Algebraic combinatorics and quantum groups

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A L G E B R A I C COMBINATORICS AND QUANTUM GROUPS

edited by

Naihuan Jing

A L G E B R A I C COMBINATORICS AND QUANTUM GROUPS

This page is intentionally left blank

A L G E B R A I C COMBINATORICS AND QUANTUM GROUPS

edited by

Naihuan Jing North Carolina State University, USA

b World Scientific b

Sir New Jersey • London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 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.

ALGEBRAIC COMBINATORICS AND QUANTUM GROUPS Copyright © 2003 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 981-238-446-4

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface

Combinatorial methods have become increasingly important in various re search in representation theory, geometry and mathematical physics. Many interesting problems of algebra and geometry are closely related with symmetric groups and Hecke algebras, which have made algebraic combina torics an indispensable tool in mathematics during the last several decades. For example, the development of Macdonald polynomials provides exam ples and stimulates interactive research both inside and outside algebraic combinatorics. Recent advances in representations of Hecke algebras and quantum affine algebras have partly relied on corresponding research in algebraic combinatorics. The current volume consists of papers devoted to algebraic combina torics, quantum groups and related topics by invited speakers in the CBMS conference "Algebraic Combinatorics" held at North Carolina State Univer sity during June 4-8, 2001. The conference's goal was to introduce graduate students and young researchers to this active field and survey some of cur rent researches in related areas. Alain Lascoux gave ten featured lectures on Schur polynomials, Schubert polynomials and their applications. His lecture notes will be published by American Mathematical Society in a separate volume. Three related areas are emphasized in this volume: 1) The theory of symmetric functions. Schur functions, Schur Q-functions (or P-functions) and Macdonald functions are studied from various angles. 2) Quantum affine algebras, Hecke algebras and quiver varieties. 3) Combinatorial formulas in statistical mechanics. I would like to thank the support from National Science Foundation and also the mathematics department at NCSU for such an activity, and in particular I am grateful to Alain Lascoux for giving special lectures at the CBMS conference which made the gathering possible. Many thanks are due to anonymous referees who have read the papers and offered constructive suggestions. And last, but not least I would like to thank the authors who have contributed to this volume. Naihuan Jing Raleigh, North Carolina February 2003 V

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Contents

Preface

v

Uno's Conjecture on Representation Types of Hecke Algebras S. Ariki

1

Quiver Varieties, Affine Lie Algebras, Algebras of BPS States, and Semicanonical Basis /. Frenkel, A. Malkin and M. Vybornov

11

Divided Differences of Type D and the Grassmannian of Complex Structures H. Duan and P. Pragacz

31

Tableaux Statistics for Two Part Macdonald Polynomials L. Lapointe and J. Morse A Crystal to Rigged Configuration Bijection for Nonexceptional Affine Algebras M. Okado, A. Schilling and M. Shimozono Littlewood's Formulas for Characters of Orthogonal and Symplectic Groups A. Lascoux A q-Analog of Schur's Q-Functions G. Tudose and M. Zabrocki

61

85

125

135

vii

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UNO'S C O N J E C T U R E O N R E P R E S E N T A T I O N T Y P E S OP HECKE ALGEBRAS SUSUMU ARIKI ABSTRACT. Based on a recent result of the author and A.Mathas, we prove that Uno's conjecture on representation types of Hecke algebras is true for all Hecke algebras of classical type.

1. INTRODUCTION

Let if be a field of characteristic I, A a, finite dimensional K-algebra. We always assume that if is a splitting field of A. We say that A is of finite representation type if there are only finitely many isomorphism classes of indecomposable A-modules. If A is a group algebra, then the following theorem answers when A is of finite representation type. Theorem 1. [3,8] Let G be a finite group, A = KG its group algebra. Then A is of finite representation type if and only if the Sylow I-subgroups of G are cyclic. We restrict ourselves to the case where G is a finite Weyl group and see consequences of this result. Theorem 2. Let W be a finite Weyl group. Then KW is of finite repre sentation type if and only if I2 does not divide the order \W\. For the proof see Appendix. Note that Theorem 2 does not hold if we consider finite Coxeter groups. Dihedral groups W(l2(m)) with odd I and I2 | m are obvious counterexamples. Uno conjectured a (/-analogue of Theorem 2. Let q € K be an invertible element, (W, S) a finite Weyl group. We denote by T-lg(W) the (one param eter) Hecke algebra associated to W. The quadratic relation we choose is (Ts + 1)(TS - q) = 0 (s e S). The Poincare polynomial Pw(x) is defined by

