ln

%2n ' ' ' %n—l,n

•El,n —1

x:=

'''

: Zl2 ^22 111

%nn

%n— l,n— 1

,

91 or as a upper triangular matrix: I Xn %\2 ••• Xl,n-1 Z l n \ 0

V 0

Z22 • ' • X2
0 •••

X2r,

0

xnnJ

T h e vector x^ e R™ is called by highest weight of a triangle x £ Xn and denoted by X(x) := x ^ . T h e weight /3 := /3(x) of a triangle x € Xn is defined t o be a vector /3 = ( / 3 i , . . . , f3n) € R n such t h a t /3j = \xW\-\xU-%

l<j
where by definition \x^ \ = YH=i xij> a n d l^0^ I = 0- For given vectors A € R n and /? 6 K n we introduce t h e following subspaces of Xn: Xx = {xe x

X (P)

Xn\X(x)

= A},

x

= {x£X \P(x)=p}.

(2.1) (2.2)

Now we are going to define the Gelfand-Tsetlin patterns, see 14>5>24. By definition a triangle x e Xn is called by Gelfand-Tsetlin pattern ( G T - p a t t e r n for short) if the following inequalities are satisfied: Xij > xi+lij+1, x^

> Xij-i,

x^

> 0,

l < i < j < n - l , 1 < i < j < n,

1 < i < j < n.

Follow 2 4 we denote by K := Kn C Xn the set of all G T - p a t t e r n s , and by Kx and Kx{(3) the subsets consisting of G T - p a t t e r n s t h a t belong to the subspaces (2.1) and (2.2) correspondingly. It is easy to see t h a t Kx and Kx{(3) J/ are convex compact polytopes in the space R+ . T h e latter polytope is commonly called by Gelfand-Tsetlin polytope. A G T - p a t t e r n x = (xij) is called integral if all its entries x^, 1 < i < j < n, are (non-negative) integer numbers. T h e following result goes back to the original paper by I.M. Gelfand and M.L. Tsetlin (Doklady Akad. Nauk SSSR (N.S.) 7 1 (1950), 825-828), and was rediscovered many times. P r o p o s i t i o n 2.1 Let X be a partition and (3 be a composition, then the number of integer points in the convex polytope Kx (respectively in Kx (p)) is equal to the dimension of irreducible highest weight X representation V\ of the Lie algebra gl(n) (respectively, to the dimension of weight 0 subspace V\{/3) c V\).

92 2.3

Gelfand-Tsetlin patterns and Young tableaux

It is well-known, see e.g. 14, that the dimension of weight subspace V\((3) C V* admits a pure combinatorial description as the number of semistandard Young tableaux of shape A and weight (or content) /3: \KX n Z"( n+1 >/ 2 | = dimVA = \STY(X, < n)\, \KX((3) n Z"( n + 1 )/ 2 | = dimVx(/3) = \STY{\, /3)|.

(2.3) (2.4)

One way to explain and prove the equalities (2.3), (2.4) is based on an observation that there exist several realizations of the irreducible highest weight A representation V\ of the Lie algebra gl(n). One such a realization of the gZ(n)-representation V\ goes back to I. Schur and H. Weyl (see e.g., 1 2 ). The representation space in this case is called by Weyl module, and its basis happens to be parameterized by the set of semistandard Young tableaux STY(\, < n). Another realization of the same representation V\ is due to I.M. Gelfand and M.L. Tsetlin [ibid], and integral GT-patterns happen to be parameterize a basis in this representation (the so-called Gelfand-Tsetlin basis). Summarizing, integral Gelfand-Tsetlin's patterns Kx(P)r\Zn(n+1^2 and semistandard Young tableaux STY(\,(3) parameterize bases in the weight /? subspace V\((3) of the irreducible representation V\ in the Gelfand-Tsetlin and Schur-Weyl realizations of the latter correspondingly. This is an explanation of the equalities (2.3) and (2.4) from the representation theory point of view. Equalities (2.4) have an interesting corollary: the number of integer points in the Gelfand-Tsetlin polytope Kx((3) is equal to the number of semistandard Young tableaux of shape A and weight (3. There exists a simple and well-known bijective proof of this corollary. Namely, let T be a semistandard Young tableaux of shape A and weight (5 = (pi,... ,/3n). The tableau T defines (and is defined by) a sequence of partitions 0 = M(°> C M(1) C • • • C M(n) = M

(2.5)

such that each skew diagram 9^ = /zW \ n^1"1^ (1 < i < n) is a horizontal strip, see e.g. 34 , Chapter I, Section 1. Starting from the sequence of partitions (2.5), one can define a triangle x := x(T) e Xn by the following rule x ( i ) (T) =shape(/iW), i =

l...,n.

It is easy to check that the resulting triangle x(T) appears to be an integral GT-pattern and vise versa, i.e. if x G Kx((3) n 1n^n+1)l2 is a GT-pattern, then a sequence of partitions ^ :— iW, 1 < z < n, defines a tableau T which is in fact a semistandard one. This simple bijection allows to transfer certain combinatorial structures given on the set of semistandard Young tableaux to

93 those given on the set of integral Gelfand-Tsetlin patterns. Among such combinatorial structures on the set of semistandard Young tableaux we consider in subsequent Sections the following: • exponents of a Young tableau; • Bender-Knuth transformations; • Lascoux-Schutzenberger action of the symmetric group; • Schiitzenberger transformation; • Lascoux-Schiitzenberger's statistics cocharge. It happens very often that the latter combinatorial structures have natural extensions to the whole Gelfand-Tsetlin polytope Kx((3) and even to the whole space of triangles Xn. 2.4

Exponents and Littlewood-Richardson

numbers

Let x € Xn be a triangle. Follow 15 define exponents K\,..., K n -i of the triangle x by Kj = uiax{dj(x)\l < j < i}, where

d{j\x) = XI (^M+i ~ 2xk,i + xk,i-\) + xjA+i - xjti.

(2.6)

l
In the case when a triangle x corresponds to a semistandard Young tableaux x := x{T) under the bijection constructed in Section 2.3, the exponents Ki(x), 1 < i < n — 1, have a simple combinatorial interpretation in terms of the so-called descent set DES(T) of tableau T. Namely, let T be a semistandard Young tableau of shape A and weight /3, l(/3) < n. For any integer m between 1 and 7i — l, denote by dm{T) the maximal number of quadruples of positive integers ((ilt j i ) , (i[, j[)),..., ((ik,jk), {i'kJ'k)) such that i) all pairs (ir,jr) and (i'r, j' r ), 1 < r < k, are distinct; ii) ir < i'r, 1 < r < k; iii) T(ir,jr) = m and T(i'r,j'r) = m + 1. Recall that T(i,j) denotes the entry of the tableau T which is located in the box with coordinates (i, j). Finally, let us introduce numbers K»(T) = (3i+i-di(T), 1 < i < n-1. The numbers Ki(T),..., K „ _ I ( T ) are called by exponents of the tableau T. It is not difficult to see that Ki(x(T)) = Kj(T) for any semistandard Young tableau T. We refer the reader to the paper by A.D. Berenstein and A.V. Zelevinsky 7 , where interesting connections between exponents Kj(x) of GT-patterns, string bases and Lusztig's canonical basis are explained. Let A be a partition and / j b e a composition. For any partition v we denote by STY(\,n || v) the following set {T e STY(\:ti-

v)\Ki{x{T))
vi+1}.

94 Proposition 2.2 Let \,\i,v

be partitions such that |A| + \n\ = \u\. Then \STY(X,u\\fi)\=^,

where the number cV denotes the Littlewood-Richardson number, i.e. the multiplicity Mult [Vv : V\ <8> V^]. Example 2.3 Let us take A = fi = (321) and v = (4332). Thus v - \i = (1122) and the set STY(X, v — pi) consists of the following semistandard Young tableaux:

T:

123 34 4

124 33 4

133 24 4

134 24 3

DES{T) :

(233)

(23)

(133)

(123)

(1,1,0)

(1,1,1)

(1,2,0)

(0,1,1)

K(T):

Below each tableau T we put, respectively, its descent set DES(T) and a set of exponents K(T) = ( K I ( T ) , . . . , K n _i(T)). For example, let us compute /133\ K2(X I 2 4 ). It is easy to see that 4

V

/

133

3

2 3

T:=

24

—

x(T):=

4

i

\ \ °

1 0 n ,

1

and /3 K2(£

3

2

1

1

V

1 0\ ° )=max(42)(x),42)(a;))

1

i

) = max(xi 3 - Xi2,xi3 - 2xi 2 + i n + x23 - X22) = 2.

Therefore, the set STY(X, v\\n)

< m - ^ + i = 1, i = 1,2,3} 1 33 consists of all the above semistandard tableaux, except that 2 4 . Hence 4 «M =

3

-

= {T€ STY(X, v -

/J)|/^(T)

95 2.5

Bender-Knuth

transformations

It is an easy consequence of the representation theory of the Lie algebra gl(n) that the number of semistandard Young tableaux of a given shape A and weight (3 = (/?i,..., pn) does not depend on the order of components of the weight p. In other words, if a £ Sn be a permutation, then \STY(\,P)\

= |Sry(A,(/3 f f ( 1 ) l ...,/? f f ( n ) ))|.

A pure combinatorial proof of this statement had been given for the first time by A. Bender and D. Knuth, 2 . The key step in their proof is a combinatorial construction of certain transformations on the set of semistandard Young tableaux. Follow 24 we will call these transformations by Bender-Knuth's involutions. In this Section we will recall the definition of Bender-Knuth's transformations, and, follow 24 , we are going to rewrite the latter in terms of Gelfand-Tsetlin patterns. We recall also one of the basic results of the paper 24 which describes relations between Bender-Knuth's involutions. Given a tableau T e STY(X,P), consider the part Tj of tableau T filled by the letters j and j +1. It is clear that Tj is a disjoint union of the following fragments:

3 3

3

3+1

3 + 13 + 1

3

3

3

3+1

3+1

3

3+1

3+1

3+1

We replace every such fragment by the following one

3 3

3

3+1

3 + 13 + 1

5

3

3+1

The part T \ Tj of the tableau T remains unchanged. As a result, we obtain a new tableau T G STY(X, (j,j + l)/3), where (j,j + 1) denotes the simple transposition that interchanges j and j + 1, and fixes all other elements of [1,n]; for any sequence /?=(/?!,...,Pj,(3j+i,...,/?„), we put by definition (j,j + l)P=(Pl,...,PJ

+

l,Pj,...,Pn)-

96 Denote the new tableau T by tj(T).

T h e transformation

tj : STY (A, /?) -> S T F (A, (j, j + 1)/?)

T->f is called by the B e n d e r - K n u t h transformation. It was introduced by A. Bender and D. K n u t h 2 . P r o p o s i t i o n 2.4 2 4 Let T £ STY(X,p) andx = x(T) be the corresponding integral GT-pattern. Then the GT-pattern x corresponding to the tableau T = tj (T) has the following coordinates Xik = Xik,

if k ^ j ;

(2.7)

Xij = mw.(xitj+i,Xi-itj-i) where by definition

+ m&x(xitj-i,xi+ij+i)

we put XQJ := + 0 0 , Xjj-i

-

xtj,

= —00 for all j , 1 < j < n — 1.

D e f i n i t i o n 2.5 The Bender-Knuth transformations ti, 1 Xn, 1 2;

if i > 3, where ^ = ti t2h

• • • UU-i • • • t2h .

It is clear from the very definition t h a t q? = 1, 1 < i < n — 1. T h e transformation qn-\ of the space of triangles Xn is called by Schiitzenberger involution, and will denoted also by S n . R e m a r k 2.7 It is convenient to introduce a new basis {yij\l 2, and X\j = y\j. In this new basis {yij} the action of t h e B e n d e r - K n u t h transformations tj, 1 < j < n — 1, can be written as follows: if tj(y) = y, then • y%k = yik, if k y^j; • y^ = m a x ( j / j j _ i + yitj+i,yi-ij-i

+ yi+ij+i)

• yij = m a x ( y i j _ i + yi,j+i,y2,j+i) • Vjj = Vj-Uj-i • yn = 2/22

+ Vj+i,j+i ~ Viv -yn-

- yij, if

-yij,

if j > 2;

2 < j < n - 1;

if 2 < i < j < n - 1;

97 2.6

Schiitzenberger involution

Let A be a partition and /u = (fix,... ,fin) be a composition of the same size. Consider a Young tableau T of shape A and weight \x. Remove the entry, say t\ from the upper left corner of T, and apply the Schiitzenberger "jeu de taquin" algorithm n ' 3 8 to the skew tableau that is left. This gives a tableau that we will denote by AT, whose diagram has one box removed from the diagram of T. Put the number n + 1 — ti in this box. Repeat the algorithm on AT, getting a smaller tableau A 2 T, and putting n + 1 — ti in the box removed, where ti is the number in the upper left corner of AT. Continue until all the entries of T have been removed. As a result we will obtain a new Young tableau T* of the same shape A and weight % = (/x„,/U n _i,... ,^2, Mi)The map STY(X,fi) T is called by Schiitzenberger transformation or evacuation map, and denoted by S. The map S defines the map (2.8)

S„ : STY(X, < n) -» STY(X, < n),

which is called by Schiitzenberger involution on the set of semistandard Young tableaux STY(A, < n). 1123 Example 2.8 Consider semistandard Young tableau T = 2 2 3 , and apply 4 the algorithm for construction of the Schiitzenberger transform T* of the tableau T as it was described above. We display the construction of T* step by step: ©123 2 23 4

®2

2 3

2 3g] 4

3 non]

4 [£

[I]

(2) 2 2 3

3 HE 4

2 3 [3] 4 4 4

(2) 2 3 [3]

3 HH

3H

4

—1

@ _2_ 3 2 4 4 _3_

®

(2) 3 4

a

—»

1 2 3 2 4 4 3

®

98 1233 Hence, T* = §(T) = 2 4 4 . It is not difficult to check that T* = q3{T), 4 where q3 = t^tit^h. Theorem 2.9 ( 13 ' 24 ) The involution qn-\ given on the space of triangles Xn being restricted to the set of semistandard Young tableaux STY(X, < n), coincides with the Schiitzenberger involution Sn as it was defined above, see (2.8). 2.7

Action of the symmetric group on the set of Young tableaux

In this Section we recall the definition of an action of the symmetric group Sn on the set of semistandard Young tableaux STY (A, < n) due to A. Lascoux and M.-P. Schiitzenberger 30 . Given a positive integer j , 1 < j < n — 1, and a tableau T € STY(\,f3), l((3) < n, consider the word w(T) that corresponds to the tableau T. Recall, that by definition, the word w(T) can be obtained from the tableau T by reading the elements of T from the right to the left in consecutive rows, starting from the top one, see e.g. 34 , Chapter I, Section 9, p.142. For convenience, we will replace j by a and j + 1 by b. As a first step we extract from the word w(T) a subword w' that contains the letters a and b only. As a second step, we consequently remove from the word w' all pairs ab. As a result, we obtain a subword of type bman. We replace it by the word bnam and, after this, recover all the removed pairs ab and all the letters which are different from a and b. Finally, we obtain a_new word w. This word corresponds to some semistandard Young tableau T e STY(\,(j,j + 1)/?)We will denote the latter tableau by Tj(T) := T. Proposition 2.10 (A. Lascoux and M.-P. Schiitzenberger 30 ) Transformations Tj, 1 < j < n — 1, satisfy the defining relations of the symmetric group 1) T? = 1,

l<j
2) (TjTj+1f = 1, 1 < j < n - 2; 3) (TiTj)2 = 1, l
|i-j|>2.

In other words, the involutions Tj, 1 < j < n — 1, define a representation of the symmetric group on the set STY (A, < n). Theorem 2.11 (A.D. Berenstein and A.N. Kirillov 24 ) On the set of semistandard Young tableaux STY(X, < n) one has Tj

=qit1qj.

99 Remark 2.12 The fact that involutions Tj, 1 < j < n — 1, generate the symmetric group Sn has been proved by A. Lascoux and M.-P. Schiitzenberger 30 by subtle analysis of the properties of the plactic monoid and the RobinsonSchensted correspondence. The same action was rediscovered by M. Kashiwara 20 ' 21 in the context of the theory of crystal bases, as an action of the Weyl group W (in our case W ~ Sn) on the crystal basis corresponding to the highest weight A irreducible representation V\ of the Lie algebra gln. Example 2.13 Let us illustrate on example the difference between the Bender-Knuth transformations tj and those of Lascoux and Schiitzenberger Sj. 11223 11224 Take

T = ^

4

-t h e n

4

^(T) = 3 3 4

, but

4

S3{T) = tit2tlt3t2tit2t3tit2ti(T)

11224 2233 = ^ 4

Indeed, w(T)3A

= 3(3(34)4)(34) ^

4(3(34)4)(34),

but w(T) 3 , 4 = 33(34)4(34) ^ 2.8

44(34)3(34).

Plane partitions

Let A be a partition, a reverse plane partition of shape A is an array 7r = (xij) of positive integers of shape A (i.e. 1 < i < l(X), 1 < j < Aj) that is weakly increasing in every row and every column. In the sequel, we will assume that A has a rectangular shape (mn). If IT = (x^) is a reverse plain partition of shape (m™), then TT — (xij := x n _j + i i m _,, + i) is a plane partition of shape (m n ), i.e. an array of positive integers of shape (mn) that is weakly decreasing in every row and in every column. With any plane partition n of square shape (n") one can associate a pair of GT-patterns (P,Q) such that P and Q have the same highest weights. Namely, starting from a square shape plane partition n, let us define two triangles P = {xij\n > i > j > 1} and Q = {xij\l < i < j < n}. It is easy to see that P and Q are in fact GT-patterns and X(P) = X(Q) = (£11,2:22, • • • ,xnn).

100

Using natural bijection between integral GT-patterns and semistandard Young tableaux described in Section 2.2, we see that the set of square shape plane partitions is in one-to-one correspondence with pairs of semistandard Young tableaux of the same shape. It is convenient to display a square shape plane partition n = (2?y), 1 < i, j < n, as a diamond

•E\n — \

Xl2

/ \ %ij\ \

*^2n

=

I n

/

•En— In X22

%n—ln—l

^21

XnrL Znn—1

3
*^n2 Xn\

or by

2.9

(

) , where P and Q are the GT-patterns defined above.

Charge and cocharge

Statistics charge was introduced by A. Lascoux and M.-P. Schiitzenberger 29 in their proof of Foulkes' conjecture. More precisely, let A and /i be partitions of the same size, denote by K\^ (q) the corresponding Kostka-Foulkes polynomial. Recall that by definition, the Kostka-Foulkes polynomials K\IJj(q) are matrix elements of the transition matrix between the Schur functions s\ (x) and the Hall-Littlewood ones:

sx(x) = J2K^(
(2-9)

We refer the reader to 3 4 , Chapter III, Sections 2 and 6 for more details. Proposition 2.14 (Lascoux-Schiitzenberger 29 ) Let A and /j, be partitions, then

KxM=

E

1C{T)'

( 2J °)

T€STY(X,fj.)

summed over the set STY(X,/j.) of semistandard Young tableaux of shape A and weight /J,, and c(T) denotes the charge of tableau T.

101

We refer the reader to 34 , Chapter III, Section 6, for a definition of the statistic charge. Follow 28 , see also 8 , we define two statistics plus and minus charges c*(») on the set of triangles Xn, namely, if x G Xn, then c

n(x)

•=

max m i n

-l

\ ^2(n-j)Kj{s(x))\seSn := (su ..., s„_i)

On the set of dominant weight tableaux STY+(< n) the statistics c~, coincides with statistics charge introduced by A. Lascoux and M.-P. Schiitzenberger 29 , see 28 for the proof and further details. One can show that on the set STY+(< n) both statistics c~ and c+ are equidistributed. 2.10

Transportation matrices and transportation polytope

Let A and LI be compositions of the same size such that l(X) < n, 1(LI) < n. Denote by M(A, LI) the set of n by n matrices with non-negative real entries which have prescribed sums of rows A and those of columns /J,. This is a convex integral polytope which is called by transportation polytope of type (A; LI). We denote by Mz(A,/i)=M(A,M)nZ"2 the set of integer points of the polytope M(X,LI), and call matrices from the set MI{\,LI) by transportation matrices of type (A;/i). Let d = (dij) € M n x n ( K ) be a matrix, follow 7 ' 4 2 denote by n

n

I

!

i

Y^da+ij+ij=k n

n

Y.di+iMii=k

£ daJ+l\l
(2.11)

^

J2 di+i,b\l
(2.12)

)

the plus and minus row and column exponents of matrix d. Here and further we assume that the sum i + I is considered by modulo n. In other words, we assume that dij = di'j< if either i = i'(mod n), or j = j'(mod n). We define also the plus and minus indexes of matrix d G M{„ x n ( R ) , denoted by ind±(d), ind±{d) = ^2(nk)h ^±(d). to be fc=i

102

Finally, for two compositions A and fi, and two partitions v and 77 (all of lengths < n) denote by M^J the rational convex polytope AO)

d £ M{\

-U,fM-T])

hiti(d) ^vi~ ^i+i. 1 < i < n - 1; :(o) h)X{d)
In the sequel we will display a matrix d = (dy-) e M n X n or by a square with the main diagonal (dti)i
by a square , or by a

where A, B, C, D

Action of the extended affine symmetric group W{An_1) space of triangles Xn and on the set of GT-patterns

on the

Let t\,t2, • • • ,tn-i be the Bender-Knuth involutions acting on the space of triangles, see Section 2.5. Follow 24 , define transformations Si : = qihqi,

1 < i < n - 1,

(2.13)

which act on the space of triangles Xn. It is easy to see that Sj=l,

1

1,

and it follows from Theorem 2.6 that involutions Si, 1 2.

Therefore, the involutions Si,l
— ^11 - e-

103

Finally, follow

24

let us define transformations s^

= qiT-eqi,

l
•K = s n _js n _2 • • -S2Si, so := 4

£)

and

1

=nTesnr~ .

T h e o r e m 2.16 (A.D. Berenstein and A.N. Kirillov 24 ) Transformations U, 1 3 , (Yang — Baxter relation)

• •

t2Tct2T-e+st2Ts = Tst2T^s+ct2T-et2, V e, 5 £ R; (Tet2tiTst2t1f = 1, Ve,£<sR; 2 (hT.qjtiTsqj) = 1 and TeqjTsqj = qjTsqjTc if j > 3

Corollary 2.17 Transformations s\ .

S

WSW=S(^);

.

SWSW=SWSW

•

(Yang — Baxter

TeTs=Tt+s;

e,5eR.

satisfy the following relations

l
e^GR;

|i_j|>2)

e,r,€R;

tf

and

equation)

sPs&sP^s&s^sgv

l
e,r?GM.

(2.14)

Note that if x e X„, and /?(a;) = (/3i,..., /3n), then 0(Si(x)) = Si(P(x)) = (/?!,..., A+l, A , •••,Pn) and /3(4 e) (rr)) = 4 £ ) (/?W) = (Pn + e,P2, • . • ,/3„-i,/?i - e). Corollary 2.18 Transformations Sj, 0 2 ,

0
In other words, involutions Sj, 0 < i < n — 1, satisfy the relations of the affine symmetric group W ^ J , ^ ) . Therefore, the involutions s,, 0 < i < n—1, define a piece-wise linear representation of the afHne symmetric group W(A\l_l) on the space of triangles Xn.

104 R e m a r k 2.19 Follow 24 let us define transformations &) — „.„(«) 1, i ' j

M

(e) __-Ae)

- — "n

" £

~{ £ )

— bn-lbn-2

e

3< > := S< '«> =

. ..-(eWe) s

2

s

l

'

TT^T^CTTW)-1,

where e,«J e R.

Using Theorem 2.16 one can show 24 that transformations s] , 0 2 ,

Ve,??eR.

It follows from these relations that for any e,S 6 R the involutions sf , 0
= (Pn + nS + e,fo, • • •, A»-i,ft -n5-

e).

We conclude this Section by a construction of the action of the extended affine symmetric group W(A)li1) on the space of triangles Xn. To this end, let us introduce the following transformations: gi ••= gf

= ^~lTttn:l-\

l < i < n,

and define go '•= QnTheorem 2.20 Transformations go,gi, • • • ,gn~i o/nd ix satisfy the relations of the extended affine symmetric group W(A„_ 1 ). In other words, • • • •

g\ = 1, i = 2

0,l,...,n-l; if

(9i9j) = 1, \i ~ J\ > 2- 0 < i, j < n - 1; (9i9i+i)) 3 = !. if 0 < i < n - 1; 7r
Corollary 2.21 The elements T\ = Trgn-ign-2 Tk=

• • • gi,

gk-i9k-2---9iTgn-i---9k

Tn = gn-ign-2

if

2 < f c < n - l ,

•••9291^

generate a commutative subgroup in the group W(A]n_1). r i - - - r n = i.

Furthermore,

105 2.12

Action

of the symmetric

group Sn on the space of

matrices

In this Section we define an action of the extended affine symmetric group WiA^) on the space of matrices Mn = R n . Let d — (dij) £ Mnxn(R). In t h e sequel it is convenient t o impose t h e periodicity conditions for matrix elements =

dj+n,j

dij = dij-\-n,

1 < i < n,

j G IL.

Let us introduce transformations <7i, 1 < i < n, acting on the space of matrices Mn: <Ji(djk) = djk, ai(dij)

if j ^ i , i + l; h^+(d)-h^1\d);

= di+i<j +

al(di+1^

h^1)(d)-h^+(d).

= dlj +

See Section 2.10 for t h e definition of functions

nf±{d).

P r o p o s i t i o n 2 . 2 2 If n > 3, the ransformations defining relations • • •

(o-icn+i)

of the affine symmetric

o~i, 1 2,

(o-io-n)3 = 1; and either

(i,j)

^ ( l , n ) , or (n, 1).

Therefore the involutions Oi, 1 < i < n, define a piece—wise linear representation of the affine symmetric group W ( J 4 „ _ J ) on the space of matrices Mn. R e m a r k 2 . 2 3 It is clear from the definition of transformation Oi t h a t o-i{dij) + ai(di+itj) n

^2(Ti(dij) j=i

= d^ + di+ij,

1 < i < n,

j G Z;

(2.15)

n

=

^dj+ij. j=i

It is less obvious t h a t one has the following splitting of the relations (2.15): max(o-i(dij),<Ti(di+ij))

=

mm(o-i(dij),o-i(di+1}j))

-

max(dij,di+ij), min(dy,di+ij).

Let us define also transformations 5^, 1 < i < n, acting on the space of matrices Mn by t h e rule 0i{d) = o-i(d*),

106

where d1 denotes the matrix which is transposed to d with respect to the main diagonal. It is clear that the transformations 5j, 1 < i < n, also define a piece—wise linear representation of the affine symmetric group Sn on the space of matrices Mn. In order to define an action of the extended affine Weyl group on the space of matrices Mn, let us introduce the shift operator -K : Mn —> Mn by the following rule 7T(dy) =

di+ij.

It is easy to see that

*(h
=

<7 a + i7T.

In other words we came to the following T h e o r e m 2.24 If n > 3, the involutions aa, 1 < a < n, and the shift operator 7r give rise to a piece-wise linear representation of the extended affine symmetric group W{Arn_x) on the space of matrices Mn. Finally, for any matrix d G M „ x n ( R ) we define its left plus and minus c^'L(d) and right plus and minus c^'R(d) charges to be c L d

t' ( ) (d)

2.13

= minV™^ 0 "^)) max i n Und(a(d)) min

Robinson-Schensted-Knuth

o-€ <5n := (cri,CT2,...,cr n _i)|, a e Sn := ( ? i , 5 2 , . . • , ? „ _ i ) | . correspondence

The Robinson-Schensted-Knuth correspondence is a bijection between the set of n by n matrices with non-negative integer entries, and the set of reverse plane partitions of square shape (n n ). We will display this correspondence as follows RSK

or

RSK

We refer the reader to the original paper by D. Knuth 26 , or to u , for the definition and basic properties of the Robinson-Schensted-Knuth correspon-

107

dence. Note that in the literature there are several variants of the RobinsonSchensted-Knuth correspondence. In the sequel we will use the original construction due to Knuth 26 . We are not going to describe this construction explicitly, but illustrate it by example. /413\ E x a m p l e 2.25 Take a transportation matrix m = I 2 2 1 . Starting \350y from the matrix m, define the corresponding two-rowed array, or multipermutation, /111111112222233333333 \ w(m) ^111123331122311122222 J Using the row bumping procedure, see e.g. n , Section 4, or 26 , we will come to the pair of semistandard Young tableaux of the same shape: 11111111122222 P = 22233 , 33

11111111233333 Q = 222223 33

This pair of Young tableaux defines the corresponding diamond plane partition n and plane partition w: 9 =14

4 5

14

2, 3

2 4 8 3 5 9, 9 14 14

n=

14 9 8 14 5 4. 9 32

9 P r o p o s i t i o n 2.26 The Robinson-Schensted-Knuth for short) has the following properties:

RSK

1. if

2. if

B

RSK

correspondence

RSK

then

then

where P = S P and Q = §<3, and S = qn-i formation;

(RSK

By

RSK

denotes the Schiitzenberger trans-

108

RSK

3. let

then

Kj{P) = h)/+(d) := max } ^

diJ+1 -

^ i=k

^

dij\l < k < n \ , 1 < j < n - 1,

i=k+l

Ki(Q) = h\°l(d) := max < ^ d i + l j - - ^ d y | l
RSK

]JSTY{X,fi

Robinson-Schensted-Knuth

|| v) x STY(X,n' || i/').

A

Corollary 2.27 i) izI)

# { m e M(^,/i)|m* = m} = £ A

# { m e M(/i,/i)|m* =m,mT=m}

\STY(X,/J,)\.

= J2x |Tab^(A,//)|,

where mT denotes the matrix which is transposed to m with respect to the secondary diagonal, and Tab^2'(X,fi) denotes the set of semistandard domino tableaux of shape X and content /J,. Theorem 2.28 For any a such that 1 < a < n — 1, the following diagrams of maps are commutative

RSK

RSK

dij

l*Uup a

O-a I RSK

O-a I RSK

where P (resp. Q) denotes the image sa{P) of the GT-pattern P (resp. Q) under the action of involution sa.

109 Corollary 2.29 The involutions aa and aa commute for all integer a, 1 < a < n. Exercises. 1. Show that i) Si = ti-itiSi-itiU-i;

(2.16) l

H) Qi = Qi-iPi = P^ Qi-\,

where pi = tiU-i • • • t2t\;

Hi) qn-iSiqn-i

= «n-i, Qn-isf'Wi

it,)

1

s<

= a'-^xCT -*,

e)

s\

1

= «„_»

if

1 < i < n - 1;

1

= a'- ^^ -^,

( 2 -l 7 ) (2.18) (2.19)

:

where a = fi*2 • • -*n-i = Pn-iProofs of these statements may be found in 24 , Proposition 1.4. 2. Define transformations Vi = ijii+i • • • i n - i and Wi = wn_iwn_2 • • • « n -iShow that • u\ = 1; • Cr" = 9 n - l W n - i ;

• on the set of semistandard Young tableaux STY(X, < n) the involution un-\ coincides with the so-called dual Schiitzenberger involution 38 . 3. a. Show that if A is a rectangular diagram, /(A) < n, then on the set of semistandard Young tableaux STY(\, < n) the Schiitzenberger and dual Schiitzenberger involutions coincide: b . Show that if A is the staircase diagram A = (n — 1, n — 2 , . . . , 2,1), and T is a standard Young tableau of shape A, then un-iqn-i(T)

= (3>n_iMn_i(T) = cr™_i(T) = T',

where T' denotes the standard Young tableau which is conjugated to T. The statement 3a is due to M.-P. Schiitzenberger 38 , and the statement 3b is due to P. Edelman and C. Greene (Adv. Math. 63 (1987), 42-99). 4. a. Let m be an odd non-negative integer and ( e i , . . . ,e m ) € K m be a vector such that e\-\ + em = 0. Show that {TClt2TC2t2---TCmt2)2

= 1.

b . Let m be a non-negative integer such that m = 6k, k = l(mod 2), and e = ( e i , . . . , e m ) e Rm be a vector such that for all Z, 1 < Z < 3fc + 3, the sum /; :— e/ + e; +3 + • • • + ei+3k-3 depends only on the parity of Z, i.e. / i = /3 = • • • = /3fc+2 and h = fi = •• • = fzk+z-

110

Show that T^t\tiT^t\t2

• • -Ttmt\t2

= 1.

5. Action of the transformations a = t\t<2---tn-i and pn-i = er _1 = tn-itn-2 • • • titi on the set of Young tableaux STY{\, < n). a. Let A be a partition and /x = (/xi,... ,/xn) be a composition of the same size. We assume that /xi > 0. Given a Young tableau T e STF(A,/x), remove the entry 1 from the upper left corner of T and apply Schutzenberger's "jeu de taquin" algorithm n ' 3 8 to the skew tableau that is left. This gives a tableau, say AT, whose diagram has one box removed from the diagram of T. If the upper left corner of A T still contains 1, repeat the algorithm on AT, getting a smaller tableau A 2 T, whose diagram has one box removed from the diagram of AT. Continue until all the entries of T which are equal to 1, have been removed. As a result we obtain a new tableau T" of content (0,/x2, • • • ,Mn) and a horizontal strip 0n, \8n\ = xxi. Let us subtract 1 from all the entries of tableau T". We obtain a tableau T of content (/X2, • • •, Mn)Finally, let us fill in the horizontal strip 8n with the numbers n. The tableau T and the filled horizontal strip 9n determine a tableau T G STY{\,Ji), where Ji= (/i 2 ,...,/i„,Ml)Show that (j-

1

( T ) = p n _ i ( T ) = T.

b . Let A be a partition and n = (/xi,..., /i n ) be a composition of the same size. We assume that /x„ > 0. Given a Young tableau T £ STY(X,fi), remove the entry n from the bottom right corner of T, and apply Schutzenberger's "jeu de taquin" algorithm n ' 3 8 to the tableau that is left. This gives a skew tableau, say, v ^ i w r t n * n e same right border strip, and whose left border strip has one box less. Repeating this algorithm, we remove all the entries equal to n from the tableau T. As a result we obtain a skew tableau T" of content (0,/i2, • • • ,Mn-i) a n d a horizontal strip 9\, \6-\_\ = nn. Let us fill in the strip 6\ with l's, and then add 1 to all the entries of tableau T". Let T be the resulting tableau. The tableau T and the filled horizontal strip 9\ determine a tableau T s STY(\,fi), where /x = (iXn>Mi>M2, • • • ,Mn-i)Show that
111 where

§ : STY(\,n)

-*

STY(\,*ji)

denotes t h e Schiitzenberger transformation, see Section 2.6. All these results are due to E. Gansner 1 3 . 6. Lusztig's transformations. For each triple of integers (ijk), 1 < i < j < k < n, let us define the transformation

of t h e space of triangles by the following rule: if Rijk(x) Xij = Xij + Xik Xjj — Xjj

xafj = xa0

Xi^k—i

mm^jfc

Xj^k—iiXij

Xik T" Xitk—l -\- mm^Xjfc

if either (a,P)

= x, t h e n Xij—i),

Xj^k—1, Xij

^ (i, j)

Xi}j_ij,

or (a, (3) ^

(j,j).

Show t h a t transformations Rijk satisfy the following relations i) ( f l y * ) 2 = 1.

(3(Rijk{x))

ii) if \(ijk) n (i'j'k')\

= P(x);

^ 2, then RljkRilfk'

=

R^rk'Rijk;

iii) (tetrahedral relations) RijkRijiRikiRjki iv) RijiRikiRtji

= RjkiRikiRijlRijk = RikiRijiRiki

for

for

all 1 < i < j < k < I < n;

all 1 < i < j < k < I < n.

T h e s t a t e m e n t s i)-iii) are essentially due to G. Lusztig 3 3 . 7. Factorization formula for involution Sj. Define transformation 8j of the space of triangles Xn rule: if x G Xn, then (ej(x))jj

= Xjj + Pj+i{x)

{Qj{x))af} = xa0,

-

if (a,/3) /

by the following

(3j(x), (i,j).

Show t h a t Sj:= Rj-ijRj-2j

• • • RijBjRij••• • • Rj_2,jRj-ij,

where Rjk := Rjkk+i, 1 < j
24

for

, Appendix.

all l<j
1,

112 3

E x p l i c i t f o r m u l a e : piece—wise linear v e r s i o n

3.1

Piece-wise

linear action of involutions

Sk on the space of

triangles

Follow 2 4 let us introduce the following linear functions on t h e space of triangles Xn: (fij := Xi-ij-i

+ Xij+i

- Xi^ij

- Xij,

2
ipij-.— xij+i+Xj+i^+i-xij-Xjj,

— l;

2<j

Let us introduce also a piece-wise linear functions Q*, 1 < k < n, given on t h e space R n : Qn(ai,..

.,an)=ai+

max(0, a2,a2

+ a3,...

,a2 H

h an)

- max(0, a i , a i + 02, • • • , a i H Q*(ai,...,an) = <5^(afc,...,a„,ai,a2,...,afc-i),

+an_i);

2 < k < n.

T h e o r e m 3.1 ( 2 4 ) For a positive integer k, 2 < k < n — 1, and a x G X n , assume that Sk(x) = x, ske'(x)

triangular

= x. TTien

i) xik = xik + Ql( where f3(x) = (0\(x),...

•••> fk,k(x)'

,/3n(x))

Vi,k{x)

(3- 1 ) + Pk+i(x) - Pk(x) + e), (3.2)

denotes the weight of the triangle

E x a m p l e s 3.2 1) If n = 2, then £12 = xu ~ X22 + £23 + max(xi2 + £22,2:13 + ^33) - max(a;i2 + £22, £11 + ^23) = J/12 + I/11 — 2/22 + m a x ( y i 3 + y 3 3 , y22 + 2/23) - max(j/n + 2/23,2/13 + 2/22)2) If n = 3, t h e n £13 = ^13 + max(v5i 3 , -(£33, 0) + min(v? 33 , 13 = Xu + X44 - I13 - X33,
x.

113

Exercises. 1. Show that functions Q„{ai,.. .,an) tions • Q\{a±) = ai; • Qi(ai,.--,an)

satisfy the following recurrence rela-

=min(0,<2i_1(ai,...,an_i)) + max(0,Ql l _ 1 (a 2 ,a 3 ,... ,an)).

(3.3)

2. Let Sk '• Xn —* Xn, l < f c < n — 1, be piece-wise linear transformations of the space of triangles given by the formula (3.1). Show that recurrence relations (3.3) imply that the transformation Sk satisfy the following relation (cf. (2.15)): Sk =

tk-itkSk-itktk-i-

3. Let j be an integer, 2 < j < n — 1. Show that the following piece—wise linear functions on the space of triangles Xn are s 7 -invariant: ipij(x) = Xjj - Xj+ij+i + miru>ij(a:),0), tpij(x) = Xi-itj - Z i - i j - i + mm((pij(x),0), i>ijix) = max(mm(ipij(x),

-
rpijix) = max(min((/: i j (x),-^_i, j (x)),0). This result is due to A.D. Berenstein and A.N. Kirillov 3.2

24

, Theorem 3.2.

Schiitzenberger involution g„_i

We are going to describe the action of Schutzenberger's involution qn-\ the space of triangles Xn. It is convenient to introduce new parameters dij = Xij — Xij-i,

if 1 < i < j < n, and du = xu.

on (3.4)

In order to state the main result of this Section, let us introduce a couple of notation. We will identify the index (i, j), 1 < i, j < n, with the point on the plane A 2 with coordinates (i,j). A lattice path p between two points (a, b) and (c, d) is a sequence of points (a, b) = (h,ji),(ii,j2),---,(iN,JN)

= (c,d)

such that for each k, 1 < k < N — 1, the point (ik+i,jk+i) is equal to either (ik + l,jk) or (ik,jk + 1)- Let us define the weight wt(p) of a lattice path p to be N wt

(p) = ^2diktjk. fc=i

114

Two lattice paths pi and p 2 are called non-crossing (or non-intersecting) if they do not have common points. For given 1 < j < k < n denote by TZ-1 the set of all j-tuples p = ( p i , . . . ,pj) of non-crossing lattice paths between the points (n - k +1, n - k +1) and (l,n), 1
jk

: =

(Qn-i(x))jk,

l<j
T h e o r e m 3.3 Let 1 < j < k < n, then (

x

(") .=

j

(")« _ i „ )O \^,„+(„ TpW\ x™(d) = mmin wt(p )\ I| f„\ {p} CL € U™

T

s

,

(3.5)

where the minimum is taken over all sequences {p} = { p i , . . . , pj} of noncrossing lattice paths from the set TVh,', and parameters d = {djk)\<j
Continuous analog of the Robinson-Schensted-Knuth

correspondence

In what follows it is convenient to use new parameterization t = (tij), 1 < i < j < n, of the space of triangles Xn, and introduce new parameters min(i— l,j — 1) Uij '.= Uij{t) =

/

j

^i—k,j — k-

Let us denote by Vn,m the rectangular [l;n] x [l;m] = {(i,j) G Z2 | 1 < i < n, 1 < j < m}. For any integer k, 1 < k < min(n, m), denote by 7lntm the set consisting of all fc-tuples p = ( p i , . . . , p r , . . . ,Pk) of non-crossing lattice paths, where p r is a lattice path between the points ( l , r ) and (n,m — k + r), 1 < r < k. For any fc-tuples of lattice paths p = (pi, • • • ,Pk) G TZn,m, denote by Supp(p) = {{i, j) € Vn,m\(i, j) e pi for some 1 b.

(a,6)eP„, m \Supp(p)

115

Theorem 3.5 (Berenstein A.D. and Kirillov A.N. 4 ) Let d = (dij) be a n by n matrix with non-negative integer coefficients, and t := t(d) = (t^) be the reverse plane partition that corresponds to the matrix d under the RobinsonSchensted-Knuth correspondence. *Ifl
= m i n { £ w t ( p ) | p e <"+-£•,„},

(3.6)

p

• tjn =

\l

dik-

1<*<J. l
• If 1 < i < 3 < n, then Uji(t(d)) = Ujj(£(d*)), where d* denotes the matrix which is transposed to d with respect to the main diagonal. 4

Explicit formulae: tropification

The main goal of this Section is to describe the tropical images of transformations defined in Sections 2 and 3. 4-1

Tropification and ultradiscretization

A function
]T

affxij,

<4fc) e Z for all 1 < k < n + 1,

l
and

mk(x) =

Yi,

b )x

ij H>

h<

S

eZ

f o r a11

l
+l

l
such that (p(x) = max(Ii(x),... ,IN(X))

— max(mi(a;),...,mjv(x)).

We denote by CPL(n + 1) the set of all cpl-functions on the set Xn+\. Follow M.-P. Schutzenberger (private communication) and G. Lusztig 31 32 33 ' ' , define a map 6 : CPL(n + 1) -> Q(xij |1 < i < j < n + 1) from the set of cpl-functions on Xn+\ to the field of rational functions in the variables {xij}\
116

Z-linear form l(x) = J2aijxij t o max(!(a;),..., IN(X)) to the sum

tne

product 0(l(x)) = Ylx^',

and function

N

6{max(l(x),...,

lN(x)) = ^

9{lj(x)).

i=i

Follow M.-P. Schiitzenberger (private communication) we are calling the map 8 by tropification map, and the image 9(ip) of any cpl-function ip under the tropification map 9 by its tropical image. As far as I am aware, the name ultradiscretization was invented by B. Grammaticos for certain limiting procedure (see below) which allows to construct Cellular Automata (CAs) from continuous equations (cf. the 9-th problem posed by S. Wolfram 4 1 ) . Recall that a cellular automaton is a discrete dynamical system in which dependent variables take on a finite set of discrete values, see e.g. 41>39>40 for details. One of the most well-developed examples of CAs is the soliton cellular automaton (SCA) proposed by D. Takahashi and J. Satsuma 39 . Now this SCA and its numerous generalizations are known in literature as Ball-Box Systems (BBS), see e.g. 40 . Recently it was discovered by several groups of people, see e.g. 17 and the literature quoted therein, that BBS are closely related to Combinatorics of transportation matrices, Young tableaux, RSK, crystal bases and so on. Finally, ultradiscretization of a certain object <&(A, B,C,...) depending on variables A,B,C,... (say, function, differential equation, ...) is the following procedure: first of all let us put A = e a / e , B = ehl\ C = e c / c , ..., and then take the limit (if exists) lim $ ( e a / e , e 6 / £ , e c / e , . . . ) . €—•+0

The basic rules are lim e(A + B + C H

) — max(a, 6, c,...),

e-»+0

lim eea/ceb/,Lec/e

• • • = a+ b + c+• • •.

e—+0

Thus, ultradiscretization (if exists) is the inverse procedure to tropification. 4-2

Tropical image of involutions Si andsf

It is convenient to change slightly notation and use variables instead of those {xij}i
= Q{yij\l
+ l)

{yij}i
117

be the field of rational functions in the variables y^, 1 < % < j < n + 1, with rational coefficients. Define "elementary" birational transformations tj of the field fn+V• (tj(y))ik = yik if k ^ j ;

(y2,j+i + yi,j-m,j+i)/yij, (fj(y))»j = { (yij-m,j+i + yi-i,j-m+i,j+i)/yij, yi-i,j-m+i,j+i/yij,

if i = l; if 1 < « < j ; if i = 3-

(4.1)

In the latter formulae we assume that yoj = 1 if j > 0, and J/JO = <$i,oLet e be a new variable, and consider the field Jrn+l(e)=J7n+l®Q(c).

where
t2ti

•••tjtj-i

•••ti,

s\e' = qiTeqit

Si = qihqi,

1 < % < n.

In order to state our main theorem of this Section, let us introduce the Laurent polynomials

Qitiy) -1 V3T'T+l J=1

yj-l,nyj,n

+e

£ V~?" + 1 ' (4-2)

j=k+l

yj-l,nyj,n

where 1 < k < n, and as before we assume that yoj = 1, if j > 1. Note that Laurent's polynomial Qn,n(y) doesn't depend on e, so we will denote it by <9n,n(y)R e m a r k 4.1 It is not difficult to see that if x G Xn is a triangle and j is an integer between 1 and n — 1, then Laurent's polynomial Qjj(y) coincides with the tropical image of the j-th exponents Kj(x) of the triangle x, see Section 2.4, formula (2.6), when the former is written using the y-coordinates. For example, if x e X&, then K 3 ( x ) = m a x ( x i 4 - I 1 3 , £14 ~ 2X13 + X\2 + 224 ~ ^ 2 3 , X14 - 2X13 + Zl2 + X 2 4 - 2X23 + ^22 + ^34 - X33) = m a x ( y i 4 - y13, yl2 + 2/24 - 2/13 - 2/23,2/22 + 2/23 - 2/33 - 2/34)-

118

The tropical image of the latter max(...) is exactly Laurent's polynomial i N J/14 , J/122/24 , 2/222/23 Qzzyy) = — H 1 • 2/13 2/132/23 2/332/34 n

Theorem 4.2 (Tropical image of transformation s„ ) (4£)(2/)hn=e-12/fcn^f(,

l
(4.3)

QnAv) Corollary 4.3 / / \\ (Sn(y))kn

= Vkn

Qk,n(y) ,.

2/n-l,n-l2/n+l,n+l ! ^ ,

, . .. (4.4)

2

where Qk,n{y) = Q(kv)n{y) with r, =

^

.

2/n-l,n-lJ/n+l,n+l

Corollary 4.4 (3i£)(2/))fcn = ykn-~-r^,

(4.5)

2

where 5 =

—

. Recall that s£' := s„s„ =

qnTetiqn-

2/ra-l,n-l2/n+l,n+l

Our proof of Theorem 4.2 is based on some remarkable properties of the Laurent polynomials <3fc„(2/) a n d Qkn(y) which we will state in Theorem 4.5 below. Theorem 4.5 Let sn, s„ and s„ be birational transformations of the field Fn+i{e,rj) given by formulae (4-3) and (4-4)- Then • Qnn{s^{y))=eQnn{y)(4.6)

• Q%(s%Hy)) = vQie:v)(y)Qnn(y)/Qi%),

(4.7)

e

where we put by definition e * r\ := es^ >(rj); • if n ± 1 > 1, then for any integer k between 1 and n the Laurent polynomials y2 n an±L

(y)

'^

(4.8)

2/n±2,n±22/n,n

are invariant with respect to the action of involution sn. For example, if e = yln/yn-i,n-iyn+i,n+i, then s (e) (e) = e - 1 , and e*e = l. In particular, it follows from (4.7) that

• Qkn(Sn(y)) = 7 ^ 4 Wkn\y)

— 2/n-l,n-l2/n+l,n+l

•

(4-9)

119

Corollary 4.6 1) For any integer j > 1 the Laurent polynomial yjjQjj(y) is a Sj -invariant. In particular, the latter Laurent polynomial is invariant with respect to the action of involution Sj. 2) The following Laurent's polynomials yijQij(v)/yjj,

1<*<J,

are Sj-invariants. Remark 4.7 1) It is clear from the very definition (4.2) that Laurent's polynomials Qkn(y), I < k < n, belong to the ring % * * _ ! , y ^ . y ^ + i L and therefore they are Sj-invariants if 1 < j < n — 2. 2) It follows from (4.4) that if y € XN, where N > n +1, then sn(y) = y if and only if the following condition holds: 2 2 / n + l , n + l 2 / n - l , n - l — Vn,n-

Therefore, for a triangle y e Xn+i the condition Sj(y) = y for all j , 1 < j < n, is equivalent to the following one: ykk = Vn, 1 < fc < n + 1. The latter condition means that the triangle y has zero weight. Summarizing, the action of the symmetric group Sn on the set of Young tableaux STY(< n + 1) constructed by A. Lascoux and M.-P. Schiitzenberger 30 , as well as that of its tropical version constructed in this Section, see (4.4), acts trivially on the zero weight subspace, as well as on the zero weight set Xn+1(0)

= {ye Xn+1\ykk

= y^,

l
+ l}.