J2xl(w)-

pw{x) =

w€W

1991 Mathematics Subject Classification. 20C08, 16G60. Key words and phrases. Hecke algebra, finite representation type. 1

2

SUSUMU ARIKI

Let e be the smallest positive integer such that qe~l + • ■ ■ + 1 = 0. If q = 1 then e = I, and if q ^ 1 then e is the multiplicative order of q. As the condition that I2 does not divide \W\ is the same as ( ^ ) 2 evaluated at x — 1 does not divide PwO-), the following is a reasonable guess. We call this Uno's conjecture. Conjecture 3. Assume that q y^ I and K a splitting field of Hg(W). Then %q{W) is of finite representation type and not semisimple if and only if q is a simple root of P\y{x) = 0, that is, if and only if (x — q)2 does not divide Pw{x). It is well-known that rlq{W) is semisimple if and only if PW(Q) ^ 0. See [7, Proposition 2.3] for example. This fact will be used later. Remark 4. In [11], it is proved that the conjecture is true for rig(l2(m)). So, unlike the case of q = 1, we may ask the same question for finite Coxeter groups instead of finite Weyl groups. The following theorem is proved in [11]. T h e o r e m 5. [11, Proposition 3.7,Theorem 3.8] Suppose that q^\ and de note its multiplicative order by e. Then %q{An-i) is of finite representation type if and only if n < 2e. As Pw(x) = n r = i IT^t m t m s c a s e » a primitive eth root of unity is a simple root if and only if n < 2e. In particular, the conjecture is true if W = W(An-i). The purpose of this article is to prove. T h e o r e m 6. (Main T h e o r e m ) Assume that W is of classical type and that K is a splitting field ofHq(W). Then riq{W) is of finite representation type and not semisimple if and only if q is a simple root of P\v(x) = 0. We remark that the exceptional cases are settled recently by Miyachi [10] under the assumption that the characteristic I of the base field K is not too small. 2. REDUCTION TO HECKE ALGEBRAS ASSOCIATED TO IRREDUCIBLE W E Y L GROUPS

This is proved by using the complexity of modules. Let A be a selfinjective finite dimensional if-algebra, M a finite dimensional A-module, P' —> M be its minimal projective resolution. Then the complexity CA{M) is the smallest integer s > 0 such that d i m ^ i 3 * ) / ^ + l ) s _ 1 (t = 0 , 1 , . . . ) is bounded. The following lemma is fundamental. L e m m a 7. Suppose that A is self-injective as above. Then (1) CA(M) = 0 if and only if M is a projective A-module.