3) It is easy to see that

Qit(y) = QnAy{k])i where for any triangle y G Xn+i,

the triangle y^

i
(y{k})ij = ! / « , if j ^ n + 1, („{k}\ _ j Vi,n+i, Hi
Qkn(y) = Qnn(y{v}), where 77 =

yln/yn-i,n-iyn+i,n+i-

i
120

Finally, let us define the following birational automorphisms of the field •Fn(e)=Q(e)(s/y|l
•••

so ••= 4

_1

S2Si;

= 7rT e ii7r ;

9i-=9i = 7 r l _ 1 T € t i 7 r 1 _ I , 1 < i < n, g0 := gn Tk = gk-igk-2 • • -gingn-i • • -gk, if 2 < k
• • -92gi,

Tn = gn-\gn-2

and

• ••92911*.

Corollary 4.8 1) Birational transformations Si, i = 0 , 1 , . . . , n — 1, define a birational representation of the affine symmetric group W{A\l_^) on the field •^n(e); 2) birational transformations
s\e\y)s(ilV\y)s(r\y)

= -i?i(tf)«ie+,,)(tf)*Si(y),

where matrices s\£'(y) and s^j(y) are given by (4-3) with n = i and n = i +1 correspondingly. We conclude this Section by describing a certain set of simple invariants of the involution sn : Tn+i —> Fn+iT h e o r e m 4.9 (cf. 24 , Theorem 3.2, (3.19)) The following Laurent polynomials i

i \

Vln

• vi,n(y) '•=

, Vn— l , n j / n + l , n + l

H J/l,n+l

• iPi,n{y) '•=

. VnnVn,n+\

+ yi-2,n-iyi-l,n

'

2

yi-l,nyi,n+l

are invariant with respect to the action of the involution sn. Other examples of s„-invariants which are given by Laurent's polynomials, one can find in Theorem 4.5, formula (4.8), and Corollary 4.8, as well as in Remark 4.10 and Theorem 4.32. It looks a challenging task to describe a system of fundamental s n -invariants for the birational action of the symmetric group Sn+i on the field J-n+i- In other words to find a minimal set of

121

s„-invariants such that any s„-invariant can be written as a rational function of the former. Remark 4.10 Let n > 2 be an integer. It is easy to see that Sj(yjj/Vj-i,j-i)

=Vj+i,j+i/yj,j,

l < j < n - l ,

2/00 : = 1,

and therefore the coefficients 6k (y) of the expansion of the product , l
.

V^-M-l

n(n-l)/2

Vj-l,l-l/

k=Q

are invariants with respect to the birational action of the symmetric group Sn constructed in this Section. Note that the product IJ

(Vi,i/yi-l,i-l

+ Vhi/Vj-hJ-l) '•= O^-i)

(y)

l
is equal to the tropification of the function

«(/%/)) = E ™HPi(y),Pj(y)) l
given on the space of triangles Xn, where for any triangle y £ Xn, /3(y) := (Pi(y)i • • • >Pn(y)) £ K™ denotes its weight, cf 34 , Chapter I, formulae (1.5) and (1.6). 4-3

Tropical image of involution aa

Let us denote by Mn = Q{dij)i
1 k=l i=k

( 4 - 10 )

i=fc+l

where 0 < j < n — 1 and 1 < a < n. Remark 4.11 i) It is clear from the very definition (4.10) of Laurent's polynomial ha that the former depends only on 2 by n matrix da+l,l> • • • >d a+l,n

122

It is natural therefore to define for any 2 by n matrix I , 1 ' ' ' ' ' n I its expo\t>i,... ,o„ J nents h^(a, b), 0 (a, b) coincides with tropical image of the "local energy" #(*&", D ) of states t r and b , see Section 4.9. Here for any sequence a = (a\,..., an) we denote by Cn ® C n ( a i , . . . , an) ® (h,...,

bn) -> (a[,..., a'n) (&i,..., 6JJ,

where b'k = a fc L (fc) (a,6)/L (fc - 1) (a,6),

1 < fc < n,

and n

fc

1

n j-1

1

L^(a,6) = M )(a- ,6-

)naJ'

n

a

= E I I ^ II 6*+*

j=l

j —l i = l

i=^'-|-i

Recall that n

n

n

a

ft<'>(a,&):=EII '+' I I ft.7+l> 1 < ^ « . denotes the exponents of the following 2 by n matrix I '•••>« \ a i , . . . ,an Final remark, the interpretation of tropical version of the combinatorial .R-matrix presented above, I had learned from the lecture given by Prof. Y. Yamada during the International Workshop " Physics and Combinatorics", August 21-26, 2000, Graduate School of Mathematics, Nagoya University. Now let us define birational transformations aa = <7ai„, 1 < a < n, of the field Mn (or affine space A™ ): •

(aa(d))ij

= dij if i ^ a,a + 1;

123

(aa(d))a,=da+1,,^-1)(d)/^)(d);

.

(4.11)

1

.

{- \d).

Theorem 4.12 (Tropical image of transformations aa) i) If n > 3, the transformations aa, 1 < a < n, given by formulae (4-11) define a birational representation of the affine symmetric group Sn = W^vl^ij) on the field Mn (or affine space An ) . ii) The ultradiscretization of this action coincides with the cpl-action of the affine symmetric group Sn on the space of matrices Mn introduced in Section 2.11. Proposition 4.13 Involutions aa, 1 < a < n, have the following properties: •

o-a(dajda+ij)

=

dajda+ij;

• o-a(d~]+l + d~lhi) = d"J.+1 + d~lhj,

j € Z.

In other words, for any integer a, 1 < a < n, the "linear" forms d"a\+i + ^a+i i> and the products dajda+\j, j € Z, are invariant with respect to the action of involution aa. Corollary 4.14 For any integer a, 1 < a < n, the "linear" forms n-l

Ha:=Ha{&)

= Y.d~n-k,a+k fc=0

are invariants with respect to the action of all involutions Oj, 1 < j < n. Our proof of Theorem 4.12 is based on the following remarkable properties of Laurent's polynomials hj (d): Theorem 4.15 For any integers a and j between 1 and n the Laurent polynomials hj (d) and ha3 (c r a±i,n(d))ft^ 1 (d) are invariant with respect to the action of the involution aa:

i)

hP(
Remark 4.16 i) The Laurent polynomials haJ (d), given by formulae (4.10), coincide with the tropical images of the row exponents h^'+(d)oi matrix d, see definition in Section 2.10. It is convenient also to introduce polynomial analog Lv^(d) of functions hv^(d), see Remark 4.11. Namely,

L^(d):=L\a\d) = h^n(d-i)f[da+1J = J2f[da+uJ+l f[ da,i+i, j=l

fe=l

j=l

i=fc+l

124

where by definition we set (d 1)ij = d^1, I < i,j < n. It follows from Theorem 4.12 that birational transformations a* — a* n, 1 < a < n, of the 2

n

affine space A a

given by the following formulae

•

( !(d))ij

= dij if i ^ a,a + 1;

. .

(a*a(d))aJ=da+hjL{;\d)/Lf\(dy: « ( d ) ) a + M = da,J-L
(4.12)

also define a birational representation of the affine symmetric group W(A„_ 1 ) on the field M.nii) It follows from the definition (4.10) of polynomials ha (d) that if for some k, 1 < k < n, one has dk,ldk,2

• • • dk,n = dfc+l,ldfc+l,2 • • ' <^fc+l,n,

then crfc(d) = d; in other words in the case under consideration the action of involution Uk on the field Ain is trivial. Therefore, in the case when the products n ? = i da,j a r e the same for all a, 1 < a < n, the action of affine Weyl group W(An_i) = (<7i,...,
= di+ij,

1 < i,j < n.

For example, Tr(dn,j) = d\j, 1 < j < n. It is easy to see that *(h$n(d))

=

ft^lin(7r(d)),

and therefore 7r 3, the involutions aa, 1 < a < n, and the shift operator •K give rise to a birational representation of the extended affine Weyl group W{A^_X) on the field of rational functions Mn. 4-4

Tropical image of Schiitzenberger involution

qn-i

Let {i/ij 11 < % < j < n} be the set of independent variables, consider the field of rational functions Tn = Q(yij)- Let us introduce also the set of variables {dij\l I. Now consider the set 7c£) of all collections p = ( p i , . . . ,pj) of j-tuples of non-crossing lattice paths p i , . . . ,pi,.. .,Pj between the points (n — k +1, n — k + l) and (l,n), 1
(»a,ja)eSupp(p)

where Supp(p) = {(a,6) e Z 2 |(a,6) G pi for some 1 < I < j} denotes the support of the family of non-crossing lattice paths p. Theorem 4.18 (Tropical image of Schutzenberger involution qn-\) Assume that qn-i(y) = (y^.), then

y»=

E

(4-13)

«*(p)-

Remark 4.19 The number of terms in the RHS (4.13) is equal to the number of plane partitions contained in the box j x (k — j) x (n — k), and is equal also to the dimension of highest weight A = ((k — j ) J ) irreducible representation of the general linear group gl(n— k+ j). Remark 4.20 Formula (4.13) for y^/ can be written as a determinant. To this end, let x = (a, b) and y = (c, d) be two points with integer coordinates in the plane A 2 . Denote by V(x;y) the set of lattice paths from the point x to that y, and denote by E(x;y) the generating function E(x;y):=E((a,b);(c,d))=

^

wt(p).

peV(x;y)

Theorem 4.21 (Determinantal formula) y$\d)

= det \E((n - k + a,n-

k + a);(b,n))|i
(4.14)

Finally we are going to prove that polynomials y{ik := v{ik\d)e

z,[
satisfy the Hirota type discrete bilinear equations. Theorem 4.22 Polynomials y^] := y%\d) = (qn-i(y))ik, the following quadratic relations (n) (n+l) _

and "boundary conditions"

(n)

(n+1)

(n)

V e Xn+1,

(n+1)

satisfy , . -, ^

126

• y$ = yik •=

J] l
<<» 1 < i < A: < n + 1 ,

(4.16)

l
• 1 / ^ = 1, ^ < n .

(4.17) 24

Proof. First of all let us recall, see e.g. , Proposition 1.4, or the present paper, formula (2.15), that qn = p^qn-i, where p " 1 = ti< 2 ••-*n- Secondly, let us observe that the Bender-Knuth involution tj, 1 < j < n, changes entries in the j - t h row of a triangle y £ Xn+i only. Finally, applying the formula (4.1) for the action of involution tk to the triangle y = (yij), where y^ = y\n+ , if j > k, and y^ = y\j if j < k, we are coming to the recurrence relation (4.15).

• It is easy to see that the recurrence relations (4.15) together with boundary conditions (4.16) and (4.17) define y£/ uniquely as a function of y^k, l 2, then (n)

2/n—l,n —1 ~.

, x

yn-2,n-2

4-5

Tropical image of the Robinson-Schensted-Knuth

correspondence

The tropical image of the Robinson-Schensted-Knuth correspondence is a certain birational transformation of the affine space A n ; in other words it is a birational automorphism of the field of rational functions M.n — Q(d), where d = (dij), 1 < i, j < n, denotes the set of variables. In order to describe this birational automorphism, it is convenient to introduce new sets of variables (xij), (zij), and (ify-), 1 < i,j < n, which are connected by birational transformations min(n—i,n—j) Zij '-

=

ZijyX)

=

II k=0

Xi+k,j+ki

127 min(i — l , j — 1)

Vij(x)

n

*•i—k,j — k->

k=0 X^f

— XijyZ)

— ^ij/

*^U

==

=

^JA^/

-^i+ljj + l j

Vij/Vi

— l , j — 1-

Here and further we assume that x^ ".; - 1, if ( M ) £ [l;n] x [l;n] To continue, let us denote by P^m the set of points (i,j) € Z 2 such that l < i < n — rn + 1 and 1 < j < n. For each integer k such that (k)

1 < k < n — m + 1, denote by Vk,m the set of all fc-tuples of non-crossing lattice paths p = (pi,... ,Pk), where the path pr, 1 < r < k, is a lattice path between the points (1, r) and (n — m + 1, n — k + r). For any path p between two points in the square [1; n] x [1; n] we define its weight wt(p) to be

wt(p)=

Y[

dikjk.

(ik,jk)€p

Theorem 4.23 Tropical image of the Robinson-Schensted-Knuth dence RSK

aa

is given by the following formulae: • assume that x := x(d) = RSK{d),

j2 P

z

tK

correspon-

then

n wtM>if * ^{^ 2>

(4.18)

n,j-i+l

x

l,k{ ) =

di

II

z

kAx)

i>

• n - k + l l << ii < 1 < j < n l:

n

u>ij ]

1
• if' i < j , then Zji(x(d))

=

Zij(x{dt)).

(4.19)

(k)

Remark 4.24 The cardinality of the set Vn,rn is equal to 2n — m — k\ .(in — m — k I\ k—a n—a a=l

n

The latter product is equal to the dimension of irreducible representation V\ of the Lie algebra Ql(n) corresponding to the highest weight A = ((n—m—k+l) h ).

128

Corollary 4.25 (Determinantal formula) Assume that x := :r(d) = RSK(d). If j >i >2, then Zij(x) = det |£((1, a);(n-j

+ i-l,b + j - l))| 1 < a , b < i _ i + i.

See Remark 4.20 for definition of function E((a,b); (c,d)). Remark 4.26 For any diagram A denote by STY(\, < n) the set of semistandard Young tableaux of shape A filled by integer numbers 1,2,... ,n. Using well-known bijection between the set of certain non-crossing paths and that of semistandard (i.e. column strict) Young tableaux, see e.g. 16 , one can write the tropical image of the formula (3.6) in the following form: if i < j , then i-l 1|

Zi-fcj-fc =

fc=0

2 ^

\ [

^T(x)+c(x),n+l-T(x),

(4.20)

TeSTY{(P),
where the sum is taken over the set of semistandard Young tableaux T of rectangular shape ( i , . . . , i), filled by integer numbers from the interval [1; n]; 3

the product is taken over all cells in the rectangular (P); T{x) denotes the entry of tableau T located in the cell x, and if x = (i,j), then c(x) = j — i denotes the content of the cell x. Note that if i < j , then ^(x(d))=%(x(dt)).

(4.21)

To state our next result which describes interrelations between RSK and certain birational actions of the symmetric group Sn on the affine space A™ , we need a few additional definitions. First of all, let us define a birational transformation aa : A n —> An by the following rule: ?„(d) = <7a(d*), where d* denotes the matrix which is transposed to d with respect to the main diagonal. Next, let us define a birational map s^p (corr. s

a°wn)

as

f°ll° w s :

consider a matrix of size n by n

1 < i, j £ n, are the set of variables. Let \^~) sponding to the matrix

X-i-i

where l y ,

be the diamond corre-

see Section 2.8. Recall that P and Q stand

129 for the following triangles %-nni

%n— l , n — 1 J

%n—

l,n;

• • • •>

^22,

Xn

^12

• • • j

Q = %2n,

X\ n - 1

•ElyTl

•^nrii

*^n—l,n~l?

%n,n— 1)

• • • •>

Z22, aru

•• * )

X21

*^n2i 3?n - 1 , 1 ^nl

Then, by definition,

u

ij

ndown

V where P = saP and Q = saQ stand for the images of triangles P and Q under the involution sa, 1 < a < n — 1. Now we are ready to state our result: Theorem 4.27 (Tropical version of Theorem 2.28) For any a, 1 < a < n — 1, ift.e following diagrams of birational maps are commutative RSK

RSK

is

O-a I RSK

up a >

isdadown

O-a I RSK

130

Corollary 4.28 (Tropical version of "conserved quantities for Ball-Box systems") If 1 < i < j < n, then Laurent's polynomials Zji(x(d)), given by formulae (4-18) and (4-19), are invariant with respect to the action of symmetric group Sn = (ai,..., cr„_i). We conclude this Section by stating the result which is a tropical version of Proposition 2.26, item 3. To this end, let n > 2 be an integer and d = (d^) be n by n matrix, where we treat the entries dy of matrix d, 1 < i, j < n, as the set of independent variables. For these data in Section 4.3 we introduced Laurent's polynomials n

n

n a+l,j+l

k=lj=k

IId-,i+«'

1 a n

(4-22)

<<-

k+1

Note that the latter Laurent polynomial coincides with the tropification of the row exponent ha+(d) of matrix d € M „ x n ( K ) . To state our result we need a couple of notation. First of all, for a given n by n matrix x = (x^) we denote by x+ the following triangle: Xnn

£n_i)Tl_i

. ..

%n — l,n

X22

' ••

X+=

£ll 2-12

; 3-2,n

3-1,n— 1

Note that the triangle x+ coincides with that Q from Theorem 4.27. Finally, let us consider the birational map RSK : A™ —> A™ defined by formulae (4.18) and (4.19), see Theorem 4.23. Theorem 4.29 Let d = (d^) and x = (xij), 1 < i,j < n, be two sets of variables connected by the birational map RSK: x = RSK(d). Then for any integer a, 1 < a < n — 1, we have

/iffi-(d) = Qaa(y(x+)).

(4.23)

Example 4.30 Let us take n = 3, and d = (dy) and x = (x^) be 3 by 3 matrices. Then

*S(d) = d23 + d22d23/di3 + d^xd-xid-ii Id\2d\z, 4°l(d) = d33 + d32d /d23 + d id32d33/d22d23, 4°l(d) = di3 + d i 2 d i / d 3 + d d i 2 d i / d 3 2 d , 33 3

3

3

u

3

33

131 X33 x

=

Z22 X23

Xll £12

X33 '

2/

=

£333:22 £23

£13

XnX22X33 Z12Z23

'

^13

<9ll(y + ) = ^23/2/13 = Z23/Z13, Q22(y + ) = yt3/y 23 + Vi^V 22^^11

= x33/x23

+

x13x22X33/x12xj3.

Now we are going to use our assumption that x = RSK(d). In the case n — 3 the corresponding formulae can be found in Appendix 4, Subsection 2, item ii). First of all, it is easy to see from these formulae that Q n ( y + ) = *23(d)/*i3(d) = h?X(d). The second equality, namely Q22{y+) = x33(d)/x23(d)

+ x13(d)x22{d)x33(d)

/xi2(d)xl3(d)

= /i 2 °+( d )

is more difficult to check, and we leave details to the reader. 4-6

Tropical image of the inverse transformation

Robinson-Schensted-Knuth

Given natural numbers n, i and j such that 1 < i < j < n, denote by SS™' the set of all tableau T of rectangular shape (i,..., i) filled by integer numbers from the interval [i + 1; 2n — j] such that • part of the tableau T filled by numbers from the interval [i + 1; n] is a columns strict one; • part of the tableau T filled by numbers from the interval [n + 1,2n — j] is a rows strict one. For any tableau T G SSmL(n-j i) j define its weight wt(T) to be WUJ<\

_

TT s =

*T(j)-fc,max(n-i,j)+t-fc+l

(k,l)

tT(a)-/e,raax(n-i,j")+/-fc

1 < fc < j, 1 < Z < min(i, n — j) T(s) < n

n

':max(Ti—i,j')+( —fc+l,max(n—i,j)+l— k+n+1—T(s)

s = (k IV 1 < it < ' 1 < Z < min(i, n — j) T(s) > n + 1

J_J_ ti+j — n — k,j — k' k>0

max(n-i,j)+/-fc,max(n—ij')+(-fc+n+l-T(s)

132 Here t = (tij)i
n l
dkl{t)= l
wt(

Y,

?)>

T e S S ^ ^ ^

• if 1 < i < j < n, then n 4- 7

dki{t)=

n

dkidUj)*).

Tropical image of cocharge

Follow 2 4 denote by P n C KnnZn(n+1^2 the subset of the set of G T - p a t t e r n s Kn consisting of all triangles p = (py) G Xn n Z > Q such t h a t i)

Pnn =Pn-l,n-l

= 0;

H) Pi,n - 1 < Pa < Pi+i,n, Hi) pij - Pi+ij+i iv)

p^ - pi+ij

< 1,

< 1,

1 < i < n - 1;

1 < i < j < n - 1;

1 < i < j < n.

It is easy to see t h a t for each i,l
there exists a unique index

Pin ~ ' ' ' = Pi,n — ki ^ Pi,n—ki — l — ' ' ' ^ PiiWe will denote this indexfcjby fci(p). For example, fci(p) = n — i if and only if pin = pi^n^.i = • • • = Pu- It is easy to see t h a t the correspondence P->(fci(p),...,fc„-i(p)) defines a bijection between t h e set P n and t h e set of all (n — l ) - t u p l e s of integers ( f c i , . . . , fcn-i) such t h a t 0 < ki < n — i for all i = 1 , . . . , n — 1. I n particular, | P n | = n!. For any p € P„ and any triangle y e X n + i let us define p-weight of y, denoted by wtp(y), as follows wt

ty\

=

rr

/yi.j+iyj-i.jyj+u+iy^

TT

/^-i,j-i^-2,jyi,j+iy,J

133

where p^ := pn+\-j 1. Finally, follow 24 , formulae (3.5)-(3.7), for any triangle y e Xn+\ define Laurent polynomials cn+i(y) and cn+i(y): cn+l{y)

= \ ^

t/rtpfo) 1 1 ] - ^ - ,

[per.

J i=i

if n > 2;

we

(4.24)

yj,ri+1

ci(y) = c 2 (y) = 1, n+l c„+i(j/) = J J Cj(y), if n > 2. 3=2

Theorem 4.32 Let y € Xn+\ n-2

Sn+i(y)

E

£

fc=0

P e P„_i , fci(p) = A;

be a triangle, then

e^-^wtpiy)

* Qk+i,n(y)

(4.25)

yi,n • • • J/n-l.n 2/n-l,n2/n+l,n+l J/l,n+l • • -yn,n+l Vn,;

where en

I / n - l . n - i y n + l . n + l ;. 2 Vn.n

• i/ie cocharge function c„+i(y) is an invariant with respect to the action of the affine symmetric group W(An )• In other words, for any i, 0 < i < n, Cn+i(si{y))

=cn+1(y);

• (stability) assume that the first row (yk,n+i)k=i °f a triangle y € Xn+\ such that yk,n+i = yk,n ifl
is

CTfin+tiy)] = 1; • the ultra-discretization of the function cn+\ being restricted to the set of semistandard Young tableaux STY(X, < n + 1) coincides with the cocharge statistics c introduced by A. Lascoux and M.-P. Schiitzenberger. Here for any Laurent polynomial /(iJeC^1,..,^], the symbol CT[f{x)\ denotes the constant term of f(x). Let us remark that Laurent's polynomial c n +i(y), y € Xn+i, contains exactly n!? = l! • 2!---n! non-zero monomials, if n > 2. Note also, that

134 cardinality of the set {p e P n | fci(p) = k} is equal t o (n — 1)! for all integer k between 0 and n — 1. Our proof of Theorem 4.32 and, in particular, the proof of statement t h a t t h e function cocharge cn+i{y) is invariant under the action of the affine Weyl group W(^4n ), is based on two results which we are going to state now. One result is an explicit expression for the sum in the braces of formula (4.25), and another one is inductive formulae for the function c,(y). T h e o r e m 4 . 3 3 Given integers n > 4 and k, 0 < k < n — 2, and a triangle y € Xn+i, then

#.-tiMy)=2n-^ [fci )!H4

E ~f

yfc+i

-^-n-2

Pn-2{y)

yn-l,nyk,n-iyk+l,n-l

P 6 r„_i «i(p) = k -2 — k Ai-2,n-22/n+l,n+l ' y-n — l , n — l ? / n , n

where • the triangle y^ € X n _ i , 0 < k < n — 2, is defined as follows: (y[%j

=Vij,

if 1

i [fcl\ (Vi,n-i, (yl ] ) i , n - i = < y »~-2,n- a yn+i,.+i

if i
• for any integer k, 1 < k < n + 1, and a triangle y e Xn+i we set by definition (3k(y) = yiky2k •••ykkT h e o r e m 4 . 3 4 Let n > 2 be an integer, y = ( y y | l < i < j < n + 1) be the set of variables andc~n+i{y) be Laurent's polynomial defined by formula (4-24) Then , N

~ I

<\\)

Qn,n(y)

Cn+l(2/) = C„(s n (j/)) ^

'•

[yn+l,n+l

TT Qj,n(y)

M -= f^Qn^y)

TT J/n-l,n-l TT J/n+l,n+l

— M fj,

II Vj.n-l

fj[

Vj,n+l

(4.26) = -2fLQj,n(y), 1 < 3 < nyn,n Note t h a t according t o Corollary 4.6, the Laurent polynomials Qj,n(y), 1 < j < w, are invariant under the action of involution s„. Therefore the product in the braces of the RHS(4.26), is s„-invariant as well. C o r o l l a r y 4 . 3 5 (Inductive formula) Let n > 2 be an integer, then where Qj,n{y)

•\

~

I \

*) cn+1(y)

~ (

I \ \ \ yn+l,n+lQnn{y)Pn{y)

= Cn(sn(y))

TT/

<—p -i{y)Pn+i{y) T—Z r-—[[{Sn(y))kn n fc=l

/,»

I

> ,

lA

ov>,

(4.27)

135 where as before f3k{y) denotes the product Pk{y) '•= 2/ifc • • • ykki a)

cn+1(y)

= &^-i(vlk])yk+1'nyk'n-2£-2-kQk+iM) Pn-i{y)Pn{y) pn-2(y)Pn+i(y)

where

£n = — — -

\

(4-28)

(yn+i,n+iK y\fn,n

—-— =

sn-i(sn(ynn)/ynn);

2M—l,n—l2/n,n 444\ ~ f„\ Cn(y)Cn(sn(y)) III) Cn+1{y) = —7

Cn-l{y)

\ g n + l , n + l < 9 n , n ( l / ) / ? n ( y ) T T / - /„NN < -z 7 - r - r J— \\{Sn{y))k,n

1 > •

[

J (4.29)

Pn-l{y)Pn+l(y)

^

C o r o l l a r y 4 . 3 6 Let n > 2 be an integer, then 1) Laurent's polynomial cn+i(y)cn(y) is sn-invariant; 2) Laurent's

polynomial

cn(y)

is Sj-invariant

for all j , 1 < j < n — 2.

At the end of this Section we are going t o study in more details t h e function cocharge cn+i(y) in t h e case when the triangle y has weight zero, i.e. y e Xn+1(0)

= {ye Xn+l\ykk

= y^, 1 < k < n + 1}.

First of all let us recall, see Remark 4.7, 2), t h a t if the triangle y € Xn+\ satisfies t h e following condition: J/fe+i,fc+i2/fc-i,fc-i = 2/fe,fc for some k, 1 < k < n, then the involution Sk acts on the triangle y trivially, i.e. Sk(y) = y- In this case t h e formula (4.26) takes the form: ~ r„\ ~ f..\yk+hk+i ck+i(y) = ck(y) ykk

Qk,k(y)Pliy)

/ , ™\ (4-30)

pk_l{m+i{yy

Furthermore, if y € X n + i ( 0 ) , then the equality (4.30) holds for all k, 1 < k < n, and therefore ~

f \

TT^>

I \2/n+l,n+l^n(y)

cn+1(y) = UQJAy)

Pn+i{y)

•

From the latter equality one can deduce the following expression for the value of function cocharge cn+i(y) on a zero weight triangle y € X n + i ( 0 ) :

{

n

)

(n+l)(n+2)/2

n««'»»"*'ft*r'

<431)

136

The formula (4.31) is the tropical version or tropification, of the following wellknown formula for the cocharge of a zero-weight tableau T E STY(X, (ln)): n-l

c(T) =

Y,(*i-*i(T))(n-J)-l

; •

or in equivalent form for the value of statistics charge n-l

Cn(T) =

^(n-j)Kj(T), J=l

where A = (Ai, A2, • • •, An) is the shape of tableau T, and I = |A|/n, see e.g., A. Lascoux, B. Leclerc and J.-Y. Thibon 28 , or A.N. Kirillov, Journ. Geom. and Phys. 5 (1988), 365-389. 4-8

Tropical image of charge

Let n > 2 be an integer. Recall that statistics charge cn(T) on the set of dominant weight semistandard Young tableaux T € STY+(< n) was discovered and invented by A. Lascoux and M.P.-Schutzenberger 29 , who had used it to give a combinatorial description of the Kostka-Foulkes polynomials Kx^q). On the set of tableaux STY+(< n) statistics charge cn and cocharge cn are connected by the following relation: for partitions A and /j,, and tableau T € STY(A,/j,) one has cn(T)+cn(T)

=

n(ji)-n(\),

where for any partition v, n{y) = ]Cj>i(* — 1)^It seems natural to ask: how one can define a tropical version of statistics charge ? We know, see Theorem 4.27, that the tropical version of statistics cocharge cn is ^ ( A n - ^ - i n v a r i a n t . We would like to keep this property for tropical version of statistics charge as well. Also we would like to have for the former an analog of formula (4.31). Keeping in mind these restrictions, we are coming to the following Definition 4.37 i) The function tropical index ind(y) on the space of triangles Xn is defined to be n-l

ind(y)=H(Qkk(y))n-k; fe=i

(4.32)

137

ii) the function tropical charge cn(y) on the space of triangles Xn is defined to be Cn(y) = Yl ™d(w(y)).

(4.33)

t«6S„

Theorem 4.38 The ultradiscritization of the function tropical charge cn, see (4-33), being restricted to the set of dominant weight semistandard Young tableaux STY+(< n), coincides with statistics charge introduced by A. Lascoux and M.P.-Schutzenberger 29 . 4-9

Energy function and charge

Let a = ( a i , . . . , an) and b = ( 6 1 , . . . , bn) be two n-tuples of real numbers, follow 19>10, (6.1), define "the local energy of states" a and b to be ff(a,b)

= max{/ii(a,b)}f =1 ,

where hi = ax -\ h at — 61 - • • • - 6j_i = 01 H h a* + bi H + bn — |b|. We assume here that 60 = 0, and by definition |b| = b\ + • • • + bn. Now for any matrix d € M„ x n (R) we define its energy E(d) to be 19 n-l

E(d) = J2(n-j)H(Sj\d(j+V),

(4.34)

where d^ = (dji,... ,djn), 1 < j < n. Theorem 4.39 Assume that for alii, 1 - *)«*i(Q) - Cn(Q), and cn(Q) stands for the cocharge of Q as it was defined in Section 2.9. ii) More precisely, if 1 < j < n — 1, then F(dO))d(j+D) =

K

.(Q)i

where Kj(Q) denotes the j-th exponent of triangle Q.

138

Let us remark that in the case under consideration if 1 < j < n — 1, then

H(dU\d^)=h(°l(d), where h,+{d) denotes the j-th row exponent of matrix d, see Section 2.10. To treat the general case, let us define for a matrix d G Mnxn(WL) and integer j between 1 and n — 1, the "local energy" functions

^(d)

=

^ C { ^ ( ( ^ ° ' ) . ( ^ ° ' + 1 ) ) k e Sn = <*!,...,*„_!>},

and the energy functions n-l

£±(d):=$>-j)£f(d). Denote also by £±(d) =

(Et(d),...,Et1(d))

the vectors of local energy functions, and

h^,<°) = ,(«)

(o)

(h^(d),...,h^lt±(d))

the vectors of exponents. Theorem 4.40 For any matrix d G M n x n (M) there exist permutations a

G Sn := ( o i , . . . ,
suc/i £/ia£ (0)/

#(d) = /4Xd)). Corollary 4.41 (cf.

35

) Le£d G M n x n ( K ) , and

RSK Ct^'

, then

Et(d) = 4(Q), in other words the energy of matrix d G Mn is equal to the charge of the corresponding "recording triangle" Q. We conclude this Section by stating some interesting property of the energy function E. To this end, let us define an involution In : Xn —* Xn by setting for any triangle x £ Xn {In{x))ij = xln - Xj+i-ij,

1< i <j <

139

Proposition 4.42 If x G Xn, then i) cn(In(x)) = cn(x), Vx e Xn; RSK

ii) assume that

,

and for alii, 1 < i < n, we

have Y^=i dij = I, then E(d) =

ind(In(Q)),

ind(Q) = Y^j=i{n ~ J)Kj(Q)

where by definition

:=

ind(d).

Acknowledgments This notes grew out of series of lectures delivered at the Workshop "Combinatorial methods in representation theory", RIMS, Kyoto University (July 1998), organized by S. Okada et al., at the Workshop "Representation Theory and Integrable Systems", Okinawa (December 1998), organized by M. Okado et al., at the Erwin Schrodinger International Institute for Mathematical Physics, Vienna (July 2000), and at the International Workshop "Physics and Combinatorics" (August 2000), Graduate School of Mathematics, Nagoya University. I would like to thank those who attended for helpful comments, suggestions and support. I would like to thank Professors V.G. Kac and A.A. Kirillov for their hospitality during my visit to ESI, Vienna. I wish to thank with much gratitude Dr. N.A. Liskova for inestimable help and support on all stages of the paper writing.

Appendix:

Explicit formulae for small n

1. Tropical image of Bender-Knuth's involutions tn, n = 2,3: / 2/112/13+2/23 \ ' 2/n; , 2/13 * 2/12 *2 |

2/u 2/12 2/13 0 J/22 2/23 I = 0 0 2/33

2/112/33 2/22

\o,

0,

2/23

2/33/

140 /

*3

/ j / l l 2/12 2/13 2/14 \ 0 J/22 2/23 2/24 0 0 j/33 2/34

V 0

0

2/122/14 + 2/24

2/11, 2/12,

„ 2/122/34 + 2/222/24 2/22, 7 , 2/24

n

0,

0 j/44/

\ 2/14

2/13

2/23

0,

2/222/44

0,

2/34 2/33

V 0, o,

0,

2/44/

2. Tropical image of involution sn, n = 2,3:

(

/ 2/112/12(2/132/33 + 2/222/23) ' 2/11, ; ; r1, 2/22(2/112/23 + J/13J/22) 2/ii 2/12 2/13 0 2/22 2/23 0 0 t/33,

2/112/33

S3

2/23

2/22

V 0,

\ 2/13 »

0,

2/33/

/ j / l l 2/12 2/13 2/14 \ 0 2/22 2/23 2/24 0 0 J/33 J/34 \ 0 0 0 j/44/

/

J/11, 2/12,

0,

J/22,

0,

0,

2/13(2/332/24 J/12 + 2/222/142/442/23 + J/332/34J/22J/13) 2/33(2/332/142/23 + 2/33 J/242/12 + 2/342/222/13) 2/23(2/442/142/23 + 2/332/132/34 + 2/442/242/12)2/22 2/33(2/332/142/23 + 2/332/242/12 + 2/342/222/13)

ni

2/14

\

2/24

2/44 2/22 2/34 2/33

V 0,

0,

0,

2/44/

For the reader's convenience we list below explicit formulae for the Laurent polynomials Qkn(y), I < k < n, n = 2,3: „

,

N

<2l2(j/) =

J/13 . 2/22J/23 2/12

1

2/122/33

^

/ x

, <222(j/) =

2/13 , 2/112/23 2/12

2/122/22

141 „ , v 2/14 , 2/122/242/33 Ql3(2/J = 1 2/13 2/132/222/232/44

, 2/332/34 1 1 2/232/44

^ / N 2/14 , 2/122/24 . 2/332/34 <923(2/) = + + > 2/13 2/132/23 2/232/44 ,-. / x 2/14 , 2/122/24 , 2/222/34
3. Tropical image of Schiitzenberger involution qn, n = 2,3: / 2/33

2/112/23 + 2/132/22

/ —; 2/22

2/11 2/12 2/13 92 I

0 J/22 2/23 | 0 0 J/33

\ ; yi3

\

2/112/12 2/33

=

2/23

2/ii'

V o>

2/33/

0,

/ 2/44 (4) (4) \ ' ' 2/12% 2/13 ' 2/14 ^ 2/33 / 2 / n 2/12 2/13 2/14 \ 0 J/22 2/23 2/24 93 0 0 j/33 j/34

V 0

0

0 j/44/

n °-

2/44

0,

0,

2/22

(4) . 2/23 . 2/24 2/44

, J/34

2/11

\ 0,

0,

0,

y4iJ

where (4) _ 2/122/242/33 + 2/142/232/33 + 2/132/222/34 2/l2 — 2/132/222/23 = ^33^23^24^14 + ^33^23^13^14 + ^33^34^24^14! (4) 2/112/142/23 + 2/132/142/22 + J/112/122/24 2/13 = 2/112/122/13 = ^22^23^13^14 + ^22^12^13^14 + ^22^23^24^14; (4) 2/112/232/34 + 2/132/222/34 + 2/122/242/33 , , , , , , , 2/23 = = «22a23ai3«14«33a34a24 2/112/122/23

l

.££pZ£pX£p£ZpZZpXZp£XpZXplXp

'ZEpZZpZXpXEpXZpXXp = ZE^lZy

= ££^ZZ^XX^

'£ZpZZpXZp£XpZXpXXp — E S ^ I ?

'£Zp£XpXZ^ -f Z£pX£pZX^ — ZEpXEpZZpXZp -|_ Z£pX£pXZp£Xp -f XZp£ZpXZp£Xp -|_ Z£pX£p£XpZXp -|- X£p£Zp£lpZlp

-f £ZpZZp£lpZXp

'ISpTZpTIp = 12^

— ZZjTIj

<£XpZXpXXp =

EIj

ZI

'(»P)f)7 = ^P P + WpZXp + XEpXZp = IZ? '(p)f )7 = eePTZP + lzp£lp + zlpzlp

= ZH

<X£p + ZEp -f £Xp = XX} U0lSI8A-(+ ' m i l l ) (l ZZx •

ZZX I

xzx

=

zxx

„„ t-Sp i T_r

XZx _ zZpXZp

'

esaiTix

'XZX = XZpXXp y

ZXx

y

= ZZpZXp r

zzziio; tzx six

'ZXX = ZXpXXp

'

1

=

r

XXp

^ F xxx xxx ^ gouapuodsauoo q^nu^j-pa^suaqog-uosuiqo'y; asjaAuj • !(Teip + 3Tp)Z3pTIp = 3Zx

'XZpXXp = XZX

'ZXpXXp = ZXX

'ZZpXZpZXpXXp

=

KZxTIx

UOISI9A-(+ 'XT2Ul) (ll •

XZ-i

1

ZXn

= ZZp 'TZj = TSnTIp

'ZXi = Z I » T I »

111 Til '—1 J 1 = XXp

aotrapuodsguoo q^nu^-pa^sugqog-uosuiqoy asjaAuj

•ZZpXZpZXpXXp

~ ZZ^XX^

'XZpXXp = XZ^ 'ZXpXXp — ZXy ' XZp + ZXp = XX^ UOISJ9A-( + 'UTUl) (l

•8SJ8AUI s%\ p i r e a o u a p u o d s g j j o o q ^ n u j j - p s ^ s u a q D g - u o s u t q o y zx\% j o 9 § B U I I j B o i d a i x *^ •X>Zp£Zp££pnp£XpZ\pZZp

+ PZpf'£p££pVXp£XpZXpZZp _|_

ZH

143 • Inverse Robinson-Schensted-Knuth correspondence: ,

_ *13 *12

,

,

*31

fr23

*32

*21 *22 *22 ^13 , *13*21 . *21*23 . *21*32

111 , , *31 , dn
*11*12 111*22 111*22 *31*12 . *12*32 . *12*23 7—— + 7—7- + — T - , 111*21 *11*22 *11*22

d l l d l 2 d l 3 = *13>

^11^21^31 = *31,

, , , , *12*21*23 , *12*21*32 a i i a i 2 a 2 i a 2 2 = ——7 h -7-7 , *11*22 *11*22 ^11^12^13^21^22^23 = *12*23,

^11^12^21^22^31^32 = *21*32,

^11^12^13^21^22^23^31^32^33 = *11*22*33ii) ( m a x , + ) - v e r s i o n X11X22X33 = ^11^12^13^21^22^23^31^32^33, X12X23 = ^11^12^13^21^22^23, X21X32 = ^11^21^31^12^22^32, Xl3 = d n d i 2 d i 3 , X31 = d n d 2 i d 3 i , £22233 = ^11^12^13^21^22^23^31^32^33(1/^13 + 1/^22 + l / ^ l ) , X23 = ^11^12^13^23 + ^11^12^22^23 + ^11^21^22^23, ^32 = ^11^21^31^32 + ^11^21^22^32 + ^11^12^22^32, X33 = ^11^21^31^32^33 + ^11^21^22^32^33 + ^11^21(2*22^23^33 + ^11^12^22^32^33 + (^11(^12^22^23^33 + ^11^12^13^23^33 -

( x 2 3 +2:32)d33-

• Inverse Robinson-Schensted-Knuth , _ i _ £12 X21 x22 "11 — I I Xl3 2=31 X23 ,-1,-1 xn x\\Xx2 "11 " 1 2 = 1 2:13 2:13X21 ,-1,-1 2:11 X11X21

d\\«2i

=

1

correspondence

222 I

, 2:32 , 2:110:22 , 2:112:22 1 1 > 2:21X23 2:212:32 , 2:11X22 , 2:11X22

1

1

,

2:31 2:31X12 2:122:32 2:12X23 d i i d i 2 d i 3 = X13, d i i d 2 i d 3 i = x 3 i , j - i .-I ,-1 j - i _ 2:11X22 X11X22 11 12 21 22 — ' ) X12X21X23 2:12x21x32 ^11^12^13^21^22^23 = X i 2 X 2 3 , ^11(^21^31^12^22^32 = X21X32,

144 ^11^12^13^21^22^23^31^32^33 = ^11^22^33-

It is not difficult to see t h a t the (min, +)-version and the (max, +)-version of tropical image of t h e inverse Robinson-Schensted-Knuth correspondence are connected by the following change of variables: da —> d" 1 , ta —> x~l for all admissible i and j . 5 . Tropical image of involutions aa := cra,n, 1 < a < n, and n = 2,3. 1) In the case n = 2 one has h i iA

\

:= ha(&)

J

L0/L1

— da+i,i(da+lti+i

+da,i)/da,u

and

^11^22(^12+^21) "12(«11 + O 2 2 )

IJ

\

j

1,1 /LO

^ i ( d i 2 ) = d22hjKi

^12^2l(<^ll+^22)

=

',

/j

\

J

j.i/1,0

«ll(«12 +O21) ^12^21(^11+^22) (^22(^12+^21)

^j

\

J

uO/ui

dnd22(di2

+

d2i(dn 02=

d2\)

+d22)

o\.

Note t h a t O-l K 2 1 + rf 21 X ) = ^ 1 2 1

+ d

O'lC^U1 + ^22*) = ^U 1 +

21 1 ' d

22-

2) In t h e case n = 3 one has , (()

/.Wfyil — ^ a + 1 -'(^ a -'^ a >'+ 2 ~*~ da,l+2dg+l,l+2 + ^a+l,t+l^o+l,i+2) da,ida,i+2

c-a(datj)=da+h3hiJ-V/hlJ\ a0(d0+i,J-)=da,i^")/^'"1). and / T

N_

rfoj^o+l,j-l(^o,j-l^a,j+l da,j+l(da,jdaj+2

+ ^q,j+l^a+l,j+l +

rfq+l,j^o+l,j+l)

+ ^q,j'+2<^a+l,j+2 + <^q+l ,j + l d q + l , j + 2 )

Note t h a t o-i(d13 + d22 ) = d 1 3 + ^ l f e 1 + ^ l 1 ) = di2 + d J i \ ^1 {dii + d^) = d^ 1 + djg1. See Proposition 4.13 for general statement.

145 6 . Tropical image of cocharge cn+i,

„ •<,

n = 2,3.

fy™

- , N 2/12 , 2/122/33 .c I • c3(y) = + , if 2/ = 2/13 2/222/23 I In other words,

_ / s

y33

y^ 2/12

2/22 2/u

C3(x) = m i n ( x i 3 — X12, X22 — X33), if x €

2/132/232/44 /2/12

c4(2/) =

. 2/122/33 \

— +

. 2/122/33 . 2/132/34 , 2/142/23

— +

2/142/332/34 \ y i 3

In other words, if x € X j , then

/1/12

X3.

2/222/23/ \2/l3

H

H

2/222/23

2/232/24

2/132/24

2/332/34

2/142/44 \

2/242/44

2/242/33 / '

24

c 4 (x) = min(xi3 - Xi 2 , x 2 2 - x 3 3 ) + Xi 4 - x i 3 + x 3 3 - x 4 4 + min(x 2 3 - 2:34, x 2 4 - 2:23, ^13 - 2:12,2:22 - 2:33, Pi(x) - x 3 4 , x 2 4 -

Pi{xj),

where (/?i(x),/?2(x),/?3(x),/34(x)) = wt(x) denotes weight of triangle x. As an illustration of the formula (4.25), let us compute t h e function 05(2/). First of all, t h e set P3 contains 6 integral G T - p a t t e r n s :

P3

fci(p)

1 1 0 = ^ 1 0 , 0 1

1

0 0 1 0 0 2 1 0 , 0 0 , 0 0 1

0

1 0 1 1 0 ,

1 0 0 1 0 , 1 1

0

0 0 0 0

0

As t h e second step, let us compute the corresponding weights wtp(y). definition, if p G P3, is a G T - p a t t e r n , then

wtp{y)

= (yimiY23 V 2/222/23/

^142/232/44V 1 3 (ynmV22 V 2/132/332/34/ \ 2/132/22/

/ 2/122/24 V 1 2 /2/132/222/34V11 V 2/142/23/

V 2/122/242/33/

By

146 In particular one has wt

2/122/242/332/44.

(y) = 1 1 1 0°

wt

( )

2/222/232/332/34'

^"(j"

0

2/132/332/34'

0

2/142/232/44, 2/132/332/34'

( ) y)

Wt

yumyu.

=

\°/

1 1 1 0°

" W W "

2/232/33

2/142/44. ,23^'

°0°0

1

0

As t h e third step, let us compute the sums

Bk:=l

ePl^wtp(y)

Y. kPePs,

fcl(p)=fc

2/142/242/34

2/342/55

' 2/152/252/352/45 2/442/fc+l,4'

for A; = 0 , 1 , 2 . After simple calculations one can find B

=

/ ^ 2 3 , 2/55 ^ 2/13

£

= (yil

+

V2/13 5

=

2/142/242/342/55

2/44/ 2/152/252/352/452/44' V^V33\

2/142/242/342/55

2/222/23/ 2/l52/252/352/452/<

| ^44 , ^13 \ 2/142/242/342/55 2/55 2/23/ 2/152/252/352/452/44

Finally, we obtain t h e following formula for t h e function 05(3/): -/ N C5(2/) = +

2/142/242/342/55 | Y 2/23 . 2/55 \ , ^ , ^ ^V H — + ~ " (yi4<9i4(y)) 2/152/252/352/452/44 IV 2/13 2/44/ fy* V2/13

+

2/122/3A 2/222/23/

{

Q

M

)

+(*±

+

V2/55

m) 2/23/

{ y u Q M

where r. , v 2/15 , 2/132/442/25 , 2/232/352/44 , 2/442/45
2/132/25 , 2/232/442/35 , 2/442/45 1 1 1 2/142/24 2/242/332/342/55 2/342/55 2/132/25 . 2/232/35 . 2/442/45 + 1" , 2/142/24 2/242/34 2/342/55

147 „

, s

2/15 , 2/132/25 2/14

2/142/24

, 2/232/35

, 2/332/45

2/242/34

2/342/44

Recall, that Laurent's polynomials Qkn(y) = Qfc„(y)> where 77 = 2/nn/2/n-i,n-i2/n+i,n+i and 1
/ \\

_

Qknjy)

2/n-l,n-l2fa+l,n+l

{sn{y))kn-yknQnn{yy

y2^

see Corollary 4.3. For the reader's convenience and for references we give below the explicit formula for the cocharge cs{x) of a triangle x € l s which is obtained from the latter formula for c$(y) by means of the ultradiscretization map: Cs(x)

= Ci(x) + 2^15 - 2^14 + £25 - £24 + £34 - £45 + £44 ~ £55 + m i n { £ 1 3 - £ 1 2 , £22 - £ 3 3 , £23 - £34, £24 - £ 2 3 , £22 - £ 3 4 + £45 - £ 4 4 , £13 - £12 + £33 - £34 + £45 - £ 4 4 , £13 - £12 + £14 - £15 + £44 - £ 5 5 , £14 - £13 + £24 - £23 + £45 - £ 3 4 , £14 - £13 + £35 - 2 £ 3 4 + £ 4 5 , £14 - £15 + £22 - £33 + £44 - £ 5 5 , £14 - £15 + £23 - £34 + £44 - £ 5 5 , £14 - £15 + £24 - £23 + £44 - £ 5 5 , £24 - £23 + £33 - £34 + £45 - £ 4 4 , £25 - £24 + £35 - 2£34 + £ 4 5 , £33 - 2£34 + £35 - £44 + £ 4 5 , 2 £ i 4 - £ 1 2 + £24 - £15 + £44 - £25 - £ 5 5 , £14 - £15 + £23 - £25 + £24 - £34 + £33 - £ 5 5 , £14 - £15 + 2£24 - £25 + £33 - £35 - £ 5 5 , 2 £ 1 4 - £13 + £ 2 2 - £15 + £24 - £25 + £44 - £33 - £ 5 5 , 2 £ 1 4 - £13 + £ 2 3 - £15 + £24 - £25 + £44 - £34 - £ 3 5 , 2 £ i 4 - £13 + 2 £ 2 4 - £15 + £44 - £25 - £35 - £ 5 5 , £14 - £15 + 2 £ 2 4 - £25 + £34 - £35 + £44 - £45 - £ 5 5 , £15 - £14 + £25 - £24 + £35 - 2£34 + £45 - £44 + £ 5 5 ,

2^14 - 2 x i 5 + 2 x 2 4 - £25 + £34 ~ ^35 + 2a; 4 4 - X45 -

2x55}.

It is very interesting problem to describe explicitly the linearity domains ("fan") of the continuous piece-wise linear function cn(x) on the space of triangles Xn ~ R n ( n + 1 )/ 2 , and investigate the corresponding toric variety. References 1. Baccelli F., Cohen G., Olsder G.J. and Quadrat S.-P., Synchronization and linearity; an algebra for discrete event systems, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Chichester, 1992; 2. Bender A. and Knuth D., Enumeration of plain partitions, Journ. Comb. Theory, ser. A 13 (1972), 40-54; 3. Berenstein A.D., Fomin S.V. and Zelevinsky A.V., Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), 49-149; 4. Berenstein A.D. and Kirillov A.N., The Robinson-Schensted-Knuth bisection, quantum matrices and piece-wise linear combinatorics, in preparation; 5. Berenstein A.D. and Zelevinsky A.V., Tensor product multiplicities and convex polytopes in partition space, J. Geom. Phys. 5 (1988), 453-472; 6. Berenstein A.D. and Zelevinsky A.V., Triple multiplicities for sl(r + 1) and the spectrum of the exterior algebra of the adjoint representation, Journ. Algebraic Comb. 1 (1992), 7-22. 7. Berenstein A.D. and Zelevinsky A.V., Canonical bases for the quantum group of type Ar and piece-wise linear combinatorics, Duke Math. Journ. 82 (1996), 473-502; 8. Donin I.F., Multiplicities of Sn-modules and the index and charge of tableaux, Funct. Anal. Appl. 27 (1993), no. 4, 280-282; 9. Eilenberg S., Automata, languages and machines, vol.A and vol.B, Academic Press, New York, 1974, 1976; 10. Frenkel E. and Szenes A., Crystal bases, dilogarithm identities and torsion in algebraic K-groups, Journ. Amer. Math. Soc. 8 (1995), no. 3, 629664; 11. Fulton W., Young Tableaux, Cambridge Univ. Press, Cambridge, 1997. 12. Fulton W. and Harris G., Representation theory, Graduate Texts in Math., vol 129, Springer-Verlag, 1999; 13. Gansner E.R., On the equality of two plane partition correspondences, Discrete Math. 30 (1980), 121-132;

149

14. Gelfand I.M. and Zelevinsky A.V., Polyhedra in a space of diagrams and the canonical basis in irreducible representations of gln, Punkt. Analis i Prilozhen. 19 (1985), No. 2, 72-75 (in Russian); 15. Gelfand I.M. and Zelevinsky A.V., Multiplicities and proper bases for gln, Group Theoretical Methods in Physics (Yurmala, 1985), vol.2, Nauka, Moscow, 1986, pp.22-45; English translation: VNU Sci. Press, Utrecht, 1986, pp.127-159; 16. Gessel I. and Viennot G., Binomial determinants, paths and hook length formulae, Adv. Math. 58 (1985), 300-321; 17. Hatayama G., Hikami K., Inoue R., Kuniba A., Takagi T. and Tokihiro T., The A\J automata related to crystals of symmetric tensors, Preprint math.QA/9912209, 45 p.; 18. Golan J.S., Semirings and their applications, Kluwer Academic Publ., Dordrecht, 1999; 19. Jimbo M., Misra K.C., Miwa T. and Okado M., Combinatorics of representations of Uq(sl(n)) at q = 0, Commun. Math. Phys. 136 (1991), 543-566; 20. Kashiwara M., On crystal bases of the q-analogue of the universal enveloping algebra, Duke Math. J. 63 (1991), 465-516; 21. Kashiwara M. and Nakashima T., Crystal graphs for representations of the q-analog of classical Lie algebras, J. Algebra 165 (1994), 295-345; 22. Kirillov A.N., Ubiquity of Kostka polynomials, Preprint math.QA/ 9912094, 104p.; 23. Kirillov A.N., Discrete bilinear Hirota equations, Schiitzenberger involutions and rational representations of Symmetric group, Lectures given in Okinawa, December 1998, in preparation, 1998, 20p.; 24. Kirillov A.N. and Berenstein A.D., Groups generated by involutions, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux, St. Petersburg Math. J. 7 (1996), no.l, 77-127; 25. Knight H. and Zelevinsky A.V., Representations of quivers of type A and the multisegment duality, Adv. Math. 117 (1996), 273-293; 26. Knuth D., Permutations, matrices, and generalized Young tableaux, Pacific Journ. Math. 34 (1970), 709-727; 27. Krattenthaler C., The major counting of nonintersecting lattice paths and generating functions for tableaux, Memoirs of AMS 115 (1995); 28. Lascoux A., Leclerc B. and J.-Y. Thibon, Polynomes de Kostka-Foulkes et graphes cristallins des groupes quantiques de type An, C.R. Acad. Sci. Paris 320 (1995), 131-134; 29. Lascoux A. and Schiitzenberger M.-P., Sur une conjecture de H.O. Foulkes, C. R. Acad. Sci. Paris 286A (1978), 323-324;

150

30. Lascoux A. and Schiitzenberger M.-P., La monoid plaxique, in "Noncommutative structures in Algebra and Geometric Combinatorics", ROMA, CNR, 1981, 129-156; 31. Lusztig G., Total positivity in reductive groups, in: Lie theory and geometry, Progr. Math. 123, Birkhauser, Boston, MA, 1994, pp.531-568; 32. Lusztig G., Total positivity and canonical bases, in: Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser. 9, Cambridge Univ. Press, Cambridge, 1997, pp.281-295; 33. Lusztig G., Introduction to Quantum groups, Progress in Math. 110, Birkhauser, Boston. MA, 1993; 34. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford, 1995; 35. Nakayashiki A. and Yamada Y., Kostka polynomials and energy functions in solvable lattice models, Selecta Math., New Ser. 3 (1997), 547-599; 36. Noumi M. and Yamada Y., Affine Weyl groups, discrete dynamical systems and Painleve equations, Comm. Math. Phys. 199 (1998), 281-295; 37. Pin J.-E., Tropical semirings, Idempotency Analysis (J. Grunawardena ed.), Publ. Newton Inst. 11, Cambridge Univ. Press, Cambridge, 1998, 50-69; 38. Schiitzenberger M.-P., Promotion des morphisms d'ensembles ordonnes, Discr. Math. 2 (1972), 73-94; 39. Takahashi D. and Satsuma J., A soliton celluar automaton, Journ. of the Physical Society of Japan 59 (1990), 3514-3519; 40. Tokihiro T., Takahashi D. and Matsukidaira J., Box and ball system as a realization of ultradiscrete nonautonomous KP equation, J. Phys. A 33 (2000), 607-619; 41. Wolfram S., ed. Theory and applications of cellular automata, World Scientific, Singapore, 1986; 42. Zelevinsky A.V., A generalization of the Littlewood-Richardson rule and the Robinson-Schensted-Knuth correspondence, Journ. of Algebra 69 (1981), 82-94.

BETHE'S STATES FOR GENERALIZED A N D XXZ MODELS

XXX

ANATOL N. KIRILLOV & NADEJDA LISKOVA Graduate School of Mathematics, Nagoya University Chikusa-ku, Nagoya 486-8602, Japan and Steklov Mathematical Institute at St.Petersburg Fontanka 27, St.Petersburg 191011, Russia E-mail: [email protected] For any rational number po > 1 we prove an identity of Rogers-RamanujanGordon-Andrews' type. Bijection between the space of states for XXZ model and that of XXX model is constructed. 1

Introduction

The main goal of our paper is to study a combinatorial relationship between the space of states for generalized XXZ model and that for XXX one. In our previous paper 4 we gave a combinatorial description of states for generalized XXZ model in terms of the so-called sl(2)-XXZ rigged configurations. On the other hand it is well-known that when the anisotropy parameter po of XXZ model goes to infinity then the XXZ model under consideration transforms to the XXX one. We are going to describe this transformation from combinatorial point of view in the case when po is an integer. A combinatorial completeness of Bethe's states for generalized XXXmodel had been proved in 1 and appeared to be a starting point for numerous applications to combinatorics of Young tableaux and representation theory of symmetric and general linear groups, see e.g. 2 . Here we mention only a "fermionic" formula for the Kostka-Foulkes polynomials, see e.g. 2 , and the relationship of the latter with s£(2)-branching functions 6*A°(q), see e.g. 3 . We will show in Section 1, Theorem 2.3 and Remark 2.5, that g-counting of the number of XXZ states using Bethe's ansatz approach 5 ' 6 , gives rise to the Rogers-Ramanujan-Gordon-Andrews' type identity for any rational number P0>1. It seems an interesting problem to find a polynomial version of the RogersRamanujan type identity (2.12) from our Theorem 2.3. Another question which we are interested in is to understand a combinatorial nature of the limit XXZ

— Po-v+oo

151

XXX.

152 In Section 3 we describe a combinatorial rule which shows how the XX Zconfigurations fall to the XXX pieces. For simplicity we consider in our paper only the case po > J2m s™- General case will be considered elsewhere. 2

Rogers—Ramanujan's type identity

This paper is a continuation of our previous work 4 . Let us remind the main definitions, notation and results from 4 . For fixed po 6 R, po > 1 let us define (cf. 5 ) a sequence of real numbers Pi and sequences of integer numbers i;,, mj, j/j, z^: PO -=P0,

Pi

= 1,

Vi

=

Pi

Pi+i = Pi-i ~ Vi-\Pu i = 1,2,... (2.1)

Pi+i

y - i = 0,2/o = l, J/i = fo, Z/*+i =Vi-i +^iVi, i = 0 , 1 , 2 , . . . ,z_i = 0 , z0 = 1, zi=vi, zi+i = Zi-i + Vi+iZi, i = 0 , 1 , 2 , . . . m 0 = 0, mi = f0, mi+i =rrii + i/i, i = 0 , 1 , 2 , . . . r(j) =i, if mi < j < mi+i, j = 0 , 1 , 2 , . . .

(2.2) (2.3) (2.4) (2.5)

It is clear that integer numbers i>; define the decomposition of po into continuous fraction i

Po = [vo, v\, v2, • • •} = "0 H

1

Vl + V2 + . . .

Let us define (see Fig. 1) a piecewise linear function rij, j > 0, rij := yt-i + (j - rrii)yi, if m ; < j < mi+1.

(2.6)

It is clear that for any integer n > 1 there exists the unique rational number t such that n = ntLet us introduce additionally the following functions (see 4 ) qj - (-lY(Pi

~ U ~ mi)Pi+i),

7r~{
where 2s = n v — 1.

if rrii<j<

mi+1,

if nk > 2s,

tyo I (_l)r(*)-l r— (qk - qxnk) + -1 ^ , if "* < 2s, 2p0 I

(2.7)

153

Vi+i-Vi

rrii-i mj_i+l m— 1 m*

rrii+1 mi+i-l

mi+1

j

Fig.l. Image of piecewise linear function rij in the interval [mj_i,mj + i]

In what follows we assume that the anisotropy parameter po > 1 is a rational number, and po = [VQ, V\, ..., ua] denotes its decomposition into continuous fraction. It is not difficult to see that Po - Wo,vi,---,^a] Po := [v0,vi,...,va-i]

=

Va+l

=

Za-l '

where the numbers {j/j}"=0 and {z,-}?=0 are defined by (2.2) and (2.3) correspondingly. We assume that if a > 0, then va > 2. In order to formulate our main result of the paper 4 about the number of Bethe's states for generalized XXZ model, let us consider the following symmetric matrix 0 _ 1 = (cy)i
if \i — j \ > 2.

154

ii) Cj-ij = (—1)J

, if rtii < j < m »+i-

2(—1)*, if rrii < j < m*+i — 1, i < a, «i) CJJ = { (-1) 4 , if j = mi+i - 1, j < a, ( - l ) a + 1 , if j = ma+1. Example 1 Let us take po = —, then po = [2,3,2], a = 2, p = [2, 3] = - , i/0 = 2, i / i = 3, i/2 = 2; mo = 0 , mi — 2, 7712 = 5 , mz — 7; 2/o = 1, 2/1 = 2, 2/2 = 7, 2/3 = 16; z0 = 1, 21 = 3, z2 = 7. Therefore, p 0 = ——, Po

, and Za-l

f i, l + 2(i-2), 2 + 70"-5), U+16(j-7),

if 0 < j < 2, if2<j<5, if5<j<7, if 7 < j .

Finally,

0

and

5 3 2 1 1\ /9 7 7 - 7 -5 - 3 - 2 - 1 - 1 5 - 5 15 - 9 - 6 - 3 - 3 1 3-3 -9-15-10-5-5 16 2 - 2 - 6 - 1 0 4 2 2 1 -1 -3 -5 2 9 9 \ 1 - 1 -3 -5 2 9-7/

0-x

/I 1 1 -2 1 1 -2 1 = 1 -1 -1 -1 2 -1 -1 1

V

\

1

1-1/

detl©"1^^.

Note, see 4 , Theorem 4.7, that the absolute value of the determinant d e t ( 0 - 1 ) is equal to ya+i, i-e. the numerator of poTo continue, let us consider the matrix E = (ejk)i<j,k<ma+i, e

jk

= (~l)r

(&j,k - Sj,ma+1-l

• Sk,ma+1

+ ^j,ma+1

"

where

Sk,ma+i-l)-

155

One can check that the vacancy numbers P,-(A), see 4 , (3.9), can be computed as follows Pj(\) + \j = {{E - 20)A* + 6 %

1< j <

where the j - t h component of vector b = (bi,... ,bma+1) following formula

s, = (-1)*> (»A ^~iy-a

l-V2* (n,{^^)^

ma+1, is defined by the

„.jv,

i l m • llm Vj,2s

,

and for any sequence of integer numbers A = ( A i , . . . , A mQ+1 ) we denote by A the sequence ( A i , . . . , Xma+1), where Xj — (—l)r^'Xj, 1 < j < ma+i. Theorem 2.1 (4) The number of Bethe's states Zxxz(N,s ralized XX Z model is equal to

A

| /) of the gene-

•3

j

where summation is taken over all sequences of non-negative integer numbers A = {Xk}™^?1 such that ma+i

^2 nkXk = 1 , Xk> 0; fc=i

A = (Ai,....A m a + 1 ), Xj = (-l)rij)\j,

£ = 20.

Recall, see 4 , or Section 3, that N and s in the above formula for the number of states ZXXZ(N, s\l) denote vectors N = (Ni,..., Nk) and s=

(si,...,si,...,sk,...,sk),

>

v

'

«

v

'

i.e. Nm is equal to the number of spins in the XXZ-chain which are equal to sm. One of the main goal of the present paper is to consider a natural gr-analog for (2.8). Namely, let us define the following g-analog of the sum (2.8)

A

where

ej = ( - l ) r W .

j

((E - B)Xt + bl)j A,'

(2.9)

156

Let us recall that

M N

M N

is the Gaussian q'-binomial coefficient: (9; q) M -,iiO
= <

0,

<M,

otherwise.

Remark 2.2 In our previous paper 4 , see (5.1) and (5.2), we had considered another g-analog of (2.8). It turned out however that the g-analog (5.1) from 4 , probably, does not possess good combinatorial properties. One of the main results of the present paper is the following: Theorem 2.3 Assume that po > 1 be a rational number, and consider a rational function

V|(9):=^,(M)(
\\B\'

n^^v

(2.10)

summation in (2.10) is taken over all sequences of non-negative integer numbers A = {Afc}^"1"1 such that

l = ^2nk\k,

Afc > 0.

fc>i

Then we have l2

Y,Q"V<»>\q)

(2.11)

l>0

1+

£

(-D

iiM^-fc+m

{kya+1+mya)(kza+mza-i)+Aa(k,m)

,

{ ir) Q^k,m L (q)

(q;q)k

k >m >0 (*,m) # (0,0)

where

Q^(q)--=Qi±l(q)

=

q1¥{k+m)

fc-i m

+

l±i(2fc-m)

fc-1

m —1

A a ( * , m ) = 1 + ( 0 - 1 ) a r*- f " > l + , 1 - ( - 1 ) t t ^ r o2 I "

157

Let us emphasize that polynomials Q\ ^(q) are the same for all rational numbers po > 1Taking the sum with respect to the index m in the RHS(2.11), we obtain Corollary 2.4 (Rogers-Ramanujan-Gordon-Andrews' type identity)

= l + Yl{-l)1=i^V'(ya+1+y'){'a+Za-l)-^f^,

Y^q^V^iq) L(*a\q)

where

(2-12)

=

\~"* (_-\\™a™2y*'o,-i-km(ya+iza-i+2yc,za-1+yaza)+Aa(k,k-m)

Q^~1^°'^

(a)

m>0

A proof of identity (2.11) is a "g-version" of that given in 4 , Theorem 4.1. Remark 2.5 (Gordon-Andrews' type identity) Let po > 1 be an integer, then a = 0, y\ = po, yo — 1, ZQ = 1, z_i = 0, and the RHS of (2.11) takes the following form 1 ,

V

f - 1 \k+mtkp0+m)k+

(*-~)(fc-"-'>

Qfc.m(g)

fc > m > 0 (*,m)#(0,0)

J^+M^II

r *

- i + D - i ) * W2 - ^ - E ( - » r « " * , ' 0 . ) • (»•«)

fc>0 '^* lm=0 J It is not difficult to see that if k > 0, the sum in the brackets (2.13) is equal to (1 + qk)(q;q)k- Hence, if po > 1 is an integer, then we come to the following identity: £«&V,^?) = l + £(-l)V l>0

a

*

+ i

^ ( l +
(2-14)

k>0

Using the Jacobi triple identity, one can rewrite (2.14) in the following forms

J2lkVl{po\q)

(2.15)

l>0 =

T T ( 1 - g(2Po + l ) n ) Q _

( 2 p 0 + l ) n - p 0 - l \ Q _ ~(2po + l ) n - p o )

n>l

X)««^i (P0) (9)= '^°

II n^0,p 0 ,po+l(mod 2p 0 +l)

(1-9T1-

(2-16)

158 It looks very challenging task to find for any rational number p0 > 1 an explicit product formula (Weyl's denominator identity) for the right hand side of identity (2.11). We consider the identity (2.12) as an identity between bosonic and fermionic formulae for the character of "vacuum representation" of the generalized Kac-Moody algebra corresponding to the matrix 0 _ 1 . 3

XXZ

-> XXX

bijection

In this section we are going to describe a bijection between the space of states for XXZ-model and that of XXX-model. Let us formulate the corresponding combinatorial problem more explicitly. First of all as it follows from the results of our previous paper, the combinatorial completeness of Bethe's states for the XXZ model is equivalent to the following identity

J[(2sm + 1)N™ = J2 ZXXZ(N, s\l), m

(3.1)

(=0

where N = ^ 2 s m i V m and the numbers Zxxz(N,s

| I) are given by (2.8).

771

On the other hand it follows from the combinatorial completeness of Bethe's states for XXX model (see x ) that 1JV

N

Y[(2sm + 1) ™ = £ ( 7 V -21 + 1)ZXXX(N, m

s | 0,

(3.2)

1=0

where the number Zxxx(N,s | I) stands for the multiplicity of ( y — Z)-spin irreducible representation of s/(2) in the tensor product V®N'®---®V®N™. Let us remark that both numbers Zxxz(N,s | /) and Zxxx(N,s | I) admit a combinatorial interpretation in terms of rigged configurations. The difference between the space of states of XXX model and that of XXZ model is the availability of the so-called 1~-configurations (or 1~ string) in the space of states for the latter model. The presence of 1 _ -strings in the space of states for XXZ-model is a consequence of broken s/(2)-symmetry of the XXZmodel. Our goal in this section is to understand from a combinatorial point of view how the anisotropy of XXZ model breaks the symmetry of the XXX chain. More exactly, we suppose to describe a bijection between XXZ-rigged configurations and A'A'A'-rigged configurations. Let us start with recalling a definition of rigged configurations.

159

We consider at first the case of sl(2) XXX-magnet. Given a composition /i = (HI, \ii,...) and a natural integer /, by definition a s/(2)-configuration of type (/, fi) is a partition v \-1 such that all vacancy numbers Pn{u; H) := ] T min(n, fik) - 2 ^

(3.3)

v'vkk

k
are nonnegative. Here v' denotes the conjugate partition to that v. A rigged configuration of type (I, fi) is a configuration v of type (I, fi) together with the collection of integer numbers { Ja}™2i which satisfy the following inequalities 0<Ji<J2<---<Jm„w<

Pn{v, /i).

Here mn(v) is equal to the number of parts of the partition v which are equal to n. It is clear that the total number of rigged configurations of type (l,fi) is equal to the following number

Z (Mri:=En( v\-l n>\

p (

" "t;r (, ' ) y

K

x

'

'

x

The following result had been proved in . Theorem 3.1 Multiplicity of (N — 21 + 1)-dimensional irreducible representation of si (2) in the tensor product y®Wi ® • • • ® SI

is equal to the number

Z

v.9NUrn

I | 2 s i , . . . , 2s\,...,

Example 2 One can check that y®5 = ev0 + i5Fi

2sm,...,

2sn

+ i5y2 + ioy3 + m + v5.

In our case we have /i = (2 5 ). Let us consider 1 = 5. It turns out that there exist three configurations of type (3, (2 5 )), namely

10

0 2

Hence, Z(3 | (25)) = 1 + 2 + 3 = 6 = Mult Vo (V® 5 ). Now let us give a definition of sl(2)-XXZ configuration. We consider in this Section only the case when the anisotropy parameter po is an integer,

160 po € Z>2- Under this assumption the formulae (2.6) and (2.7) take the following form: nj = j , if 1 < j < po, Vj = + 1 ; n

Po

=

•*•> vpo

2*/fc,2s =

~

~ ^i

2sk

min(A;, 2s), if 1 < k < p0, 2s + 1 < p0; Po 2s 2$p0,2s = —, if 2 s + 1
= 1> if 1 < k
aj := a,j(l \ fi) = ^ m i n ( j , ^ m ) -21-

j E

m

m

/*fi» - 2/ Po

if i < j
Em^m-21 a

P0

(l\

/•»)

=

PO

Definition 3.2 A sl(2)-XXZ-configuration of type (Z,/i) is a pair (A, A Po ), = where A is a composition with all parts strictly less thanpo, V^ j^j+^p0 h j
and such that all vacancy numbers Pj (A | n) are nonnegative. Let us recall 4 that if the anisotropy parameter p0 > 2 is an integer, then

Pj(\\li) := aj(l \n) + 2 J2 (k' ^')A* + A P ° >

if

3
j
P P o -i(A | /x) := apo-i(l P

PO(X

| jt) + AP0;

I M) == apoC I M) + V - i -

In the sequel we are displaying a configuration (A, APo) = (Ai, A2,..., A Po _i, APo) as the diagram of the following partition (1 A I + A PO , 2 A a , . . . , (p0 - l) A p °- 1 ). Example 3 Let us consider po = 6 , s = | , i V = 5 ) / = 5. The total number of type (5, (35)) sl{2)-XXZ configurations is equal to 12.

161

3 * 0 * *

l*l 0 * * * *

* *

H

m

l*l

0

2

The total number of type (5, (35)) rigged configurations is equal to Zxxz(5

| (35)) = 101 = 1 + 4 + 3 + 7 + 10 + 10 + 6 + 16 + 8 + 12 + 18 + 6.

Here we have used the symbol £ to mark the 1 _ -strings. Now we are ready to describe a map II from the space of states for XXZ model to that of XXX one. More exactly we are going to describe a rule how a XXZ-configuration falls to the A'A'A'-pieces. At first we describe this rule schematically:

*

k<

r

/J

A

n}

A

A ~^\m—\

+

+

J

X

~^\m—\ + . • +

^

lm-1

k<

•

SI This decomposition corresponds to the well-known identity m+k k

k

3=0

m+ j j

In what follows we will assume that po > J2m s™Theorem 3.3 The map U is well-defined and gives rise to a bijection between the space of states of XXZ -model and that of XXX one.

162

Proof. Let us start with rewriting the formulae (3.4) for the XXZ-vacancy numbers in more convenient form, namely,

pfxz{v i M) = E

Em2Sm-2/

min

0'> 2sm) - 2 xy fc - J k<j

m

Po

if 1 < j < Po - 1;

Em2sm-2/

^i Z (^lM)=Po
'l

*Po

J

+ APo;

Po

(3.5)

Em2sm-2/

+ mP0-i(v). Po Here ^ = ( 2 s i , . . . , 2sm) and v is a pair v = (z/, APo), where v is a partition such that l(y) < p0 - 1 , |i/| + APo = /. Relationship between A from Definition 1 and v is the following {v\n)

=

mj{v) = \j,

i.e. u= (l A l 2 A 2 ...(p0 - l ) ^ - 1 ) .

Next, let us consider an integer / < J^ s m and let v \- I be a configuration. Let Apo be an integer such that

XXX-

2 ^2 sm - 21 - p0 < APo

771

and consider the pair v — {v, APo). It is easy to check that

Pxxz(u

| ii) = *£mm(j,2sm) m

- 2 j > i = PfXX{v

\ /*) > 0,

k<j

if 1 < j < po - 1;

Ppo-ii^ I li) = E

2s

™-

2l

+ V > 0;

^ X Z ( £ I /*) = AP0_! > 0. Thus the pair v = [v, APo) is a XXZ-configuration. Furthermore it follows from our assumptions (namely, ^2msm < p0, APo > 0) that APo_i = 0 and both l~-strings and (po — l)-strings do not give a contribution to the space of XXZ-states. Thus we see that both XXX-configuration v and XXZ-configuration v — (u, APo) define the same number of states. Now, \iv = (v,^) is a XXZ-configuration then v is a XXX configuration as well. This is clear because (see (3.5)) Pxxx(u

| M) > Pxxz(u

\fi),

l<j
1.

163

By the similar reasons if {u, APo) is a XXZ-configuration, then for any integer k, 0 < k < APo, the pair (i/, Xpg - A;) is also a XXZ-configuration. It follows from the above considerations that II is a well-defined map. Furthermore, there exists one to one correspondence between the space of XXXconfigurations and that of XXZ-configurations, namely, v <->£=

(v,\P0),

where AP0 = [ E r a s m - WWAll others XXZ-configurations (u, k) with 0 < k < J2m a contribution to the space of descendants for v 44 v.

s

1 give

References 1. Kirillov A.N., Combinatorial identities and completeness of states for the generalized Heisenberg magnet, Zap. Nauch. Sem. LOMI 131 (1984), 88-105. 2. Kirillov A.N., On the Kostka-Green-Foulkes polynomials and ClebschGordan numbers, Journ. Geom. and Phys. 5 (1988), no. 3, 365-389. 3. Kirillov A.N., Dilogarithm identities, Progress of Theor. Phys. Suppl. 118 (1995), 61-142. 4. Kirillov A.N. and Liskova N.A., Completeness of Bethe's states for generalized XX Z model, Int. Jour. Mod. Phys. 7 (1992), 611-621. 5. Takahashi M. and Suzuki M., One dimensional anisotropic Heisenberg model at finite temperature, Progr. of Theor. Phys. 48 (1972), no. 6B, 2187-2209. 6. Kirillov A.N. and Reshetikhin N.Yu., Exact solution of the integrable XXZ Heisenberg model with arbitrary spin, J. Phys. A.: Math. Gen. 20 (1987), 1565-1597.

T R A N S I T I O N O N G R O T H E N D I E C K POLYNOMIALS A. LASCOUX CNRS, Institut Gaspard Monge, Universite de Marne-la- Vallee 77454 Marne-la-Vallee Cedex, France E-mail: [email protected] Grothendieck polynomials represent Schubert varieties in the Grothendieck ring of the flag manifold (for Gl(n,C)). We describe how general Grothendieck polynomials are related to those for Grasmann manifolds, which themselves are deformations of Schur functions.

1

Introduction

Given a vector bundle V of rank n, the Grothendieck ring of classes of vector bundles of the relative flag manifold T{V) is generated by the classes o i , . . . ,an of the so-called tautological line bundles on .F(V). The structure sheaf of a Schubert variety in TiV), having a finite resolution by vector bundles, can be expressed as a (Laurent) polynomial in the ai, 1/aj. Explicit representatives G>, a 6 6 n were defined in 16 , 17 under the name "Grothendieck polynomials " (we recall their definition in Eq. 16). We shall also use codes of permutations rather than permutations, and in that case, note G[J], J &W1 (codes are defined in section 2). I gave in 12 , 13 two different expressions of the G[J], J partition, that I recall in the last section. As usual in the theory of symmetric functions, it is appropriate to let the number of indeterminates be infinite and to obtain the finite case by specializing xn+i = 0 = xn+2 = • • • • When using these conventions, we shall write partitions decreasingly and write g\ for the symmetric function in an infinite number of variables obtained from the symmetric function G[J] (in x x , . . . ,xn), with J = [0 < j i < • • • < jn], X = [j„,... ,jt]. In general, introducing some equivalence (taking the "stable part") that can be defined in four different manners which will be described in section 5, and that we shall note ~ , one can decompose any Grothendieck polynomial into a sum of Grasmannian ones

G„ S ^]>><7 A

(1)

The similar decomposition for cohomology classes (represented by Schubert polynomials) embeds into the tradition of "Schubert calculus" (for example, intersection multiplicities of Schubert varieties, generalizing the so-called Littlewood-Richardson rule). 164

165 Buch 1 studies the algebra generated by the g\, and the decomposition (1) that he explicits for certain permutations. One motivation to write this text was to answer his question about positivity of the coefficients in (1) (once taking the correct normalization), and I thank him for having sent me his preprint. Adapting what has been done for Schubert polynomials Xa, a 6 6 „ , in 15 and 18 , we shall give in Eq. (32) a formula which can be translated into a tree which describes the multiplicities in (1). Moreover, this tree, as in the case of cohomology, gives a way to explicit the multiplicative structure of the Grothendieck ring of the Grassmannian, which is the problem considered by Buch 1 . To generate the tree, one needs only to describe the multiplication of Grothendieck polynomials by l — l/ai, i = 1 , . . . , n, i.e. to describe the class of hyperplane sections of Schubert varieties. The similar rule for the cohomology ring is due to Monk (it was previously obtained by Chevalley) and leads to the explicit expression of Schubert polynomials as (positive) polynomials in the Chern classes of the tautological line bundles, as well as their stable part. The cohomological computations can easily be extended to two sets of indeterminates, due to the extension of Monk's rule (cf. 10 ), or translated into planar combinatorial constructions. The same is true for Grothendieck polynomials , but for simplicity, we shall restrict here to only one set of indeterminates. Grothendieck polynomials are obtained from the action of the O-Hecke algebra on the ring of polynomials. Let us emphasize that this action is not unique and depends upon several arbitrary parameters that we shall have to put into use (the action of the Iwahori-Hecke algebra of the symmetric group also depends upon several parameters, cf. 19 ). There exists quantum analogies of Grothendieck polynomials and we refer to the work of Kirillov8 for their definition and properties. The geometry of Grassmann varieties is well understood thanks to Schubert, Giambelli and their successors9. It is hoped that relating the Schubert subvarieties of a flag manifold to those of a Grassmannian will be of some help to understand those varieties, the singularities of which we still do not know how to describe 2 . In the present article, we do not consider Schubert varieties, but only the classes of their structure sheaves. Much of the combinatorics of the symmetric group that appear in my work with M.P. Schiitzenberger finds application to Schubert calculus, and I mentioned in this text the connections with 1 5 - 2 2 .

166

2

O-Hecke algebra

The O-Hecke algebra H of the symmetric group 6 „ is the algebra generated by T i , . . . ,T n _i satisfying the braid relations TiTi+iTi = Ti+iTiTi+i

& TtTj = TjTt , \i - j \ ^ 1

together with T? = Ti. Given arbitrary parameters a,fi,j,6 € C, such that aS — /?7 = 1, one gets an action of % on the ring C[af,... , a±] by setting ( a a » + ft)(7Q«+i + 8)f ~ (QOi+i + Fiinai + S)fSi o-i — a»+i

T--ft-+fT--=

where Sj exchanges Xi,Xi+\ (cf.18,19 for a more general statement; operators on polynomials will be denoted on the right). Moreover, the elements 1 - Ti also satisfy the braid relations together with (1 - Ti) 2 = 1 - Ti. The operators / ^ fdi :=

/-/Si o»+i

id differ* are Newton's divided differences. The operators , v , / Q a « / _ ai+ifs f »-> /7Ti := fa,idi = — : — a»+i and the operators / (->• /7fj := fdi Oj+i =

aj+i/-Oj+i/s Zj — Oj+i

3

are due to Demazure and called isobaric divided differences in 16 . Let us also use the variables Xi := 1 — 1/aj, and the corresponding operators that we shall distinguish by superscripts (for example, df sends x\ onto 1, and ai = 1/(1 — xi) onto 0102). Since braid relations are satisfied, to any permutation a are associated operators To-,7T£, 7r^, <9", d*, 7?", 7?* which can be obtained by taking any reduced decomposition of a and evaluating the associated product of elementary operators. In particular, the operators associated to the maximal permutation u) S &n, CJ :=[n,... ,1] play a fundamental role. We shall also need the isobaric divided differences corresponding to the shifted variables ai—1,02—1,.-• ,a n —1 that we shall denote X?i,... ,Dn-i to avoid any confusion with the preceding ones.

167 All these operators can be expressed in terms of Newton's divided differences only, but it greatly simplify computations, and comprehension, to introduce them instead of restricting to divided differences. Most of their properties directly arise from the following relations, which can be easily checked from their concrete action on polynomials or using the braid relations together with Leibniz formula: / gdi- f (gdi) + (/ <%) gSi. For example, Eq. (2)-(7) are statements involving only two indices {i, i+l}. The operators commuting with multiplication by a function invariant under Sj, it is sufficient to check their action on the two polynomials 1 a n d Xj.

L e m m a 1. Divided differences satisfy the following relations :

7rf = d? =

(l-xi+1)df ; 5rf = df (1 -

Xi)

(2)

(i-xi)(i-xi+1)d? = df (1 - Xi)(l - Xi+1)

(3)

Trf = ( 0 i - l K + i df ; 5rf = 5?ai(a i + i - 1)

(4)

df = a{ai+1 df = ai+1 nf = 7?° cii = dfa,iai+i Di = Xi(l - xi+i) df =

(ai-l)d?

(6) -

i—i

xi+1) ; (xi - 1) 7T? = 5rf (xi+1

•i irf = nf xi+i + (1 -

(5)

(7)

< £> n -i • • • Z?i= I? n _i •••Dx nf+1 , i < n, - 1 ^ 1 • • ' T T n - l TTi = TTj+l 7?1 • • • 7T„_i , i < n •( a i

_ l

)

n - l . . .

( a n

_

1 )

O

a

a

<

— xl =

'''

1

x 7v

n u

( l _ a ; 2 ) l . . . ( l _ a ; n ) « - l QX

(8) (9) (10) (11)

Only the last two relations require a comment. Indeed d^ can be charaterized as the &ym(ai,... ,a n )-linear operator which annihilates all monomials ax := aXl • • • a*", A < p := [ n - 1 , . . . ,0] and sends ap onto 1 (the preceding monomials are a basis of C[af,... , a„] as a free module over the ring &ym:= ring of symmetric functions in the a^s or Xj's). Using this fact, and the remark that all the operators that we have written are &ym-\ineax, one checks that

as = £ (-D^* n J^a.) = n (I-a-) £ *

(12)

168

< = ofdu = = - — i

^

£ a

(13)

and, by change of variables a^ t-> Oj — 1,

^=

E-=nH^ a = ( a i - i ) n _ 1 --- ( a "- i ) 0 ^

^ Ht<j

aj-1

i<j

3

(14) More generally, one has, for aS — @j = 1, ^

= n

(aai

+

0)(,aj+6)

^

^=

+

^

+

^

^

In the above factorization, we consider Tu as an operator on polynomials, not as an element of the Hecke algebra. Otherwise, we should have to work in the affine Hecke algebra. The Grothendieck polynomials Ga, a G Gn, are defined as follows:

r GW = * r 1 •.. *» = (i - i ) - i • • • a - ijo

(i6)

^ G^ — Gw 7r2CT Since Schubert polynomials are defined as the images under the d% of xp, and since 7rf = (1 — Xi+i)df, one sees that the term of smallest degree of Ga is the Schubert polynomial (in the a^'s) of the same index. Grothendieck polynomials , as well as Schubert polynomials, are a &ymb a s i s o f C [ a f , . . . ,a±] = C[xf,... ,x±]. It is also convenient to index Grothendieck polynomials by codes of permutations, instead of permutations. The code of a permutation a = [cr 1 ,... , «Tn] € 6 „ is the vector J — \ji,... ,jn] such that j , := #{A; > i,ak < a1}. We shall write Ga or G[J] indifferently. Grothendieck polynomials are stable with respect to. the embeddings ©n = ©n x ©l c_> 6 n + i , and accordingly, one can freely add or suppress terminal O's to codes without changing the polynomials : G[J] — G[J0.. .0). Instead of starting from G[p] = xp for a given n, and letting n vary, one can as well rewrite the preceding definition as follows. Say that a vector Av= [Ai,... , A*] is dominant if Ai > • • • > A* > 0. Then J G[A] = xx A dominant \ jk+l This convention allows to have all n's at the same time.

, .

169

3

Symmetric Grothendieck Polynomials

On the Grothendieck ring of classes of vector bundles over a manifold act operators induced from the exterior powers and symmetric powers of vector bundles, that we shall denote A1 and S1. Explicitly, taking an extra variable z, the generating function of the Sl is such that oo

£ z* S'(B - B') = H(l - zb')l ]J(1 - zb)

(18)

i=0

if 1 = 53 6, B' = 53 b1 are finite sums of classes of line bundles. More generally, given any partition K = [hi,... , kn], 0 < fci < • • • < fcn, one has a Schur functor SK, the action of which on sums B = 53™= l ^« °^ u n e bundles is SK(B) = det Isr^+i-^B)]l l

^i'^"

= det 6 ^ + n " ' *

l<»,j
/ det

L"-J

l
"

("19)

13

Still a little more generally again , Schur functors indexed by partitions in N" can be considered as morphisms from the n-th tensor power of a A-ring into itself : SK(M\...

,B") ^ d e t l S ^ ' - W \l
(20)

where B 1 , . . . , B n are arbitrary elements of the A-ring. 4

Stable part by restriction or symmetrization.

Let k, n S N, k > n, I £ N*. Then G[0fc_11] is symmetrical in x\,... therefore the restriction

, Xk and

G[ok-1r\\ is a symmetric function F[I] of fc variables. Similarly to the case of Schubert polynomials (cf. 15 , 25 ) we shall adopt the common terminology and call F[I] the stable part of G[I], writing also Fa if a is any permutation of code [JO . . . 0]. One needs to check some stability with respect to k, since one has only required k > n. This will be clarified by the following property.

170

T h e o r e m 2. 1) Let I EN1.

Then

G[I]Dn---Dl

= G[QL]

(21)

2) Let u) be the maximal permutation of &n.

Then

G[I]DU = F[I] G &ym{Xl,...

, xn)

(22)

Proof. G[I] is the image of some G[J], J € W dominant, under some product °f "IN i < n — 1. However, G[J] can be written as a determinant (with Afc :=: ai + • • • + afc) : G[J] = G[J0] = (ai • • • o n + i ) - n 5 n - + i ( A n + i - 0 > A „ - i n > . . . , A x - j i ) . Since Vi,j,fc G N, one has 5fc(Aj - j) Dt = S*(Aj + i - j - 1 ) ( A being the isobaric divided difference with respect to the pair ( a j - l , a ; + i - l ) ) , each Di in the product Dn • • • D\ operates in turn on a single column of the determinant, the others being symmetrical in (ai — 1, Oj+i — 1). Therefore the image of G[ JO] under Dn • • • D\ is (ai •••a n + i)~"5„n+i(A n + i - 0, A„ + i - j „ - l , . . . , A„ + 1 - j i - 1 ) i.e. is equal to the image of G[ji + 1 , . . . , j n + 1,0] under TT° • • • irf which by definition is G[0J]. Now, if G[I] = G[J]nfTTj • • • , then the commutations (8) show that G[0I] = G[0J] nf+1n]+1

••• = G[J]Dn • • • D^+i^^

' ' ' = GWD"

- ^

and this proves Eq. (21). To simplify indices, let us take k = n = 4 to attack Eq. (22) The morphism G[I] •-> G[000I], 7 G N 4 is D„ := (D4 • • • Dx) (D6 • • • DX) (D6 • • • Dx).

The

specialization x$ = XQ = xt of G [000.7] is the same as the specialization of G[000I]Dn, with rj = S6S5S4S6S5S6. However, un = u~i (= the maximal element of 67), and thus is equal to W4 (W4W7). Now, given any symmetric function of # i , . . . ,x4, applying to it any sequence of D, and specializing x 5 — 0 = xe = • • • does not change the function. Therefore G[I] DWi, which indeed is symmetrical in x\,... ,x±, coincides with G[/]JD W 7 | _ 0 = _ D Notice that if I is the code of a vexillary permutation, then G[/0 n _ 1 ] has a determinantal expression in terms of the Sfc(Aj — j), and the morphism G[/0 n _ 1 ] i->- G[0" _1 7] consists in replacing in the determinantal expression of G[/0 n _ 1 ] each term Sk(Ai - j) by S ,fc (A i+n _i - j-n+1). Specializing xn+i,xn+2,. • • to 0 , that is, an+i,an+2,... to 1, is obtained by putting A j + n = An + i, and thus the stable part of G[I] is a determinant in the Sk(An-j) (which is recognized to be the one expressing G[J], with J— increasing reordering of I).

171

In other words, if 7 e N" is the code of a vexillary permutation, then G[I] Du — G[J]. This leads to define, for a general I £ N71, a G-key polynomial KG[I] by KG[I] KG[... ,jk,jk+1,...]Dk

= G[I] = KG[... ,jk+i,jk,---]

/dominant jk> jk+i

.^

Using that DiDu = Du,i
(24)

induces a decomposition G[I]DU

= YJCLJKG[J]D„

(25)

of the stable part of G[I] into Grassmannian Grothendieck polynomials. In the case of Schubert polynomials, I gave with Schiitzenberger such a decomposition in terms of "Demazure characters", but it involves a combinatorics of tableaux which will be avoided here (cf. 21 , 26 ). A Grothendieck polynomial indexed by a permutation a = v x r/ belonging to a Young subgroup &m x &n '-> &m+n factorizes into two Grothendieck polynomials (proposition 6.7 of 1 4 ). It is clear, on the definition by restriction, that the stable part of Ga factorizes into the product of the stable parts of Gv and G,,. In particular, if v and n are vexillary, of respective codes J, K, which reorders into the partitions A, fi, then Ga^bG[J]G[K]st^gxg,, and therefore, as in the case of cohomology, any algorithm producing the stable part of a Grothendieck polynomial allows to describe products of Grassmannian Grothendieck polynomials. Instead of symmetrizing operators, or restrictions, we shall prefer, in the next section, using transitions. Grothendieck polynomials satisfy a recursion with respect to C[a;i,... , xn] = C[a;i] C[x2,... ,xn] (cf.17) that Fomin and Kirillov6 have written as a generating function in the O-Hecke algebra. This allows them to define in the same fashion stable Grothendieck polynomials : Let Ti be the generators of the O-Hecke algebra (with T 2 = — Tj), and for any n,r: n > r, let Antr{x) := (1 + zT„_i) • • • (1 + xTr)

172 Definition 6 . For any n, and any a 6 6 „ , the stable Grothendieck polynomials Fa are the coefficients of the generating function An,l(Xi) • • • An,l(xn)

= V

FaTa .

(26)

On the other hand, the generating function6 of Grothendieck polynomials G
Now, if TiTj-Tk is equal to T„, a G 6 „ , a Ti+n-iTj+n-i •••Tk+n-i is equal to Ta>, a' having fore, the coefficient of Ta< in the specialization T\ xn+i = 0 = • • • = X2n-i of £/2n-i is the same as Ai.ifai) • • • An,i(xn), that is, one has the identity Fa = G[On~H] l ^ ^ . . . ^ ^

(27)

having code / , then code [0 n _ 1 J]. There= 0 = • • • = T„_i; the coefficient of Ta in

= F. .

(28)

In fact, Grothendieck polynomials already appear at the level of the "twisted" group algebra of the symmetric group (i.e. the algebra generated by the Sj's, Xj's), as coefficients in the expansion of isobaric divided differences 7iv flV in the basis of permutations (cf. 22 ). Yang-Baxter equation in the O-Hecke algebra also lead to similar expansion (6, n ) . 5

T r a n s i t i o n on G r o t h e n d i e c k polynomials

Let u b e a permutation J its code, and r be the length of J (i.e. j r > 0, ji — 0 Vi > r). Let v be the permutation of code [ji,.-. ,jr-i,jr — !]• Recall that for what concerns Schubert polynomials in x\, X2, • • •, one has the equation * * = SrX* + £ i € l ( ( r ) * T „ , ,

(29)

sum over all transpositions T,-* : j := vr,i € 1(a) (i.e. i such that such that l(TjiV) = 1{V) + 1). These equations, called transition in 15 , refine the Ehresmann-Bruhat order on the symmetric group and give the expression of Schubert polynomials as a positive sum of monomials; they also furnish the stable part of them as a sum of Schur functions, if one cancels the successive terms xrXv, when iterating Eq. (29). One has similar transitions for Grothendieck polynomials . But first, one needs to explicit products of Ga by £ r 's. In 1 4 , 1 described the Gaj{a\ •••ar). It requires a small adaptation to get products by l/ar = 1 — xr instead. On

173

the other hand, I cannot relate products G ^ l - xr) to the products a^G^, A dominant, considered in 7 under the name Fieri formula. Geometrically these last products are classes of restriction of ample line bundles over Schubert subvarieties, and not intersections as in our case. Let us keep the same hypotheses as in Eq. (29), ordering the values i\ < %2 < • • • < h, i G Z( appearing in the summation. Given any a 6 &n, and any pair of integers j,i < n, we shall write Tji*Ga for GTjia, considering that a transposition r,-, acts on indices of Grothendieck polynomials present on its right. Proposition 3. Let a be a permutation. With the hypothesis of Eq. (29), one has Ga = Gv + {Xr - 1) (1 - 7-,-iJ • • • • * (1 - Tfr) *G„ .

(30)

Proof. Suppose first that o\ > ••• 0. By induction on r, we know that G\jl,- • • ,jr-2,jr

+ 1] = G\J!,. . . ,jr-2,jr]

+

(31)

(X r _i - 1)(1 - TjjJ * • • • * (1 - Tji2) *G\ji, ., . ,jr~2,jr] the transpositions appearing in this formula being the same as in Eq. (30), but for TJI which is missing ( t h e hypothesis j r _ 2 > 0 implies ix = 1). Taking the image of (32) under 7r£_i with the help of (7), one gets G[jl,-..

,jr-2,0,jr]

= G\ji,...

+ (1-T*ij * • • • * (1-T ji2 ) *G\jlt = G[jl,... ,jr-2,0,jr-l]

,jr_2,0,jr-l] . . . , jr-2,ir,0]7T^_ 1 (a; P -l)

+ ( 1 - 7 * J * • • • * (1 - Tji2) * (G\ji, . . . , > _ 2 , 0,jT - 1] - G[jl, • • • , j r - 2 , jr.O]) = G[jl,...

,jr-2,0,jr-l]

+ (1 - Tjik) • • • * ( ! - Tji2) * (1 - Tji) *G\jU . . . ,jr-2,0,3r

~ 1] (xr - 1)

which is the required expression for those special J. A general G[J'}, J' eW, will be obtained from a G[J], ji > • • • > jr-i by repeated use of the 7r£, i < r — 2. Suppose that Eq. (30) is true for a permutation a. Let p < r - 2 be an integer such that vp > up+1. Then, except when both up and vp+i belong to 1(a), the image of (30) under 7r° is a decomposition G(rsp = GVSp + {xr - 1) (1 - Tjik) • • • • * (1 -

Tjh)

* GV8p

174

of the type required by the theorem. On the other hand, if vp = /?, i/ p+1 = a € 1(a), one writes the RHS of (30) as a summation (xr - 1) Y,V' * (1 - fj/3)(l - Tja)*r]" *G„ , sum over certain pairs of permutations r)',r)". Thus terms occur in four-tuples (xr - l)(G...pa...j... —G...ja...p... -G...pj...a...