UNO'S CONJECTURE

3

(2) A is semisimple if and only if CA{M) = 0 for all indecomposable Amodules M. (3) If A is of finite representation type and not semisimple then CA(M) < 1 for all indecomposable A-modules M and the equality holds for some M. Proposition 8. Let S be a set of irreducible finite Weyl groups. If Uno's conjecture is true for all ~Hq(W) with W e S then the conjecture is true for *Hq(Wi x • • • x Wr) with Wx,...,WreS. Proof. Write W = W1x---xWr. Then Hq{W) = n^W^ • • • Hq{Wr) and Pw{x) = Pwx {x) ■ ■ ■ Pwr(x). First assume that q ^ 1 is a simple root of Pw(%) = 0. Then q is a simple root of Pwi{%) = 0 and Pwj{q) ^ 0 for all j ^ i. Then 'Hq(Wj) for j / i are all semisimple and Hq(Wi) is of finite representation type and not semisimple. Thus 1iq(W) is of finite representation type and not semisimple. Next assume that q is a multiple root of Pw{x) = 0 . If q is a multiple root of P\Vi(x) = 0, for some i, then %q(Wi) is of infinite representation type by assumption. Thus Hq{W) is of infinite representation type. If q is a simple root of Pwi(x) = 0 and PWJ(X) = 0, for some i ^ j , then 'rlqiWi) and T-Lq{Wj) are of finite representation type and not semisimple. By Lemma 7(3), there exist an indecomposable 7^ g (H^)-module M, and an indecomposable ?^q(W^)-module Mj such that c« (w4)(Mi) = 1 and c u (w)(Mj) = 1. Write M = Mi®Mj. We shall prove that the complexity of M as an indecomposable Hq(Wi) 'Hq(Wj)-modu\e is equal to 2; if we use the fact that c^ (WixW-){M) is the growth rate of Exf (M,M) then the Kunneth formula implies the result. In a more concrete manner, the proof of C-H (WixW-)(-W) = 2 is as follows. Let P: and P- be minimal projective resolutions of M; and Mj respec tively. Then c^ ^^(Mi) — 1 and cH (w-)(Mj) = 1 imply that there exists a constant C such that 1 < dim/c(P/) < C and 1 < dimx(Pj) < C for all t. As P' = Pi ® Pj is a minimal projective resolution of M, we have t

t + l< dim K (P') = ^ d i m K ( P / ) d i m ^ ( P J - s ) < C2{t + 1). Therefore, the complexity of M is exactly 2. As a result, rHq(Wi)®'Hq(Wj) is of infinite representation type by Lemma 7(2) and (3). Thus Hq(W) is also of infinite representation type. □ 3. T Y P E B AND TYPE D

To prove Theorem 6, it is enough to consider type B and type D by virtue of Theorem 5 and Proposition 8.

4

SUSUMU ARIKI

Let q and Q be invertible elements of K. The (two parameter) Hecke algebra T-Lq 2 as above. IfQ ^ —qf for any f € Z then rlqtQ(Bn) is of finite representation type if and only if n < 2e. Proof. If n > 2e then l-Lq(An-i) is of infinite representation type by The orem 5. Thus, rHq^Q{Bn) is of inifinite representation type by Theorem 9. If n < 2e then one of m and n — m is smaller than e for each m. Thus, one of %q(Am^i) and 7iq(An-m^i) is semisimple and the other is of finite representation type for each m. Thus, HqtQ(Bn) is of finite representation type by Theorem 9. □ Theorem 11. [1, Theorem 1.4] Suppose that —Q = qf (0 < / < e) and e > 3. Then %q!Q(Bn) is of finite representation type if and only if n < min{e, 2 min{/, e — / } + 4}. It is also easy to prove that Theorem 11 is valid for e = 2; see [2]. Corollary 12. Uno's conjecture is true ifW =

W(Bn).

Proof. See the argument of [1, Introduction].

□

Recall that 7iq(Dn) is the iif-algebra defined by generators T®,..., and relations

T®_1

(TiD + l)(TP-q) =0(0 < i < n - l ) , T0 T2DT0D = T2DT0DT2D, T0DT° = T°T0D (i± 2), T&TPT& = TPTP+XTP (1 < i < n - 2), D

TDTD

=

TDTD

(1 < j < j _ 1 <

n

_ 2).

Now assume that Q = 1 and denote the generators of 1-tqii(Bn) by T 0 B ,..., Tn_Y. Then we have an algebra homomorphism :Hg(Dn)—+Hq,i(Bn) denned by TD ^ T^T^T^

and TD >-> Tf for 1 i(Bn) by 7r(lf) = i f i f i f

and 7r(lf) = i f

for i ^ 1.