+G...jp...a...)

,

writing only the indices at places p,p + l,r. Now, the image of such a term under 7r£ is (xr - l)[G...a0...j... - G...aj...p...) and therefore the decomposition of G>Sp is obtained from the one of G„ by replacing the double factor (1 - Tjp)(l - Tja) by the single one (1 — Tjp), and exchanging v for vsp. D Specializing xr to 0, one finally gets T h e o r e m 4. 1) Let a be a permutation, o\ = 1. Using conventions (29), the stable part of Ga is equal to that of {l-(l-Tjik)*---*(l-Tjil))*Gv

.

(32)

2) There exists positive integers rrij such that Ga

^b

Y,^_l)lM-\J\

maG[J]

>

(33)

sum over all partitions J. Proof. One gets 1) from a transition by specializing xr to 0. Instead of having to embed &n into 6 i x &n, we have choosen the hypothesis o\ = 1. Prom the theory of Schubert polynomials 18 , one knows that all the permutations appearing when iterating sufficiently many times a Schubert-transition are vexillary. Now, in the case where a is vexillary, the set 1(a) is of cardinality 1, and every long sequence of transitions starting from a encounter a grassmannian permutation (with code obtained by reordering increasingly the code of a). There are more permutations appearing in Eq. (30) than Eq. (29). To adapt the argument of 18 , one has to modify the weight function (m — r)£(a), where m is the first index such am > i^l3i+m, where [ji,J2,~] is the code of a. This function vanishes for grasmannian permutations, and strictly decreases in a transition a (->• n, n being any permutation appearing on the RHS of (30). In summary, the stable part of Ga belongs to the space generated by the stable parts of the G 7 , 7 grassmannian. However, each product by a

175

transposition appearing in a transition increases length (of permutations) by 1, and signs are as stated, since £(a) = \code(a)\ for any permutation a (this positivity was conjectured by Buch 1, as we already mentioned in the introduction). • As an example of decomposition given by theorem 4, let us take a = [1,5,2,4,3,9,6,7,8]; this implies J = [0,3,0,1,0,3], r = 6. One has -Xl52439678 = ^6^152438679 + ^15248367 + -^15283467 + -^18243567 ,

and therefore 1 = {3,4,5}, Gi52439678 = Gl52438679 + (x6 - 1 ) ( 1 - T85) * ( 1 - T84) * (1 ~ T83) •Gi52438679 •

Expression (32) develops in this case into (1 - T85) * (1 - 7"84) * (1 - TS3) * Gi52438679 = (1 — T85) * (1 — Tsi) (Gi52438679 ~ Gi52483679j = (1 — T85) * (Gi52438679 — Gi52483679 — Gi52834679 + Gi52843679) = Gi52438679 — Gi52483679 — Gi52834679 + Gi52843679 — Gi82435679 + Gl82453679 + Gig2835679 - Gi82543679 •

Iterating transitions, and using codes now, one finally gets a sum G[030103] st~b G[0601] + G[0502] + G[034] - G[053] - G[0602] + G[05011] + G[0412] + G[0331] - G[0431] - G[05021] - G[06011] - G[0512] - G[0341] + G[0531] + G[06021] of vexillary Grothendieck polynomials , which implies, by sorting indices, G[030103]

s

~

g61 + g52 + #43 - 553 - 962 + 9511 + #421 + 3331 - 25431

_

2(?52i - 5611 + 5531 + 5621-

One has a similar decomposition of the stable part of Schubert polynomials in terms of Schur functions (cf. 17 , 4 ). This decomposition is a direct by-product (just sorting increasingly all indices) of the decomposition of Schubert polynomials in terms of key polynomials (cf. 20 , 26 ), which involves finding all Young tableaux (in the alphabet si, S2,...) which are reduced decompositions of a given permutation. Fomin and Greene 5 obtain a decomposition of the stable part of Ga in terms of Schur functions, by enumerating tableaux in the TVs which are decompositions, in the 0-Hecke algebra, of a given Ta. Reconstructing Eq. (33) from their result, however, is not possible since many cancellations occur :

176

Schur functions are not the appropriate symmetric functions for what concerns the Grothendieck ring. We shall give elsewhere a decomposition of Grothendieck polynomials in terms of G-key polynomials. Let us mention that Grothendieck polynomials are given by a simple statistics on alternating sign matrices. This also will be written elsewhere. 6

Variations among determinantal expressions

There are many determinantal expressions of classes of Schubert subvarieties of Grassmannians, that is, of g\, A partition, or of G[J], J increasing. Let us illustrate them on a concrete example, the case of a determinantal variety defined by the vanishing of minors of order 2 of a matrix of order 4. With the notations used in this text, the Grothendieck polynomial which is to be explicited is G3412 = G[0022]. By definition, it is (1 - l / a i ) 5 ( l - l / a 2 ) 4 ( l - 1/as)17r4°321

(34)

= (l-l/a1)4(l-l/a2)47r4a321

(35)

M a

1

- l ) ^ a

2

- l ) ^ a ^

? 3 2 1

= (01 - l) 4 (a 2 - l ) 4 a | a 4 a\a\a\ 3 4 a 321

det I oftax - l ) 4 , !•?(«! - l ) 4 , a? , a 4 | ^

^

(36)

I

\

(37)

^

(38)

with A(A) ~Ui<j(ai-aj)The equality between Eq. (34) and Eq. (35) comes from the fact that (1 - l/ax)nf = 1, and that n%321 ni - n%321. Since d^i *s given by an alternating summation on the symmetric group, (37) is the expansion of (38). The quotient is a multi-Schur function 5 4 (A - 4) 5 5 (A - 4) 5 6 (A - 4) 5 7 (A - 4) 5 3 (A - 4) SA{k - 4) 5 5 (A - 4) 5 6 (A - 4) 5 2 (A) 5 3 (A) 5 4 (A) 5 5 (A)

5HA)

2

5 (A)

3

5 (A)

(oia 2 a 3 a4) 4

(39)

177

and this the expression that is given in 12 (specialized to the case one of the vector bundles is trivial of rank 4). In chapter 2 of my thesis 1 3 ,1 used determinants with entries more general than symmetric powers. In the case considered, it gives:

I aia2A3d4

G[0022] = aia2CL3d4

(4-

Ol + 0,2 + a 3 + 04 - 4

J

1

1

1\

a\

a.2

(13

a4'

1

i

• (40)

One can see, more easily than using Grothendieck polynomials , that this last determinant is equal to the class of the complex resolving submaximal minors (Eagon-Northcott complex). Using now variables Xj's, and relation (11), one has: G[0022] = x\x\(l

— U5 „4 X\ X?(l

Xi)

- x2)x3(l

Xi(l ~ Xif

- x 3 ) 2 (l - x4)6 d%321

{l-Xif

A(X)

S2(X) S3(X) SA(X) S5(X) 3 4 5 S (X - 1) S (X - 1) S (X - 1) S (X - 1) S°(X - 2) S^X - 2) S2(X - 2) S3(X - 2) S°(X - 3) S1(X - 3) S2(X - 3) S3(X - 3)

(41)

(42)

2

=±

(43)

The equivalence of (38) and (42), or (39) and (43) is in fact a mere change of variables Xj \-t 1 — ^ , using A(X) = n « l - h

- (1 - h ) = ( - 1 ) 6 ) A(A)( 0 l • • • an)~n+\

The expression (43) is given by Lenart 2 4 (th. 2.4). Proposition 3.11 of 14 states that, more generally, Grothendieck polynomials are still given by determinants in the case of vexillary permutations. Note, however, that when the partition J has repeated parts, then one can slightly modify the determinant expressing G[J] without changing its value. This is already seen at the level of the equality of (34) and (35).

178

References 1. A. Buch, A Littlewood-Richardson rule for the K-theory of Grassmannians, math.AG/0004137 (2000). 2. M. Demazure, Desingularisation des varietes de Schubert generalisees, Ann. Sci. Ecole Norm. Sup. (4) 7, 53-88 (1974). 3. M. Demazure, Une nouvelle formule des caracteres, Bull. Sc. M. 98, 163-172 (1974). 4. P.H. Edelman, C. Greene, Balanced tableaux, Adv. in Math. 63, 42-99 (1987). 5. S. Fomin and C. Greene, Noncommutative Schur functions and their applications, Discrete Math. 193, 179-200 (1998). 6. S. Fomin and A. Kirillov, Grothendieck polynomials and the Yang-Baxter equation, 6th Conf. on Formal Power Series and Alg. Comb., DIMACS, Rutgers, p. 183-190 (1994). 7. W. Fulton, A. Lascoux, A Fieri formula in the Grothendieck ring of a flag bundle, Duke Math. 76, 711-729 (1994). 8. A. Kirillov, Quantum Grothendieck polynomials, CRM Proc. Lecture Notes 22, p. 215-226, Amer. Math. Soc. (1999). 9. S. Kleiman, D. Laksov , Schubert calculus, Am. Math. Monthly 79, 1061-1082 (1972). 10. A. Konhert, S. Veigneau, Using Schubert basis to compute with multivariate polynomials, Adv. Appl. Math. 19, 45-60 (1997). 11. A. Lascoux, B. Leclerc and J.Y. Thibon, Flag Varieties and the YangBaxter Equation , Letters in Math. Phys. 40, 75-90 (1997). 12. A. Lascoux, Puissances exterieures, determinants et cycles de Schubert, Bull S.M.F. 102, 161-179 (1974). 13. A. Lascoux, Polynomes symetriques, Foncteurs de Schur et Grassmanniennes, These, Universite Paris 7 (1977). 14. A. Lascoux, Anneau de Grothendieck de la variete de drapeaux, in "The Grothendieck Festchrift", vol III, 1-34, Birkhauser(1990). 15. A. Lascoux & M.P. Schiitzenberger, Polynomes de Schubert, Comptes Rendus 294, 447 (1982). 16. A. Lascoux & M.P. Schiitzenberger, Symmetry and Flag manifolds, in Invariant Theory, Springer L.N. 996, 118-144 (1983). 17. A. Lascoux & M.P. Schiitzenberger, Structure de Hopf de I'anneau de cohomologie et de I'anneau de Grothendieck d'une variete de drapeaux, Comptes Rendus 295, 629 (1982). 18. A. Lascoux & M.P. Schiitzenberger, Schubert polynomials and the Littlewood-Richardson rule, Letters in Math.Phys. 10, 111-124 (1985).

179

19. A. Lascoux & M.P. Schiitzenberger, Symmetrization operators on polynomial rings, Funct.Anal. 21, 77-78 (1987). 20. A. Lascoux & M.P. Schiitzenberger, Tableaux and non commutative Schubert Polynomials, Funk.Anal 23, 63-64 (1989). 21. A. Lascoux & M.P. Schiitzenberger, Keys and Standard Bases, "Invariant Theory and Tableaux", IMA Vol. in Maths and Appl. 19,125-144, Springer (1990). 22. A. Lascoux & M.P. Schiitzenberger, Algebre des differences divisees, Discrete Maths 99, 165-179 (1992). 23. C. Lenart, Noncommutative Schubert calculus and Grothendieck polynomials, Adv. in Math. 143, 159-183 (1999). 24. C. Lenart, Combinatorial aspects of the K-theory of Grassmannians Annals of Comb 4, 67-82 (2000). 25. I. G. Macdonald, Notes on Schubert polynomials, Pub. du LACIM, Universite du Quebec a Montreal, (1991). 26. V. Reiner, M. Shimozono, Key polynomials and a flagged LittlewoodRichardson rule, J. Comb. Th. A 70, 107-143 (1995).

TABLEAU R E P R E S E N T A T I O N FOR M A C D O N A L D ' S N I N T H VARIATION OF S C H U R F U N C T I O N S JUN NAKAGAWA Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan E-mail: [email protected] MASATOSHI NOUMI Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan E-mail: noumi6math.sci.kobe-u.ac.jp MIKI SHIRAKAWA Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan E-mail: [email protected] YASUHIKO YAMADA Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan E-mail: [email protected] The ninth variation of Schur functions associated with a general N x N matrix is studied. In particular, two types of tableau representations for them are constructed by using decomposition of matrices and properties of minor determinants.

Introduction The purpose of this paper is to give several constructions for Macdonald's ninth variation for Schur functions 4 such as the Weyl formula, the JacobiTrudi formula and the tableau representation. As for the terminology of partitions and symmetric functions, we basically follow the textbook of Macdonald 5 (see also Chapter I, §3, Examples 21 and §5, Example 30, as for the ninth variation). Let n£' be a set of variables indexed by k = 1,2,... and r e Z . For convenience, define h^ = 1 and fi£' = 0 (k < 0) for all r € Z . For a given partition A = (Ai,A2,...), I.G. Macdonald 4 defined the ninth variation of Schur function S\ by the Jacobi-Trudi formula Sx = det{h^-_%J)1
(0-1)

181

where l(X) stands for the number of parts of A. It is convenient to extend this definition to S?

= d e t ( / l K ; f )!
(r G Z ) ,

(0.2)

so that S\ = S® . By specializing the variables h^', S^' provide with many variations of Schur functions 4 . In this paper, we define the ninth variation of Schur functions Sjf associated with a general matrix X in terms of the Weyl formula. We then prove the Jacobi-Trudi formula and the Giambelli formula by using appropriate decompositions of matrices. By means of elementary properties of minor determinants, we also construct tableau representations for S^ . We remark that such tableau representations as we discuss in Section 3 appeared for the first time in the work of A.N. Kirillov and N.Y. Reshetikhin on spin chain models (see also the paper by V.V. Bazhanov and N.Y. Reshetikhin 2 ). Throughout this paper, we use the following notation of minor determinants for a n i V x J V matrix X = (£rj)i
(0.3)

{ 1 , . . . , N} with \I\ = | J\ = r, we define £j(X) to be

#(*) = £::::£(*),

(0.4)

where i\ < i2 < • •. < ir and ii < J2 < • • • < jr are the sequences obtained by arranging the elements of / and J in the increasing order, respectively. When / = { l , . . . , r } , we simply write O ( ^ ) = €ji,...,j*{X) f° r £} ''"' r '(X) = (jiZ'^iX)Recall that, as for the product of matrices, the minor determinants are determined by the expansion tf(A^)

= £&P0tf(

y

)-

(0.5)

K

This formula plays the key role in constructing tableau representations for c(r) °X •

1

Weyl formula and Jacobi-Trudi formula

In this section, we define the ninth variation of Schur functions for a general N x N matrix X, and prove two types of Jacobi-Trudi formulas and the Giambelli formula for them. (We remark that a similar argument can be carried out for a general matrix X with the infinite indexing set N or Z , with appropriate modifications.)

182

In the following, we fix an N x N matrix X = (xij)i
j

a

= K+l-a

+ a

(a = l , . . . , r ) .

(1.6)

This subset J, or the increasing sequence of indices (ji,... ,jr), is called the Maya diagram for A. Note that this correspondence gives rise to a bijection between the partitions A contained in the r x s rectangle (sr) and the subsets of { 1 , . . . , TV} of cardinality r. We define the ninth variation of Schur function Sjf for X in terms of the Weyl formula

as the ratio of minor determinants of X, specified by the Maya diagram J = {ji < • •• < jr} for A. As for the partitions A of one row and of one column, we use the notation u(r) _ Mr) _ £l,'::.',r-l,r+fcPO

lb - {) I

£1'""V^L > ) _ Mr) _ e k - D(i*>) -

S

si

(1.8)

(X)

l,...,r-*+l,-,r+l £1'-'r(X)

V

fjfc - 0 1 r) ~ ' '""•' ''

respectively. For convenience we define h^' = e^ — 0 for k < 0. Theorem 1.1 (Jacobi-Trudi formula) For each partition A contained in the rectangle (sr), the ninth variation of Schur function 5^ for X, defined by Eq. (1.1), has the Jacobi-Trudi formulas S? Proof.

= det (htX^jsr'

S T)

i

= d e t (et-itf

)r
(L9)

Consider the Gauss decomposition of X: X = X- X0 X+,

(1.10)

where X_, XQ and X+ are lower unitriangular, diagonal and upper unitriangular, respectively. Note that these three matrices are determined uniquely

183

as matrices with entries in the field of rational functions in the variables Xij (1 3 < N). Furthermore, the entries of these matrices are given explicitly as follows:

(

)l3

{1 J)

~ tt&x)

- '

(X0)u = jCi{*l.

(i = l,2,...,N),

(1.11)

Si',...',i-lKx)

(x+h = 'C,i'

(* < J)-

?i,...,iiA;

Also the inverse matrix X+1 of the upper triangular component is given by ti,...,j-i C'-i~.(x) i,...,?,...,j J

(

x+%

=( - i r « - ^ ^ — -

(i<j).

(i.i2)

It is worthwhile to notice that (X?)ij = (-iy-ie&)

(X+)ij=hfli,

(i<j).

(1.13)

r

We take a partition A c (s ) and the increasing sequence of indices ( j i , . . . , j r ) defined by Eq. (1.6). Then the minor determinant ^".''^liX) is computed via the Gauss decomposition as

fcrr(^o)4-;::::ir(Jf+).

&';::•;,•,(*) =

(1.14)

Since

fcvp(*.) = n fcffiL = fc;w, a-is) i=i

?i,...,i-iW

we have

A

" fc;w -^-•J-(x+)-

By Eq. (1.13), the right-hand side is rewritten as £ji',..'.,>(^+)

=

det((X + ) 0)J -Ji< 0> (,< r det(/l^L 0 )l
-

det /l

( jl +1 _ i ,-(r+l-a)) 1 <°. ! '<'-

= det(/iJHL16+°))1
(L16)

184

Hence we have the first Jacobi-Trudi formula s ]

x = 4';::::,v (*+) = ^^h+X%<^
(i.is)

We now consider the increasing sequence (fci, fc2,..., ks) of indices, complementary tO (jl,...,jr)ka=r

+ a-\'a

(o = l , . . . , a).

(1.19)

Note that 0'i, • • • ,jr} U {fci, ...,ks}

= {l,...,r,r

+ l,...,r

+ s = N}.

(1.20)

1

The minor determinant ^ i ' ' ' ' ^ ( X + ) is then expressed as

47.::y*+) = (-!)|A| &:.fc,;+5(^+1)

(1.21)

in terms of the inverse matrix, where |A| = ^ o = i Ja — a = S a = i T + a — ka. Again by Eq. (1.13), we compute £r+i','.'..,r+s(-X+ ) = det((X+ )ka,r+b)l
= det((-ir+*-*» e g ^ ) , ^ ,

(1.22)

= (-1)1*1 det(e^_6-+16))1
(1.23) Q.E.D.

Let / i b e a partition contained in A, and let I be the corresponding Maya diagram with |7| = r. The ninth variation of skew Schur function S^J is then defined by

s[% = #(x + ) = (-I)IVMI c

tfrpr-i),

(124)

c

where I , J are the complements of 7, J in { 1 , . . . ,N}, respectively. This definition can be written as ^A/M

_

a e t

l^Ai-Mi-t+j )i
a e t

\eK-^-i+j

)i
y]-zb)

By a similar method, one can also prove the Giambelli formula for SA . We introduce the following notation for the hooks (p lq):

C'-'L^. (X) 1 rtl = sfc\t)--;T,'::;;: , y

a
(1.26)

185

For a given partition A C (s r ), we define the two sequences of indices Pi > p 2 > ••• >Pt > 0 ,

qi >q2 > ••• >qt > 0

(1.27)

by the condition {Ji."-,Jr}\{l,.--,»-} = { r + p t , . . . > r + p i } {l,...,T-}\{ji,...,>} = {r-
(1-29)

5

in the Frobenius notation . Theorem 1.2 (Giambelli formula) With the two sequences and (qi,.-., qt) of indices defined as above, S^' is expressed as ^)=«Jet(^r!w)1
(j>i,---,Pt)

(1-30)

Proof. We consider the r x N matrix Y = (xi,j)i
= (_1)r_,

£ 1 ""W-y

XX) = (

_l)-^Wrr_.

(i.3i)

for 1 w =

fcvr(r)4';::'T>(z)-

( L32 )

Since £i ,'.','.'£ CO = £i,'.'.'.','rP0, we have

= (-i) l ' l+<(t - 1)/2 « 1 ;:::,v p ; P 'I(^) = (-1) 1 ' 1 det(Z r _ ? i i r + P 3 .)i<jj< t

(1.33)

= det(/4,v 7i )i
Q.E.D.

Remark. Let us take the Vandermonde matrix X = (x3i~1)i
=sx(Xl,...,xr),

(1.34)

186

for 0 < r < N and A C (sr). In particular, we have h< ]

k =hk(x1,...,xr)

(0
=ek(xi,...,xr)

(0
() e r k

(1.35)

where hk and ek denotes the complete homogeneous symmetric functions and the elementary symmetric functions in the usual sense, respectively. Then Theorem 1.1 implies ) = det (hxi-i+j(xi,...

, Xr-j+i ))l
(L36)

Fix a positive integer n and set / = l(X) and m = l(X') = Ai. We take r and s so that r > n + l — l and s > m +1 — 1. Then by setting xn+i = • • • = XN = 0 (N = r + s) in Eq. (1.36), we obtain the usual Jacobi-Trudi formula s\(xi,...

^xn))1
,xn) = det [h\i-i+j{x\,...

(1.37)

for Schur functions in n variables. Similarly, from sx(xi,...,xr)

= d e t (eAj_i+j(xi,...,a; r + j_i)) 1 ^ i ) j ^ J ( v ) ,

(1.38)

we obtain sx(xi,.

..,xr)

= det (ey.-i+j(xi,...,

Xr+j-i))^..^.

(1.39)

(Take r > max{Z + m — 1, n}, s >m and set a; n+ i = • • • = XN = 0.) We also remark that, from Eq. (1.25), we have S

V/i

=

^

(h^-H-i+j(Xl>

• • • >XH+r-3 + l))l

(L4°)

hence, S

\/v\xn

+ 1 = -=xN=0

= det

ih>«-H-i+i(xl>-

— s\/ii\^11

• ••>

••

>X"))l
x

n)

for any /i C A, provided that r > n + I — 1 and s > m + I — 1. 2

Tableau representation

As we have seen in the previous section, the ninth variation of Schur function 5 A r for a general matrix X is represented by a minor determinant £j(X+) of the upper triangular component X+ in the Gauss decomposition of X. Suppose furthermore that X+ is decomposed into the product X+ = UlU2---Um

(2.42)

187

of several upper unitriangular matrices Uk of particular form. Let / , J be two subsets of { 1 , . . . , N} of cardinality r. Then for each I, J with |/| = \J\ = r, the minor determinant £j(X+) is expressed as a sum eAX+)

=

eKl (Ui)&2 (U2) • • • tf-1 (C/m)

£

(2.43)

ifi,...,A-m-i

over all sequences (Ki,..., ular, we have

S^=

ATm_i) of indexing sets of cardinality r. In partic-

£

« } K l u W ••«&»_,./,

(2-44)

where I = {1,... ,r} and u ^ B = £^(£4). If the matrices Uk are chosen so that the minor determinants ukAB can be described nicely, the formula Eq. (2.44) will possibly give rise to a combinatorial expression for S^ . In this way, one can provide with various combinatorial formulas for S^ , depending on the choice of the decomposition Eq. (2.42). Note also that, as is often the case, the sequences of indexing sets, over which the summation is taken, can be interpreted in terms of some other types of combinatorial objects such as non-intersecting paths (in the sense of I. Gessel and G. Viennot 3 ) and Young tableaux. What we will discuss in this section concerns only the upper triangular matrix X+, and does not depend on the components X_, XQ of the Gauss decomposition Eq. (1.10) of X. We now consider two types of upper unitriangular matrices for the building blocks of the decomposition Eq. (2.42) of X+: (1) : U = (1 + ui£a 2 )(l + u2E23) •••(! + UAr-iBjv-LJv), (2) : V = (1 + VN-iErf-w) •••(!+ v2E23)(l + VlE12), where E^ = {&ia&ib)\
V = l+ £

ViE^i+i.

(2.46)

»=i

Let / , J be two subsets of {1, ...,JV} with |/| = \J\ = r and /J, A the corresponding partitions. Consider first an elementary matrix 1 + u £ i ] j + 1 (1 < i < iV — 1). Then there are only two cases where the minor determinant £j(l + uEiti+i) is nonzero: (1) I = J; or (2) i G I, i + 1 ^ I and J is obtained from I by replacing i + 1 for i. In these two cases, £5(1 + w-E»,t+i) is equal to l o r w , respectively. Prom this fact, one can easily prove

188

Lemma 2.1 Let I, J and fi, X as above. (1) The minor determinant £j(U) ofU in Eq. (2-45) vanish unless /x C A and the skew diagram A//x is a horizontal strip. If A//z is a horizontal strip, one has

tip) = n

ur„a+b.

(2.47)

(a,6)€A/j*

(2) The minor determinant £,j{V) ofV in Eq. (2.45) vanish unless fj, c X and the skew diagram X/fi is a vertical strip. If X/fi is a vertical strip, one has (2.48) (a,b)e\/lt

The conditions on the skew diagram X//J, in Lemma 2.1 can be interpreted in terms of the Maya diagrams I = {ii < • • • < ir} and J = {ji < • • • < j r } . Firstly, the condition that fi C X and that A//i is a horizontal strip holds if and only if H < h

< *2 < h

< • • • < ir <

(2.49)

jv

This condition can also be represented graphically as follows. ja-1

J I

Ul

1

«2

2

ja

jo+1 UN.

«ia

ia-l

ia

ia + 1

N

(2.50)

The minor determinant £j(U) also has an alternative expression

£j(U) = I I

Ui Ui +1

° »

(2.51)

•••Wja-i.

o=l

as the product of the u-variables attached to the horizontal line segments that the paths pass through in the figure above. Similarly, the condition for the skew diagram X/fi to be a vertical strip is equivalent to j a -ia

(2.52)

or ia + 1 for a = 1 , . . . , r.

By using the graphical expression Ja-1 L

7

I

/v\

1

/V2

2

....I.., w\

'Ja

J

a+1

S° la+l

N

(2.53)

189

the minor determinant £j(V) is determined as the product of the w-variables attached to the slanted line segments lying on the paths:

l
+l

We now suppose that X+ is expressed as the product X+ = U1U2---Um,

(2.55)

where Uk = (1 + u{k1]El2){\ + uk2)E23) •••(1 + U^-^EN-!^)

(2.56)

for k = 1 , . . . , m. It is convenient to arrange the variables uk graphically as in Fig. 1. For two sets J, J of indices with \I\ = \J\ = r, we have the expression of the minor determinant #(*+) =

&1(U1)$l(U2)...g'»-1{Um).

£

(2.57)

Ki,...,Km_i

Let n and A be the partitions corresponding to the Maya diagrams i" and J, respectively. Also, let X^ be the partition corresponding to Kk for k = 0 , 1 , . . . , m , setting KQ = / and Km = J. Note that a sequence (Ko, K\,..., Km) with \Kk\ = r contributes nontrivially to the sum Eq. (2.57) only if each pair (Kk-i, Kk) satisfies the condition of Lemma 2.1, (1) for each k = 1 , . . . ,TO.Such sequences of subsets (.Ko, K\,..., Km) are in one-to-one correspondence with the sequences of partitions ft = A(°) C A(1) C • • • C \{m) = A,

(2.58)

such that XW /\(k~1'> (k = 1 , . . . ,m) are horizontal strips. Representing such sequences of partitions in terms of the semi-standard tableaux (column strict tableaux 5 ) of shape X/fi with labels { 1 , . . . , m } , we obtain the tableau representation for the ninth variation of skew Schur function S\T/ :

4/i=

E TeSSTab m (A/M)

n
(^

(a,b)e\/fi

We recall that the semi-standard tableau for the sequence Eq. (2.58) is defined as the mapping T : A//i —• { 1 , . . . , m} such that T{a,b) = k

<=> (a,6)eAW/A ( f c " 1 ) ,

(2.60)

190 J — K„

(2)

utt

u£>

Km-l

72

71 (2) «3

<>

t4 1 } u 2

Ki.

u(1) 1

r=

4

K„

(

(3) «2(

(2)

K2

73

3)

(2)

U^

,

1 2

3

4

5

•••

K3 = {2,4,6}

A<3> = - J - 1 tr

^ 2 = {1,4, 5}

A<2>=EE

i^x = { 1 , 2 , 5 }

A ( D

K0 = { 1 , 2 , 4 }

A(°'=D

= C D

N

Figure 1. Non-intersecting paths (1)

for each (a, b) G A//J. In terms of the graphical expression Eq. (2.50), it is rewritten as the summation S

H=

E

"-» •••"7..

(2-61)

(7l,---,7r)

over all r-tuple of non-intersecting paths ( 7 1 , . . . ,7 r ) connecting («i,... ,ir) to ( j i , . . . ,jr), as in Fig. 1. Here, for a path 7, w7 denotes the product of all u-variables u^' attached to the horizontal line segments lying on 7. We also formulate the other type of tableau representation associated with the matrices of the form V of Eq. (2.45). Suppose that X+ is decomposed into X+=V1V2

•••Vmi

(2.62)

where Vi = (1 + V^-^EK^N)

• • • (1 + v^E2S)(l

+ v^E12),

(2.63)

From this decomposition, we obtain the tableau representation

s

H=

E TeSSTab m (A'//i')

n <Mb)-

(2-64)

(a,b)€\'/n'

It is also represented as a sum

4 1 = E «-n"-«7, (71. —>7r)

(2-65)

191 N-l

N

A<3> = Kz = {2,4,5} K 2 = {2,3,4} ^

= {1,2,4}

#,, = {1,2,3}

A(2)

A

=

(D

= n

A<°> = 0

Figure 2. Non-intersecting paths (2)

over all r-tuple of non-intersecting paths ( 7 1 , . . . ,7,.) connecting ( i i , . . . ,ir) to ( j i , . . . , j r ) , as in Fig. 2. Here, u 7 denotes the product of all u-variables v£ attached to the slanted line segments lying on 7. 3

Specifying the tableau variables

In this section, we discuss a particular choice of decompositions Eq. (2.55) and Eq. (2.62) such that the variables u^' and v)?' for the tableau representations can be specified in terms of the matrix X or X+ itself. For each pair (a, b) of nonnegative integers with a + b < N, define

Ql = &'.°.,6+aP0 = ^ t K»+J-)l
(3.66)

so that QS = i,

QO-V)'

(3.67)

We also define the variables (b) _ ^a

°

Va-1

Q"at\ Q\

(3.68)

for a > 1 b > 0 with a + b < N - 1. We set x{b) = 0 for b < 0. (In these definitions, we have tacitly assumed that the denominators do not vanish.)

*i 0 ) 4 "

xT (2)

r(0)

r(D

l

T (0)

*rt

2)

x2

3)

1 2

3

4

5

•••

,

F2

xC1) x(0)

Pi

2

1

PN-2 PN-I

N

Figure 3. Arrangement of x-variables (1)

Proposition 3.1 The upper triangular matrix X+ product

is decomposed into the

X+=PN-1---P1,

(3.69)

where Pk = (l + 4 0) £fc,*+i)(l + x^Ek+ltk+2)

•••(1 + x<j»-k-x)EN-i,N).

(3.70)

This type of decomposition of matrices is investigated extensively in the work of A. Berenstein, S. Fomin and A. Zelevinsky x . Here we indicate an idea for a proof of the proposition above. The fact that a generic upper unitriangular matrix has a decomposition of this form can be shown by induction on N, for example. Supposing that X+ has a decomposition in the form above, consider the tableau representation Eq. (2.59) for the diagram Fig. 3. Then one sees immediately that, for J = { 1 , . . . , a} and J = {b + 1 , . . . , b + a}, there is only one r-tuple of non-intersecting paths connecting I to J:

S=^ ) ) =n»i°M i, -«r ) .

(3-71)

t=l

^o

so that Qba

•.x^xV-xW.

Qba-1

Hence we have the expression Eq . (3.68) for the x-variables.

(3-72)

193

Theorem 3.2 With the x-variables defined as above, the ninth variation of Schur function S{ and of skew Schur function Sw have the tableau representations

#>=

n »»r'-

E

T6SSTabr(A)

(a,6)eA

^

and ^A/M -

2^

11

T 6 S S T a b r + > i l (A/M)

(a,6)eA/j»

X

r+m + l-T(a,b)

^-'^

respectively, where fi C A C (s r ). Proof. This theorem follows from the general tableau representation Eq. (2.59) for the arrangement of x-variables indicated in Fig. 3. Formula Eq. (2.59) for these x-variables reads as

*&=

n *i&&r r ~^

E TeSSTab N -i(A//i)

<3-75)

(a,t)eA//i

-

since ujj. = x^_ f c . Note that, in this expression, the sum is taken over all tableaux T € SSTabjv_i such that T{a, b)>N-r

+ a-b

for any (a, 6) G A//x.

(3.76)

This condition implies T(a, b)>N-r-ni

for any (a, 6) e A//i.

(3.77)

By renaming the labels {N — r — fj,i,...,N — l} as { 1 , . . . , r + /ii}, we obtain Eq. (3.74), hence, Eq. (3.73) as the special case when /zi = 0. Q.E.D. We remark that, in Eq. (3.73), one has T(a,b) — a + b — 1 > Ofor any T 6 SSTabr(A) and (a, b) £ A, since T(a,b) > a. It means that the tableau representation Eq. (3.73) for S^' has as many monomial terms as the semistandard tableaux in SSTab r (A). In Eq. (3.74) for the case of skew-tableaux, not all T 6 SSTab r + M l (\//J) contribute in general; the summation is taken over all T such that T(a, b) > ^i + a — b + 1 for all (a, b) 6 A//x. Remark.

In the special case where X = (xJi~1)i
JJ

(xj-Xi),

x^=xa

one has (3.78)

l
for all a, b. Hence, Eq. (3.73) recovers the well-known tableau representation for the Schur function s\(xr,xr-i,... , £i). Also, from Eq. (3.74) together

194 1

2

3

4

•••

JV-1 N -2)

VN-i

v3 2

Vi 1

1 2

3

4

5

•••

N

Figure 4. Arrangement of rc-variables (2)

with Eq. (1.41), we obtain the tableau representation for the skew Schur function S\/n(Xl,...,Xn)=

^

IJ

TeSSTab„(A/M)

Zn+1-T(a,&)

(3.79)

(a,6)6A//j

by shifting the labels appropriately. By rearranging the order of elementary matrices in the decomposition Eq. (3.69), we obtain the following decomposition of X+: X+ = V1V2---VN.1,

(3.80)

where • • • (1 + x^~1]Ek+hk+2){l

Vk = (1 + x^EN-llN)

+ x^-1)Ekjk+1)

(3.81)

for k — 1 , . . . , N — 1, with the same x-variables Eq. (3.68). From the tableau representation of Eq. (2.64), we now obtain

*&=

E

n

TeSSTabN-ifA'/M')

(a,fr)eA'//i'

*z{:%?-Tla.>y

(3-82)

This formula can be regarded as the dual version of Eq. (3.73), Eq. (3.74). Note that the summation is in fact taken over all T G SSTab/v_i(A'///) such that T(a,b)
+ a-b

for all {a,b) £ \'/fi'.

(3.83)

195

References 1. A. Berenstein, S. Fomin and A. Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. in Math. 122(1996), 49-149. 2. V.V. Bazhanov and N.Y. Reshetikhin, Scattering amplitudes in offcritical models and RSOS integrable models, in Common trends in mathematics and quantum field theories, Kyoto, 1990, Progr. Theoret. Phys. Suppl. 102(1990), 301-318. 3. I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae. Adv. in Math. 58(1985), 300-321. 4. I.G. Macdonald, Schur functions: Theme and variations, Publ. I.R.M.A. Strasbourg, Acte 28 e , Seminaire Lotharingien, 5-39, 1992. 5. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition, Oxford University Press, 1995.

T-ANALOGUE OF T H E Q-CHARACTERS OF F I N I T E D I M E N S I O N A L REPRESENTATIONS OF Q U A N T U M A F F I N E ALGEBRAS HIRAKU NAKAJIMA Department of Mathematics, Kyoto University Kyoto 606-8502, Japan Email: [email protected] Prenkel-Reshetikhin introduced g-characters of finite dimensional representations of quantum affine algebras [6]. We give a combinatorial algorithm to compute them for all simple modules. Our tool is t-analogue of the q-characters, which is similar to Kazhdan-Lusztig polynomials, and our algorithm has a resemblance with their definition. We need the theory of quiver varieties for the definition of 4-analogues and the proof. But it appear only in the last section. The rest of the paper is devoted to an explanation of the algorithm, which one can read without the knowledge about quiver varieties. A proof is given only in part. A full proof will appear elsewhere.

1

T h e q u a n t u m loop algebra

Let g be a simple Lie algebra of type ADE over C, Lg = g ® C[z,z _ 1 ] be its loop algebra, and U g (Lg) be its quantum universal enveloping algebra, or the quantum loop algebra for short. It is a subquotient of the quantum affine algebra U g (g), i.e., without central extension and degree operator. Let I be the set of simple roots, P be the weight lattice, and P* be its dual lattice (all for g). The algebra has the so-called Drinfeld's new realization: It is a C(q)algebra with generators qh, ek,r, fk,r, ^fc,n (h € P*, k £ I, r eZ, n e Z\ {0}) with certain relations (see e.g., [1, 12.2]). The algebra U g (Lg) is a Hopf algebra, where the coproduct is defined using the Drinfeld-Jimbo realization of XJq(Lg). So a tensor product M ®c(g) M' of U g (Lg)-modules M, M' has a structure of a U g (Lg)-module. Let U £ (Lg) be its specialization at q = e £ C*. For precise definition of the specialization, we first introduce an integral form U^ (Lg) of U g (Lg) and set U £ (L 0 ) = U^(L 0 ) ® z[ ,,,-i] C, where Z[q, q~1} -» C is given by q±l ^ e±l. See [3] for detail. But we assume e is not a root of unity in this paper. So we just replace q by e in the definition of U q (Lg). The quantum loop algebra U g (Lg) contains the quantum enveloping algebra U Q (g) for the finite dimensional Lie algebra g as a subalgebra. The specialization U £ (Lg) contains the specialization U e (jj) of U,(g). 196

197

1.1

Finite dimensional representations o/U e (Lg)

The algebra U e (Lg) contains a commutative subalgebra generated by qh, hkyTl (heP*,keI,neZ\ {0}). Let us introduce generating functions tp^(z) (k G I) by

V£(*) d =' Q±hk exp ±(9 - q-1) £ \

hk,±mz=Fm

m=l

/

A U £ (Lg)-module M is called of type 1 if M has a weight space decomposition as a U £ (g)-module:

M=0M(A),

M(A) = {mGM qh * m = £{h'x) m

ASP

We will only consider type 1 modules in this paper. A type 1 module M is an l-highest weight module ( T stands for the loop) if there exists a vector mo G M such that efc.r * m 0 = 0,

U e (Lfl)~ * m 0 = M,

ip^{z) * m 0 = ^ ( z ) m 0

for A; G I

for some y£(z) G
*&-<**{?&)

(1L2)

for some polynomials Pk{u) G C[M] with Pfc(0) = 1. Here ( ) G C[[zT]] denotes the expansion at z = ex and 0 respectively. (2) Conversely, for given Pk{u) as above, there exists a finite-dimensional simple l-highest weight U e (Lg)-module M of type 1 such that the l-highest weight is given by the above formula. Assigning to M the I-tuple P = (-Pfc)fce/ G C[u] 7 (-Pfc(O) = 1) defines a bijection between the set of all P 's and the set of isomorphism classes of finite-dimensional simple U e (Lg) -modules of type 1. We denote by Lp the simple U £ (Lg)-module associated to P. We call P the Drinfeld polynomial. For the abuse of terminology, we also say ' P is the /-highest weight of Lp\

198 Since C{qh, hk,n) ls a commutative subalgebra of U£(Lfl), any XJe(Lg)module M decomposes into a direct sum M = 0 M ( $ + , *&~) of generalized eigenspaces, where M(*+,*-) d

='

{m e M | (tpk(z) ~ ^k(z)

Id N

)

* m = 0 for A; G / and sufficiently large TV} ,

for tf ±(z) € C[[z=F]]. The pair of the /-tuple ( * + , * - ) = (*+, * ^ ) f c e / is called an l-weight, and M ( * + , * _ ) is called an l-weight space of M if M ( $ + , \I>~) ^ 0. Theorem 1.1.3 (Prenkel-Reshetikhin [6]). Any l-weight of any finite dimensional Ue(L{j) -module M of type 1 has the following form

/or some polynomials

Rk u =

»=i

( J no-

_ b u

j=i

kj )-

Again for the abuse of terminology, we also say 'Q/R is an /-weight of M\ We denote the /-weight space Af (*+,*"") by M(Q/R). Prenkel-Reshetikhin [6] defined the (/-character of M by Xq(M)

^ £ dimM(Q/R) II ft ft Y^Yk,6kiQ/R

k€li=lj=l

Theorem 1.1.5 (Frenkel-Reshetikhin [6]). (1) Xq defines an injective ring homomorphism from the Grothendieck ring RepU e (Lg) of finite dimensional U e (Lg)-modules of type 1 to Z[3^ 0]fc£/)agc* (a ring of Laurent polynomials in infinitely many variables). (2) If we compose a map Y^a — i > y^ (forgetting 'spectral parameters'), it gives the usual character of the restriction of M to a Ue(fl) -module. Definition 1.1.6. A monomial | J TT TT Yk,aki YkT^

appearing in the q-

keli=lj=l

character Xq is called l-dominant if rk = 0 for all k, i.e., a product of positive powers of Yjc,c's or 1. If Lp is the simple U £ (Lg)-module with /-highest weight P , its (/-character contains an /-dominant monomial corresponding to the /-highest weight. We denote it by mp. Its coefficient in Xq(Lp) is 1.

199

Since {LP}P forms a basis of RepU e (Lg), we have the following useful condition for the simplicity of a finite dimensional U e (Lg)-module M of type 1: If Xq(M) contains only one /-dominant term, then M is simple. 1.2

(1-1-7)

Example

We give examples of ^-characters. If g = An, we have an evaluation homomorphism ev a : U e (Lg) -> U e (g) corresponding to Lg —> g; z i-» a (Jimbo). Hence pullbacks of simple U e (g)modules are simple U e (Lg)-modules. E x a m p l e 1.2.1. Let g = A\ = sfe and V be the 2-dimensional simple U e (Lg)-module. Then the g-character of Ma = ev a (M) is given by a Xq(Ma)

Since Xq is Xq(Ma

a rm

= Ylta + Y-Je2.

g homomorphism, we have

® Mb) = ( y l i a + Y-^2) ( y l i b + Y-b\2) = Y1>aY1>b + Y~ae3Y1>b + Y i . a V ] ^ +

Y'^Y'^.

If b ^ ae2, ae~2, then Ma ® Mb is simple by the criterion (1.1.7). If b = ae2 or ae~2, then the second or third term becomes 1. In fact, it is known that Ma Maei decomposes (in RepU e (Lg)) to a sum M'a ® M", where M' is the 3-dimensional simple U e (Lg)-module, and M" is the trivial module. Thus we have Xq(Ma

® Ma£2) = Xq(M'a) +

Xq(M")

= YltaYlia* + YliaY-^

+ Y-^Y-^

+ 1.

See also Examples 4.1.6, 6.1.3, 7.2.1. 2

Standard modules

2.1 In [15] we defined a family of finite dimensional U £ (Lg)-modules of type 1 and called them standard modules. They are parametrized by the /-tuples P = (-Pfc)fceJ € C[u] J exactly as simple modules. We denote by MP associated to P. The definition will be recalled in §8, but we give here their algebraic identification due to Varagnolo-Vasserot [16]. "This can be checked directly. But it also follows from Theorem 5.2.1 below.

200

Definition 2.1.1. We say Lp an l-fundamental representation if p , s _ \ 1 - su 11

if k = k0, otherwise,

for some s e C* and fco € /• We denote Lp by L(Afc0)5. (A& is the k-th fundamental weight of fj.) For s e C * and a finite sequence (fca)Q = (ki, ki,...) in I and a sequence (na)a — (ni > ri2 > . . . ) of integers, we set A f ( « ; ( & a ) c n ("<*)<*)

= ' - ^ ( A f c J ^ i s (g) i ( A A ; : 2 ) £ n 2 « <8> • • • .

Note that U £ (Lg) is not cocommutative Hopf algebra, so the tensor product depends on the ordering of factors. Theorem 2.1.2 (Varagnolo-Vasserot [16]). (1) A standard module M is isomorphic to a module of the form (finite tensor product)

i

= Mis1; ( 0 « i > « ) " i ) ® M ( s 2 ; (kZa)a2,(nl2)aa)

®•••

such that s%/s^ £ ez for i ^ j and n\ > n\ > •.. for each i. (2) The above tensor product is independent of the ordering of the factors (3) The I-tuple of polynomials P corresponding to M is the product of Drinfeld polynomials of l-fundamental representations appearing as factors of M. Note that if P is given, we can define a module M of the above form by decomposing P into a product of Drinfeld polynomials of ^-fundamental representations. Thus we may denote the above module by Mp. The following properties of Mp were shown in [15]: (1) {Mp} is a basis of RepU £ (Ljj). (2) Mp is an /-highest weight module with /-highest weight P (i.e., given by (1.1.2)). (3) Lp is the unique simple quotient of Mp. (4) Mp depends 'continuously' on P in a certain sense. For example, dim Mp is independent of P. (5) for a generic P, Mp = Lp.

201

Conjecturally Mp is isomorphic to the specialization of the module V max (A), introduced by Kashiwara [8], and further studied by ChariPressley [4]. 3

^-analogues of g-characters

A main tool in this paper is a ^-analogue of the g-character: Xq,t:

RepU £ (L f l ) - • Z t M - ^ J / t e i . a e c -

This is a homomorphism of additive groups, not of rings, and has the property Xq,t=i — Xq- We define Xq,t f° r a n standard modules Mp. Since {Mp}p is a basis of RepU e (Lg), we can extend it linearly to any finite dimensional U £ (L{j)-modules. For the definition we need geometric constructions of standard modules, so we will postpone it to §8.3. We give an alternative definition, which is conjecturally the same as the geometric definition. 3.1

A conjectural definition

Let M — Mp be a standard module, Q/R be an /-weight of M, M(Q/R) the corresponding /-weight space. Define a filtration on M(Q/R) by 0 = M~l(Q/R) n

M (Q/R)

d

c M°(Q/R)

c M\Q/R)

be

c •••

^ p|Ker(V£(*) - *£(z) id)" + 1 . k

C o n j e c t u r e 3.1.1. The t-analogue Xq,t(Mp), defined geometrically in §8.3, is equal to XqAMp)

= Y,Y,t2n~d{QIR'P)A™(Mn(QIR)lMn~\QIR))

mQ/R,

Q/R n

where d{Q/R,P) is an integer (determined explicitly from Q/R, P by (5.1.2) below), and mQ/R is a monomial in Yfc a corresponding to the l-weight space M(Q/R). This definition makes sense for any finite dimensional modules, but is not well-defined on the Grothendieck group RepU £ (Lg). Thus the above does not hold for simple modules.

202

3.2 A main result of this paper is a combinatorial algorithm for computing Xq,t{Mp) and [Mp : LQ}. It is divided into three steps: Step 1 Compute Xq,t f° r all /-fundamental representations. Step 2 Compute Xq,t(Mp) for all standard modules Mp. Step 3 Express the multiplicity [Mp : LQ) in terms of Xq,t(Mp_) for various R. Step 1 is a modification of Prenkel-Mukhin's algorithm [5] for computing Xq of /-fundamental representations. Step 2 is nothing but a study of Xq,t of tensor products of /-fundamental representations. Although Xq,t is not a ring homomorphism, Xq,t of tensor products is given by a simply modified multiplication. For the proof we use an idea in [13]. Step 3 was essentially done in [15]. 4

Step 3

We start with Step 3. The algorithm is similar to the definition of KazhdanLusztig polynomials [9]. It is also similar to the algorithm for computing the transition matrix between the canonical basis and the PBW basis of type ADE [10]. 4.1 Let Ak,a

=

ifc.aeifc.ae- 1 J ^

*l,a '

where c^i is the (k, /)-entry of the Cartan matrix. Definition 4.1.1. (1) Let m, m! be monomials in Y^a (k e I, a e C*). We define an ordering < among monomials by m' m < m' <*=^ — is a monimial in Az\k a (k € I, a € C*). ~ m ' Here a monomial in A^\ means a product of nonnegative powers of A^a. It does not contain any factors Ak,a(2) If \Ef±,
203

Recall that \q{Lp) contains an /-dominant monomial mp corresponding to the highest weight vector. It is known that any monomial m appearing Xq(Lp), Xg(Mp) satisfies m<mP ([5, 4.1], [15, 13.5.2]). Let CQp(t) = the coefficient of TUQ in

\q,t{Mp).

Then (CQP^^P^Q is upper-triangular and cpp(t) = 1 by the above mentioned result. Let (c®p(t)) be the inverse matrix (CQP^))-1. Let

uRP{t)A^

Ys^r^QPit). Q 1

Let be the involution on Z[i, t" ] given by t^1 >—> t^1. Lemma 4.1.2 (Lusztig [10, 7.10]). There exists a unique solution ZQp{t) G

At'1}

(Q < P) of ZRP{t)=

Yl

ZRQ(t)UQp{t),

(4.1.3)

Q:R
ZPP(t)

= 1,

ZQP(t)

G t~lZ\t-1]

for Q
(4.1.4)

This lemma is proved by induction, and holds in a general setting. Lusztig has been using this (or its variant) in many places. Theorem 4.1.5. The multiplicity [Mp : LQ] of a simple module LQ in a standard module Mp is equal to ZQP(1). The proof will be given in §8.4. Example 4.1.6. Let g = A\, Ma = ev*(M) where M is the 2-dimensional simple U £ (g)-module as before. By steps 1,2 explained below6 we have X 9 ,t(M a£2 ® Ma) = YltaYUae2 + Y^Y-^

+ Y-^Y-^4

+rl\.

(4.1.7)

Let P{u) = (1 -au){\ -ae2u) (i.e., MP = Mae2 M0), Q{u) = 1 (i.e. MQ = trivial module). Then the above algorithm gives us ZQp(t) =t_1. 5 5.1

Step 1 Some definitions

Let Mp be a standard module. Let mp be the monomial corresponding to the /-highest weight vector. Let Mp(Q/R) be an /-weight space ''or direct calculation for the definition (8.3.1)

204

as before. We denote by mQ/R the corresponding monomial. wk,a{P),vk,a{Q/R,P) € Z>o,uk,a(Q/R) e Z by

Y

mP= n mQ/R=mP

n

{p

kT

\

kei,aec* kAQ,R p)

AZ

We define

< =

y^{Q/R)-

n

fc6/,aec* fce/.aec* Suppose two standard modules MPi, MPi and /-weight MPi{QxIRl) C M P i , MP2{Q2/R2) c M p 2 are given. We define

spaces

d(Q1/R1,Pl;Q2/R2,p2) d

=-

2 K a W V ^ . - P ' K o e - ^ g fc,a

2

/ ^

2

) +™fc,ae(^1Ka(Q2/-R2,^2)) .

(5.1.1) We also define d(Q/R, P) d = r d(Q/R, P; Q/R, P). (5.1.2) 2 2 2 We denote d(Q /R ,P ;Q /R ,P ) also by d(mQi/Ri,mPi;mQ2fR2,mP2). We need the following modification of Xq,tWrite Xq,t{Mp) = X/m am(t) m: where m is a monomial and am(t) is its coefficient. Let 1

1

1

X^t(MP) ^ J2td^m^am(t)

m,

(5.1.3)

m

where d(m,mp)

is defined in (5.1.2).c

5.2 Frenkel-Mukhin [5, 5.1,5.2] proved that the image of the g-character Xq is contained in fee/ We have the t-analogue of this result, replacing (1 + A^\e)n

( i + ^ ) ; ^ E *r =r0 ( B -

-r)

by

A~r

where [™]t is the i-binomial coefficient. More precisely, we have c

In fact, d(m,mp) is determined from am(t) so that td<-m'mp^am(t) is a polynomial in t with nonzero constant term.

205

Theorem 5.2.1. (1) For each k £ I, Xq,t{Mp) is expressed as a linear combination of i

with coefficients in Z{i][Y^](^fc,aeC') where bi € C*, n* € Z > 0 wiift h ^ bj fori ^ j . (2) / / L p is an l-fundamental representation (and hence Mp = Lp), then Xq,t(Mp) contains no l-dominant monomials other than mp and the condition above uniquely determines Xq,t(Mp). Remark 5.2.2. The statements (1),(2) for t = 1 was proved by FrenkelMukhin [5, Theorem 5.9]. The proof of (2) is the same for t = 1 and the general case, as illustrated in the following examples. In this sense, (2) should also be creditted to them. 5.3

Graph

We give few examples of Xq,t of /-fundamental representations determined by the above theorem. We attach to each standard module Mp, an oriented colored graph Tp. (It is a slight modification of the graph in [6, 5.3].) The vertices are monomials in Xq,t(Mp). We draw an colored edge —— ' > from mi to m^ if m^ = miA~^\. We also write the multiplicity of the monomials in Xq,t(Mp). Example 5.3.1. Let g = A3 = sU and Mp = L(A2)\. Then the corresponding graph Tp is ^2,1

[

• YitEY2e2Y3>e

'• >

3,£ 2

Y1,eY~e\

Yle3Y3>£ 3,e 2

- i ^ U Y-£\Y2^Y-£\

-

^

Y~e\.

Let us explain how we determine this graph inductively. We start with the /-highest weight 3^,1 • We know that its coefficient is 1. Applying Theorem 5.2.1(1) with k = 2, we get Yit£Y~^2Y3ie with coefficient 1. Then we apply Theorem 5.2.1(1) with k = 1 to get Y~^3Y3tS. And so on. All multiplicities are 1 in this case. For g = An, it is known that the coefficients of Xq,t(L(Ak)a) are all l.d T h u s Xg,t(£(Afc)a) = X g , t = l ( £ ( A f c ) a ) . d

More generally, if t h e coefficients of a^ in the highest root is 1, then the same holds. This

206

Example 5.3.2. Let g = D 4 and MP = L(A 2 )i. The graph TP is Figure 1. It is known that the restriction of Mp to a U £ (fl)-module is a direct sum of the adjoint representation and the trivial representation. This fact is reflected in Xq,t(Mp) where Y^e 2 ^"/ 4 ^ a s *^ e coefficient [2]t and all others has 1. Note that the number of monomials is 28, which is the dimension of the adjoint representation. See also Example 7.2.3 below. Let us give a more complicated example. Example 5.3.3. Let 0 = ^2 and MP = L(A 2 )f 2 L(Ai)i. Although this is not an /-fundamental representation, Xq,t(Mp) has no /-dominant terms other than mp, so the condition Theorem 5.2.1(1) gives us Xq,t- The graph is Figure 2. Remark 5.3.4- As we can see in above examples, the crystal graphs are subgraphs of Tp. The set of vertices is the same, but the set of arrows is smaller. We would like to discuss this further elsewhere. 6

Step 2

6.1 Let MP = M(s1; (k^J, (n^x))<8)M(s2; (fc 2 J, (n 2 ^))®- • • be a standard module with si/sj i ez as in Theorem 2.1.2. Proposition 6.1.1. We have Xq,t(Mp)

= X,,t(M(s 1 ; {kxai)ai,

« ) Q l ) ) X q , t ( M ( S 2 ; ( * £ ) „ , , (n 2 2 ) a 2 )) • • •

ifsi/s^ezfori^j. Thus it is enough to study Xq,t(M(s'>

(ka)a,

(na)a))

= Xg,t(£(Afci ) e " i * ® L(Afc 2 ) e " 2 S « > • • • ) •

Let Xq,t(L(Aka)e"<*s))

=Y2amatra(t)ma:ra,

where maira is a monomial in Yfc a and amatra(t) G Z[£,£_1] is its coefficient. If t = 1, Xq,i i s a r m g homomorphism, hence we have Xq,i(M(a;(ka)a,{

)) = Yl IIam'->r°(1)ma'r<»ri,r2,...

a

result easily follows from the theory of quiver varieties. E x e r c i s e : Check this using the above algorithm.

207 Y2-

Y!,cY-\Y3,cYiiC

YheY3,eY-J3

YllSY-J3Y4,c

5/

2c4

y

4,eV4e3

^.« y «:i

Y

2,^Yil3Y4,lf

n..an:.4 y 3,«3Y-'

Y~*Y-*

Y

l,t5Y3,C3\t5

Y3tc3Y4]e3

l,e3Y3,eSY4,e*

r

- 1

A\—1 V i,e5r2,e4r3 5 r 4 5

2,e6

Figure 1. T h e graph for L(A2)i

i.e3y2,e4y3|e5y4,e3

^l.i^S.i^.e 3

Y

Y^1 Y

y

208

YIAY?. l.e

%e'

y-1 y3

PltF!,!^^^,,^-!3 2,e

[2]tYi,xY-^Y2,c 2,ea

Mt^i^^i,^^"

[3]t^l,£2^2,e^2£3

Plt^Liil^^K^s l.e

\ A 2 \^ —3 •rl,£2'r2,e3

([3]t +

l)Y-s\Y2,eY-'3

l,ea

^ U ^ l,e

^J^lj^.e 2,e2

y-2y-l

Figure 2. The graph for L ( A 2 ) ? 2 (g> L(Ai)i

209 Theorem 6.1.2. Let Pa be the Drinfeld polynomial of L(Aka)e"cShave

Then we

Xq,t{M(a;{ka),(na)))

=

t^^±d{m^'mpa'm^^m^)l[amrira(t)matra,

J2 7-1^2,...

Oc

where the sign for d(matra, mp*; mp:ri3, mp?) is — if a < (3 and + otherwise. Example 6.1.3. For g = Ai, we have

d(Y-la,Yi,a;Y1>a,Ylta)

= 1,

d^Y^Y^^^-,)

=1

and all others are 0. Then we get (4.1.7). If P — (1 — au)n, we get Xq,t(MP)

= £

ytl-ry-r M,a J l , a e 2

r=0

from Xq,t(L(A.i)a) = Yi>a + V^ 1 ^- This also follows directly from the definition (8.3.1) below. The ^-binomial coefficients appear as Poincare polynomials of Grassmann manifolds. 7

Restriction to U e (g)

Finite dimensional simple U e (g)-modules are classified by highest weights. Let Res Mp be the restriction of a standard module Mp to a U e (g)-module. It decomposes into a sum of various simple modules. Once \q{Mp) is computed, the character of Res Mp is given by replacing Vfc a by yj: (Theorem 1.1.5(2)). Combining with the knowledge of characters of simple finite dimensional U e (g)-modules, we can determine the multiplicity of simple modules in Res Mp. Characters of simple finite dimensional U e (g)-modules are the same as that of simple g-modules, hence are known. However, we express them in terms of Xq,t in this section. 7.1 For a dominant weight w = J2wk^-k we denote by Lw the simple highest weight U £ (g)-module with the highest weight w. We consider a standard module Mp with deg Pfc = tUfc. By the 'continuity' of Mp on P, Res Mp depends only on Wk = deg Pfc, and not on P itself. Let

210

us denote the multiplicity of Lw> in Res Mp by Z W ' W , i.e.,

ResMp = 0 I ^ ' w'

We will give a formula expressing Zw>iV/ in terms of \q,t{Mp). we can give algorithm for arbitrary P in principle, the following make the formula simple. Choose and fix orientations of edges in the Dynkin diagram. integer m{k) for each vertex k so that m{k) — m(l) = 1 if we have edge from k to I, i.e., k —> I. Then we define P by Pfc(u) = (1 -

Although choice will We define an oriented

uem^k))Wk.

Let Xq7t{MP) as in (5.1.3). Let xt(MP) € Z[t] <8>Z[tf]kei be a ^-analogue of the ordinary character which is obtained from x~qt(Mp) by sending Y^ to For another dominant weight w' = Ylwk^k>

^

(t) = ' the coefficient of TTyfe''' in Xt(Mp). The matrix (c W ' )W (t)) W ' >w is upper-triangular with respect to the usual order on weights, and diagonal entries are all 1. Theorem 7.1.1. c w ' iW (0) is the weight multiplicity of w' in the highest weight module L w with the highest weight w. This is just a simple rephrasing of a main result in [12,14]. The proof will be given in §8.5. Note that Cw']W(l) gives the weight multiplicity of w' in Res Mp since Xt^i is the ordinary character. Thus we have c w ",w(l) = 2__,Cw".w'(0) ^ w , w This equation determines the multiplicity Z w ' , w only from the knowledge of Xq,t-

According to a conjecture of Lusztig [11] together with a formula (8.5.1) below, Cw'>w(t) should be written by ferminonic form of Hatayama el al. [7]. More precisely, we should have 5ZW/ c w ' w '(0) c W ', w (i) = M(w, w", t2), where (c w ' w (0)) is the inverse matrix of (cw" ]W '(0)). See [11] for the definition of M ( w , w ' , g ) . Although this formula can be checked in many examples, the complexity of the combinatorics prevent us from proving it in full generality. Conjecturally M(w, w',q = 1) gives us the multiplicities of the restriction of Mp (Kirillov-Reshetikhine). Thus the conjecture is compatible with our e

In fact, they consider more general modules, not necessarily standard modules.

211

result in this section. 7.2 E x a m p l e 7.2.1. Let g = Ai and w = 2AX. We take P = (1 - u)2 by the above choice. By Example 6.1.3, we have X t (Mp) = ^ + ( l + i 2 ) + y i - 2 . Thus Zo, w = 1- Since Res(iWp) = Z,Al ® L A j = -£<2Ai © L0, this is the correct answer ! Example 7.2.2. Let g = A3 and w = A2. By Example 5.3.1 all the coefficients of Xq,t{L{h.2)i) are 1. Hence Res Mp — Res.L(A2)i is simple as a U £ (g)-module. E x a m p l e 7.2.3. Let g = D4, w = A2, w' = 0. By Example 5.3.2 we have c W ',wW = 4 + t2. Thus Z W ', W = 1, i.e. Res(£(A 2 )i) = LA 2 © L0. 8

Quiver varieties

In this section, we give the definition of \q,t and prove Theorems 4.1.5, 7.1.1. As we mentioned, those proofs are essentially given in [12,14] and [15] respectively. The only things we do here are translation of results into the language °f Xq,t- We believe that this section gives good introductions to [12,14,15]. 8.1 Let w = 'Y^Wk^-k (u>k £ Z>o) be a dominant weight of the finite dimensional Lie algebra g. In [12,14,15], we have attached to each w, a map n: 9JT(w) —> 9Jlo(oo,w) with the following properties: (1) 9Tt(w) is a finite disjoint union of nonsingular quasi-projective varieties of various dimensions. (2) 9Jto(co,w) is an affine algebraic variety. (3) n is a projective morphism. (4) There exist actions of G w x C* on 3Jt(w) and 9Jl0(oo, w) such that tr is equivariant. (5) 9Jl 0 (oo,w) is a cone, and the vertex (denoted by 0) is the unique fixed point of the C*-action (restriction of G w x C*-action to the second factor).

212

H e r e G w = n f c e /GL(u, f e ,C). We consider the fiber product Z(w) d ^ 2tt(w) x ^ ^ )

£K(w).

The convolution product makes the (Borel-Moore) homology group H*(Z'(w),C) into an associative (noncommutative) algebra. One of main results in [14] is a construction of a surjective algebra homomorphism U(B)^fftop(Z(w),C), where U(g) is the universal enveloping algebra of jj (NB: not a 'quantum' version). Here Htop( ) means the degree = dimjjZ(w) part of the homology group. More precisely, we take degree = dimension part on each connected components of Z(w), and then make the direct sum. Note that the the dimension differs on various components. Let £(w) = 7r -1 (0). It is known that SDT(w) has a holomorphic symplectic form such that £(w) is a lagrangian subvariety. The convolution makes #top(£( w )>C) (the top degree part of the Borel-Moore homology group, in the same sense as above) into an Ht0p(Z(w), C)-module. It is a U(g)-module by the above homomorphism. By [14, 10.2] it is the simple finite dimensional U(£|)-module L w with highest weight w. And connected components 9Jt(v, w) of SDT(w) are parametrized by vectors v = J2 vk&k (c*fc is the fcth simple root of g) so that Hto P (£(w),C) = 0 f f t o P ( 2 r t ( v , w ) fl£(w),C) V

is the weight space decomposition of the simple highest weight module L w , where i7 top (9JI(v, w) n £(w),C) has weight w - v. In particular, v = 0 corresponds to the highest weight vector. In fact, 9Jt(0, w) is consisting of a single point. The space 9Jto(oo,w) has a stratification 9Jlo(oo,w) = U ^ e g ( v , w ) , where v runs over the set of vectors such that w — v is a weight of L w which is dominant [12, §3]. 8.2 Let us give the U 9 (Lg)-version of the construction of the previous subsection. We use the following notation: Let R(G) denote the representation ring of a linear algebraic group G. If G acts a quasi-projective variety X, KG(X) denotes the Grothendieck group of G-equivariant coherent sheaves on X.

213

The representation ring R(GW x C*) of G w x C* is isomorphic to the tensor product R(GW) ®% R(C*). Moreover, R(C*) is isomorphic to Z[q, g _ 1 ], where q is the canonical 1-dimensional representation of C*. The convolution makes the Grothendieck group KGv,xC* (Z(w)) into a R(GW x C*) = i?(Gw)[<7,<7-1]-algebra. One of main results in [15] is a construction of an algebra homomorphism U^(L fl ) ® z i?(Gw) ->

ffG~*c*(Z(w))/torsion.

By the equivariance of n, £(w) = 7r _1 (0) is invariant under G w x C*. The convolution makes K G w X C *(£(w)) into a i^ GwXC *(Z(w))-module. Moreover, it is free of finite rank over R{GW x C*) [15, §7]. It is a U^(Lfl) ® z R{GW)module by the above homomorphism. By [15, §13], it contains a vector mo such that ek,r * m 0 = 0,

(Uj(Lfl)- ®z R(GW)) * m 0 = # G ~ x C * ( £ ( w ) ) , (K

1/ Q^WA*

(8.2.1)

The right hand side of the third equation needs an explanation: First Wk is the vector representation of GL(wk,C), considered as a G w x C*-module. Then A „ ^ = E ^ A ^ - S i n c e A-q/zl^Wk is 1 - (l/z)Wk + ... (1 is the trivial module), we can define [f\_qizq~lWk\ \/z.

This gives us the case ( )

+

as a formal power series in

of the above formula. In the case ( )~,

we expand as A - , / * ? - 1 ^ = {-l/z)Wk

(A™* W* -

Z/^^WH

+ • • • ) . Then

/\WkWk is an invertible element, we can also define (f\_qizq~1Wk) G

• The

xC

vector mo is the canonical generator of K ™ (9Jt(0,w)). (Recall 9JT(0,w) is a point.) The module K G w X C *(£(w)) should be considered as a 'universal' standard module since standard modules are obtained from it by specializations as we explain now. Let a = (s,e) € Gw x C* be a semisimple element. It defines a homomorphism Xa '• R(Gv, x C*) —> C by sending a representation to the value of the character at a. Then KG»xC*(£(w))®K(GwXC,)C 1

(8.2.2)

is a module over U £ (Lg) = Ug(Lg) ®z[g,g- ] C. By (8.2.1) it is a finitedimensional /-highest weight module. This is the standard module Mp, where Pk(u) = Xa{l\-uq~1Wk)Note that the set of conjugacy classes of a = (s,e)

214

bijectively corresponds to the set of /-tuple of polynomials P with deg P^ =

8.3 Let A be the Zariski closure of az in Gw x C*. It is an abelian reductive group. We have tfG~xC*(£(w)) ®fl (GwXC .) R(A) S KA(2(w)) [15, §7]. Since Xa factors through R(A), the standard module Mp is isomorphic to KA(£(w)) ®fl(A) C By Thomason's localization theorem, it is isomorphic to K(£(w)A) z C, where £(w)A is the fixed point set. Furthermore, it is isomorphic to H*(£(vr)A, C) via the Chern character homomorphism [15, §7]. Let -KA : VJl(w)A —> 97l0(oo, w)A be the restriction of the map n: SDT(w) —> SDTo(oo,w) to the fixed point set. Let 9Jl(w)A = \_\p9R(p) be the decomposition into connected components. Each Wl(p) is a nonsingular quasi-projective variety. Then we have the direct sum decomposition Mp<*H*(Z(w)A,C)

=

®H.(*m(p)nZ(-w),C)p

In [15, §13, §14] we have shown that this is the /-weight space decomposition of Mp. In particular, the index p can be considered as an /-weight of Mp. Thus we have arrived at a geometric interpretation of \q'Xq{MP) = £)dimff„(2K(p) n £ ( w ) , C ) mp, p

where mp is the monomial corresponding to the /-weight p. Now we define the i-analogue \q,t by

Xg,t(Mp) d=J2J2dimH^m(p)n£(w)'C) P

tk~dimcm(p) mp.

(8.3.1)

k

By [15, §14] we have a stratification 27lo(oo,w)^ = l J a J l - g ( / , ) , p

consisting of nonsingular locally closed subvarieties. Here the index set {p} is the subset of the above index set consisting of l-dominant /-weights. 8.4

Proof of Theorem 4-1-5

The /-highest weight P is fixed throughout the proof. Thus the dominant weight vector w and the element a = (s, e) e Gw are fixed.

215

We change the notation now. If p corresponds to an /-weight space Mp(Q/R), we denote above Wl(p) by 9Jl(Q/R,P). We also denote by DJtTQS(Q,P) for above TtTQB(p) if p corresponds to an /-dominant /-weight Q. Thus we have 2rt(w) A = [ J <0l(Q/R,P),

\JmT0es(Q,P).

9Ko(oo,w)^ =

Q/R

Q

In this notation if*(97t(P, P) n £(w),C) is the /-highest weight space. Since 9Jt(0,w) is a single point as we explained, we have 9Jl(P,P) — 9Jt(0,w). We also have 2J^ e g (P,P) = {0}. L e m m a 8.4.1. (1) dim c 0JI(Q/R,P) = d{Q/R,P). dim c m T 0 e g (Q,P) = d(Q,P). (2) IfTlr0eK(Q,P) C 9Jto eg (fl,P), then R P) = dime %R(Q, P), which is clear from the definition [15, §4]. (2),(3) The results are known or trivial for Q = P. Now use the transversal slice at x G 9JITQS(Q,P) [15, §3] to reduce a general case to this case. • Let Db(DJlo(oo,w)A) be the bounded derived category of complexes of sheaves such that cohomology sheaves are constant along each stratum 9J?Qeg(Q,P). Let IC(VJlT0ee(Q,P)) be the intersection homology complex associated with the constant local system CmregtQp\ on 9Jt™s(Q, P). By using the transversal slice [15, §3], one can check that it is an object in D6(SOT0(oo, w ) A ) . Let C OT (Q/ R p) be the constant local system on Wt(Q/R, P). Then irA (Cyn(Q I R,P)) is an object of Db(Tlo(oo,w)A) again by the transversal slice argument. Using the decomposition theorem of Beilinson-BernsteinDeligne, we have shown that there exists an isomorphism in Db(OJlo(oo, w)A): TvA(Cm{RtP)[dimc<m(R,P)})

^0LQ]t(i},P)®rc(lJes(Q,P))[fc]

(8.4.2)

Q,k

for some vector space LQik(R,P) [15, 14.3.2]. Since irA(Wl(R,P)) c r S 9Jl Q (R,P) by definition [15, §4], the summation runs over Q > R by Lemma 8.4.1. Let

W * ) d=' 5>imLQife(P,P)*-fc. k

216

Applying the Verdier duality to the both hand side of (8.4.2) and using the self-duality of Tr?(Cm{R,P)[dimc Wl{R, P)]) and IC(VJlrQe6(Q,P)), we find LRQ(t) = LRQ(t). Choose a point XQ from 9JtrQS(Q,P) for each stratum. Let iXQ : {XQ} —> SDTo(oo,w)'4 denote the inclusion. Consider tffe(4Q^COT(fl,P)[dimOJt(#,P)])

=

Hdimcm{RtP)_k((TrA)-1(xQ)nm(R,P),C)-

By Lemma 8.4.1(3) this is isomorphic to Hdimc m(R,P)-k(Wl(R, Q) n £ ( w ) , C). Therefore we have J2 dim Hk{i'XQ7rXm(R,p)

[dim 501(12, P)}) t d i m c

OT

«^>-*

k

= Y,dimHd(^(R,Q)

^ &M,c)td+dimcmiQ'p)~dimcm{R'p)

= cRQ{t), (8.4.3)

where we used dim c VJl(R, P) - dim c Wl{Q, P) = dim c M(R, Q) in the last equality. By [15, 14.3.10], we have [MQ : LR] = d i m f f * ( 4 Q J C ( 0 J C ( i ? , P ) ) ) . (In fact, we defined the standard module MQ as i7*((7r yl ) _1 (xQ),C) in [15, §13], which apriori depends on P . Thus the definition coincides only when Q = P. However, by using the transversal slice, we can show that the right hand side is the same for both definitions, cf. Lemma 8.4.1.) Let

zRQ(t) d^

J2dimHk(^Qic(mies(R,p)))tdimVnoe^^-k. k

We have [MQ : LR] = ZRQ(1). By the defining property of the intersection homology, ZRQ{t) satisfies (4.1.4). Substituting (8.4.2) into (8.4.3), we get

csQ(t) =

Y,LSR(t)ZRQ(t). R

Now LsR(t) rem 4.1.5.

= LsR(t)

implies (4.1.3). This completes the proof of Theo-

217

8.5

Proof of Theorem 7.1.1

By the result explained in §8.1, the weight multiplicity of w' in L w is equal to dim Htop(Wl(w - w', w) n £(w), C). The assertion follows from more general formula

xt(Mp) = J2J2 ^ ( ^ ( w - w ' ' w ) n £ ( w )> c ) t d i m c OT(w-w'-w)-d Yi vtk • w'

d

k

(8.5.1) Note that 9Jl(w — w', w) n £(w) is a lagrangian subvariety in SDT(w — w', w), so we have top = dime 97t(w — w', w). In order to prove (8.5.1), we use [12, 5.7], where the Betti numbers are given in terms of those of fixed point components. It looks almost the same as above. However, there is one significant difference. The C*-action used there is different from our C*-action used here, defined in [15, §2]. This is the reason why we choose P and corresponding a = (s, e) as explained in §7. Then A = a z is isomorphic to C* and the action is the same as the C*-action considered in [12, §5]. We decompose 9Jl(w)A = \_\Wl(p) into connected components as before. By [12, 5.7] we have' dim# d (9Jt(w - w', w) n £(w), C) = ^ d i m H d i m c 9Ui(w—w',w)—d p

where the summation runs over the set of p such that the corresponding monomial mp is sent to n * ^ ^ a ^ e r ^ M —* Vk- The C*-action makes SDT0(oo,w)c* = {0}, so Wl(p) — SOT(p) n -C(w). Hence the above expression coincides with the definition of the coefficients \t • Acknowledgements We would like to thank E. Frenkel and E. Mukhin for explanations of their algorithm computing g-characters of fundamental representations. The author are partially supported by by the Grant-in-aid for Scientific Research (No.11740011), the Ministry of Education, Japan. •^In fact, this formula even holds for general P if we replace dime 3Jt(w — w ' , w) — d by a suitable degree. However, this degree shift is given by a complicated expression in p. So our choice of P is most economical.

References 1. V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. 2. , Quantum afjine algebras and their representations, Representations of groups (Banff, AB, 1994), Amer. Math. Soc, Providence, RI, 1995, pp. 59-78. 3. , Quantum afjine algebras at roots of unity, Represent. Theory 1 (1997), 280-328 (electronic). 4. , Integrable and Weyl modules for quantum afjine sfo, preprint, arXiv:math.QA/0007123. 5. E. Frenkel and E. Mukhin, Combinatorics of q-characters of finitedimensional representations of quantum afjine algebras, preprint, arXiv:math.QA/9911112. 6. E. Frenkel and N. Reshetikhin, The q-characters of representations of quantum afjine algebras and deformations of W-algebras, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math., 248, Amer. Math. Soc, Providence, RI, 1999, pp. 163-205. 7. G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Remarks on fermionic formula, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math., 248, Amer. Math. Soc, Providence, RI, 1999, pp. 243-291. 8. M. Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383-413. 9. D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. 10. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. 11. , Ferminonic form and Betti numbers, preprint, arXiv:math.QA/0005010. 12. H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365-416. 13. , Homology of moduli spaces instantons on ALE spaces I, J. Differential Geom. 40 (1994), 105-127. 14. , Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515-560. 15. , Quiver varieties and finite dimensional representations of quantum afjine algebras, preprint, arXiv:math.QA/9912158, to appear in J. Amer. Math. Soc.

219

16. M. Varagnolo and E. Vasserot, Standard modules of quantum affine algebras, preprint, arXiv:math.QA/0006084.

GENERALIZED HOLDER'S T H E O R E M FOR MULTIPLE GAMMA FUNCTION MICHITOMO NISHIZAWA Department of Mathematics,School of Science and Engineering Waseda University 3-4-1, Ohkubo Shinjuku-ku, Tokyo 169-8555, Japan E-mail: [email protected] We improve Barnes' proof of nonexistence of algebraic differential equations for his multiple gamma function. It is also proved that multiple g-gamma function satisfies no algebraic differential equation over a certain field. Furthermore, we consider an example of differential relations between several Barnes' multiple gamma functions in a special case.

1

Introduction

In 1904, Barnes 2 introduced his multiple gamma function which satisfies functional relation Tr(z

+ L}j\u)

=

rP(*|w)

Pr-ljUjjj))

iv^iwOO)

m

l ;

where w = (CJI,W2, - • -w r ) is a set of complex parameters, w(j) := (CJI, • • • ,<jjj-i,Uj+i,- • • ,u>r) and pr(ui(j)) depends only on w(j). In the case r = 1, Ti(z\(oJi)) is equal to Euler's gamma function T(z/ui) up to an elementary factor. After his discovery, many mathematicians have studied this function: Hardy 4 ' 5 studied this function from his viewpoint of theory of elliptic functions, and Shintani 20,21 applied it to the study on the Kronecker limit formula for zeta functions attached to certain algebraic fields. Recently, this function plays an important role in studies on quantum integral systems 8 ' 10 ' 11 ' 12,19 and on ^-analysis with \q\ — l14>16>22. In this article, we consider differential relations between these functions. It is well known that Euler's gamma function satisfies no algebraic differential equation. It is called Holder's theorem. As a generalization of it, Barnes 3 gave a proof that his multiple gamma function satisfies no algebraic differential equation. He introduced an weight for algebraic differential equations and showed that any algebraic differential equation with nonzero weight can be transformed to some equation with weight 1 by using the functional relation for the multiple gamma function. His result, which corresponds to Lemma 5 in this article, is very important. However it should be considered as the 220

221

first step of induction. In order to prove the generalized Holder's theorem, we have to construct the induction completely. Furthermore, on defining this weight, he did not make clear the function field which he considered. There would remain some ambiguous parts in his proof. In this article, we reconstruct a proof of the generalized Holder's theorem by using a weight for multivariable polynomials over C(z). This method can be applied to its generalization for a ^-analogue of multiple gamma function introduced by the author 13 . We can see that this g-analogue satisfies no algebraic differential equation over C(z,qz). Although there is no differential relation for a single multiple gamma function, there exists differential relation between several multiple gamma functions in a special case. In the case when all Wj (j = l , - - - , r ) are equal, we can derive an differential relation between multiple gamma function by using an infinite sum representation of the logarithmic derivative of the multiple gamma function. As a corollary of this relation, we can prove generalized Holder's theorem more easily in this case. We remark that this case is closely related Vigneras multiple gamma function 25 . She introduced meromorphic functions Gr(z) which satisfy the following relation: Gr{z + l) = Gr-i{z)Gr{z), d2 —^ logGr(z + 1) > 0 for

G P (1) = 1 (2) z > 0,

G0(z) = z.

This function have been applied to represent the determinant of the Laplacians 24 ' 25 ' 26 . Gr(z) is related with Tr(z) by the relation Gr(z) = (elementary factor) x r ( z | ( l , • • •, 1)). This article is organized as follows: In Section 2, we give a survey on Barnes' multiple gamma function. In Section 3, we give a proof of generalized Holder's theorem and remark its generalization to g-analogue. In Section 4, we derive a differential relation between Tr(z\(iJi, • • • ,w r )) and r r + i ( z | ( w i , • • • ,w r + 1 )) in the case when uj\ = • • • = u>r = w r + i . 2

Barnes' Multiple Gamma Function

We summarize some facts about Barnes' multiple gamma function. We follow the notation in Barnes 2 . Let w := (wi, w2, • • •, uir) be a set of complex numbers lying the same side of some straight line across the origin on the complex plane.

222

Definition 1 We define £(s,z\w) and Tr(z\ui) by the following formula: (r(s,z\w):=

^2

m-u)~s,

(z +

me(Z>0)

rr(z\uS) := p r (w) exp

—

Cr{s,z\w) s=0

where m •= (mi, 7712,- • • ,mr), rn-i^:— miLJi + logp r (w) := — lim

»{

8 ds

\-mTur,

and

+ k>gZ

(r(s,z\w)

I

s=0

We recall the basic facts concerning Barnes' multiple gamma function. Proposition 2 (Barnes) (i) Tr(z\uf) satisfies afunctional relation (1) where w(i) := (wi,u2,- • -Ui-i,Ui+i,- • • >Ur)(ii) r r (^|a;) have an infinite product representation

r^r^exp £ ^ W n '

m-u)

\k=l

where jTik (k = 1, •••,!•) are scalars and FJ nonnegative integer except m= (0,0, • • •, 0).

[f^11 (m• <*>)'

means that m runs over all

(iii) r r (z|o;) is symmetric with respect to UJ := (wi, • • •, w r ).

|

We define ipr(z\u) as the logarithmic derivative of rr(z\u/). From Proposition 2 (ii), it follows that d ipr(z\uf) ~ — logTr(2:|a;) az lr,k

E-(fc-1)!

fc-i

_ J- _ V ^ /

z+m•u

+^

ft*tL

1

(m ••hiV LU) £—/ (m.

where £) means that m runs over all elements of (Z> 0 ) r — {(0,0, • • • ,0)}. From Proposition 2 (i), ipr(z\u) satisfies the relation 1pr(z\uL) - 1pr(z + CJj\u) =

1pr-l(z\uJ.(j)).

(3) (n),

We observe that i/>r(z\w) has simple poles at z = m • LO and V>r {z\u) '•= j^i>r(z\w) has poles of order n at z = m • w, where m € (Z< 0 ) r .

223

For the following argument, we introduce several notations. We denote w<:7by W<j

and define ifjj(z) := tpj(z\u^j),

3

:

=

(wi,W2,---,Wj),

and ipj(z)

:=

^ipj(z).

Generalized Holder's Theorem

In this section, we give a proof of generalized Holder's theorem. Theorem 3 (i) There exists no n + 1-variable polynomial g(to,ti,- • • ,tn) G C(z)[t0,ti,•

• • ,tn]

such that

g(Mz),41\z),---4n)W) = 0-

(4)

(ii) There exists no n + 1-variable polynomial f(to,h,

• • • ,tn) S C(z)[to,ti,

•••*„]

such that

f(rr(z\u),rV(z\u),---rin\z\ui)) = o. In other words, Barnes' multiple gamma function satisfies no algebraic differential equation. | We know the fact that if some function satisfies an algebraic differential equation, then its logarithmic derivative also satisfies an algebraic differential equation, (cf. Komatsu 7 , Pastro 17 ). Thus, it is sufficient to show the following theorem since Theorem 3 (i) corresponds to the case when p = r:

Theorem 4 For p £ {1, •••?•} and arbitrary fixed nx 6 (lp+i,lp+2, • • -lr) G Z r - P ) , there is no polynomial

/CO G C(z) T = (ti,fc) I = (lp+i,lp+2,--

• ,lr) S Z r

P

Z> 0 (/

:=

, 0 < k < Til,

such that it vanishes when we substitute ipp(z + X)[=p+i h^i) for i;^

/ ( U(PjH(z+ £ '**))) =°-

(5)

224

Proof: : We prove this theorem by induction. In the case when r = 1, we can prove it in a similar way to the original Holder's theorem for Euler's gamma function. We suppose that this claim holds for p — 1. First, we fix some notations. We define Ep(z) as the ring generated by z and V'p-i (z+52i=P hwr), (j G Z> 0 , h G Z )

Ep(z) := C z^l^z+

^2hu)i) j G Z> 0 , k £ Z

/,(i) and Ap(z) as the field generated by z and Vp-i(2 + 2[=p'iwi)> 0 Z;GZ)

kp{z):=C\z^(J\{z

e

Z>o,

+ Y,k"r) j G Z > 0 , /j G Z t=p

and A r (z) := C(z). That is, any element of A p (z) is represented as Q\{z)IQi{z), where Qi(z), £2(2) € Hp(,z) and Q 2 (z) 7^ 0. Key of proof is to show the following lemma: Lemma 5 If there exist some polynomial E{T) G Ap(z) T = (tu)

0<j<mh

le Zr-p

such that E

\

\Tp{pi)(z+ E

^ ) l I

= 0

(6)

then, we have

i=p+l r p

for some I G Z . in otAer words, for some I G Z r element ipp{z + X)[=P+i ^ w i) as A(z)i>p(z+

p

, tue can represent the

^T ZjWi) = B(z),

(7)

i=p+l

where A(z), B(z) G H p (z).

225

Proof of Lemma: We prove this proposition by induction. In the case when p = 1, we can prove the claim by the same argument as Holder's theorem for Euler's gamma function. We suppose that there is no algebraic relation between Vp_i(z + Ei= P '» w ») 0' e z > o , I G Zr~p+1). We introduce weight wt for element of Ap(z)[T]. For a monomial FJ; ITj=o ^ j ' > w e define M

^(n,n"ioC)

/ j=0

J k=0

For polynomials, we define its weight by the maximum of the weights of monomials included in the polynomial. We often abbreviates indeterminates (ipp(z + Ei= P +i k^i)) by t&piz). We suppose that there exists a polynomial EZ(T) satisfying Ez(*p(z))=0.

(8)

Without loss of generality, we can suppose that Ez (T) has the lowest weight s among such polynomials. We denote kv + 1 by the number of monomials of weight v (0 < v < s). For 0 Qiti / Qij- We define A s as the sum of terms whose weight is lower than s. By using these notation, we can see that EZ(T) has the following representation: k,

Ex(T):=YiRt,i(z)Qt,i(T)

+ At,

i=0

where Rs,i(z) G Ap(z). Without loss of generality, we can suppose that one of nonzero coefficients, say RSfi(z), is equal to 1. We remark that Ez+U (T) satisfies the relation Bz+ Up (*p(z + w p ) ) = 0 , and that wt(Ez+u>p(T)) Ez+Up(yp(z

= s. From relation (3), it follows that + ujp)) = Ez+Wp(Vp(z) k,

i=0

- * p _i(«))

226

where

Furthermore, we can regard an element of Ap(z) [9p(z + up)] as an element of Ap(z)[&p(z)] by using (3). In this sense, we may regard that Ez+Up (* p (z + Up)) as an element of Ap(z)[^p(z)]. We define it as Ez+U}p(^p(z)). Let us put FZ{T) = EZ(T) - Ez+Up(T) G AP(z)[T]. We can see that Fz{T):=Ez{T)-Ez+ulp{T) k,

= £

(RBAZ

+ up) - R.,i{z)) Q8,i{T) + A . .

i=0

Prom the fact that Rs,o(z + wp) = RSto(z) = 1, it follows that the number of monomial with weight s in Fz(T) is fcs. Thus, FZ(T) = 0 because the minimal number of monomial of weight s is ks + 1. Therefore, we have that Rs,i(z + OJP) - Rs,i(z) = 0

for

1 < t < ks.

(9)

However, if (9) have a nontrivial solution, then there exist an algebraic relation between rppj\{z + YlUph^i) (I € Zr~P+1, 0 < j < m ). This contradict to the assumption of the induction. Thus, Rs,i{z) := a s ,j G C and FZ(T) can be represented by the formula k,

Fz(Vp(z))

=£

a,,i (QSti{Vp{z)) - Qs,i{^p{z + up))) + A,.

i=l

From the definition of weight, it follows that ~ QSA*P(Z

Q.,i(*P(z))

+ WP)) r

= 6s,i1pp-i(z + £

r

kUi)Qs,i(®p(z))/1Pp(z+

i=p+l

'i w i) + A s - l ,

£ i=p-fl

r_p

for some I G Z , where 5Sii is a constant. Therefore, Fz(^p(z)) sented by the formula, ks-i

/

F z (# p (z)) = ] T »=o y

is repre-

r

4S/_1)^-!^ + £

\

'*"<) + R-u(z)

- Rs-i,i(z + wP)

*=P+I

xg,-i,i(*,,(«)) + A . - i ,

/

(io)

227

where S\SjS (1 < j
is such a constant that it is equal to zero if there exists no j satisfying

Q s ,j(* P (2)) - QsA^P(z

+ W P)) = Qs-i,j(yP(z))

+ AS_L

If (10) is a nontrivial representation of Fz(^p(z)), then there exists such a polynomial in g{T) <E A p (z)[T]\{0} with weight s - 1 that g(Vp{z)) = 0. It is contradiction to the assumption of the induction. Thus, all coefficients on the right hand side of (10) vanish r

Rs-iti{z

=S\y1'i(jp^1(z+

+ u)p) -Rs^.lA{z)

^

hut)

i=p+l

for I € Z r ~ p . Therefore, from the assumption of induction, the following facts hold: (i) <5-iS/_1) = 0 for j = 0,1, • • • ks and i = 0,1, • • •, ks-i, (ii) Rs-xti(z)

= o s _i,j € C.

If we define rjj(s,i) as the sum of all /xjj(s,i) over I G zr~p Vj{sJ)

:=^2ni,j(s,i). i

It follows from (i) that T]O(s,i) = 0 for

i = 0,1, • • -,fcs.

Repeating this procedure successively, we can prove that T]J(V,W)

— 0 for

0 < j < max.ni, v = 1, • • • ,s, w = 0, • • • ,kv.

Thus, fiitj(s,i) = 0 for VZ e Z r ~ p since fj,ij(s,i) are nonnegative integers. It means that wt(Ez(T)) = 1. Therefore, we conclude that r

A(z)ipp(z + ] T

liWi)=B(z),

»=P+I

for some I e Zr~p where A(z), B(z) £ Ep(z).

|

Once, we have this lemma, we can prove Theorem 4 easily. If there is no polynomial E(T)EAp(z)

T=(tltj) I e

Zr~p,0<j
228

such that

then, of course, there is no differential relation like formula (5). Thus, it is sufficient to consider the claim in the case when there exist such differential relations. If there exists a differential relation (6), then, from Lemma 5, it follows that the relation is transformed to a formula represented as r

A(z)ipp(z + ^2 ^i)

= B(z)

for s o m e

l e

zr_P

-

i=p+l

By using it, we can reduce any algebraic relation over C(z) ipp (z + X)i=p+i h^i) (' e Z r _ p , 0 < j < ni) to some algebraic over C(z) between ip^z + E [ = „ ^ ) (/ 6 z r - " + 1 , j e Z> 0 ). a contradiction to the assumption of the induction. Therefore, no algebraic relation over C(z) between ipp {z + J2l=P+i h^i) {I o<j<m). |

between relation This is there is € Zr~p,

We remark a generalization to its (/-analogue. The author 14 introduced multiple g-gamma function Gr(z;q) which satisfies a (/-analogue of (2) Gr(z + l;q) ^Gr-1(z;q)Gr(z;q),

Gr{l;q) = 1,

-^-\ogGr(z + l;q)>0 for z > 0, G0(z;q) = \ ^ . az 1— q On the other hand, Pastro 17 proved that (/-gamma function satisfies no algebraic differential equation over C(z,qz). By using the same argument as above, we can prove the following theorem: Theorem 6 When 0 < q < 1, the multiple q-gamma function Gr{z; q) satisfies no algebraic differential equation. In other words, there exists no n + 1variable polynomial f(to,tl,---,tn)

£ C{z,qZ)[to,tl,

•••tn]

such that f(Gr(z;q),GP(z;q),---Gin\z-q))=0.

|

229 4

Differential Relation in a Special Case

As we have seen in the previous section, there is no algebraic differential equation for a multiple gamma function. However, it is not always true that there is no algebraic relation between several multiple gamma functions with different r. For example, we can derive a differential relation between Tr(z$u)i, • ••, uir)) and r r + i(2:|(u;i, • • •,w r +i)) m the case when Wl = 0J2 = • • • = U)r = U!r+1 = W.

(11)

Let Vv(z) be the infinite sum part of ipr(z)

^w==-f;{jTfej(r!1)+^*i«)}. fc=0

v

x

where
(-zy-i miH+ "-\-m \-mrr—k =k 1=1 m\

v

—

—'

Then, we have the following relation: Lemma 7 z + (r-l)u7 ruj

~ -1pr\Z) = -1pr+l{z).

Proof: First, we notice the following identity: z + (r - l)w / k \ rw(z + ku) \r — $

1 fk\ z + kw \rj

I ( k rui \r — 1

for r, k £ Z>o- We have z + (r — l)w -ipr(z) ru

From the absolute convergence of tpr(z), it follows that the sum of the right hand side is absolutely convergent. The convergent factor of this sum have to be equal to r+i(z,k\w). In fact, the factor is uniquely determined when its degree of k is larger than — 2. |

230

Remark 8 We can compute r(z,k\us) explicitly. We have the following formula by the above argument: (j>r+1(z,k\u)

= — (

1

) - -

—r(z,k\w).

rui \r — 1/ ru> We can calculate the convergent factor by using this formula as a recurrence formula, {cf. Nishizawa 15 ). | From Lemma 7, we have relations between logarithmic derivatives. Proposition 9 (i) ipr(z) satisfies a recurrence formula .

. .

z + (r — 1)CJ , . . .. roj where pT(z) is a polynomial with degree less than equal r. (ii) Vv+i (z) is transformed by the following formula:

Tpr+1(Z)=(-ir(»+rr~iy1(z)+pr(Z), where Pr{z) is a polynomial with degree less than equal r.

|

Corollary 10 In the case when UJ\ — • • • = ur = ojr+i = w, we have

j^r r+1 (z)}r r (2) = -z

+ ir

1)oj

r~

|ArP(^)|rr+1(z)+p(z)rP+1(z)rI.(z),

where rr(z):=rr(z\(w,u},---,w)).

I

We remark that Proposition 9 (i) corresponds to the formula (7) in Lemma 5. Thus, in this case, we can see a differential relation between logarithmic derivatives and we can easily prove Theorem 3. If some polynomial satisfies (4), then there exists an algebraic differential equation for i(>r(z). However, from Proposition 9 (ii), it follows that there exists an algebraic differential equation for ipi(z) over C(z). This contradict to Holder's theorem for the gamma function.

231

Acknowledgment: The author expresses his deep gratitude to the organizing committee of International Workshop on Physics and Combinatorics 2000 and the staff of Graduate School of Mathematical Sciences of Nagoya University. He also thanks a referee for his careful reading of the manuscript and for valuable comments. The author is partially supported by Grant-inAid for Scientific Research from the Ministry of Education (12640046) and Waseda University Grant for Special Research Project 2000A-172.

References 1. E.W. Barnes, The Theory of G-Function, Quat. J. Math. 31(1899), 264314. 2. E.W. Barnes, The theory of the Multiple Gamma Function, Trans. Cambridge Phil. Soc. 19 (1904), 374-425. 3. E.W. Barnes, On Functions Generated by Linear Difference Equations of the First Order, Proc. London Math. Soc. (2) 2(1905), 280-292 . 4. G.H. Hardy, On the expression of the double zeta-function and double gamma functions, Trans. Cambridge Phil. Soc. 20 (1905), 395-437. 5. G.H. Hardy, On double Fourier series and especially these which represent the double zeta-function and incommensurable parameters, Quart. J. Math. 37 (1906). 53-79 6. K. Katayama and M. Ohtsuki, On the Multiple Gamma Functions, Tokyo J. Math, bf 21 (1998), 159-182 7. Y. Komotsu Special Functions (in Japanese), Asakura-Shoten. 8. M. Jimbo and T. Miwa, Quantum KZ equation with \q\ = 1 and correlation functions of the XXZ models in the gapless regime, J. Phys. A: Math. Gen. 29 (1996) 2923-2958 . 9. K. Matsumoto, Asymptotic series for the double zeta, double gamma and Hecke L-functions, Math. Proc. Camb. Phil. Soc, 123 (1998), 385-405 10. T. Miwa and Y. Takeyama, Determinant Formula for the Solutions of the Quantum Kniznik-Zamolodchikov Equation with \q\ = 1, RIMS preprint 1220. 11. T. Miwa and Y.Takeyama, The integral formula for the solutions of the quantum Kniznik-Zamolodchikov equation associated with Ug(sln) with |g| = 1, RIMS preprint 1226. 12. T. Miwa, Y. Takeyama and V. Tarasov, Determinant Formula for Solutions of the Uq(sln) qKZ Equation with \q\ = 1, RIMS preprint 1232. 13. M. Nishizawa, On a q-Analogue of the Multiple Gamma Function, Lett.