2

We have 7r = 1 and n induces the Dynkin automorphism of T-Lq{Dn) defined by I f H-J. TP_t for i = 0,1 and I f n- I f for t > 2. Lemma 13. (1) 7/e is odd then 'Hq,i(Bn) is of finite representation type if and only if n < 2e. (2) If e is even then 'Hq,i(Bn) is of finite representation type if and only if n < e. (3) If M is a semisimple Wqti(Bn)-module then so is the T-Lq(Dn)-module Res(M). Proof. (1)(2) These are consequences of Corollary 10 and Theorem 11. (3) We may assume that M is a simple ?^ gi i(B n )-submodule without loss of generality. Let iV be a simple "H g (D n )-submodule of Res(M). Then i f iV is also a simple % ? (-D n )-submodule whose action is the twist of the action of JV by IT. Since iV + l f i V is W gi i(B n )-stable, it coincides with M. Hence Res(M) is semisimple. □ Let D be a 'Hg(£'e)~niodule which affords the sign representation i f H-» — 1, for 0 3 is even then the projective cover Px of the simple ?^ q? i(5 e )-module Dx has the property that RadP A /Rad 2 P A does not contain Dx and R a d 2 P A / R a d 3 P A contains Dx © Dx. See the proof of [1, Theorem 4.1] in p.12. Dx affords the representation i f >-> — 1 (0 < i < n— 1). Lemma 14. Assume that e is even. Then P ~ Res(P A ). Proof. First note that the characteristic I of the base field is odd since e is even. Let D^ be the simple 7^ 9i i(5 e ) _ module which affords the represen tation I f h-> l and I f i-» - 1 (1 * since the left hand side is given by

and T f is diagonalizable because I is odd. Now the surjection P —>• D induces surjective homomorphisms Ind(P) -» Dx andlnd(P) —>• D^. Hence these induce surjective homomorphisms Ind(P) —> Px and Ind(P) —» P ^ . We have that both Px and P M are direct summands of Ind(P). On the other hand, Mackey's formula implies that Res(Ind(P))~P©'TP,

6

SUSUMU ARIKI

where "P is the indecomposable projective W g (£) e )-module which is the twist of P by IT. As the twist of D is D itself, we have Res(Ind(P)) ~ P®P. As Res(P A ) and Res(P M ) are direct summands of Res(Ind(P)), we conclude that Res(P A ) and Res(P M ) are isomorphic to P . D Next lemma is obvious. Lemma 15. Let A be a finite dimensional K-algebra andB a K-subalgebra such that B is a direct summand of A as a (B, B)-bimodule. If A is of finite representation type then so is B. Recall that Pw{x) = (xn — 1) Ylj=i xx-i m tyPe Dn- Thus q is a simple root of P\y (z) = 0 if and only if either e is odd and e < n < 2e or e is even and | + 1 < n < e. Proposition 16. Uno's conjecture is true ifW =

W{Dn).

Proof. First assume that e is odd. If n < 2e then Lemma 15 implies that Hq(Dn) is of finite representation type since 'Hqti(Bn) is of finite represen tation type by Lemma 13(1). If n > 2e then it is enough to prove that tLqiPie) is of infinite representation type by Lemma 15. Using the same lemma again, we further know that it is enough to prove that Hq{Aie.-\) is of infinite representation type. However, this is nothing but the result of Theorem 5. Next assume that e is even. If n < e then Lemma 15 implies that Hq(Dn) is of finite representation type since Hq,i(Bn) is of finite representation type by Lemma 13(2). If n > e then it is enough to prove that Hq(De) is of infinite representation type by Lemma 15. Note that W(D2) = W(Ai) x VF(J4I) and the conjecture is true in this case by Proposition 8. Thus we may assume that e > 4. In particular, we have q ^ — 1, and this implies that E x t ^ P , D) = 0; if we write

then Ti — q is invertible and thus a, = 0. Let P^ Res(P A /Rad 3 P A ). Lemma_13(3) and E x t ^ D . D ) = 0 im ply that P has Loewy length 3. Since P has unique head D by Lemma 14, there exists a surjective homomorphism P ^ P. Further, as P con tains D f f i D a s a 7i ? (£> e )-submodule, R a d 2 P / R a d 3 P contains D®D. On the other hand, Ext 1 (D,D) = 0 implies that R a d P / R a d 2 P does not con tain D. Hence we conclude that End w (£> e )(P/Rad 3 P) is isomorphic to K[X,Y]/(X2,XY,Y2), which is not isomorphic to any of the truncated polynomial rings K[X]/(XN) (N = 1,2,...). As we assume that the base field is a splitting field of %q{De), this implies that V.q{De) is of infinite representation type. □

UNO'S CONJECTURE 4.