232

Math. Phys. 37 (1996), 201-208 14. M. Nishizawa, On a Solution of q-Difference Analogue of Lauricella's Hypergeometric Equation with \q\ = 1, Publ. RIMS. Kyoto Univ. 34 (1998), 277-290. 15. M. Nishizawa, Generalized Holder's Theorem for Vigneras' Multiple Gamma Function, To appear in Tokyo J. Math. 16. M. Nishizawa and K. Ueno, Integral Solution of q-Difference equation of the Hypergeometric Type with \q\ = 1, HAS Report No.1997-001, "Infinite Analysis - Integral Systems and Representation Theory", 247-255. 17. P.I. Pastro, The q-Analogue of Holder's Theorem for the Gamma Function, Rend Sem. Mat. Univ. Padova, 79 (1983), 47-53. 18. S.N.M. Ruijsenaars, First order difference equations and integrable quantum systems, J. Math. Phys. 38 (1997) 1069-1146. 19. S.N.M. Ruijsenaars, A Generalized Hypergeometric Function Satisfying Four Analytic Difference Equations of Askey-Wilson Type, Comm. Math. Phys. 206 (1999), 639-690. 20. T. Shintani, On a Kronecker limit formula for real quadratic fields, J. Fac. Sci. Univ. Tokyo, 24 (1977), 167-199. 21. T. Shintani, A poof of classical Kronecker limit formula, Tokyo J. Math. 3(1980), 191-199. 22. Y. Takeyama, The q-twisted Cohomology and The q-hypergeometric Function at \q\ = 1, RIMS preprint 1278. 23. K. Ueno and M. Nishizawa, Multiple Gamma Functions and Multiple q-Gamma Functions, Publ. RIMS. Kyoto Univ. 33 (1997), 813-838. 24. I.Vardi, Determinants of Laplacians and multiple gamma functions, SIAM. J. Math. Anal 19 (1988), 493-507. 25. M.F.Vigneras, L'equation fonctionalie de la fonction zeta de Selberg de groupe modulaire PSL(2,Z), Asterisque. 61 (1979), 235-249. 26. A.Voros, Spectral functions, Special functions and the Selberg zeta functions, Comm. Math. Phys. 110 (1987), 431-465 27. E.T. Whittacker and G.N. Watson A Course of Modern Analysis, Fourth Edition, Camb. Univ. Press, 1927

Q U A N T U M CALOGERO-MOSER MODELS: C O M P L E T E I N T E G R A B I L I T Y F O R ALL T H E R O O T SYSTEMS

RYU SASAKI Yukawa

Institute

for Theoretical Physics, Kyoto Kyoto 606-8502, Japan E-mail: [email protected]

University

Complete integrability of Calogero-Moser models is discussed from various angles. Calogero-Moser models and Toda models are best known examples of solvable many-particle dynamics on a line which are based on root systems. At the classical level, the former (C-M models) is integrable for elliptic potentials (Weierstrafi p function) and their various degenerate limits. The latter (Toda) has exponential potentials, which is obtained from the former as a special limit of the elliptic potential. Here we discuss quantum Calogero-Moser models based on any root system. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quant u m conserved quantities in terms of a Lax pair, (ii) Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum, (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalizations of the results known for the models based on the A series, i.e. su(N) type, root systems, which are treated rather explicitly in the Introduction. The independence of the conserved quantities (necessary for the Liouville integrability) obtained from the universal Lax pair is discussed in some detail.

1

Introduction

Calogero-Moser models 1>2'3 are one-dimensional multi-particle dynamical systems with long range interactions. They are integrable at both classical and quantum levels. In this paper we consider quantum models. The simplest example is described by the following Hamiltonian with r degrees of freedom:

in which g is a real coupling constant (assumed to be g > 1 for simplicity) and q = (qi,..., qr) are the coordinates and p = (pi,...,pr) are the conjugate canonical momenta obeying the canonical commutation relations: kj,Pk]=iSjk,

[qj,qk] = \Pj,Pk] = 0 , 233

j,k = l,...,r.

(1.2)

234

The Heisenberg equations of motion read qj =i[H,qj]

=pj,

Qj = Pj = i[H,Pj] = 29(9 - 1) Yl

(Qj

~Qkf

(1.3)

The repulsive 1/(distance) 2 potential cannot be surmounted classically and the relative position of the particles on the line is not changed during the time evolution. It is remarkable that these equations of motion can be expressed in a matrix form (Lax pair) dL i[H, L] = — = LM-ML

= [L, M],

t Qj=Pj, Pi = 2 * ( * - l ) g ( ^ ) 3 .

(1.4)

in which L and M are defined by /

JS_

Pi J3_ 92 - g i

L =

-JS—\

Q1—Q2

qi-Qr

Pi

qz-q-r

ja_

'

(1.5)

Pr J

and (

mi

(«i-92)2

ig (q2-Qi)2

M =

V

m2 _*fi_

(9i--gi)a

(?r--«2) 2

ig (92-9r-) 2

(1.6)

mr

The diagonal element rrij of M is given by m

i=i9

^w^-

(1.7)

kitj

The matrix M has a special property (1.8) fc=i

235

It is straightforward to derive quantum conserved quantities from the total sum of powers of Lax matrix L: Qn = Ts(Ln) = Yl(Ln)jk,

[H,Qn] = 0,

n = l,...,

(1.9)

since

-§ $>");* = i J > , (in)i*l = EtL"'M li* di j,k

j,k

j,k

J2 ((Ln)jiMik - Mjt{Ln)lk\

= 0,

(1. 10)

in which (1.8) is used at the last equality. It is elementary to verify that the Hamiltonian is equivalent to Q2, H oc Qi + const. In other words, the Lax matrix L is a kind of "square root" of the Hamiltonian. These are fundamental results for the integrability of the model (1.1). Similar integrable quantum many-particle dynamics are obtained by replacing the potential in (1.1) by sinii2{qj-qk)

1 (
Ik)2

1

(1.11)

inn.

[_ sin*(qj-q k )

The quantum equations of motion are again expressed by Lax pairs if the following replacements are made: 1 {qj qk)

coth(gj- - qk)

- ~*\cot(qj-qk)

sinh' 2 (9j-g fe )

in L,

in M.

(Qj ~ Qk)2 .

sin*{qj-qk)

(1.12) Since the property (1.8) holds also in these cases, we obtain quantum conserved quantities in the same manner as above (1.9) for the models with the trigonometric and hyperbolic interactions. The main goal is to find all the eigenvalues {A} and eigenfunctions {ip} of the above Hamiltonian: Hip = \ip.

236

For the rational model Hamiltonian (1.1), it reads

4§iH9fa-i)£te^Fr=A*

(LI3>

which is a second order Fuchsian differential equation for each variable {qj} with regular singularity at each hyperplane qj = qk whose exponents are g, 1 — g. Any solution tp of (1.13) is regular at all points except those on the union of hyperplanes qj = qk • Since the structure of the singularity is the same for the hyperbolic and trigonometric potentials (1.11), the same assertion for the regularity and singularity of the solution rp holds for these cases. For the trigonometric case there are other singularities at qj — qk= in, £ £ Z, due to the periodicity of the potential. The integrability or more precisely the triangularity of the quantum Hamiltonian was first discovered by Calogero x for the model with inverse square potential plus a confining harmonic force and by Sutherland 2 for the particles on a circle with the trigonometric potential. Later, classical integrability of the models in terms of Lax pairs was proved by Moser 3 . Olshanetsky and Perelomov 4 showed that these models were based on Ar-i root systems, i.e. qj — qk = a • q and a is one of the root vectors of Ar^\ root system (A.6). They also introduced generalizations of the models 5 based on other root systems including the non-crystallographic ones. In this paper we discuss quantum Calogero-Moser models with degenerate potentials, that is the rational with/without harmonic force, the hyperbolic and the trigonometric potentials based on any root system. We demonstrate that various results known for the quantum AT models can be generalized universally to the models based on any other root systems as well 6>7>8>9>10. They are (i) Construction of a complete set of quantum conserved quantities in terms of quantum Lax pairs and other methods, (ii) Universal proof of quantum Liouville integrability for the rational, hyperbolic and trigonometric potential models. Namely, these quantum conserved quantities commute among themselves, (iii) Triangularity of the quantum Hamiltonian for all the models. In other words, the Hamiltonian is demonstrated, in certain bases, to be decomposed into a sum of finite dimensional triangular matrices. Thus any eigenvalue equation can be solved by finite steps of linear algebraic processes only. This also gives the entire discrete spectrum of the models, (iv) Equivalence of the quantum Lax pair method and that of socalled differential-reflection (Dunkl) operators u . (v) Algebraic construction of all excited states in terms of creation (annihilation) operators for rational models with harmonic confining force. For the classical Liouville integrability

237

see Ref.12. Solution methods of quantum Calogero-Moser models are expected to give some clues for understanding dynamical symmetries, non-perturbative methods for field theories, etc. For the Ar models, the Lax pairs, conserved quantities and their involution were discussed by many authors with varied degrees of completeness and rigor, see for example Refs. 5 ' 1 3 - 2 2 . The point (iv) was shown by Wadati and collaborators 20 and point (v) was initiated by Perelomov 14 and developed by Brink and collaborators 19 and Wadati and collaborators 20 . A rather different approach by Heckman and Opdam 23 ' 24 to Calogero-Moser models with degenerate potentials based on any root systems should also be mentioned in this connection. For physical applications of the Calogero-Moser models with various potentials to lower dimensional physics, ranging from solid state to particle physics and supersymmetric Yang-Mills theory, we refer to recent papers 7 ' 8 ' 9 and references therein. This paper is organized as follows. In section two, quantum CalogeroMoser Hamiltonian with degenerate potentials is introduced as a factorized form (2.5). Connection with root systems and the Coxeter invariance is emphasized. Some rudimentary facts of the root systems and reflections are summarized in Appendix A. A universal Coxeter invariant ground state wavefunction and the ground state energy are derived as simple consequences of the factorized Hamiltonian. In section three we show that all the excited states are also Coxeter invariant and that the Hamiltonian is triangular in certain bases. Complete sets of quantum conserved quantities are derived from quantum Lax operator L in section four. Instead of the trace, the total sum of Ln is conserved. That is Ts(Ln) = ^CMj„e-R.(£nW> m which TZ is a set of R r vectors invariant under the action of the Coxeter group. They form a single Coxeter orbit. Independence (and thus completeness) of the conserved quantities is explained in some detail. A sufficient choice of the set TZ for each root system for the completeness of the conserved quantities is given in Appendix B. In section five the creation and annihilation operators for the rational models with harmonic force are derived. In section six, the equivalence of the Lax pair operator formalism and the so-called differential-reflection (Dunkl) operators is demonstrated. Another form of the quantum conserved quantities are given in terms of the differential-reflection (Dunkl) operators. In section seven an algebraic construction of excited states in terms of the differential-reflection (Dunkl) operators for rational models with harmonic force is presented. The complete sets of explicit eigenfunctions for the rank two models are derived. Section eight gives a universal proof of the Liouville integrability for models with rational (without the confining force), hyperbolic and trigonometric potentials. The final section is for summary, comments and

238

outlook. 2

Quantum Calogero—Moser Models

A Calogero-Moser model is a Hamiltonian system associated with a root system A of rank r, which is a set of vectors in R r with its standard inner product. A brief review of the properties of the root systems and the associated reflections together with explicit realization of all the root systems will be found in the Appendix A. 2.1

Factorized Hamiltonian

The dynamical variables of the quantum Calogero-Moser model are the coordinates {qj} and their canonically conjugate momenta {pj}, with the canonical commutation relations: [Qj,Pk] = iSjk,

[qj,qk] = \Pj,Pk] = 0,

These will be denoted by vectors in R

j,k = l,...,r.

(2.1)

r

q=(qi,...,qr),

P = (pi, • • • ,Pr)-

(2.2)

The momentum operator pj acts as differential operators . d dqj As for the interactions we consider only the degenerate potentials, that is the rational (with/without harmonic force), hyperbolic and trigonometric potentials: f l/(p-q)2, type I, 2 V(p • q) = < a /sinh 2 a(p • q), type II, [ a2/sin2 a(p • q), type III,

p G A,

(2.3)

in which a is an arbitrary real positive constant, determining the period of the trigonometric potentials. They imply integrability for all of the CalogeroMoser models based on the crystallographic root systems. Those models based on the non-crystallographic root systems, the dihedral group him), H3, and Hi, are integrable only for the rational potential. The rational potential models are also integrable if a confining harmonic potential -wV,

^>0,

typeV,

(2.4)

239

is added to the Hamiltonian. Since we will discuss the universal properties and solutions applicable to all the interaction types as well as those for specific interaction potentials, let us adopt the conventional nomenclature for them. We call the models with rational, hyperbolic, trigonometric and rational with harmonic force the type I, II, III and V models, respectively. (Type IV models have elliptic potentials which will not be discussed in this paper.) The Hamiltonian for the quantum Calogero-Moser model can be written in a 'factorized form': „,

1-A/

i ' /

.dW\

(

fdw\2\

2

= \P2 + \ E

.dW\

i^d2w

9\P\(9\P\ ~ 1)M2 V(p • q) + ( y a2) - So

(2.5)

(2.6)

(2.7)

p€A+

It should be noted that the above factorized Hamiltonian (2.7) consists of an operator part 7i, which is the Hamiltonian in the usual definition, and a constant £Q which is the ground state energy to be discussed later: H = H-£0,

* = \P2 + \ E

(2.8)

9\P\{9\P\ ~ l)|p| 2 V(p • q) + (^«z 2 ).

(2.9)

The real positive coupling constants g\p\ are defined on orbits of the corresponding Coxeter group, i.e. they are identical for roots in the same orbit. That is, for the simple Lie algebra cases one coupling constant g\p\ = g for all roots in simply-laced models and two independent coupling constants, 9\P\ — 9L for long roots and p| p | = gs for short roots in non-simply laced models. For the h{m) models, there is one coupling if m is odd, and two independent ones if m is even. Let us call them ge for even roots and g0 for odd roots. The Hz and H4 models have one coupling constant g\p\ = g, since these root systems are simply-laced. It should be noted that the operator part of the Hamiltonian fl is strictly positive for g\p\ > 1, which we will assume throughout this paper for simplicity of the arguments. Throughout this paper we consider the coupling constants at generic values.

240

The simplest way to introduce the factorized form is through supersymmetry 10>25; in which function W is called a superpotential:

W(q)=

J2 9\P\^\w(P-q)\ + (--q2),

g\p\>l,

u > 0.

(2-10)

peA+

The potential V(u) (2.3) and the function w(u) are related by V U

^ '

=

d duX^U',

dw{u) du 'W^U'

=

{

Z

^' 0 rational, — 1 hyperbolic, 1 trigonometric.

(2.11) (2.12)

The following Table I gives these functions for each potential: potential rational hyperbolic trigonometric

type I&V II III

w(u) u sinh au sinau

x(u) 1/u acoth au a cot au

y(u) -1/u2 -a2/ sinh2 au -a2 J sin2 au

Table I: Functions appearing in the Lax pair and superpotential. For proofs that the factorized Hamiltonian (2.5) actually gives the quantum Hamiltonian (2.7) for any root system and potential see Refs. 5 ' 13 ' 10 . It is easy to verify that for any potential V(u), the Hamiltonian is invariant under reflection of the phase space variables in the hyperplane perpendicular to any root H(sa(p),sa(q))=H(p,q),

VaeA

(2.13)

with sa denned by (A.2). The main problem is to find all the eigenvalues {A} and eigenfunctions {ift} of the above Hamiltonian: Hip = \ip.

(2.14)

Some remarks are in order. For any root system and for any choice of potential (2.3), the Calogero-Moser model has a hard repulsive potential ~ l / ( a • q)2 near the reflection hyperplane Ha = {q G R r , a • q = 0}. The strength of the singularity is given by the coupling constant <7|a|(<7|o;| — 1) which is independent of the choice of the normaliszation of the roots. (Thus for rational models with/without harmonic force there is equivalence, A2 = ^ ( 3 ) , B2 S 72(4), G2 = / 2 (6), Br^Cr^ BCr.) In other words, (2.14) is a second

241

order Fuchsian differential equation with regular singularity at each reflection hyperplane Ha. That is any solution of (2.14) is regular at all points except those on the union of reflection hyperplanes Ua£A+Ha (and possibly at infinity for type V models). Near the reflection hyperplane Ha, the solution behaves tp ~ (a • <7)pi°i(l + regular terms),

or

if) ~ (a • g) 1 _ 5 | o , | (l + regular terms).

The former solution is chosen for the square integrability. This determines the form of the ground state wavefunction, as we will see in subsection 2.2. This repulsive potential is classically insurmountable. Thus the motion is always confined within one Weyl chamber. This feature allows us to constrain the configuration space to the principal Weyl chamber (II: set of simple roots, see Appendix A) P^={«6Rr|Q'g>0,

a en},

(2.15)

without loss of generality. In the case of the trigonometric potential, the configuration space is further limited due to the periodicity of the potential to PWT = {? 6 R r | a -q>0,

a <E II,

ah • q < 7r/a},

(2.16)

where ah is the highest root. The fact that the classical motions are confined in the Weyl chamber (alcove) PW(PWT) does not mean that the corresponding quantum wavefunctions vanish identically outside of the region a. On the contrary, as we will see soon, the ground state wavefunction (subsection 2.2) and all the other excited states wavefunctions (section 3) are Coxeter invariant, reflecting the Coxeter invariance of the Hamiltonian (2.13). In the early years of Calogero-Moser models in which those based on the Ar root system were mainly discussed, these Coxeter invariant solutions were considered as totally symmetric states of bosonic systems. We will not, however, adopt this interpretation, for in the models based on the other root systems the reflection is not the same as particle interchange. The quantum theory we are discussing is the so-called first quantized theory. That is, the notions of identical particles and the associated statistics are non-existent. "Remember the quantum mechanical system defined in a finite interval on a line with the periodic boundary condition. It is described by a Fourier series which does not vanish outside of the interval.

242

2.2

Ground state wavefunction and energy

One straightforward outcome of the factorized Hamiltonian (2.5) is the universal ground state wavefunction which is given by $ 0 ( g ) = e ^ M = Y[ Hp • q)\9^ e~^\

(2.17)

peA+

The exponential factor e~^q exists only for the rational potential case with the harmonic confining force. It is easy to see that it is an eigenstate of the Hamiltonian (2.5) with zero eigenvalue: W*o(*) = \ ±

(PJ - i ^ )

(Pj + i ^ )

*o(«) = 0,

(2.18)

since it satisfies (

P j

+ ^ ) e ^ > = 0 ,

j = l,...,r.

(2.19)

By using the decomposition of the factorized Hamiltonian into the operator Hamiltonian (2.9) and a constant, we obtain Hew=

\\p2

9\P\(9\P\-m2V{p-q) +

+\ £

\

M

2

(Y«2)\eW=£oeW

PEA+

(2.20) In other words, the above solution (2.17) provides an eigenstate of the Hamiltonian operator % with energy £Q. The fact that it is a ground state (for type I, III and V) can be easily shown within the framework of the supersymmetric model 10 thanks to the positivity of the supersymmetric Hamiltonian. It should be stressed that So is determined purely algebraically 10 , without really applying the operator on the left hand side of (2.20) to the wavefunction. This type of ground states has been known for some time. It is derived by various methods, see for example Refs. 3,5 , and also by using supersymmetric quantum mechanics for the models based on classical root systems 25>18. The ground state energy for the rational potential cases are f £

°-\"{$

0 + ZPeA+9M)

type I, typeV.

The same for the hyperbolic and trigonometric potential cases are £0 = 2a 2 , 2 x ( - 1 hyperbolic II trigonometric,

^ ^

243

in which 8 = 2 Y.

9\P\P

(2-23)

can be considered as a 'deformed Weyl vector' 5 ' 2 4 . Again these formulas are universal. That is they apply to all of the Calogero-Moser models based on any root system. A negative £o for the obviously positive Hamiltonian of the hyperbolic potential model indicates that the interpretation of ew as an eigenfunction is not correct. This function diverges as \q\ -> oo for the hyperbolic and the rational potential cases, destroying the hermiticity of the Hamiltonian. Obviously we have f

e™{q) /

JPW(PWT)

,„_\

oo oo : : type type I and ana II, 11 [ finite : type III and V,

v

'

in which PW and PWT denote that the integration is over the regions defined in (2.15) and (2.16). Naturally, most existing results in quantum CalogeroMoser models are for the models with trigonometric potential and the rational potential with harmonic force which have square integrable states and discrete spectra. It should be remarked that the universal ground state wavefunction $o and W are characterized as Coxeter invariant: s p $o = $o,

spW = W,

VpeA,

(2.25)

in which sp is the representation of the reflection in the function space. For an arbitrary function / of q € R r , its action is defined by (sPf)(q) = f(sP(q)).

(2.26)

This definition can be generalized to the entire Coxeter group G A : for an arbitrary element g of G&, g is defined by: (£/)(
Vff£GA.

(2.27)

In the rest of this paper we mainly discuss the type III and V models which have normalisable states and discrete spectra. 3

Coxeter invariant excited states, triangularity and spectrum

In this section we show that all the excited states wavefunctions are Coxeter invariant, too. In other words, the Fock space consists of Coxeter invariant functions only. With the knowledge of the ground state wavefunction ew, the other states of the Calogero-Moser models can be easily obtained as

244

eigenfunctions of a differential operator H obtained from ~H by a similarity transformation:

n = =

e-wHew e~w

[\P2 + \ £ 9M(9M-l)\p\2V(p-q) + •HVx = M\

<=>

n*xew

(^-q2)-£o)eW,(^)

=A*Aevv.

(3.2)

Since all the singularities of the Fuchsian differential equation (2.14) are contained in the ground state wavefunction ew the function $ A above must be regular at finite q including all the reflection boundaries. Thanks to the factorized form of the Hamiltonian % (2.5), (2.6), the transformed Hamiltonian % takes a simple form:

n^-\±(f2^f\.

( 3.3,

The Coxeter invariance of W implies those of H and T-L: spHsp = n,

SpiiSp =?{.,

V/9 G A.

(3.4)

For type V and III models we introduce proper bases of Fock space consisting of Coxeter invariant functions and show that the above Hamiltonian ti. (3.3) is triangular in these bases. This establishes the integrability of the type V and III models universally b and also gives the entire spectrum of the Hamiltonian, see (3.12), (3.13) and (3.42).

3.1

Rational potential with harmonic force

First, let us determine the structure of the set of eigenfunctions of the transformed Hamiltonian ti, for the type V models: -

d

I

, A d2

^

gw

d

As remarked above, the eigenfunctions of % have no singularities at finite q. Thus we look for polynomial (in q) eigenfunctions. The situation is very ^Triangularity of the Ar type V and III Hamiltonians was noted in the original papers of Calogero 1 and Sutherland 2 . That of rank two models in the Coxeter invariant bases was shown in Refs. 2 4 ' 2 6 .

245

similar to that of the harmonic oscillators to which the type V CalogeroMoser models reduce in the free limit g\p\ ->• 0. Obviously a constant and a quadratic polynomial uiq2 - £o/ w a r e i t s eigenfunctions with eigenvalue 0 and 2u, respectively. Let us suppose that a polynomial P(q) is an eigenfunction of ft: HP(q) = \P(q).

(3.6)

Due to the Coxeter invariance of ti (3.4), we know that spP together with the difference Q = (1 - sP)P are also eigenfunctions with the same eigenvalue: -HQ(q) = AQ(g),

(3.7)

if the latter is not identically zero. Since Q is a polynomial which is odd under reflection sp SPQ(q) = -Q{q), it can be factorized as Q(q) = (p-q)2n+1Q(q),

Q

7^0,

(3.8)

pq=0

with a non-negative integer n and a polynomial Q. By substituting (3.8) into (3.7) and using the explicit form of ti near the reflection hyperplane p • q = 0, we obtain - ( p • q)2n-1(2n

+ l)(n + gM)p2Q + 0[(p • q)2n] = X(p • q)2n+1Q,

(3.9)

which would imply the vanishing of Q on the reflection hyperplane Q

p-q=0

an obvious contradiction. Thus we are led to the conclusion that the eigenfunctions are Coxeter invariant polynomials and that the Hamiltonian H (3.5) maps a Coxeter invariant polynomial to another. An obvious basis in the space of Coxeter invariant polynomials is the homogeneous polynomials of various degrees. This basis has a natural order given by the degree. For a given degree the space of homogeneous Coxeter invariant polynomials is finite-dimensional. The explicit form of % (3.5) shows that it is lower triangular in this basis and the diagonal elements are u x degree as given by the first term. Independent Coxeter invariant polynomials exist at the degrees /_,• listed in Table II:

246

fj = \ + ej,

j = l,...,r,

(3.10)

in which {e,}, j = 1 , . . . , r, are the exponents of A. Let us denote them by Zj(Kq)=Kf'Zj(q).

zi(q),---,zr(q);

A J\.f

Br Cr Dr Eg E7

fj = 1 + ej 2,3,4,...,r + l 2,4,6,...,2r 2,4,6,...,2r 2,4,...,2r-2;r 2,5,6,8,9,12 2,6,8,10,12,14,18

A Eg Fi G2 h{m) H3 H4

(3.11)

fj = l + ej 2,8,12,14,18,20,24,30 2,6,8,12 2,6 2,m 2,6,10 2,12,20,30

Table II: The degrees fj in which independent Coxeter invariant polynomials exist. Thus we arrive at: the quantum Calogero-Moser models with the rational potential and the harmonic confining force is algebraically solvable for any (crystallographic and non-crystallographic) root system A. (See Ref.27 for algebraic linearisability theorem of the classical Calogero-Moser system.) The spectrum of the operator Hamiltonian "H is uN + S0,

(3.12)

with a non-negative integer N which can be expressed as r N

= Y,n>fi>

n GZ

;

+>

( 3 - 13 )

i=i

and the degeneracy of the above eigenvalue (3.12) is the number of different solutions of (3.13) for given N. This is generalization of Calogero's original argument for the Ar model x to the models based on any root system. Now let us denote by N the set of non-negative integers in (3.13): N=(n1,n2,...,nr), (3.14) and by ${q) the homogeneous Coxeter invariant polynomial determined by N and the above basis {ZJ} (3.11): r

247

As shown above, there exists a unique eigenstate Vtf (?) f° r •>l>f}(q) = fi(q) + 5 Z d-N^N'tv)'

eacn

d

^fiiq)'-

N> '• c o n s t >

n^n{q)=uNiPx{q).

( 3 - 16 ) (3.17)

It satisfies the orthogonality relation Wtt,0rf,) = O,

Af'
(3.18)

with respect to the inner product in PW: (,¥>)=/"

r(qMq)e2W^dq.

(3.19)

JPW

These polynomials {ipfi{q)} are generalizations of the multivariable Laguerre (Hermite) polynomials 16 known for the AT (Br, Dr) root systems to any root system. Some remarks are in order. 1. There is no Coxeter invariant linear function in q. The quadratic invariant polynomial q2 = q • q exists in all the root systems. This corresponds to the universal fact that / i = 2, (ei = 1) for all the root systems. Moreover, this is related to the fact that a special sub-series of the excited states with N = 2ni = n i / i , rij = 0, j > 2, can be expressed universally in terms of Laguerre polynomials in q2. This will be discussed at the end of section 5 and in subsection 7.2. 2. The other Coxeter invariants corresponding to the degrees / 2 , - . . , / r could be interpreted as special 'angular' variables of a unit sphere S r _ 1 (q2 = 1), with the first Coxeter invariant y/cj* being the radial coordinate. These would provide proper variables for describing solutions. Solutions in terms of separation of variables are in general possible only for the simplest cases, that is the rank two models, which will be demonstrated in subsection 7.3. 3. For A = Ai, the simplest root system of rank one, the Hamiltonian % can be rewritten in terms of a Coxeter invariant variable u = ojq2 as:

The Laguerre polynomial satisfying the differential equation

248

provides an eigenfunction with eigenvalue 2wn, which corresponds to the eigenvalue 2wn + £o of H. This is a well-known result. 3.2

Trigonometric potential

Here we consider those root systems associated with Lie algebras. In order to determine the excited states of the type III models, we have to consider the periodicity. The superpotential W and the Hamiltonian % are invariant under the following translation: W(q') = W(q),

U(P,q')='H(p,q),

q' = q + l^/a,

(3.22)

in which lv is an element of the dual weight lattice, that is r 2 lv = ^2lj—*j,

lj £ Z,

ctj G II,

a / - Afc = 8jk.

(3.23)

As is well known in quantum mechanics with periodic potentials, the (Bloch) wavefunctions diagonalizing the translation operators are expressed as e2ia„-q

I

^

bae2iaaq

\ew,

ba: const,

L(A) : root lattice,

(3.24)

in which a vector fj, G R r is as yet unspecified. In other words, up to the overall phase factor e 2 j o ' i ? j this is a Fourier expansion in terms of the simple roots. Let Pr(q) be a polynomial in e ± 2 i a a J " 9 , ctj € II and suppose that a function cj>(q) «/>(<,) = e2ia^PT(q),

M€Rr,

(3.25)

is an eigenfunction of ~H: H<j>{q) = A0(g),

(3.26)

in which the explicit form of H is given by 1

r

f\

P&

^ = - 2 E ^ 2 - f l E 9Mcot(ap-q)p--. j=i

q

i

(3.27)

peA+

As above, due to the Coxeter invariance of ti (3.4), we know that sp(f> together with the difference f = (1 - « P ) 0

249

are also eigenfunctions with the same eigenvalue: Htp(q) = \
g>\p.q=0 ± o,

in a neighbourhood of the reflection hyperplane p- q = 0. In this neighbourhood, the singularity structure of H for the trigonometric potential is the same as that of ti for the rational potential discussed in the previous subsection. Thus we obtain, as before, a contradiction / (3.25) must be Coxeter invariant. This in turn requires that the unspecified vector \i in (3.25) to be an element of the weight lattice H G A(A).

(3.28)

Since sp(j)(q) = e2ias"^-npPT{q)

e2ia^q-pV^p-q)spPT(q),

=

the following condition is necessary, but not sufficient, pv-/iGZ,

VpGA

(3.29)

for the Coxeter invariance of <j>. Thus we arrive at (3.28). Let us introduce a basis for the Coxeter invariant functions of the form (3.25). Let A be a dominant weight r

A = 'Y^m.jXj,

m,j £ Z+,

(3.30)

and W\ be the orbit of A by the action of the Weyl group: Wx = {»€A(A)\n

= g(\),

V9GG

A

}.

(3.31)

We define

MQ) = Y,

e2iaii q

'>

( 3 - 32 )

newx which is Coxeter invariant. The set of functions {4>\} has an order >: |A|2>|Af =• A>v. (3.33)

250

Next we show that ~H is lower triangular in this basis. By using (3.27) we obtain nx = 2a2\24>x-2ia2

J2

9\P\ cot (ap • q)(p-^e2^".

^

(3.34)

First let us fix one positive root p and a weight p in W\ such that p • p ^ 0. Then p! = sp(p) = p-(ps/-p)p£

Wx,

p-p' = -p-p.

(3.35)

Without loss of generality we may assume pv-p

= k>0,

k£Z.

(3.36)

The contribution of the pair (p,p') in the summation of (3.34) reads \p • p\e2ai^(l

- e~2aikp-q)

cot(ap • q)

e2ai"-9 + e2ai'1'-q + 2 ] T £°-^-3<>U )

= i\p-p\l

;

(3.37)

which is the generalization of Sutherland's fundamental identity eq(15) in Ref.2 to any root system. The summation in the expression correspond to (fry with A' being lower than A. Thus (3.34) reads H4>x = 2a2\2
£

gM\p • p\e2ia^

+

p€A + MGW A

£

cx>4>y,

(3.38)

|A'|<|A|

in which {cx>}'s are constants. It is easy to see that (p = g(X), 3g £ G A ) Yl

9\P\ \PmlA=

pGA +

S p£A+

9\p\ \9(j>) • Al = ( £

S\p\P) • A = 2g • A,

(3.39)

peA+

which is independent of p. Thus we have demonstrated the triangularity of H:

Hx=2a2(\2 + 2Q-\)x+ Y,

c

*"^'>

(3-4°)

|V|<|A|

or that of % %<j>xew =2a2{\ + Q)2ct>xew + £

cxl<j>x,ew,

(3.41)

|A'|<|A|

with the eigenvalue 2a2(A +
(3.42)

251

In other words, for each dominant weight A there exists an eigenstate of ~H with eigenvalue proportional to A(A + 2g). Let us denote this eigenfunction by ip\{q)-ffrxil) = HMQ)

+ 5Z W'<W

d

v<Mg),

dy : const,

= 2a2A(A + 2Q)MQ),

and call it a generalized Jack polynomial relation. (V>A,<M = 0,

(3.43) (3-44)

28 31

- . It satisfies the orthogonality

|A|'<|A|,

(3.45)

with respect to the inner product in PWT(•>/>,?)=[

P(q)
(3.46)

J PWT

In the Ar model, specifying a dominant weight A is the same as giving a Young diagram which designates a Jack polynomial. It should be emphasized, however, that {ip\} are not identical to the Jack polynomials even for the Ar root systems, because of different treatments of the center of mass coordinates. Thus we arrive at: the quantum Calogero-Moser models with the trigonometric potential are algebraically solvable for any crystallographic root system A. The spectrum of the Hamiltonian ti is given by (3.42) in which A is an arbitrary dominant weight. This is generalization of Sutherland's original argument 2 to the models based on any root system. Some remarks are in order. 1. The weights /i appearing in the lower order terms {(/>,y}'s are those weights contained in the Lie algebra representation belonging to the highest weight A. 2. As a simple corollary we find that for a minimal weight A,

Mq) = Mq) = E

^^

is an eigenfunction of H. A minimal representation 7 consists of a single Weyl orbit and all of its weights fj, satisfy pv • fi = 0,±1, V/>eA.

252

3. If A = ah, the highest root of a simply-laced root system, W\ is the set of roots itself. Then the lower order terms are constants only. We find that i>ah (?) = 2 ^2 ( c o s (2aP • l) + 9P2lah • {ah + 2g)) is an eigenstate of ti. 4. If A = a$h, the highest short root of a non-simply laced root system, W\ is the set of short roots itself. The lower order terms are constants, too. Similarly as above, we find that ipocSh (q) = 2 ^2 p£AL

cos (2ap • q) +

(cos (2ap • q) + gsp2s/ash

+2 ^2 p€As

• {ash + 2g))

+

is an eigenstate of H. Here &L(S)

1S

the

set

°f l° n g (short) roots.

5. If — A $ W\ then there is another set of functions containing the weight —A which belongs to the same eigenvalue. 6. The Coxeter invariant trigonometric polynomials specified by the fundamental weights {Xj} (j)Xj(q) = ^2

e2m,J''q,

Xj : fundamental weight,

j = l,...,r (3.47)

are expected to play the role of the fundamental variables 24>26. 7. Let us consider the well-known case A = Ai, the simplest root system of rank one. By rewriting the Hamiltonian % in terms of the Coxeter invariant variable z = cos(apq), we obtain Id2

/

xd

a2

\P? {„

2^ d2

„

„ x d \

(3.48) The Gegenbauer polynomials 5 , a special case of Jacobi polynomials Pn provide eigenfunctions: P^"i,p_i)(cos(oW)),

£ = a2\p\2(n + g)2/2,

n € Z+.

(3.49)

253

The Jacobi polynomial Pn

(z) satisfies a differential equation

+ n(n + a + p + 1)1 P^\z)

= 0.

(3.50)

Here we follow the notation of Ref.32. They form orthogonal polynomials with weight e2W = \sin(apq)\29 in the interval q € [0,n/ap], (2.16). 8. Triangularity of type II models follows from the same algebraic reasoning. 4

Quantum Lax pair and quantum conserved quantities

Historically, Lax pairs for Calogero-Moser models were presented in terms of Lie algebra representations 3 ' 5 , in particular, the vector representation of the Ar models as shown in Introduction. However, the invariance of CalogeroMoser models is that of Coxeter group but not that of the associated Lie algebra, which does not exist for the non-crystallographic root systems. Thus the universal and Coxeter covariant Lax pairs are given in terms of the representations of the Coxeter group. 4-1

General case

Here we recapitulate the essence of the quantum Lax pair operators for the Calogero-Moser models with degenerate potentials and without spectral parameter. The quantum Lax pair in this subsection applies to all the degenerate potential cases except for the case of the rational potential with the harmonic force, which will be treated separately in subsection 4.2. For details and a full exposition, see Ref.10. The Lax operators without spectral parameter are L(p, q)=p-H

+ X(q),

X(q) = i £

gM (p • H)x(p • q)sp,

(4.1)

P6A+

M(q) = l- Y,

9\P\ \P? V(P • q) (S„ - I),

(4.2)

PGA+

in which J is the identity operator and {sa, a € A} are the reflection operators of the root system. In contrast to the {sa} operators (2.26) which act in function space, {sa} act on a set of R r vectors 1i = {p,W € R r , k — 1 , . . . , d}, permuting them under the action of the reflection group. The vectors in 11

254

form a basis for the representation space V of dimension d. The operator M satisfies the relation

J2M^=J2 M^ = 0,

(4.3)

which is essential for deriving quantum conserved quantities. The matrix elements of the operators {sa, a € A} and {Hj, j = 1 , . . . ,r} are defined as follows: ( s P W = sn,sp(u) = <*•/,«„0i)>

(Hj)nv = /^<W>

Pe

A

>

/*," e

ft.

(4.4)

The form of the function x depends on the chosen potential, and the function y are defined by (2.11), (2.12). Note that these relations are only valid for the degenerate potentials (2.3). The underlying idea of the Lax operator L, (4.1), is quite simple. As seen from (4.10), L is a "square root" of the Hamiltonian. Thus one part of L containsp which is not associated with roots and another part contains x(p-q), a "square root" of the potential V(p-q), which being associated with a root p is therefore accompanied by the reflection operator sp. Another explanation is the factorized Hamiltonian % (2.5). We obtain, roughly speaking, L ~ y/% ~ p + i^p-s and the property of reflection s2 = 1 explains the sign change in the first term in (2.5). It is straightforward to show that the quantum Lax equation jtL

= i[H,L] = [L,M],

(4.5)

is equivalent to the quantum equations of motion derived from the Hamiltonian (2.7). From this it follows: jt(Ln)lil/

= i[H,(Ln)»v} = = £

((£"WMA„

[Ln,MU - M^L")^)

,

n = 1,....

(4.6)

Thanks to the property of the M operator (4.3):

we obtain quantum conserved quantities as the total sum (Ts) of all the matrix

255

elements of Ln c:

Qn = Ts(L") = Y, ( Ln W>

ln> <W = 0, n = 1, •...

(4.7)

The Lax pair is Coxeter covariant: L (sp(p), sp(q))^

= L (p, g ) M ,„, , M (sp{q))^

= MfaV„.,

1

fi' =sp(fi),

v = sp{u),

(4.8)

which ensures the Coxeter invariance of the conserved quantities. Independent conserved quantities appear at such power n that n = 1 + exponent

(4.9)

of each root system. These are the degrees at which independent Coxeter invariant polynomials exist. There are r exponents for each root system A of rank r. Thus we have r independent conserved quantities in Calogero-Moser models. We list in Table II these powers for each root system. In particular, the power 2 is universal to all the root systems and the quantum Hamiltonian (2.7) is given by H = -^-Ts(L2)

+ const,

(4.10)

where the constant Cu is the quadratic Casimir invariant, which depends on the representation. It is defined by

Tr(HjHk) = J2 (HjHk)pp = £

Nlik

= Cn 6jk.

(4.11)

Some remarks are in order. 1. Lax pairs can be written down in various representations and the quantum conserved quantities Qn do depend on the representations, in general. If necessary, we denote by Q1^ the explicit representation dependence. 2. The availability of plural representations of the Lax pair and the conserved quantities is essential for the completeness of the set of conserved quantities as polynomials of the momentum operators. For example, let us consider the case of Dr with even r, which has two independent c

This type of conserved quantities is known for AT models

18 :

'

256

conserved quantities at power r, see Table II. At least two different representations of the Lax pair are necessary in order to represent them in the form of (4.7). Those based on the vector, and the (anti)-spinor representations give two independent conserved quantities. For D4 case, we obtain 4

Ql = 2Y/P4j,

4

Qt-Qi = ^Y[Pi,

(4.12)

in which v, s and a stand for the vector, spinor and anti-spinor representations and we have set g = 0 for simplicity. If two conserved quantities are independent for zero coupling constants, surely they are so at nonvanishing couplings. 3. If a representation 1Z contains a vector /i and its negative — /z at the same time, then we have Ts(Lodd) — 0. In such a case the corresponding Lie algebra representations are called real. In order to construct the odd power conserved quantities appearing in Ar for all r, Dr for odd r, E@ and I(m) for odd m, we need a Lax pair in non-real representations. For Ar all the fundamental representations corresponding to the fundamental weights Xj, j — 1 , . . . , r except for the middle one A( r + 1 )/ 2 for odd r are non-real. For Dr with odd r, the spinor and anti-spinor representations are non-real. For E% the 27 and 27 are non-real, him) is the symmetry group of a regular m-sided polygon. The set of m vectors Rm corresponding to the vertices of the regular m-gon (B.l) provides a non-real representation when m is odd. 4. The independence of the conserved quantities can be easily verified by considering the free limit, i.e. g\p\ = 0. In this limit we have the conserved quantities Qn(p, q) - • Q°n(P) = Ts(p • £ ) " = £ ( ? • M)",

n : degree,

(4.13)

which is essentially the same result of the standard Lax pair formulation. In the latter formulation based on the weights of a Lie algebra representation, sometimes the zero weights \i = 0 are contained on top of those other weights (roots) involved in the present formulation. However, it is obvious that they give zero contribution and the results (4.13) are the same in both formulations. Since {Q° (p)} are obviously Coxeter invariant polynomials in {p}, the problem that the totality of {<9° } (4.13) provides the complete (total of r independent) set of free conserved quantities can

257

be rephrased that if and how the basis of Coxeter invariant polynomials can be obtained as power sums. The latter problem was addressed in 33 about thirty years ago. If the Jacobian -—^

^-,

fj : degree,

(4.14)

d[pi,...,Pr)

is non-vanishing the map (pi,... ,pr) -» (Q°fi,..., Q°jr) is invertible and the basis is complete. If the map is invertible at one point, it should be so in the same Weyl chamber and the invertibility breaks down at the boundary. By simple comparison of the powers in {p} we arrive at

? « k ^ _ * n > , , *,*

(4.15,

For lower rank cases, G2, h(fn), # 3 and F4 one can show (4.15) by direct calculation for the representations given in Appendix B. In 34 the affirmative answers are given for the classical root systems for the choices given in Appendix B. For E&, E-j and Eg one could invoke the theorems in 35 for demonstrating the non-vanishing of the Jacobian for those TZ listed in Appendix B. We have verified the independence of the free conserved quantities (4.15) for all the cases (except for the classical ones) listed in Appendix B with the aid of formula manipulation programs. The verification is exact in which left hand side of (4.15) is evaluated for integer (fraction) values of {pj}. 5. In Appendix B we list for each root system how the complete set of independent conserved quantities are obtained by choosing proper representations of the Lax pair. 4-2

Rational potential with harmonic force

The quantum Lax pair for the type V models needs a separate formulation. The explicit form of the Hamiltonian is

n = \p2 + \u>W + \ J2 9M(9M ~ 1 ) 7 ^ - *>. PGA+

KH

(4-16)

q!

The canonical equations of motion are equivalent to the following Lax equations for L±m. jtL±

= i[U, L±] = [L±, M] ± iujL±,

(4.17)

258

in which (see section 4 of Ref.8) M is the same as before (4.2), and L^ and Q are defined by L±=L±icuQ,

Q = q-H,

(4.18)

with L, H as earlier (4.1), (4.4). If we define hermitian operators £i and £2 by d = L+L-,

£2=L-L+,

(4.19)

they satisfy Lax-like equations Ck=i[H,Ck]

= [Ck,M],

k = l,2.

(4.20)

From these we can construct conserved quantities TsCq),

3 = 1,2,

n = l,2,...,

(4.21)

as before. Such quantum conserved quantities have been previously reported for models based on Ar root systems 18>20. It should be remarked that Ts(£2) is no longer the same as Ts(£") due to quantum corrections. It is elementary to check that the first conserved quantities give the Hamiltonian (4.16) H oc Ts(£i) = Ts(£ 2 ) + const.

(4.22)

This then completes the presentation of the quantum Lax pairs and quantum conserved quantities for all of the quantum Calogero-Moser models with nonelliptic potentials. 5

Algebraic construction of excited states I

In this section we show that all the excited states of the type V CalogeroMoser models can be constructed algebraically. Later in section 7 we show the same results in terms of the I operators to be introduced in section 6. The main result is surprisingly simple and can be stated universally: Corresponding to each partition of an integer iV which specify the energy level (3.12) into the sum of the degrees of Coxeter invariant polynomials (3.13), we have an eigenstate of the Hamiltonian V. with eigenvalue uiN + So'r

r

in which the integers {fj}, j = 1 , . . . , r are listed in Table II. They exhaust all the excited states. In other words the above states give the complete basis of the Fock space. The creation operators Bt and the corresponding

259

annihilation operators'* B. are defined in terms of the Lax operators Is*1 (4.18) as follows: Bf^TaiL*)'*,

j = l,...,r.

(5.2)

They are hermitian conjugate to each other (Bp=Bl

(5.3)

with respect to the standard hermitian inner product of the states defined in PW:

(1>!
P(qMQ)dq.

(5.4)

JPW

We will show later in section 6, (6.15) that the creation (annihilation) operators commute among themselves: [B+,B+] = [B-,B-\

= 0,

k,lE {fj\ j = 1 , . . . , r } ,

(5.5)

so that the state (5.1) does not depend on the order of the creation. The proof is very simple. By using (4.17) we obtain | ( L ± ) " = i[H, (L ± )"] = [(£*)», M] ± inu(L±)\

(5.6)

from which [H,B±] = ±nu,B±,

(5.7)

follows after taking the total sum. This simply says that B^ creates (annihilates) a state having energy nw. In other words we have

Moreover, it is trivial to show that V(L

)ilvew

=\p-n-iuq-fi

+i V

tLE \ew = fl.(p

+i

_ ) ew=0,

(5.8) which implies that the ground state is annihilated by all the annihilation operators Bj.ew=0,

j = l,...,r.

Some remarks are in order. d

W e adopt the notation by Olshanetsky and Perelomov 5 ' 1 4 .

(5.9)

In most cases the energy levels are highly degenerate. The above basis is neither orthogonal nor normalized. The independence of the creation-annihilation operators can also be shown in a similar way to that of the conserved quantities. As with the conserved quantities, plural representations are necessary to define the full set of creation-annihilation operators in some models. This aspect will be discussed in later sections in connection with the t operators. Reflecting the universality of the first exponent, / i = 2, the creation and annihilation operators of the least quanta, 2w, exist in all the models. They form an sZ(2,R) algebra together with the Hamiltonian i-L: [H,bf] = ±2ujbf,

[6+,&2-] = - W - 1 H ,

(5.10)

in which bf are normalized forms of B^:

bf=Yl (L^lJi^Cn).

(5.11)

The sZ(2,R) algebra was discussed by many authors (see, for example 14,36,19,23 ^ a n ( j others) in connection with the models based on classical root systems. We will show later in subsection 7.2 that the states created by B~2 (fcj") only can be expressed by the Laguerre polynomial: {b+)new

= n\L^-l\uq2)ew,

S0 = £0/OJ.

(5.12)

It is trivial to verify that Ln °~ (wg2) is an eigenfunction of % (3.5) HL^-^iujq2)

= 2nwL<£°-1Xujq'i).

(5.13)

The normalisation of the state |(6+) n ewf

= nW0/T(n

+ So),

AfQ = \\eW\2T(£0),

(5.14)

is also dictated by the sl(2, R) relations. The Laguerre polynomial wavefunctions appear as 'radial' wavefunctions in all the cases 37 . This will be shown explicitly for for the rank two models given in subsection 7.3. As is emphasized by Perelomov 14 and Gambardella 36 the sZ(2,R) algebra and the corresponding Laguerre wavefunctions are more universal than Calogero-Moser models. They arise when the potentials are homogeneous functions in q of degree —2 with the confining harmonic force.

261

5. The operators {Qn} and {B^} do not form a Lie algebra. They satisfy interesting non-linear relations, for example, [[B+,b£],b+] = nB+,

[[B-,b+]tb^] = nB~.

(5.15)

This tells, for example, that although £?+ and b% create different units of quanta n and 2, they are not independent

6

£ operators

In this section we will show the equivalence of the quantum conserved quantities obtained in the Lax operator formalism of section 4 and those derived in the 'commuting differential operators' formalism initiated by Dunkl n and followed by many authors. Again the equivalence is universal, applicable to the models based on any root system. We propose to call the operators in the latter approach simply '(. operators', since they are essentially the same as the L operator in the Lax pair formalism and that they are not mutually commuting, as we will show presently, when the interaction potentials are trigonometric (hyperbolic), (6.14). Although these two formalisms are formally equivalent, the I operator formalism has many advantages over the Lax pair one. Roughly speaking, the 'vector-like' objects £M's are easier to handle than the matrix L^„. Let us fix a representation 1Z of the Coxeter group G A and define for each element \i € TZ the following differential-reflection operator eii = £-ti=p-/j.

+ i ^T

5|p|

(p • n) x(p • q)sp,

fiell.

(6.1)

It is linear in fj, and Coxeter covariant W

=ln+lv,

Sp^Sp = £Spifi),

V/9 G A.

(6.2)

They are hermitian operators, fi = ^M, with respect to the standard inner product for the states (5.4). It is straightforward to show that the quantum conserved quantities Qn derived in the previous section (4.7) can be expressed as polynomials in the I operators as follows:

262

in which t/> is an arbitrary Coxeter invariant state, spip — tp. This also illustrates the Coxeter invariance of Qn clearly, since sp(^T, -nl™)sp = TiV.en(nep(li) = EMGW^S- F o r n = 1 it is trivial, since

5 3 (LWV> = [p • (i + i 5 3 g\p\ (P ' P) X^P ' ti 5 3 (*")"•' ) ^ = ip-fi + i 5 3 9\P\ (P • A») x(p • q)sp tp = t^ip, (6.4) V peA+ / in which

]C«/€TC(*P)M« /

=

1 anc ^ ^p^

=

*P a r e

use(

L Let us assume that

Xld/VlM^.

(6-5)

is correct, then we obtain

53(Z7l+V = ^ = 53 I p ' xen

\

^MA

LMA(L")A,V

= £ WW",

+ * 5 3 5 W ^ ' P) X^P ' 9)(*P)MA 1 Zli>P€A+

/

In the second summation only such A as A = sp(fx) contributes and we find

Thus we arrive at

53(L"+1)MVV = C V ,

(6-6)

and the equivalence of the two expressions of the conserved quantity (6.3) is proved. Commutation relations among I operators can be evaluated in a similar manner as those appearing in the Lax pair 8 ' 10 , that is, by decomposing the roots into two-dimensional sub-root systems. We obtain

{

0 rational, - 1 hyperbolic,

(6.7)

1 trigonometric. One important use of the I operators is the proof of involution of quantum conserved quantities. For type I models Heckman 23 gave a universal proof

263

based on the commutation relation (6.7): [Qn,Qm}4>= Y

[ ^ . C ] ^ = 0,

rational model.

(6.8)

This was the motivation for the introduction of the commuting differentialreflection operators by Dunkl n . In fact, Dunkl's and Heckman's operators were the similarity transformation of l^ by the ground state wavefunction ew: I, = e-wt,ew

=p.n

+ iY,

9\P\ fe4(*p - I)-

PeA+

(6-9)

q)

^

As for type V models, we define i^ corresponding to L^ (4.18): e±=e±-Li

= p-»±iu(q-»)+i

Y

9\P\J-Ah^ yP

p6A+

A* e f t .

(6-10)

q)

They are linear in //, Coxeter covariant and hermitian conjugate of each other with respect to the standard inner product (5.4): V£*P

(#)t=^-

= <(„)>

(6-n)

The conserved quantities are expressed as polynomials in € ± operators:

Ts(££)V = Y

(L-L+r^=Y(^W-

Likewise the creation and annihilation operators B^ (5.2) are expressed as

Bti, = Ts(L±rV = Y ( L ± W = Y (£ )"^

(6-13)

n,veK veil ± The commutation relations among £ operators are easy to evaluate, since I operators commute in the rational potential models (6.7):

[#,#] = [';;,C] = o, [e;,tt] = 2uU-v+ V

Y

9M(P-^(PV^)SP).

/ (6.14) Prom these it follows that the creation (annihilation) operators B„ do commute among themselves: [B+,B+)^ = [B-,B-]iP

peA+

= 0.

(6.15)

264

It is also clear that i^/y/2uj are the 'deformation' of the creation (annihilation) operators of the ordinary multicomponent harmonic oscillators. In fact we have £+ ew = 2iu{n • q) ew

and

l~ ew = 0.

(6.16)

In the next section we present an alternative scheme of algebraic construction of excited states of type V models by pursuing the analogy that £ ± are the creation and annihilation operators of the unit quantum. This method was applied to the Ar models by Brink et. al and others 19>17>20. 7 7.1

Algebraic construction of excited states II Operator solution of the triangular Hamiltonian

In subsection 3.1, we have shown that an eigenfunction of H with eigenvalue Nu is given by (pN(q) + PN-2(qj)

ew,

(7.1)

in which P/v (
This creates the above eigenfunction of ri from the ground state: 1±^PN(l+)

ew = (pN(q) + PN-2(q))

ew.

(7.3)

The proof is again elementary. By using the commutation relations among ^ ± operators it is straightforward to derive the explicit expression of the Hamiltonian in terms of ^ ± :

265

in which the second term vanishes upon acting on a Coxeter invariant state. Next we obtain

J - E ^ M ' ^ 1 = [et,S\±u,&

^ = E 9MsP,

(7-5)

which is an I operator version of (4.17). Since a commutator is a derivation, we obtain

2^~ EItf£.-PAKO] = [PNV*),S] + NuPN(e+),

(7.6)

in which the first term in r.h.s. vanishes due to the Coxeter invariance of Pjy. Thus we arrive at the desired commutation relation [H,PN(£+)}

= NuPN((l+)

gM "1 \P?

p

ew = NuPN(e+)

ew.

+ £ PGA+

2 (^r

^ +)

(sp -1),

(7.7)

and the eigenvalue equation HPN(£+)

(7.8)

Since the action of the creation operators on the ground state is # i •' • £tN

eW

= [ ( 2 H % • Mi) • •' (« • l*rt) + lower powers of q] ew,

(7.9)

our assertion (7.3) is proved. It should be stressed that in this formalism the Coxeter invariance of the polynomial P is important but not how it is obtained. Like the above Hamiltonian (7.4), the £ operator formulas of higher conserved quantities (6.12) contain extra terms:

Ts(£?) = J2 (L+L~)l* = E < W + VT-

(7-10)

Here VT stands for vanishing terms when they act on a Coxeter invariant state. The same is true for most formulas derived in section 6. 7.2

States Created by B%

Here we derive the explicit forms of the subseries of eigenstates obtained by multiple applications of the least quanta creation operator B% (5.2), or its normalized form b% (5.11). It is convenient to work with the similarity transformed operator

bt = e~wbt e^ =

* £ ft)' + VT>

(7-11)

266 in which (7.12) Let f(u) be an arbitrary function of u = wq2, then it is Coxeter invariant. We find I+f(u) = 2iu(q • M )(l - — )/(u), du

u = wq\

(7.13)

and b+f(u) = -

Since 6^1 = So — u — L\°

du J

V

(7.14)

/(«)•

du

(u), we assume (b+)nl =

n\L&-1Hu).

(7.15)

By using the Laguerre differential equation (3.21) and the recurrence formulas of the Laguerre polynomial L„ (u), ^(U), «^4 a) («) = »4a)(«) - (» + a)^.(«)> L^(u)

+ (u-2n-a

(7-16)

+ 1 ) 4 ^ (u) + (n + a-

l)L{n%(u)

= 0,(7.17)

we can show «(i-^:)a-«»(i-^) du du

L ^ - 1 ) ( « ) = (n + l ) L g : r 1 ) ( u ) .

(7.18)

Thus the induction is proved and we arrive at (5.12). The orthogonality of the states ((B+)»ew,(B+)mew)=0,

(7.19)

n^m

can be easily understood as the du part of the measure e2Wdrq = e~uu£°^1dudil,

dfl : angular part,

is the proper weight function for the Laguerre polynomial Ln °

(u).

267

7.3

Explicit solutions of the rank two models

For rank two models, the Liouville integrability, or the involution of conserved quantities is automatically satisfied since the second conserved quantity is already obtained. For rank two type V models, the complete set of orthogonal wavefunctions can be written down explicitly in terms of separation of variables by using the Coxeter invariant polynomials. These are based on the dihedral root systems him), with A2 ~ J 2 (3) 37 , B2 = J 2 (4) and G 2 = J 2 (6) 38 . The Coxeter invariant polynomials exist at degree 2, i.e. q2 and m which is 771

Ylivj-q),

(7.20)

i=i

where {VJ} is a set of vectors given in (B.2). dimensional polar coordinates system e for q

If we introduce the two-

q = r(sin6,cos6),

(7.21)

then the principal Weyl chamber is PW : 0 < r 2 < o o ,

0 < 6 > < ir/m.

(7.22)

The two Coxeter invariant variables read: 771

q2=r2,

Y[(vj-q)

= 2(r-rcosm6,

(7.23)

3=1

and the latter variable varies the full range, — 1 < cosm6 < 1 in the PW. Thus solving the eigenvalue equation for ti (3.6) by separation of variables in the polar coordinate system is compatible with Coxeter invariance. We adopt as two independent variables u = cur2,

z = cos m6.

(7.24)

The solutions consist of a Gegenbauer (Jacobi) polynomial in cos m6 times a Laguerre polynomial in wr 2 . The former we have encountered in the A\ Sutherland problem, subsection 3.2 and the latter in the A\ Calogero problem subsections 3.1 and 7.2. Let us demonstrate this for odd m with a single coupling constant and for even m with two independent coupling constants, in parallel. In terms of the e We believe no confusion arises here, between the radial coordinate variable r and the rank of the root system r, which in this case is 2 of l2(m).

268 Coxeter invariant variables (7.24) the I2(m) Hamiltonians take surprisingly simple forms:

m j

•U=cjr—-——(r — dr 2r dr \ dr

9 \yo+9e)} 2

)d_ dr

_ 1 _ \dId22 2g cot m6 f d_ 2+m go tan ^ + ge cot ml I d9 2r2 W \ 2 > =-2w U^r-z + (t0 — u) du2 du

*u

2 r

f l + 2p 1 z .l + 5e+5oJ Ja*

urn ( I - * 2 ) dz2 2u The z part admits polynomial solutions (l-z2)

(7.25)

d2 dz2

pl(ff<.-i.9e-J (2)

in which / is the degree of the polynomial. After substituting them, the radial part of the Hamiltonian "riT reads Ur = -2w

d? du2

~

m2

.d du

4M

V

\9e+9ojJ

(7.27)

By similarity transformation in terms of umll2 ex: rml, which is the radial part of the highest term of the polynomial Pj-a,ti>{rm cosm6), it reads u-ml/2JirUml/2

=

_2uJ

U

d2 fl

d^

+

(

Vnl

,

+

x

\ ad

S

ml ml

«-U)du-T

(7.28)

This is the main part of the differential equation for the Laguerre polynomial (3.21): d2

L £ B I + & ~ 1 ) ( U ) = O.

Thus the eigenstates of the Hamiltonian are obtained: H P u m , / 2 L j , m , + ^ - 1 ) ( « ) = w(2n + mZ)u m '/ 2 L< l m ' + * 0 ~ 1) (u).

(7.29)

269

r (s-J.s-J) 1 ^m(/2 L (mi+£o-D( u ) p^"-^"^1^)

(7.30)

= (w(2n + m O + g u " " / 2 ^ - 1 ) ^ ) ? , V . . - * , . . - ! > ' ( z ) It is instructive to note that the Hamiltonians % look also simple: 9(9-11 sin2 mO

2W

r

2r 2 56>2 + 2r 2

2r 9r V fr J

• . (7.31) g°(9°-l) 4 cos 2 2 #

+

fle(ge

5

Olshanetsky and Perelomov obtained the above solutions starting from these formulas. 8

Universal proof of involution of quantum conserved quantities for type I, II and III models

Here we present a proof of involution of quantum conserved quantities {Qn} derived from the universal Lax pair in subsection 4.1 for type I, II and III models. The proof is applicable to all models based on any root system. Though a universal proof of involution for type I models is given by Heckman 23 as recapitulated in section 6, we believe the universal proof applicable to type II and III models as well is new. It depends on a theorem by Olshanetsky and Perelomov 13 . Our own contribution is that we have provided a universal Lax pair and conserved quantities satisfying all the requirements of the theorem. Liouville's theorem states the complete integrability as the existence of an involutive set of conserved quantities as many as the degrees of freedom. We have already given conserved quantities {Qn} (4.7) independent and as many as the degrees of freedom (see Appendix B). They have the following properties: 1. Coxeter invariance Qn(sP(p), sp(q)) = Qn{p,q),

\/p€ A.

(8.1)

2- Qn(p,q) is a homogeneous polynomial of degree n in variables (pi,...,pr,x(p-q)). 3. Scaling property for those of type I models: 'Qui^P,

Kq) = K~n 7<3n(p, q)

as a consequence of the above point.

(8.2)

270

4. For type II and III models, the asymptotic behaviour near the origin: Qn(p,q) = IQn(p,q)(l

+ 0(\q\)),

for

| 9 | - • 0.

(8.3)

We need to show the vanishing of Jlm = [QhQm],

(8.4)

which is a polynomial in {p} of degree s s
(8.5)

Let us decompose Jim into the leading part and the rest: Jim = JL + J{™\

J?m = £ ^ " " ^ M P h

• • -Pi.

(8-6)

and Jf"* is a polynomial in {p} of degree less than s. From Jacobi identity and conservation \H,Qi(m)} = 0, we obtain [H, J[m] = 0.

(8.7)

Considering the explicit form of the Hamiltonian (2.7) (u = 0), the leading (i.e. of degree s + 1 in {p}) part of [H, Jim] comes only from the free part

bVL] and it vanishes if the following conditions are satisfied: ^Ac*i.....*.(g)=0,

(8.8)

where the sum is taken over all permutations of indices a(t,ki,...,ks) = C?i>- • • > js+i)- In Ref.39. it is proved (Lemma 2.5, p. 407) that the system (8.8) has only polynomial solutions. Then Olshanetsky and Perelomov argue that for type I models the scaling property tells that c*1'"•'** («q) = Since s J°m = 0 and J[m = 0. The same results follow for type II and III models by considering the asymptotic behaviour for \q\ ->• 0. Thus the involution of all the conserved quantities {Qn} is proved. This result also implies the involution of classical conserved quantities by taking the classical limit (h -» 0). See Ref.12 for the classical Liouville integrability of the most general Calogero-Moser models with elliptic potentials.

271

9

Summary, comments and outlook

Various issues related to quantum integrability of Calogero-Moser models based on any root system are presented. These are construction of quantum conserved quantities and a unified proof of their involution, the relationship between the Lax pair and the differential-reflection (Dunkl) operators formalisms, construction of excited states by creation operators, etc. The independence (completeness) of the set of conserved quantities is discussed in some detail. They are mainly generalizations of the results known for the models based on Ar root systems. Integrability of the models based on other classical root systems and the exceptional ones including the non-crystallographic models are also discussed in Refs. 3 8 > 4 1 - 4 4 . There are still many interesting problems to be addressed to: The structure and properties of the eigenfunctions of the trigonometric potential models, which are generalizations of the Jack polynomials 2 8 ~ 3 1 . Comprehensive treatment of Liouville integrability of rational models with harmonic force. Understanding the roles of supersymmetry and shape invariance in CalogeroMoser models 45>44. Formulation of various aspects of quantum CalogeroMoser models with elliptic potentials; Lax pair, the differential-reflection operators 46>47, conserved quantities, supersymmetry and excited states wavefunctions. Acknowledgements I thank K. Saito, J. Sekiguchi and T. Yano for bringing 33 ' 35 to our attention. This work is partially supported by the Grant-in-aid from the Ministry of Education, Science and Culture, priority area (#707) "Supersymmetry and unified theory of elementary particles". Appendix A: Root Systems In this Appendix we recapitulate the rudimentary facts of the root systems and reflections to be used in the main text. The set of roots A is invariant under reflections in the hyperplane perpendicular to each vector in A. In other words, sa(/?)GA,

Va,/?eA,

(A.l)

where sa(p) = P - (a v - /?)a,

av = 2a/\a\2.

(A.2)

272

The set of reflections {sa, a £ A} generates a group G A , known as a Coxeter group, or finite reflection group. The orbit of /? e A is the set of root vectors resulting from the action of the Coxeter group on it. The set of positive roots A + may be defined in terms of a vector U € R r , with a-U ^ 0, Va € A, as those roots a € A such that a • U > 0. Given A + , there is a unique set of r simple roots II = {ctj, j = 1 , . . . , r} defined such that they span the root space and the coefficients {a,} in /? = Y^j=i ajaj f° r P G A+ are all non-negative. The highest root ah, for which Y?j=\ ai ls maximal, is then also determined uniquely. The subset of reflections {sa, <* 6 II} in fact generates the Coxeter group GA- The products of sa, with a 6 II, are subject solely to the relations

(sasff)m^^ = i,

a, pen.

(A.3)

The interpretation is that sasp is a rotation in some plane by 27r/m(a,/3). The set of positive integers m(a,/3) (with m(a,a) = 1, Va € II) uniquely specify the Coxeter group. The weight lattice A (A) is defined as the Z-span of the fundamental weights {\j}, j = 1 , . . . , r, defined by Oj-\k = 5jk,

o-j e n .

(A.4)

The root systems for finite reflection groups may be divided into two types: crystallographic and non-crystallographic. Crystallographic root systems satisfy the additional condition QV'i?eZ,

Va,/?eA,

(A.5)

which implies that the Z-span of II is a lattice in R r and contains all roots in A. We call this the root lattice, which is denoted by L(A). These root systems are associated with simple Lie algebras: {Ar, r > 1}, {Br, r > 2}, {Cr, r > 2}, {Dr, r > 4}, E6, E7, Es, F 4 and G 2 . The Coxeter groups for these root systems are called Weyl groups. The remaining non-crystallographic root systems are H3, if4, whose Coxeter groups are the symmetry groups of the icosahedron and four-dimensional 600-cell, respectively, and the dihedral group of order 2m, {I^im), m > 4}. Here we give the explicit examples of root systems. In all cases but the Ar, {ej} denotes an orthonormal basis in R r , e,- 6 Rr,ej • e^ = 5jk1. Ar: This root system is related with the Lie algebra su(r + 1). It is convenient to have the r + 1 dimensional realization: A = { ±(ej

- e fe ), j y£ k = 1 , . . . , r + l|e,- e R r + 1 , ej • ek = 6jk},

n = {ej - ej+1,

j = 1,..., r}.

(A.6)

273

2. Br: This root system is associated with Lie algebra so(2r + 1). The long roots have (length) 2 = 2 and short roots have (length) 2 = 1: A = {±ej ± ek, ±ej, j ^ k = 1 , . . . ,r}, U = {ej-ej+1,

j = l,...,r-l}U{er}.

3. Cr: This root system is associated with Lie algebra sp(2r). roots have (length) 2 = 4 and short roots have (length) 2 = 2: A = {±ej±ek, ±2ej, j,k = l , . . . , r } , n = {ej - ej+1, j = 1 , . . . ,r - 1} U {2er}.

(A.7) The long

(A.8)

4. Dr: This root system is associated with Lie algebra so(2r): A = {±ej±ek,

j ^k = l , . . . , r } ,

11 = {ej -ej+1,

j = l , . . . , r - l } U { e r _ i +er}.

(A.9)

5. EQ: All the roots have the same (length) 2 , which is chosen to be 2: A = {±ej ±ek,

j ^k = 1,...,5}U { - ( ± e i •• • ± e 5 ± v^ee), (even + ) } ,

II = { - ( e i - e2 - e 3 - e4 + es - v^ee), e 4 - e 5 , e-$ - e^, e 4 + e 5 , - ( e i - e 2 - e 3 - e 4 - e 5 + V3e6),

e2 - e 3 } .

(A.10)

6. Ei: All the roots have the same (length) 2 , which is chosen to be 2: A = {±ej ± cfc, j ^ k = 1 , . . . , 6} U

{±V2e7}

U { | ( ± e i • • • ± e 6 ± v^ey), (even + ) } ,

(A.ll)

II = { e 2 - e 3 , e 3 - e 4 , e 4 - e 5 , e 5 - e 6 , - ( e i - e 2 - e 3 - e 4 - e 5 + e 6 - \/2e 7 ), \/2e 7 , e 5 + e 6 } . 7. Ss: All the roots have the same (length) 2 , which is chosen to be 2: A = {±ej ± ek, j £ k = 1 , . . . , 8} U { ^ ( i e i • • • ± e 8 ), (even + ) } , II = { ^ i

- e 2 - e 3 - e 4 - e 5 - e 6 - e 7 + e 8 ), e 7 + e 8 }

U{ej-ej+1,

j = 2,...,7}.

(A.12)

274

8. F4: The long roots ((length) 2 = 2) are those of £>4 and the short roots ((length) 2 = 1) are the union of the vector, spinor and anti-spinor weights of D4: A = {±ej ±ek,

± e J - , - ( ± e 1 - - - ± e 4 ) , j / k = 1 , . . . ,4},

II = {e 2 - e 3 , e 3 - e 4 , e 4 , - ( e i - e 2 - e 3 - e 4 ) } .

(A.13)

9. G 2 : The G 2 root system consists of six long roots and six short roots, and the sets of long and short roots have the same structure as the A2 roots, scaled | a s | 2 / | a i | 2 — 1/3 and rotated by n/6. They are

A = {(±A0), ( ± 4 ^ ) , (0,±vf>. <±£.±>. n = {(V2,0), ( - ~ , ^ ) }

(A.14)

10. him): This is a symmetry group of a regular rn-gon. For odd m A consists of a single orbit, whereas for even m it has two orbits. In both cases we have a representation in which all the roots have length unity A = {(cos((j - l)7r/m),sin((j - l)ir/m)),

j =

l,...,m},

n = {(1,0), (cos((m - l)7r/m), sin((m - l)7r/m)))}

(A.15)

11. HA: Define a = COSTT/5 = (1 + \/5)/4 , b = COS2TT/5 = ( - 1 + y/E)/4. Then the H4 roots are generated by the following simple roots 34 : (A.16)

The full set of roots of if4 in this basis may be obtained from (1,0,0,0), ( | , i , | , | ) , and (a, \,b,0) by even permutations and arbitrary sign changes of coordinates. These 120 roots form a single orbit. 12. Hy. A subset of (A.16), {ai, a 2 , a 3 } is a choice of simple roots for the H3 root system. In this basis, the full set of roots for # 3 results from even permutations and arbitrary sign changes of (1,0,0) and (a, | , 6 ) . These 30 roots also form a single orbit.

275

Appendix B: Conserved quantities Here we list for each root system how the complete set of independent conserved quantities is obtained by choosing proper representations {7?.} of the Lax pair. We choose those of the lowest dimensionality for the convenience of practical calculation. Of course there are many other choices of representations giving equally good sets of conserved quantities. 1. Ar: For all powers, the vector representation ( r + 1 dimensions) is enough. 2. Br: For all powers, the representation consisting of short roots { i e , : j = 1 , . . . , r } , (2r dimensions) is enough. 3. C r : For all powers, the representation consisting of long roots {±2ej : j = 1 , . . . , r } , (2r dimensions) is enough. 4. Dr: For all even powers, the vector representation (2r dimensions) is enough. For the additional one occurring at power r, the (anti)-spinor representation ( 2 r _ 1 dimensions) would be necessary. They are minimal representations. 5. E§: For all powers, the weights of the 27 (or 27) dimensional representation of the Lie algebra is enough. They are minimal representations. 6. E7: For all powers, the weights of the 56 dimensional representation of the Lie algebra is enough. This is a minimal representation. 7. E$: For all powers, the 240 dimensional representation consisting of all the roots is enough. This is not the same as the adjoint representation of the Lie algebra. 8. F4: For all powers, either of the 24 dimensional representation consisting of all the long roots or the short roots is enough. These are not Lie algebra representations. 9. G-2,: For all powers, either of the 6 dimensional representations consisting of all the long roots or the short roots is enough. These are not Lie algebra representations. 10. him): For both powers 2 and m, the representation consisting of the vertices Rm of the regular m-gon is enough: Rm = {(cos(2k/m + t0),sin(2k/m

+ t0)) € R 2 | k = 1 , . . . , m } ,

(B.l)

276

in which t0 = 0 (l/2m) for m even (odd). Another set Vm is used in 7.3. Here, Vm is the set of vectors with 'half angles of the roots (see (A.15)) given by Vm = {VJ = (cos((2j - l)7r/2m),sin((2j - l)7r/2m)) € R 2 | j =

l,...,m}. (B.2)

11. H3: For all powers, the representation consisting of all the 30 roots is enough. 12. H4: For all powers, the representation consisting of all the 120 roots is enough. References 1. F. Calogero, Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419-436. 2. B. Sutherland, Exact results for a quantum many-body problem in onedimension, II, Phys. Rev. A5 (1972) 1372-1376. 3. J. Moser, Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975) 197-220; J. Moser, Integrable systems of non-linear evolution equations, in Dynamical Systems, Theory and Applications; J. Moser, ed., Lecture Notes in Physics 38 (1975), Springer-Verlag; F. Calogero, C. Marchioro and O. Ragnisco, Exact solution of the classical and quantal one-dimensional many body problems with the two body potential Va{x) = g2a2/sinh2 ax, Lett. Nuovo Cim. 13 (1975) 383-387; F. Calogero, Exactly solvable one-dimensional many body problems, Lett. Nuovo Cim. 13 (1975) 411-416. 4. M. A. Olshanetsky and A. M. Perelomov, Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Inventions Math. 37 (1976), 93-108. 5. M. A. Olshanetsky and A. M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983) 313-404. 6. S. P. Khastgir, A. J. Pocklington and R. Sasaki, Quantum CalogeroMoser Models: Integrability for all Root Systems, J. Phys. A33 (2000) 9033-9064, hep-th/0005277. 7. A. J. Bordner, E. Corrigan and R. Sasaki, Calogero-Moser models I: a new formulation, Prog. Theor. Phys. 100 (1998) 1107-1129, hep-th/9805106; A. J. Bordner, R. Sasaki and K. Takasaki, CalogeroMoser models II: symmetries and foldings, Prog. Theor. Phys. 101

277

8.

9.

10.

11.

12.

13.

14.

15.

16.

17. 18. 19.

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G R E E N F U N C T I O N S ASSOCIATED TO COMPLEX REFLECTION G R O U P S G ( e , l , n ) TOSHIAKI SHOJI Department of Mathematics, Science University of Tokyo Noda, Chiba 278-8510, Japan E-mail: [email protected] Green functions of classical groups are determined by the data from Weyl groups and by certain combinatorial objects called symbols. Generalizing this, we define Green functions associated to complex reflection groups G(e, l,n). We also construct Hall-Littlewood functions and Schur functions associated to G(e, l,n), and show that the above Green functions are obtained as a transition matrix between those two symmetric functions. The details of this paper will appear elsewhere.

0

Introduction

Green polynomials Q^(q) of GLn(Fq), where \,(i are partitions of n, were first introduced by J.A. Green 2 in 1955 in a combinatorial framework of symmetric functions. They are obtained as the transition matrix between Schur functions and Hall-Littlewood functions. Those Green functions play a crucial role in 2 in describing irreducible characters of GLn(Fq). In 1976, P. Deligne and G. Lusztig constructed in 1 Green functions Qj. for finite reductive groups G(Fq) in general, as the restriction to unipotent classes of G(Fq) of Deligne-Lusztig's virtual character Rj,{6), associated to a maximal torus T of G defined over F g . The Green function Qx, for an irreducible character X of the Weyl group W of G, is denned as a certain linear combination of Qy for various T. In the case of GLn(Fq), let xX be the irreducible character of the symmetric group W = &n corresponding to A, and uM the unipotent class corresponding to /x. Then Qxx (uM) coincides with the modified Kostka polynomial K\tlJ(t), which is obtained as a certain linear combination of Green polynomials with some modification. Lusztig proved in 6 , generalizing the result in 13 , that there exists a simple algorithm of computing Green functions Qx. In other words, Green functions arise as a unique solution of a certain type of matrix equation PAtP = n

(*)

with unknown P, A. (The entries of the matrix P give Green functions Qx(u)). Note that the matrix fi is completely determined by the property of irreducible characters of W, while the .shape of P and A are determined by the 281

282

property of the Springer correspondence between unipotent classes of G and irreducible characters of W. It is known, by the geometric interpretation of Green functions in 8 , that the entries of the matrices P and A are in Z[g]. In 3 , Geek and Malle considered an analogy of the equation in (*) by making use of a-functions of irreducible characters of Weyl groups and families of unipotent characters, and showed that they have a unique solution with coefficients in Q(q). They verified in the case of exceptional groups that the matrices P, A have coefficients in Z[g], and also the matrix A provides the polynomials conjectured in 9 expressing the cardinality of special pieces in G(Fq). They conjecture that similar facts will hold for reductive groups in general. In the case of classical groups, the Springer correspondence is described in terms of certain combinatorial objects called "u-symbols" introduced by Lusztig 5 . The remarkable fact is that the matrix equation (*) is completely described by the combinatorial property of such u-symbols. The notion of u-symbols was generalized by Malle 12 so that it fits to the case of complex reflection groups G(e, 1, n) as a generalization of Weyl group of type Cn. Now the ingredients used to construct the equation (*) make sense even for this case, and one can define a matrix equation by using the combinatorics of symbols. By using a similar argument as in 3 , it is shown that this equation has a unique solution. So it would be natural to call the solution of this equation as the Green function associated to G(e, l,n). After Green 2 , the character table of GL„(Fq), especially the part corresponding to the values at unipotent elements of unipotent characters, is completely dominated by Green functions. Thanks to Lusztig 6 , similar facts hold in general; the character values at unipotent elements of finite reductive groups, say, Sp2n(Fq) or S02n+i(Fq), of unipotent characters are dominated by Green functions (or rather by generalized Green functions). Lusztig showed that his method of classifying unipotent characters of finite reductive groups makes sense even for other Coxeter groups W such as H3, if4, and determined their "degrees", even there does not exist a finite group G(Fq) whose Weyl group is W. Inspired by this, Malle 12 classified, in the case of G(e, l , n ) , the "unipotent characters" and described their degrees. One might expect that Green functions associated to G(e, 1, n) will provide us the values of unipotent characters at "unipotent elements". In this note, we try to find a combinatorial theory of symmetric functions which fits to the situation for G(e, l , n ) . In our theory, the role of partitions is replaced by symbols. In particular, we construct Hall-Littlewood functions associated to G(e, l , n ) , which are parametrized by symbols. We show that a large part of arguments used to construct Green polynomials of GLn(Fq)

283

can be generalized to our setting, and that Green functions are obtained as the transition matrix between Schur functions and Hall-Littlewood functions. Thus obtained Green functions are rational functions in Q(q). As some examples show, it is very likely that these Green functions are actually polynomials in q with positive integral coefficients. The details of this note will be found in 15 . The author is grateful for G. Malle and F. Liibeck for some useful discussions, and for Malle for the computation of examples by using computers. 1

A background from finite reductive groups

1.1 Let G be a connected reductive algebraic group defined over a finite field Fq, with Frobenius map F : G ->• G. We assume that G is split over F g . We are interested in the representation theory of the finite group GF over Q;, an algebraic closure of the Z-adic number field Q; with I ^ chFq. For an F-stable maximal torus T and a character 9 : TF —> QJ", Deligne-Lusztig's virtual character R^(9) : GF —»• Q; is defined. The Green function Qj. is defined as the restriction of R^(9) to the set of unipotent elements in GF, which is independent of the choice of 9. The functions R!f(9) play a crucial role in the representation theory of GF, and it is known that the computation of Rf (9) is reduced to the computation of various Green functions of smaller reductive groups. Let W be the Weyl group of G. Then the G F -conjugacy classes of instable maximal tori are parametrized by the conjugacy classes of W. We denote by Tw the F-stable maximal torus corresponding to w E W (up to conjugacy). Let Vuni be the space of G F -invariant functions on the set of unipotent elements in GF. Thus Qj. € Vuni- Let WA be the set of irreducible characters of W. For % £ WA, let us define a function Qx on Vuni by

Qx = \w\~1 52xMQ?w. wew We refer Qx also as Green functions. Clearly, the determination of Qj, all w G W is equivalent to that of Qx for all x £ WA.

for

1.2 Let I be the set of pairs (C, p), where C is a unipotent class in G, and p is an irreducible character of AQ{U). (Here C is F-stable, and u is an element in CF. AQ{U) is the component group ZG(U)/ZQ(U)). It is known, by a suitable choice of u € CF, that the set of G F -conjugacy classes in CF is in bijection with the set of conjugacy classes in AQ{U). We denote by ua the class in

284

CF corresponding to a £ AG(u).

For each i = (C,p), we define a function

Yt G V uni by

v.f A _ jpia) i W ~ \ 0

if v

~ u " ( G F _ - conjugate), ift^C*".

Then {Yi | i G J} forms a basis of the space VuniThere exists a natural injection / , the so-called "Springer correspondence" from WA to / . We denote by I0 the image of WA under the map / . If fix) = *, we write Qx also as Qi. Then it is known that Qx can be written as Qx = Qi = Y

Pi Y

iJ

jeio

with coefficients p y - G Q;. In fact, it is clear that Qi is written as a linear combination of Yj with j G / , and it is a deep result due to Lusztig that pij = 0 if j $: IQ. We fix a total order on 1$ which is compatible with the closure relations of unipotent classes, and consider the square matrix P = (py). The determination of Qi for various i G To is equivalent to the determination of the matrix P. Lusztig showed that P is determined as a unique solution of the matrix equation of the form PAtP = fi as in (*), where 1] is a known matrix obtained from the data on WA. In fact, we define an equivalence relation ~ on I0, by the condition that i ~ i' if i = (C,p),i' — (C',p') with C = C", and consider the block matrix with respect to this equivalence relation. Then it is shown that P is a block lower triangular matrix with diagonal blocks of the form qd"I if the block corresponds to C 3 u. Here du is the dimension of the variety Bu = {B\ G G/B \ u G B\}. Furthermore, the matrix yl is a non-singular block diagonal matrix. By this condition, P and A are uniquely determined from the equation (*). 1.3 In the case where G = Sp2n or SO^n+i, all of the ingredients involved in the above arguments are described in a purely combinatorial way; the set I is in bijection with certain combinatorial objects, "u-symbols". Then the subset To of / corresponds to the set of u-symbols of defect 1. Furthermore, the equivalence class in i" with respect to ~ corresponds to the similarity class of u-symbols. The assignment u ->• du is defined as a function a : Io ->• Z>o combinatorially. As explained in the introduction, these formalism can be extended to the case of complex reflection groups W = G(e, l , n ) . In the next section, we formulate the equation corresponding to (*) for such W.

285

2

Green functions associated to G(e,

l,n)

2.1 Let W = &„ K (Z/eZ) n be the imprimitive complex reflection group G(e, l , n ) , and V = C 1 the natural reflection representation of W. Let S(V) be the symmetric algebra of V, and 1+ the ideal of S(V) generated by the homogeneous W-invariant vectors of strictly positive degrees. We denote by R = ®Ri the coinvariant algebra of W, which is a graded algebra defined as the the quotient of S(V) by J+. Let t be an indeterminate. The Poincare polynomial P\v(t) is defined as Pw(t) = ^2i>0(dimc Ri)tl, which is explicitly given as n

+ei _

i

For any class function / on W, we define R(f) by

where dety denotes the determinant on V. If x is an irreducible character of W, then R(x) coincides with J2Jx,Rj}wt%, where (, )w denotes the inner product of class functions. Hence R(x) coincides with the fake degree of \, and R(f) € Z[t] for a generalized character / of W. Let N* be the number of complex reflections in W. Then N* is the maximal degree in R, and the VF-module RN» coincides with detv. 2.2 An e-partition a — ( a C ' , . . . , ^ 6 - 1 ' ) is an e-tuple of partitions a^. The size \a\ of a is defined as \a\ = X)i=o l a ^^l- We denote by Vn,e the set of e-partitions of size n. Then the set WA of irreducible characters of W is in bijection with the set Vn^. We denote by xa t n e irreducible character corresponding to a 6 Vn,e- In particular, the unit character corresponds to a = ((n), —,..., —), and dety corresponds to a = (— , —, (l n )). Let us fix m = (mo,... , m e _ i ) , where m^ are positive integers such that mfc > n for any k. We denote by Z^'° = Z°'°(m) the set of e-partitions a with size n, where a^ is regarded as an element in Z m * written in the form, a^ : a[' > • • • > ami •> 0- We fix integers r > s > 0, and consider an e-tuple of partitions A0 = ^i°(m) = (ylo, ...,Ae-i) defined by AQ : (m 0 - l)r > • • • > 2r > r > 0, Ai : s + {mi - l)r > • • • > s + 2r > s + r > s

(2.2.1) for 1 < i < e - 1.

We denote by Z%s = Z%s(m.) the set of e-partitions of the form A = a + A0, where a e Z°'° and the sum is taken entry-wisely. We denote by A =

286

A(a) if A is as above, and call it the e-symbol of type (r, s) corresponding to a. Let us define a shift operation Z%s{m) -> Z^s{xa!) by associating A1 = {A'0,...,A'e_1) e Zrn's(m') to A = (A0,... ,Ae-X) G Z £ s ( m ) , where m ' = (m 0 + l , . . . , m e _ i + l ) with A'0 = (A0+r)U{0} and A'k = (yl f c +r)U{s} for k = 1 , . . . , e — 1. Two elements A and A' in Z£ s are said to be similar if all the entries of them coincide each other with multiplicities. The set of symbols which are similar to a fixed symbol is called a similarity class in Z%s. We shall define a function a : Z%s ->• Z> 0 . Take A G Z£ s and let A0 be as before. We put

a(A) = ^2

min(A,A')-

^Z

min

(M,A«'),

(2.2.2)

where in the first sum, we assume that A ^ A' if A and A' are contained in the same yl;, and similarly for the second sum (this occurs only when r = 0). The function a on Z%s is invariant under the shift operation, and takes a constant value on each similarity class in Z%s. Remark 2.3 The notion of symbols already appeared in several articles in various forms. If e = 1, Z£ ,s is in bijection with the set of partitions of n. So, we consider the case where e = 2. Then the set Z^'° (up to shift operation, all the same below) was used in 4 to parameterize unipotent characters of Sp2n(Fq) or S02n+i(F a ). In that case, the a-function defined in (2.2.2) coincides with the original a-function, i.e., the exact power of q dividing the degree of the corresponding unipotent character. The set Z^'1 (resp. Z%'°) was used in 5 to describe the generalized Springer correspondence for G = Sp2n(k) (resp. G = S02n+i(k)) subject to the condition that ch k ^ 2. In this case, the similarity classes are in 1-1 correspondence with the unipotent classes of G, and the value a(A) for such a symbol A coincides with dimi?„, (see, for example [7, 4.4]). While, the set Z*'2 was used in 10 to describe the generalized Springer correspondence for Sp2n{k) with ch A; = 2. On the other hand, for arbitrary e, the set Z^'° was used in 12 to parameterize unipotent degrees associated to the complex reflection group G(e, l , n ) . 2.4 Under the identification of Z°'° with Vn<e, we consider the a function as the function on the set Vn>e- Also the similarity classes in Vn,e are defined by inheriting the classes in Z%s. We denote by a ~ 0 if a,/3 € Vn,e are in the same similarity class. We choose a total order >- on Vn,e such that a(a) < o(/3) whenever a >- /3, and that each similarity class in Vn,e forms an interval. In the following, we consider the matrices of degree \Vn,e\ with respect to this order. For each a , /3 € Vn,e, we define a polynomial uja^ G Z[i]

287

by w a , /3 = ^*-R(x a ®X / 3 .®deT v ),

(2.4.1)

and put f2 — (ua,p), the matrix of degree \Vn,e\- Let P = (pa,p) and A = (Aai(g) be matrices of degree \Vn,e\. We consider the following system of equations with respect to the unknown variables \a,p,Pa,pAQ,/3 = 0 Pa,p = 0 „

_

unless a ~ /3, unless either a >- /3 and a / /3, or a = /3,

(2.4.2)

ta(a)

t

PA p = n. 2.5 There exists a system of equations which is closely related to (2.4.2). Let fi' = {oj'a „) be a matrix denned by
(2.5.1)

where x' 3 is the complex conjugate of x'3- We consider a matrix equation of the form P M " P " = n',

(2.5.2)

where either of P' = (p'a g) and P" = (p„ «) satisfies similar conditions as in the second and the third one in (2.4.2), and A' — (Xa,p) a s m the first one. Let a = {(Ta,p) be the permutation matrix realizing the complex conjugation of irreducible characters of W, i.e., aa^ = 1 if x13 — X™ a n d aa,p = 0 otherwise. Then we have il' = Cla, and the equation PA*P = f2 implies that P-AcT-a-ltPcT

= n'.

It is easily checked that the equation (2.4.2) has a solution if and only if the equation (2.5.2) has a solution, and in that case, we have P ' = P, A' — Aa, P" — a~ltPa. In the remainder of this note, we report that the solution of (2.5.2) can be described in terms of certain symmetric functions as in the case of GLn. 3

A combinatorial setting for G(e, l , n )

3.1 For a given m = (mo,... ,m e _i) as in 2.2, we introduce indeterminates Xj (0 < k < e — 1,1 < j < mk). We denote by x the whole variables (XJ ), and also denote by x^ the variables a^ , . . . , Xmk. Power sum symmetric

288

functions and Schur functions are denned as in n , Appendix B. Let C be a primitive e-th root of unity in C. For each integer r > 1 and i such that 0 < i < e — 1, put e-l 7=0

where pr(x^) denotes the usual r - t h power sum symmetric function with respect to the variables x^\ We put pr (x) = 1 for r = 0. For an e-partition a = (a^°\... , a ( e - 1 ' ) with a^ : a\ ' > ••• > amk, we define a function Pa(x) by e — 1 m.k

pa(x)=nY[P%(x). k=0 j=l

(3.i.i)

'

(Note that our power sum symmetric function pa (x) is not exactly the same as the function given in n . Our pa(x) coincides with the complex conjugate of the one in u ) . Next, we define the Schur function sa(x) and the monomial symmetric function ma(x) by e-l

e-l

sa(x) = l[saW(xW),

ma(x) = Y[maW(xW),

fc=0

k=0

(3.1.2)

where sa(k)(x^) (resp. ma(,k)(x^)) denotes the usual Schur function (resp. monomial symmetric function) associated to the partition a^ with respect to the variables x^. 3.2. Let t be an indeterminate as before. In what follows we regard the variables x[k) defined for k € Z/eZ ~ { 0 , 1 , . . . , e - 1}. For each 0 < k < e - 1 and an integer r > 0, we define a function qlt±(x; t) by

tflfrt) = E ^ H J f ' i>l

lljjtixi

tX {k)

\k)

x

(r > 1),

(3.2.1)

j

where (J = m* - 1 - mk±\, and by q\,±(x;t) = 1 for r = 0. In the product of the denominator, an ' runs over all the variables in x^ except x\ , while in the numerator, an

runs over all the variables in a;(fc±1). Note that if

xr°' = x\k x' = Xi and m^ = irik±i, qTt±(x;t) qr(x;t)

n

coincide with the function

given in , III, 2.9. It can be shown that qlj.(x;t)

is a polynomial

289

in x, t with integral coefficients, homogeneous of degree r with respect to the variable x. For an e-partition a = ( a ' 0 ' , . . . , ^ 6 " 1 ' ) 6 Vn,e, we define a function qa,±(x)

b

y e —1 m.k

9«,±(x; «) = n i l 9% Jx; t). *=oi=i

a

(3.2.2)

'

Next, for a partition «(*) : a[h) >•••> aff > 0, (here lk = !(«(*)) is the number of parts of a^), we define a function 2a(*o(f) by

^(*)(*)=n(i-c**^))-1, Then we define z a (i) by e-l

za(t) = zaJ[zaW(t),

(3.2.3)

A=0

where za is the order of the centralizer of wa in W. Explicitly, za is given as follows. For a £ Vn<e, put 1(a) — YHZQ Ka^)For a partition a — (l n i , 2 ^ , . . . ) , put za = Ui>i *"'"
detv(t • idy -wa) = J ] JJ(t a i - Cfc). k=0j=l

In particular, we have za(t~x) = 2 a £" dety(£ • idy —iw a ) -1 We now introduce infinitely many variables a;^ , y\' for i = 1,2,... and for 0 < k < e — 1. We may regard the above functions as functions with infinitely many variables x[ , x\ ', Under this setting, the following proposition holds. Proposition 3.3 Let

"<*.*')=nn\ wS

290

Then we have Q(x,y;t)

= ^Tqa>+(x;t)ma(y)

= ^ma(a;)gai.(y;(),

a

tl{x,y,t)

(3.3.1)

a

= ^2za(t)-1pa(x)pa(y),

(3.3.2)

a

where a runs over all the e-partitions of any size. In (3.3.2), pa(y) the complex conjugate of pa(y).

denotes

3.4 We now introduce functions Q^{x; t) and P^(x; t) associated to symbols A € Z%s, which is in analogy with Q\(x;t) and P\(x;t) in n , III. Let A be the subring of Q(t) consisting of functions which have no pole at t = 0. Then A is a local ring with the unique maximal ideal tA. Hence A* = A — tA is the set of units in A. In what follows, we identify the set Z%s and Z%° by the natural map, and we fix a total order -< on Z°'° as in 1.4. Then we have the following theorem. Theorem 3.5 (i) For each A € Z%s, there exists a unique function P^(x; t) satisfying the following properties. (a) P^(x;t)

can be expressed as PA(X'^)=

5Z

where ca,p{t) € Q(t) and ca^{t) (b) P^(x;t)

c

«,/3M9/3,±(z;i),

= 0 unless /3 >- a or /3 ~ a.

can be expressed as PA&t)

= sa(x)+

^2

ua,p(t)sp(x),

where ua,p(t) 6 tA, and ua,p{t) = 0 unless /3 -< a. and /3 / (ii) For each A 6 Z^s, there exists a unique function Q^(x;t) following properties. (a) Q^(x;t)

a.

satisfying the

can be expressed as QAz(x;t)=qa>±(x;t)

+ ^

da,p(t)qp,±(x;t),

where da,p{t) € Q(t) and da,p(t) = 0 unless /3 >- a and /3 ^ a .

291

(b) Q^{x;t)

can be expressed as QA:(x;t) = ] T

wafi{t)sp(x),

where wa^(t) € A, and tva^(t) = 0 unless f3 -< a or /3 ~ a . Moreover, wa,p £ tA if @ ^ a, and u> a , a £ .4*. Remark 3.6 We call P^ the Hall-Littlewood function associated to A £ Z%s. It is likely that P^(x;t) £ Z[x;i\. In fact, in the classical case, i.e., the case where e = 2,r = 2,s = 1, the assertion follows from the known properties of Green functions. The functions P^, Q^ have the stability property for the shift operation of symbols. Hence they can be regarded as functions with infinitely many variables. As a corollary to Theorem 3.5, we have Corollary 3.7 Let fl(x,y;t) n(x,y;t)

be as in Proposition 3.3. Then we have

= J2 bA:AI(t)P+(x;t)PX,(y,t).

(3.7.1)

A,A'

0(x, !/;*) = £

Q+A{x; t)PX(y; i) = £

A

PX(x; t)Q~A{y; t).

(3.7.2)

A

where in (3.7.1), A, A' run over all the elements in U^Li ^n's> andbA.,A'(t) = 0 unless \A\ = \A'\ and A ~ A'. In (3.7.2), A runs over all the elements in

U OO n=l

yr Z

s

n •

3.8 Let 6 m = (3 m o x • • • x 6 m , . ! be a parabolic subgroup of &n of type m. We denote by H m = ® £ l 0 1 \ x \ ,..., Xmk]&m>° the ring of symmetric polynomials (with respect to 6 m ) with variables x = (x^ ). S m has a structure of a graded ring H m = 0 i > o Elm, where Elm consists of homogeneous symmetric polynomials of degree i, together with the zero polynomial. We consider the inverse limit S'=lim:4 m

with respect to homomorphisms pm',m : E.*m, -» SJ„, where m ' = (m'Q,...,m'e_1) with m'k = rrik + I for some integer I > 0, and / w , m is induced from the homomorphism 0^. Z [x[k),..., x{^\ ] -» 0 f c Z [x[h),..., x(£, ] given by sending x\ ' to 0 for i > m.k, and leaving other ar- ' unchanged. H = 0 i > o S l is called the space of symmetric functions. Schur function sa(x)

292 with infinitely many variables x[ ,... is regarded as an element in H n with n = \a\, and the set {sa(x)} with a G Z%'° forms a Z-basis of E n . Put H Q W = Q(*) ®z — J t c a n b e s h o w n t h a t t h e s e t {^1 I -A G Z £ s } (resp. the set {Q^ | 7l G Z£*}, {qa,± | a G Z£-0}) forms a Q(i)-basis of the space Z$[i\. A similar property holds if one replaces En by H^ It follows from the above discussion that {sa} gives rise to a basis of the Z[i]-module Eft] = Z[£] ®z S, hence so is {ma}. Also, we see that {qa,±}, anc {QA)> * {PA} t u r n o u t t o be bases of SQ[£]. Moreover, {p a } gives a basis of C(£)-space Ec[t] = C(i) ®z S. We now define a scalar product on EQ[<] by the condition that (qa,+ (x>t)>mp(x))=fi<*,p,

(3.8.1)

and extend it to a sesquilinear form on Ec[t]. By using a similar argument as in n Chap. I, 4, it follows from Corollary 3.7 and Proposition 3.3 that we have (ma(x),qpi-(x;t))= (PA-(x;t),QA;(x;t))

5a!0, = (Q+i{x;t),PX,(x;t))=6AiA.,

(Pa(x),Pp(x))=

(3.8.2)

Za(t)6a>0.

In the special case where e = 2, the function qat+ (resp. PJ[, Q\) coincides with qa,- (resp. P^, Q~A)- We also have pa = pa and za(t) G Q(£). Hence in this case, the scalar product (, ) turns out to be symmetric. The following result gives a characterization of PA and Q^ in terms of the inner product. Proposition 3.9 The functions P j are characterized by the following properties. (Then the function Q^ is obtained as the dual basis of PA in the sense of (3.8.2)). (i) PA(x;t)

can be expressed in terms of sp(x) as P

A=s<* + J2utl3sP>

where A = A{a), and u* B = 0 unless f3 -< a and /3 / (ii) (PA~,PA'<)= 0

unless

A~

a.

A'.

3.10 Let K±(t) = M(s, P * ) be the transition matrix between Schur functions and Hall-Littlewood functions, i.e., S0(X)=

X^aW^iajto*)A(a)

293

We call Kpa(t) Kostka functions associated to Z%s, in analogy to the case of GLn(Fq). By Theorem 3.5, K±{t) is a block lower triangular matrix with identity diagonal blocks, with entries K^tp{t) in Q(i). We put K%p{t) = t^K^pir1) and let K±(t) = (K^(t)). Then the matrix K±(t) satisfies a similar condition as P given in (2.4.2). Next we consider the matrix D(t) = (&A,yi'(*)), indexed by symbols in Z%s, where 6yi,/i'(*) is as in Corollary 3.7. We write Dit)'1 = (b'a/3(t)), and define a matrix A(t) by A(t) = ( W ^ - ^ ^ C r 1 ) ) , where d(t) = tN'~n(t - l)nPw{t). Clearly, A(t) is a block diagonal matrix. Under this setting, we have the following theorem. T h e o r e m 3.11 We have

k-(t)A(t)tK+(t) = n'. In particular P' = K-(t),P" equation (2.5.2).

= K+(t)

and A' = A(t) gives a solution for the

Theorem 3.11 shows that the solution of (2.5.2), hence that of (2.4.2) can be described in terms of Kostka functions, just as in the case of Green polynomials of GLn(Fq). In particular, the matrix P coincides with K-{t). R e m a r k s 3.12 (i) As some examples show one can conjecture the following. K^ At) is a polynomial in Z[i] with positive coefficients. Moreover, Ae

&KaA^ < a(#) - a(a). This implies that Ka^(t) is also in Z[i\. The conjecture also implies that P^(x;t) is a polynomial in Z[x;i\. (ii) Prom the discussion in 3.8, we see that {qa,±(x; t)} and {ma(x)}, and {QJi(x; t)} and {P^(x; t)} are dual bases of each other. Hence we have M(Q*,q^) = M(P±,m)* = (K^ty'R)*

= 'tf±(i)ir,

(3.12.1)

where K = M(s,m) is the Kostka matrix, and X* denotes the transposed inverse of the matrix X. Since K* is a matrix with entries in Z, we see that K±(t) is a matrix with entries in Z[t] if and only if M(QT,qT) is a matrix with entries in Z[£]. It follows that P^(x;t) lie in Z[x;t] if an only if the coefficients da^{t) in Theorem 3.5, (ii), (a) lie in Z[£] for any a, ft 6 Z°'°. 3.13. Concerning the above conjecture, we have some partial result in the case where e = 2. Now assume that e = 2,r > 1, and that mo = m + l,rai — m for some m > 0. Let A = (A0,Ai) € Z%s be a symbol. We write A0 :vx> vs> ••• > v2m+i and Ai : i/2 > v*> ••• > i/ 2m . We say that A is very special if Vi > vi+i + r for i = 1,2, Then we have the following.

294

P r o p o s i t i o n 3.14 Assume that e = 2 and that A = A(a) £ Z^s is very special. Then the coefficient dQ,p(t) of qp,± in Q^ is in Z[i\. In particular, Z KfaWj M f°r any P e Zns- Moreover, we have degK±a < a{a) - a(fl). Hence K^a(t) <E Z[i]. 3.15 In the remainder of this section, we shall give explicit examples of Green functions associated to various W = G(e, l,n). In each case, Z%° are written as a\,... , a h , and the corresponding symbols in Z%s are written as Ai = A(ai). In the matrix P of Green functions, the rows and columns are arranged along the order a x , a 2 , . . . , and the entries in the first column denote a« € Z°'°. Throughout the examples, we assume that r = 2 and s = 1. (A) W = G(2,1,2). In this case W is the Weyl group of type B2, and we give the table just for the sake of reference. Z°-° is given as a i = ( - ; l 2 ) , a 2 = ( l 2 ; - ) , a 3 = (l;l),a4 = (-;2),a5 = (2;-). The corresponding symbols and similarity classes are given as { ^ = (420; 42)},

{A2 = (31;l)},

{A3 - (30; 2), A4 = (20; 3)},

{A5 = (2; - ) } .

The matrix P of Green functions is given in Table 1. Table 1. G(2,1,2)

(-;i2) (i2;-) (i;i) (-;2) (2;-)

q4 q1 6 q +q q2 1

q* q

q

l

l

q l

(B) W = G(3,1,2). Then Z°>° is given as « i = ( - ; l 2 ; - ) , a 2 = ( - ; - ; l 2 ) , a 3 = (l 2 ; - ; - ) , a 4 = ( - ; 1; 1), a5 = (l;l;-),a6 = (l;-;l),a7 = (-;2;-),a8 = (-;-;2),a9 = (2;-;-). The corresponding symbols and similarity classes are given as {ill = (420; 42; 31), A2 = (420; 31; 42)}, {A 3 = ( 3 1 ; l ; l ) } , {A 4 = (20;2;2)} {A5 = (30; 2; 1), A 6 = (30; 1; 2), A7 = (20; 3; 1)A8 = (20; 1; 3)}, {il» = ( 2 ; - ; - ) } .

295

The matrix P of Green functions is given in Table 2. Table 2. G(3,l,2) qb (-;i2;-) 2 (-;-;i ) q6 (i2;-;-) q* (-;i;i) q4 + q (i;i;-) (i;-;i) q2 (-;2;-) q2 (-;-;2) 1 (2;-;-)

q5 q4

q* q*

q3 q2

q q2 q2 q

q q2

l

1

q q q q

l

l

(C) W = G(4,1,2). Then Z°-° is given as " i = ( - ; l 2 ; - ; -),a2 = ( - ; - ; l 2 ; - ; - ) , a 3 = ( - ; - ; - ; l 2 ), "4 = ( l 2 ; - ; - ; - ) , a 5 = ( - ; l ; l ; - ) , a 6 = (-; 1 ; - ; l ) , a 7 = ( - ; - ; l ; l ) , «8 = ( l ; l ; - ; - ) , a 9 = (1; - ; 1; -),a1Q

= (-;2;-;-),an = (l;-;-;l),

"12 = ( - ; - ; 2 ; - ) , a i 3 = ( - ; - ; - ; 2 ) , a i 4 = ( 2 ; - ; - ; - ) . The corresponding symbols and similarity classes are given as {Ax = (420; 42; 31; 31), A2 = (420; 31; 42; 31), A3 = (420; 31; 31; 42)}, {A4 = (31; 1; 1; 1)},

{A5 = (20; 2; 2; 1), A6 = (20; 2; 1; 2), A7 = (20; 1; 2; 2)},

{^ 8 = (30; 2; 1; 1), A9 = (30; 1; 2; 1), A10 = (20; 3; 1; 1), A n = (30; 1; 1; 2), A12 = (20; 1; 3; 1), A13 = (20; 1; 1; 3)} {Ali

=

(2; - ; - ; - ) } .

The matrix P of Green functions is given in Table 3. (D) W = G(3,1,3). Then Z°-° is given as « ! = ( - ld;-),a2 = (-;-;l^),a3 = (l3;-;-),a4 = (-;l2;l), C*5 = ( ~

l;l2),a6 = ( l ; l 2 ; - ) , a 7 = ( - ; 2 1 ; - ) , a 8 = (1;-;12),

"9 = (~

-;21),ai0 = ( l 2 ; l ; - ) , a u = (l2;-;l),a12 = (21;-;-),

"13 = (1

l;l),ai4 = (-;2;l),ai5 = (-;l;2),a16 = (1;2;-),

a17

= (1 - ; 2 ) , a i 8 = ( 2 ; l ; - ) , a i 9 = ( 2 ; - ; l ) , a 2 0 = ( - ; 3 ; - ) ,

<*2i = ( - ; - ; 3 ) , a 2 2 = ( 3 ; - ; - ) .

296 Table 3. G(4,1,2)

(-;la;-;-) 2

q«

H-;i ;-)

q6 q6

2

(-;-;-; i ) (i2;-;-;-) (-;l;l;-) (-;i;-;i) (-;-;l;l) (l;i;-;-) (1;-;1;-) (-;2;-;-) (l;-;-;i)

qb

4 3

q

b

q + q q2 q2 q3

9s

4

q

H-;2;-)

( - ; - ; - ; 2) (2;-;-;-)

4

q3

q q5

q3 q

ql

q2 q3

q q q2

q2 1

q q2

q3

Q2

q

q q2 q2

q q

q 1

l

l

l

The corresponding symbols and families are given as {Ai = (6420; 642; 531), A2 = (6420; 531; 642)}, {A3 = (531; 31; 31)}, {AA - (420; 42; 41), A5 = (420; 41;42)}, {A6 = (520; 42; 31), A7 = (420; 52; 3 1 ) , ^ = (520; 31; 42), 4 , = (420; 31; 52)}, {A10 = (31; 2; 1), Au = (31; 1; 2)}, {Al2 = (41; 1; 1)}, {A13 {ylj 8 {A1S {A22

= = = =

(30;2;2),ili4 = (20;3;2),A 15 = (20; 2; 3)}, (30;3;l),yli7 = (30;l;3)}, 40; 2; 1), A19 = (40; 1;2), A20 = (20;4; 1), A21 = (20; 1;4)}, (3; - ; - ) } .

The matrix P of Green functions is given in Table 4, and 5. The columns in Table 4 correspond to ai,...,ocg, and those in Table 5 correspond to Q!l0,---,a22-

297 Table 4. G ( 3 , 1 , 3 ) , the former part ( a i ~ otg)

(-;i3;-) (-;-;i3) (i 3 ;-;-) (-;i2;i) (-;i;i2) (i;!*;-) (-;2l;-) (li-H2) (-;-;2i) (la;l;-) (i2;-;i) (21;-;-) (i;i;i) (-;2;1) (-;i;2) (l;2;-) (i;-;2) (2;i;-) (2;-;i) (-;3;-) (-;-;3)

912 9a qLU + q< g8 gxl + g8 + g5 g9 + g6 q7 g ^ + g'+g 4 g8 + g5 gb + g3 gy + 2gb + g3 g7 + g4 g6 g8 + g5 + g2 94 q< + g4 + g g5 + g2 q3

(3;-;-)

99 91U gn+g8

q' q7 qb

q10+q7 g9 + g6

9° ?6

9° q5 g6

q7

q5 q5

95 q< + g4 g 8 + g5 qb + q6 qb + q6

g8 gH + gb g7 g8 + g5

94 ?5 93 gb + g 3 g4 95 qb + q'2 g4

9* 1A q* + q g5+g2

?7 + 94 g5

94 g3 93 q3

g5 g« 94

94 q3 94

94

? g3

94 + 9 ?2 q3

g2 94 + 9 q2

qA

q3 q2

g3

q2 1

1

q2

1

1

Table 5. G ( 3 , 1 , 3 ) , the latter part (otio ~ 0:22) (-;i3;-)

(-;-H3) (1 3 ;-;-) (-iiaii) (-.i;i2) (i;i*;-) (-;21;-) (i;-;i 2 ) (-;-;2i)

UV;-) (i 2 ;-;i) (21;-;-) (i;i;i) (-;2;l) (-;i;2) (1;2;-) (i;-;2) (2;i;-) (2;-;i) (-;3;-) (-;-;3) (3;-;-)

94 94 93 93

93 93 93 93

9*

9' 93

9 92

92 9 92

9 92

92 9

9 9

92

9 9

1

1

1

9 1

g3

9 1

1

298

References 1. P. Deligne and G. Lusztig; Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), 103-161. 2. J.A. Green; The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402-447. 3. M. Geek and G. Malle; On special pieces in the unipotent variety, Experimental Math. 8 (1999). 4. G. Lusztig; Irreducible representations of finite classical groups, Invent. Math. 43 (1977), 125-175 5. G. Lusztig; Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), 205-272. 6. G. Lusztig; Character sheaves V, Advances in Math. 61 (1986), 103-155. 7. G. Lusztig; On the character values of finite Chevalley groups at unipotent elements, J. Algebra, 104 (1986), 146-194. 8. G. Lusztig; Green functions and character sheaves, Annals of Mathematics, 131 (1990), 355-408. 9. G. Lusztig; Notes on unipotent classes, Asian J. Math. 1 (1997), 194-207. 10. G. Lusztig and N. Spaltenstein; On the generalized Springer correspondence for classical groups, in "Advanced Studies in Pure Math." - Algebraic groups and related topics -, 6, 1985, pp. 289-316. 11. I.G. Macdonald; Symmetric functions and Hall Polynomials, second edition. Clarendon Press. Oxford 1995. 12. G. Malle; Unipotent Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra, 177 (1995), 768-826. 13. T. Shoji; On the Green polynomials of classical groups. Inventiones Math., 74 (1983), 239 - 267. 14. T. Shoji; Character sheaves and almost characters of reductive groups, II, Adv. in Math. I l l (1995), 314 - 354. 15. T. Shoji; Green functions associated to complex reflection groups. Preprint.

SIMPLIFICATION OF T H E R M O D Y N A M I C BETHE-ANSATZ EQUATIONS MINORU TAKAHASHI Institute for Solid State Physics, University of Tokyo Kashiwanoha 5-1-5, Kashiwa, Chiba, 277-8581 Japan E-mail: [email protected] Thermodynamic Bethe ansatz equations for XXZ model at |A| > 1 is simplified to an integral equation which has one unknown function. This equation is analytically continued to |A| < 1.

1

Introduction

Thermodynamic Bethe ansatz equations for exactly solvable one-dimensional systems have many unknown functions.1 About the XXZ model at |A| > 1, Gaudin-Takahashi equation contains infinite unknown functions. 2 ' 3 Here I can simplify this set of equations to an integral equation which contains only one unknown function. For Hamiltonian N

n(J,A,h)

1 +SfSf+1 + A(SfSf+1 --)-

= -J^SfSfa 1=1

2/i£sf, 1=1

h>0,

(1)

thermodynamic Bethe ansatz equation at temperature T is . . 27rJsinh . . , ,., ... In 771(0:) = — s(x) + s * ln(l + 7ft (a)), Tip lnr}j(x) =s*ln(l+7j j _ 1 (a;))(l+77 j + i(a;)), j = 2,3,...,

lim ^

= ™

(2)

Here we put A = cosh, Q =

TT/
= -

Y^ sechf

-J,

T l = — OO

rQ

s * f(x) =

s(xJ-Q The free energy per site is

y)f(y)dy.

Q , 27rJsinh<6 fQ , , , ,, , N, „ , „, m f f = / a! (x)s(x)dx -T s(x) ln(l + 771 {x))dx,

299

(3)

300

On the contrary new equation is / \ r. i ,h. / 4> ( u(a;) = 2 c o s h ( - ) +
r „•! r 27rJsinh(i -[a: - y - 2t] exp[ — ai{y

«-n 27rJsinh0 , ..,\ 1 d« + cot - [r , - , + 2,] exp[r ^ . ^ - O ] ) — ^ ,

+t)] (5)

and free energy is given by / = -Tlnu(0).

(6)

Contour C is an arbitrary closed loop counterclockwize around 0. 2nQ, n / 0 and ±2i + 2nQ should be outside of this loop. This loop should not contain zeros of u(y). It is expected that u(y) has not zero in region |9ty| < 1. This equation can be calculated even at imaginary , namely, |A| < 1 case. This equation converges numerically at least T/J > 0.07. The results coincide with those of older equations. Then this equation unifies Gaudin-Takahashi equation for A > 1 and Takahashi-Suzuki equation for |A| < l. 4 2

Derivation

We should note that if g(x) = a*h(x),

(7)

it stands g(x + i)+g(x-i)

= h(x).

(8)

The Fourier transform of (7) is q(u>) =

h(w),

u) = — n.

Then we have ( e w + e - w ) j ( « ) = A(w)l and obtain (8). Then set of equations (2) is rewritten as Vi(x + i)m(x - i) - exp[

n

^—

J2^(x

- 2nQ)] (1 + 772(2:)),

n

r)j(x + i)f]j(x - i) = (I + T]j-i(x))(l + t]j+i(x)), lim ^

= ^.

j = 2,3,..., (9)

301

At J = 0 this set of equation becomes a difference equation and we have analytical solution, V i

_ /8inh(j + l)ft/!ZY ~\ sinhh/T )

1

We can expand perturbatinally as power series of J/T. Very surprisingly rjj (x) has singularity only at x = ±ji, ±(j + 2)i + 2nQ. So we assume that r)j(x) is univalent on the complex plane of x and 1 + rjj(x) is factorized as follows: 1 + ^(a:) = Aj(x - ji)A~(x + ji)Bj(x

- (j + 2)i)B~(x + (j + 2)i).

(10)

Here functions Aj(x), Bj(x) are periodic with periodicity 2Q and have singularity only at x = 2nQ and A~{x) = Aj(x),

B~{x) = Bj(x).

One should note that Aj(x) is not an analytic function of x but Aj(x) is analytic. Then second equation of (9) becomes Vi (x + i)Vj (x-i) = Aj-!(x - (j - l)t)^--i(a: + (j - l)t)£;-i(:c - (j + l)t)Bj-i(a; + (j + 1)*) xAj+1(x - (j + l)i)Aj+1(x + (j + l)i)Bj+1(x - (j + 3)i)Bj+1(x + {j + 3)t). (11) If we assume that rjj (x) is factorized as r)j(x) = X{x - ji)Y(x

+ ji)Z(x

- (j + 2)i)W(x + {j + 2)t),

rjj(x + i)r]j(x — i) is X(x - (j - l)i)Y(x + (j - l)i)X(x Z{x - (j + l)i)W(x + (j + l)i)Z(x

- (j + l)i)Y{x + (j + l)i) - (j + 3)i)W(x + {j + 3)t).

Comparing the singularity at x = ±(j — \)i and x = ±(j + 3)i we have X ~ Aj-i, Y ~ Aj-i, Z ~ Bj+1 and W ~ Bj+i. Then rjj(x) should be factorized as: r)j(x) = i4j_!(a: - ji)A~~[(x + ji)Bj+1

(x - (j + 2)i)Bj+1(x

+ (j + 2)i). (12)

Considering the singularity at x = ±(j + l)i we have A,_i(aQ ^ Aj+1(x) Bj-tix) Bj+1(xY Using the first equation of (9) and noting that 27r l x - it

x + iei

{L6)

302

we have ...

.Jsinh -s-^

M*) = eM-^

—i

£ x-2nQ-ie)

/ J sinh d> COt {x -it)), 2Ti 2 -

= GXP(

n

(14)

B0(x) = 1. Prom (13) we have Mx)

~A'(x) (15) A-jjjx) A0(x) A2j+1(x) B2j(x) B0(x) °W' B2j+1(x) One can show that AQ{X) — A'Q(X). Consider T]2^_^(x ~^~ $J ~^~ 1)0- This quantity approaches 0 in the limit of infinite j . Of course near the singularity at 2i, 0, — (Aj + 2)i and —(4j + 4)i the deviation from 0 becomes big. Nevertheless the region on the complex plane where \T)2J+I(X + (2j + 1)*)| > e is expected to be narrower as j goes to infinity. So we have l =

+ (4j + 2)i) U m l + tfe,-+i(a! + (2j + l)t) = U m A'0(x)A^(x j->oo rj2j+i(x + (2j + l)i) i->oo A0(x) A0{x + (4j + 2)i) B2j+i {x)B2j+i (x + (4j + 2)i) B2j+1 (x - 2i)B~2~^(x + (4j + 4)i) i B2j (x)So~(x + (4j + 2)i) ' B 2 i + 2 (x - 2i)So~^(x + (4j + 4)») -I " (16)

The first fraction in the big bracket should go to ehlT and the second should go to e~hlT. The bracket in (16) becomes 1 in the limit of infinite j . Then we have A'0(x) _ A^jioo) _ A0(x) ~ I^(ioo) As A0(ioo)/A'0(ioo) = A0(-ioo)/A'Q(—ioo) — 1/57, we have \a\ = 1. If we choose the phase factor a is 1, the ratio of Aj(x) and Bj(x) is always A$(x). Prom (10) and (12) we have Aj-! (x - j t ) A , - i (* + ji)Bj+i {x - (j + 2)i)Bj+i (x + (j + 2)t) + 1 = Aj(x - ji)~A~(x + ji)Bj(x - (j + 2)i)B~{x + (j + 2)i). (17) At j = 1 we have Bi{x - i)Bi{x + i)B1(x - 3i)Bi(x + 3i) = — 1_ + B 2 ( a ; - 3 t ) ^ ( a ; + 3t). ylo(z - «)>40(a; + i)

(18)

303 JE?I

, .02 are unknown functions. But this equation and condition lim B JDUX" — l)&l\X 1(x-i)!h(x

i)B1(x-3i)lh(x f+ %)£}\\X — Ot)I3l\X

2 + T Zi) Ol) = = (2coshh/T) (£ CUSI1/1/-1 ,

(19)

X—MOO

are sufficient to determine these functions. Put u{x) =B1(x-

2i)Bi {x + 2i).

(20)

Equation (18) is written as , ... 1 , B2{x-3i)B~2{x + 2,i) u{x + i) = -— -==— - + -, ^— • A0(x - i)A0(x + i)u(x - i) u(x-i)

(21)

The l.h.s. has singularity at i,—3i. The first term of r.h.s. has at i,—i,3i. The second term of r.h.s has at 3i, — 3i, —i. Assume that u(x) is expanded as follows ,

oo

oo

u(x) = 2 c o s h ( - ) + E E

(x

_

2

^ _

2i)j

+E

E

{x

_ 2^Q + 2iy • <W>

Consider the contour integral around x = i. Coefficients c, is determined by Cj=I

(^Stl *L=f ^ / ,4o(a;-*)^o(a; + «)w(a;-*)27ri / A0{y)A0(y

*L. + 2i)u(y) 2ni

(23)

The first sum of r.h.s. of (22) is "

r

exp[-2-^^a1(y

^i J ^

(x-

+ i)]yJ-i

2nQ - 2i)J

dy

u{y) 2m

27rJsinh

= / v ^ e xe Pxl p [ - ^Tj^ a i ( y + t)] 1 dy u(y) 2i\i J „ x - y - 2nQ — 2i 6 6, . . 27rJsinh
, y

... 1 dw — 1JL.

+ t)]

. . (24)

The second sum is calculated in similar way. Thus we get (5). From equation (10) we have (/ 27r 2nJJ sinh -> i \ <> / , .\ 1 + ?7i(a;) = expl — ai(a;)lu(a;+ z)u(a; - « ) .

~w

(25)

Substituting this into eq.(4) we get eq.(6) rQ

f = -T

s(x)[\n u(x + i)+In u(x-i)]dx J-Q

= -Tlnu(0).

(26)

304

3

Analytical solutions

Ising limit In this limit we put A -> oo, J = Jz/A. As <j> goes to oo, 2Q becomes 0. Then function 27rJ j' 0 nhl/ 'aifa) is 0 at | % | > 1 and Jz/T at | % | < 1. Function u(y) is 2 cosh h/T at |9y| > 2 and also a constant u(0) at |9y| < 2. From eq.(5) we have u(0) = 2cosh(fc/T) + l-***>(-J*IT\ So we have u(0) = cosh(h/T) + Jsmh2(h/T) energy of the Ising model.

(27)

+ e x p ( - J 2 / T ) and known free

Kuniba, Sakai and Suzuki 5 showed that Takahashi-Suzuki equation for | A | < 1, h = 0 can be derived from the quantum transfer matrix and its fusion hierarchy matrices. We are confident to derive eqs.(5, 6) from the quantum transfer matrix and its fusion hierarchies. Details will be published elsewhere.

Acknowledgments This research was supported in part by Grants-in-Aid for the Scientific Research (B) No. 11440103 from the Ministry of Education, Science and Culture, Japan. References 1. Takahashi, M. (1999) Thermodynamics of One-Dimensional Solvable Models, Cambridge University Press. 2. Takahashi, M. (1971) Prog. Theor. Phys. 46, 401. 3. Gaudin, M. (1971) Phys. Rev. Lett. 26, 1301. 4. Takahashi, M. and Suzuki, M. (1972) Prog. Theor. Phys. 46, 2187. 5. Kuniba, A. , Sakai, K. and Suzuki, J. (1998) Nucl. Phys. B 525, 597.

A BIRATIONAL R E P R E S E N T A T I O N OF W E Y L GROUP, COMBINATORIAL .R-MATRIX A N D D I S C R E T E T O D A EQUATION YASUHIKO YAMADA Department of Mathematics, Kobe University Rokko, Kobe 657-8501, Japan E-mail: [email protected] We study certain birational representation of symmetric group arising from inverse ultra discretization of the combinatorial R-matrix. This representation is related with the Weyl group action on the crystal bases and Lusztig's bijection on the canonical bases. We apply the result to the discrete Toda equation.

1

Introduction

Recently, a relation between the box and ball systems 27'28>30 and the theory of crystal bases 14 ' 16 was discovered. 11-9-4 This remarkable relation has been further studied and extended in the papers. 10>7'6'8 In these studies, the combinatorial .R-matrix of the quantum algebra 16 ' 14 plays the fundamental role. An explicit piecewise linear formula of the combinatorial .R-matrix was obtained for the case of symmetric tensors of Uq(sln) by Hatayama et al. 6 In this paper, we study the inverse ultra discretization 29 of the formula 6 and give a direct proof of their braid relations. As a result, we obtain a birational representation of the symmetric group. We construct conserved quantities (invariants) of the birational mappings which play essential role in our proof. In Sec. 2, we present the main results and their proofs. Then we discuss the relation of our representation with • the combinatorial .R-matrix, 2 4 , 6 and Weyl group action,

17

• Lusztig's bijection on parametrizations of the canonical bases, • (ultra) discrete Toda equations,

20 21 22

' .

12 13

'

in Sec. 3, 4 and 5 respectively. In the Appendix we give some determinant formulas used in the main text. 305

306

2

A birational map R

Let N, L be positive integers and K = C(x) be the field of rational functions on NL variables xf (i = 1, • • •, N, a = 1, • • •, L). We extend the index i of xf for i 6 Z by the condition xf+N = xf and define a cyclic shift operator ir : K ^ K by v(xt) = x?+1. Definition 2.1 Define algebra automorphisms Ra : K —> K (a = 1,---,L — 1) as follows: D ( a\ _

a+lQi-l(xa>xa

)

Qi-1(xa,xa+iy (b?a,a + l),

Ra{x\)=x\,

(1)

where Qj (x,y) = Qj (xi, • • •, xN, yx, • • •, yN) is given by N

/k-l

Qj(x,y)=j2 ( n

N

Xi

+j n

\

yi

+j) •

(2)

PPe afeo define operators Li (i 6 Zj by Li = (Tr + x1i)(7T + x2i)---(7r + xf),

(3)

where xf are interpreted as multiplication operators. The following theorem is our main result of this paper. Theorem 2.2 We have the following relations: RaTr = TrRa, Ra(Li)=Li,

(a = l,---,L-l,i

R2a = l, RaRa+lRa

(a = 1, • • • ,L - 1), = l,---,N),

(o=l,"-,L-l),

= -Ro+l-Ro^a+1,

(o = 1, • • • , L — 2 ) .

(4) (5) (6) (7)

In words, the automorphisms Ra (a = 1, • • •, L — 1) generate a birational representation of the symmetric group SL on AT. The operator Li, or its coefficients Ik,i, defined by h

Li = YJh,i*L-\ fc=0

(8)

307

are invariants of the SL action. The relation between Ra and Li is an analog of that between screening operators and W-currents. 5 Explicit form of Ik,i is given by J

M -

X

2^

i+(a-l)>

l
2,i =

\Xi+(a-l)Xi+(a-l)+(l>-2))

22

>

l
3,i =

(^a;"+(a_i)a;i+(a_i)+(6_2)a;i+(a-i)+(i--2)+(c-3)J >

2_/ l
iL,i=n

(9)

=$>

a=l

[Proof of the Theorem 2.2}. The relation (4) is obvious from the explicit formula. Proof of Eq. (5). Since Ra acts non-trivially only on the two factors (ir + x i)('K + xi+1) m -^»> ^ 1S enough to show in case of L = 2. So we prove that a^y* = xiVi,

x'i + y'i+i =xi + Vi+i,

(10)

where Q^fog) ' " *

ft(*,y)

, '

Qj(a;,y)

W<_X,

U

Qi-i(a:,l/)-

j

The first equation is trivial. The second one is due to the equation x =Xi

'i

~7Tf

T'

y'i=yi+n

Qi{x,y)

(l2)

7 7> Qi-i(x,y)

which follows from N

XiQi(x,y) -yiQi-i(x,y)

N

= J J x t - J | y * = 8.

(13)

Proo/ o/ #17. (6). It is also enough to show RR = 1 in case of L = 2 as above. Using the Lemma 5.3 in Appendix, we see that the polynomial Qj(x,y) can be expressed as a determinant of the following (N - 1) x (N — 1) matrix,

fxj+1+yj+2 x

i+2Vj+2

1 xj+2

+ yj+3

\ 1

(14)

1 \

Xj+N-iyj+N-1

Xj+N-1 +Vj+N

)

308 Then we have Qj(x',y')

= Qj(x,y)

and hence

RR(xi)=R(x'i)=R(mQi-lix'y) Qi(x,y) ,Qi_i(a;,,y/) Qi(x,y) Qi-i(x',y') _ Vi Xi Qi(x',y') Qi-i(x,y) Qi(x>,y>) ~ Xi>

[

^>

RR(yi) = yt is similar. Proof of Eq. (7). It is enough t o show R1R2R1 on 3N variables (xi,yi,Zi).

= R2R1R2

(16)

Consider

p p / x _ p (, Qj-i(x>y)\ _.,Qj-i(x,y') R2Ki{xJ) - Ha \Vi Q.{xy) J-Vi Q.{^yl)

_ Qj-i(y,z)Qj-i(x,y') - *i Q ^ z ) , Q ^ y l ) (17)

where (18) We have

Qi,j(x,y,z) = Qj(y,z)Qj(x,y')

xi

N

/k-1

N

\

i=k+l

J

=Qj(v,z)E n +j n y'i+j k=l \ i = l

N =

E

/k-1

(n fc=l \ i = l

N (k-\ =

=

N

Xi

+j n

N

+j) Qk+j(y,z)

i=fc+l

/

N

\

E(n*<+* n k=l \i=l

\ 2i

N /l-l

N

\

Zi

i=k+l

/k-1

k+l-1

fc,(=l \ i = l AT //fc-1

i=fc+l k+l-1

+i J2[Yiyi+j+k n *<+*+*) J 1=1 \ i = l

i=l+l

2N

J

k+N

\

»=fc+H-l

/

5Z ( n **+•» n »<+* n **+* n **+>) i=fc+iV+l \ 2JV

= E (lI X i +J I I ^ ) I I Zi+Jk,l=l \i=l

i=k+l

/ i=k+l+l

(19)

309 Again, by L e m m a 5.3 in Appendix, the quantity /fc-l

m-l

(2°)

\

n *»i n vw >

E

i
\i=l

i=k+l

can be expressed as an (N — 1) x (AT — 1) minor determinant of the following (N -l)x(N + 1) matrix: x

(xj+iVj+i

i+i +Vj+2 1 Xj+lVj+1 Xj+2 + Vj+3

\

\ 1 (21)

Xj+N-lVj+N-1

Xj+N-1 +Uj+N 1 /

where two columns, 1-st and m - t h (for 2<mv'*), Qitj(x,y,z)

RaR1{xi)

(22)

is invariant under the action of R±. From this and obvious fact R2 {XJ ) = Xj we have R2RIR2(XJ)

= RzRiixj)

= RIR2RI(XJ).

(23)

In a similar way, we have Q2,j(x,y,z) ' , 1 ', 1, = R2RIR2(ZJ), Q2,j-i(x,y,z)

R1R2R1{zj)

= a:,-^ J

(24)

where N k+l-2

Q2j(x,y,z)= Y, k,l=l Finally, RiR2Ri(yj)

2N

+i\ I I V*+i I I

i=l

V i=k+l

= R2R\R2(yj)

RiR2Ri(xjyjZj)

/k+N-1 Xi

II

i=k+N+l

\ z

i+j) •

(25)

/

follows from - XjyjZj

= R2RxR2{xjyjZj).

T h e proof of the Theorem 2.2 is completed. I R e m a r k 2 . 3 We have 'l~ , Vl,i

R1R2R1(xi)

= Zi

RiR2Ri(yi)

= yi7z jr—, Wl, i-lW2,i

(26)

310

RiR2Ri(zi)

= Xi

Q2,i

(27)

It is interesting to note the similarity with the dressed vacuum form of analytic Bethe ansatz. 5 The relation (10) can be written in matrix form as X(x)X(y)=X(x')X(y'),

(28)

where (xx

1

\

X2

1

X(x) =

(29) XN-I

V*

1 XN J

By similar computation as in the Proof of Eq. (7), we have the following formula RpRp-i

•••R1{xj)-

(30)

p(a.i>...>a.H-i) '

Xj

where pN -*!j\

x

1' ' ' )

x

)

=

pN c

x

/ , j,m\ m=p

>" ' ' j

x

)

J__|_ Xi+j i=m+l

'

(31)

The coefficient Cj im (x 1 , • • •, xp) is given explicitly as

E

n

I<*I,---,*P<JV

a

= l \ia=k1

fci+"+fc«-i

n <

+—+ka-i

+3

(32)

+l

The coefficient Cjim(a;1, • • • ,x p ) has a simple expression in terms of minor determinants of the matrix ^(a; 1 ) • • • X(xp). For N x N matrix X, let C£(X) (1 < a, b < N) be the following minor determinant: Cu(X) = det (Xi j) i
(33)

l
Note that b+N-l

c?(x(x))= n **> k=a+l

(34)

311

where Xk+N = %k, and a factor [x\ • • -XN) should be replaced with (—l)Nz when b > a. Then we have CS{X(x1)---X(x"))=Y/(-l)Nkzkca^k)N+b_a(x1,---,xn.

(35)

k=0

Remark 2.4 The representation of symmetric group SL can be extended to the (extended) affine Weyl group of type AL_^ by including the automorphism p-.K-^K such that pW)=xt+1,(a=l,---,L-l),

P(xf)=x}.

(36)

Then the translations such as I \ = PRL-I • •• R\ etc. define a system of commuting rational maps that can be viewed as a version of generalized discrete Painleve equation. 25 The conserved quantities of this system are generated by the characteristic polynomials of the matrix X(xl)X{x2)---X(xL).

(37)

It is remarkable that this system can be ultra discretized. 3

Combinatorial .R-matrix

We have obtained a representation of symmetric group SL as birational mapping Ra on LN variables x\ (a = 1, • • •, L, i = 1, • • •, N). Note that all the coefficients of the rational functions Ra and invariants Ik,i are nonnegative integers. This property is natural since this representation is expected to be an inverse ultra discretization of its combinatorial counterpart. 6 Here, we consider the ultra discrete analog of our representation obtained by the standard procedure, 29 x + y ->• mm(x,y),

xy-lx

+ y.

(38)

For objects in the previous section such as xf, Ra, Qi, we denote their ultra discretized analogs by the same symbols. For instance, in case of L = 2 we have R(xi)=yi

+ Qi-i-Qi,

R(vi) = xi + Qi-Qi-1,

(39)

where

$ > i + j + Y, yi+Ai=X

i=k+l

I

(4°)

312

Then, the ultra discretized maps Ra (a — 1, • • •, L — 1) define bijections on the set I = KeZ>o \a = l,---,L, i = l,---,N},

(41)

and satisfy the relation R2a = l,

(a=l,---,L)

RaRa+lRa

= Ra+lRaRa+1,

(a = 1, • • • , L — 1).

(42)

One can identify the set X with the crystal bases of Z,-fold tensor product of complete symmetric representations of Uq(sliv) B = Bh®---BiL,

(43)

where xf is interpreted as the i-th weight of a-th element ba 6 Bia. (la = x\ + • • • + x%). On the bases B, we have a bijection Ra that acts on the Ba Ba+i as the isomorphism of crystals. 24 Proposition 3.1 Under the identification of X and B, we have Ra = Ra,

(a = l , - - - , L - l ) .

(44)

This fact follows by direct comparison of the expression Eq. (39) for R and the expression for R in (the proof of) Proposition 4.1 of the Ref. 6 As a ultra discrete analog of Eq. (9), we obtain the following conserved quantities for the action of Ra

h

'>

=

/3,i =

1<™£
1

)

+ X

i+(a-l)+(b-2)

+ < + ( a - l ) + (6-2)+(c-3)) .

L

The last one is nothing but the total weight. There is a remarkable relation between L and N. By replacing the role of the indices i and a in the formula for Ra in Eq. (39), we define bijections = l,---,N-1). Si(i Proposition 3.2 The bijection Sj is nothing but the Weyl group action on the crystal bases B = Bh<S---B,L.

(46)

313

This fact" follows from the coincidence of the signature rule for the Weyl group action 17 and the rule for the combinatorial R. 24 As a consequence of this Proposition, we see that the actions Ra and s; are mutually commute. This commutativity seems also hold in the inverse ultra discretized version, namely Conjecture 3.3 Define birational automorphism Sj by the same formula for Ra in Eq. (1) replacing the indices a's and i 's. Then the birational actions Ra and Si mutually commute. Remark 3.4 A similar combinatorial representation of symmetric group has been also obtained by A. N. Kirillov. 18 His construction is based on BenderKnuth transformation and Schutzenberger involution. 2'3>19 4

Lusztig's bijection R\

Let X be the set of all sequences i = (ii, • • •, i„) such that s^s^ • • • s^ is a reduced element6 in a symmetric group Sn. For i, i' € X, Lusztig defined a bijection i?|' : N " -> N " as follows, 20>21 (a) If i = (••• ,i,j,i,---) and i' = (• • • ,j,i, j , • • •) with \i - j \ = 1, then R\ (• • • ,a, b, c, •••) = (••• ,b + c — min(a, c),min(a, c),a + b — min(a,c), • • •). (47) (b) If i = (• • • ,i, j , • • •) and i' = (• • • ,j,i,- • •) with \i — j \ > 1, then

RU---,a,b,---) = (---,b,a,---).

(48)

For any two reduced decompositions i' and i, there is a sequence ii, i2, • • •, i p such that ii = i, ip = i' and any consecutive pair satisfy the conditions (a) or (b), then R?=R\P

•••Rl'Rl2.

(49)

Note that R\ is independent of the sequence i i , i 2 , • • • , i p . This fact plays essential role in our construction. The action R\ is a ultra discretization of the following R\ : (a) If i = (••• ,i,j,i,---) and i' = (• • •, j,i, j , • • •) with \i - j \ = 1, then R\ (•••,a,b,c, •••) = (•••, — — , a + c,——,-••)• a+c a+c a

l would like to thank M. Okado for pointing this fact to me. ''Lusztig considered the case of longest element.

( 50 )

314

(b) If i = (• • •, i, j , • • •) and i' = (• • •, j , i, • • •) with \i - j \ > 1, then # ! ' ( • • • , a , &,-••) = ( • • • , 6 , a , - " ) .

One can interpret this in matrix form as follows. be a matrix Ei(x) - 1 + xEiti+1

22

(51)

Let Ei(x) (i — 1, • • •, n — 1)

= exp{xEi)i+i),

(52)

ai where i5y is the matrix unit. Then we have (a) for \i - j \ = 1,

Eii^Eji^Eiic)

= Ej{—^r)Ei{a

+ c)£,(^—),

(53)

(b) for |i - j \ > 1, E^Ejib)

= Ej(b)Ei(a).

(54)

That is, for two sequences i and i', we have EhOn) •••Eih{xv)

= £*- (xi) • • • EVk(x'v),

(55)

if the parameters are related as R?(xi,---,xv)

= {x'ir--,x'v).

(56)

Explicit formulas for R\ , R\ and their application for the totally positive matrices were obtained. 1 For a < b and x = (xa, •••, xb), define b

Za,b(x) = Eb(xb) • • • Ea+1(xa+1)Ea(xa)

= l + Y^XiEij+i-

(57)

i=a

Proposition 4.1 We have an exchange algebra Za,b-i(y)Za,b(x)

= Za,b(x')Za+ltb{y'),

(58)

where c

R J

,

x

R.

p =

i = p ~> y'i = iyi-ijt^~>

These parameters Xi,yi,x\, x

i

"

/k X

~

"

N

IZ ( n ^ I I ^ j •

(59)

and y\ satisfy the following relations:

iVi+i = xi+m,

x

'i+y'i=Xi+

Vi,

(60)

for i = a, • • •, b (y'a = yb = 0). This map (x,y) -¥ (x',y') is a finite (i.e. non periodic ) analog of our R operator in section 2.

315 5

Discrete Toda equation

A relation between the birational and/or combinatorial R and nonautonomous discrete KP equation was discovered by Hatayama et al. 6 Here, we consider the relation between the map R and the discrete Toda equation. 12 (61) Here i e Z and t £ Z are discrete coordinates of space and time and I\ and V* are the dependent variables. We will consider the case with periodic boundary conditions I\+N = I\ and Vt+N — V*. If we put

xi = l!+i,

yi = Vlt,

x[ = V?+\

y,i =

l?\

(62)

we have the relation Eq. (10). Hence, the time evolution of the periodic discrete Toda system is essentially given by the map R. Explicitly jt+1 _ Jrt i — i+1

^i+1 Qt •

1

v*+1 = vf

3

(63)

i+l

G\ N

lk-\

N-\

fc=l \i=l

j=k

(64) This system can be ultra discretized as follows:

li+1 = lt+1+G\+1-Gl

T A ^ = ^ + ^-G*+1,

G^^J^

+ Y,^ \i=l

(66)

j=k

It is known that the system Eq. (61) can be written as, Lt+iRt+i

(65)

=

RtLtj

12 13

'

(67)

where

V£\

/ I V{ 1 r

t

(I{ 1

n i

_

(68)

n~i

i./

\i

316

Proposition 5.1 The polynomials Pk = Ffc(/', V ) defined by N

Y, PkZN~k = det(i2*L* + zl),

(69)

k=o

are conserved quantities of the discrete Toda equation Eq. (61). One may choose tr (R(t)L(t)) as another complete set of conserved quantities. 12 The choice here is convenient for ultra discretization. Using the determinant formulas in Appendix, the explicit form of the conserved quantities Pk can be written as follows:

P

* = E f E Ii---lLmvj1---vi),

Prf = (l + I*---TtiN)(l

(k =

+ V?---Vfr),

o,...,N-i), (70)

1S

where the sum ^2u\ {j\ taken over all the indices 1 < i\ < • • • < ik-m < N and 1 < ji < • • • < j m < N such that ia ^ jb,jb + 1 for all a, b. The ultra discrete analog of these conserved quantities coincide with the result by Nagai. 23 Remark 5.2 A non-commutative analog of discrete Toda equation has been studied by Kashaev-Reshetkhin, 15 where the 2-diacjonal matrices like R(t),L(t) appear as the "minimal" representation o/f/?(sZjv)Acknowledgments The author would like to thank K. Hasegawa, G. Hatayama, R. M. Kashaev, K. Kajiwara, A. Kirillov, A. Kuniba, A. Lascoux, M. Noumi, M. Okado, T. Takagi and T. Tokihiro for valuable discussions and comments. Appendix We give determinant formulas used in the text. Let X = (xj)i
6(X)=det(a:j;)1<0,6<m.

(71)

#(xr) = X;&(x)tf(n

(72)

The basic formula is K

317 where X, Y are N x N matrix and K = (1 < h < • • • < km < N). Another useful formula is

tet(X + lz) = Y,*N-mZL(X)>

(73)

where L = (1
(y\ v\

\

1/2 Vl Y

(74)

= N-l

\

N-l

V% J

the formula Eq. (72) can be written as

#(XY) = £^(X)2/£ •••*,*

(75)

K

This simple formula is a key of our explicit computation. In particular, any minor determinant of a matrix which is a product of 2-diagonal matrices can be expressed as sum of monomials with nonnegative coefficients. The most fundamental object in soliton theory, the r function, is difficult to handle in ultra discrete formalism, since it is a determinant (or Pfaffian) with alternating signs and in the ultra discretization, terms with negative coefficient are problematic. As we have seen in the main text, the above treatment of the determinant is useful to overcome this difficulty. References 1. A. Berenstein, S. Fomin and A. Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122(1996) 49-149. 2. A. Berenstein and A. N. Kirillov, Domino tableaux, Schiitzenberger involution, and the symmetric group action q-alg/9709010. 3. A. Berenstein and A.N. Kirillov, Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux Algebra i Analiz, 7 (1995) 92-152; translation in St. Petersburg Math. J. 7 (1996) 77-127. 4. K. Fukuda, M. Okado and Y. Yamada, Energy functions in box ball systems, Int. Mod. Phys. A15 (2000) 1379-1392.

318

5. E. Frenkel and N. Reshetikhin, The q-characters of representations of quantum affine algebras and deformed W-algebras, Cont. Math. 248 (1999) 163-206. 6. G. Hatayama, K. Hikami, R. Inoue, A. Kuniba, T. Takagi and T. Tokihiro, The AM automata related to crystals of symmetric tensors, qalg/9912209. 7. G. Hatayama, A. Kuniba M. Okado and T. Takagi, Combinatorial R matrices for a family of crystals : Cn ' and >4n_i c a s es, math/9909068. 8. G. Hatayama, A. Kuniba M. Okado, T. Takagi and Y. Yamada, Scattering rules in soliton cellular automata associated with crystal bases, q-alg/0007175. 9. G. Hatayama, A. Kuniba and T. Takagi, Soliton cellular automata associated with crystal bases, solv-int/9907020. 10. G. Hatayama, A. Kuniba and T. Takagi, Factorization of combinatorial R matrices and associated cellular automata, math/0003161. 11. K. Hikami, R. Inoue and Y. Komori, Crystallization of the Bogoyavlensky lattice, J. Phys. Soc. Jpn. 68 (1999) 2234-2240. 12. R. Hirota and S. Tsujimoto, Conserved quantities of a class of nonlinear difference-difference equations, J. Phys. Soc. Japan 64 (1995), 31253127. 13. R. Hirota, S. Tsujimoto and T. Imai, Difference scheme of soliton equations in Future directions of nonlinear dynamics in physics and biological systems, ed. P. L. Christiansen, et al, (Plenum, New York, 1993) p.7. 14. S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Affine crystals and vertex models, Int. J. Mod. Phys. A 7 (suppl. 1A), 449-484 (1992). 15. R. M. Kashaev and N. Reshetikhin, Affine Toda field theory as a 3dimensional integrable system Commun. Math. Phys. 188 (1997) 251266. 16. M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63, 465-516 (1991). 17. M. Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73, 383-413 (1994). 18. A. N. Kirillov, Discrete Hirota equations, Schutzenberger involution and rational representations of extended affine symmetric group, Talk presented at the workshop at Okinawa (Dec. 1998) organized by M. Okado et al. 19. A. Lascoux, M.-P. Schutzenberger, Le monoide plaxique in Noncommutative structures in algebra and geometric combinatorics, (Naples, 1978), 129-156, Quad. "Ricerca Sci.", 109, CNR, Rome, 1981.

319

20. G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990) 447-498. 21. G. Lusztig, Canonical bases arising from quantized enveloping algebras II, Progr. Theor. Phys. Suppl. 102 (1990) 175-201. 22. G. Lusztig, Total positivity in reductive groups, in Lie theory and geometry: In honor of Bertram Kostant, Progress in Mathematics, Vol.123, Birkhauser, Basel, 1994. 23. A. Nagai, Talk presented at the meeting of Japan Math. Soc. (Hiroshima, 1999). 24. A. Nakayashiki and Y. Yamada, Kostka polynomials and energy functions in solvable lattice models, Selecta Mathematica, New Ser. 3, 547-599 (1997). 25. M. Noumi and Y. Yamada, Affine Weyl groups, discrete dynamical systems and Painleve equations, Comm. Math. Phys. 199 (1998) 281-295. 26. M. Noumi and Y. Yamada, Birational Weyl group actions arising from nilpotent Poisson algebras, preprint (2000). 27. D. Takahashi, On some soliton systems defined by using boxes and balls, Proceedings of the International Symposium on Nonlinear Theory and Its Applications (NOLTA '93), (1993) 555-558. 28. D. Takahashi and J. Satsuma, A soliton cellular automaton, J. Phys. Soc. Jpn. 59 (1990) 3514-3519. 29. T. Tokihiro, D. Takahashi, J. Matsukidaira and J. Satsuma, From soliton equations to integrable cellular automata through a limiting procedure, Phys. Rev. Lett. 76 (1996) 3247-3250. 30. T. Tokihiro, A. Nagai and J. Satsuma, Proof of solitonical nature of box and ball systems by means of inverse ultra-discretization, preprint (1999).

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