7

APPENDIX

In this section, we prove Theorem 2. If the reader is familiar with the structure of the Sylow subgroups of exceptional Weyl groups then (s)he would not need this proof to know that Theorem 2 is true. In the proof be low, we use standard facts about the structure of exceptional Weyl groups; they can be found in [4] or [9, 2.12]. First we consider irreducible Weyl groups. Type ,4 n _i; W has cyclic Sylow /-subgroups if and only if n < 21, and this is equiv alent to the condition that I2 does not divide \W\ = n\. Type BnW has cyclic Sylow /-subgroups if and only if either / > 2 and n < 21 or I = 2 and n < 2, and this is equivalent to the condition that I2 does not divide \W\ = 2"rc!. Type Dn; W has cyclic Sylow /-subgroups if and only if either / > 2 and n < 21 or / = 2 and n < 2, and this is equivalent to the condition I2 does not divide \W\ = 2n~1n\. Type F4; As |W(F 4 )| = 2 7 • 3 2 , we prove that the Sylow /-subgroups for / = 2,3 are not cyclic. Let A(F4) be the root system of type F4. The long roots form a root system which is isomorphic to A(D4). Let T be the Dynkin automorphism group of the Dynkin diagram of type D4. Then it is known that W(F4) is isomorphic to the semi-direct product of W{D4) and I\ Assume that 1 = 2. Since W(D4) contains C^ x C2, the Sylow 2-subgroup of W(F4) cannot be cyclic. Assume that / = 3. Since T is isomorphic to the symmetric group of degree 3, we can choose a G T of order 3. Let P be a Sylow 3-subgroup of WX.F4) containing a. As the Sylow 3-subgroup of W(D4) is a cyclic group of order 3, we have |VF(Z?4) n P\ < 3. On the other hand, as (a) is a Sylow 3-subgroup of T and \P\ = 9, both W(D4) n P and P/W(D4) n P are isomorphic to the cyclic group of order 3. Let r be a generator of W(D4) n P. Then we have the following split exact sequence. 1 —* (r) ~ C3 —>■ P —+ ( ~ C3 —> 1. As Aut(C 3 ) ~ C2, cr acts on (r) trivially and P — C3 x C3. Type £ n ; Recall that |W(£?6)I = 2 7 - 3 4 -5, U 2 • 3 5 • 5 2 • 7.

|W(JE? 7 )|

= 2 10 - 3 4 -5 -7 and \W(E8)\ =

8

SUSUMU ARIKI

Assume that I = 2 or I = 3. Since VK(F4) C W{En), The Sylow Zsubgroup of W(En) contains Ci x C/. Thus it cannot be a cyclic group. Assume that I = 5. Let Q be the root lattice of type E$ with scalar product normalized to (ai,ai) = 2 for simple roots a» (1 < i < 8). Then g(x) = ^ ^ (mod 2) defines a quadratic form on Q/2Q ~ F | . Note that if we choose simple roots as a basis, we can write down q(x) explicitly, and the computation of its Witt decomposition shows that its Witt index is 4. Thus, by choosing a different basis, we may assume that q(x) = J2i=ix2i-ix2iLet Os(2) be the orthogonal group associated to this form. Then it is known that there is an exact sequence 1 —► {±1} —> W(E8) —► O s (2) —>■ 1. Let q'{x) = x\X2 + x\ + x^x^ + x\ be a quadratic form on F | . If we write 0^(2) for the orthogonal group associated to this form, we know that its Sylow 5-subgroups are cyclic of order 5. Now explicit computation of Witt decomposition again shows that q' ® q' has Witt index 4. Thus Og(2) contains Cs x C5. As a result, the Sylow 5-subgroup of W(E&) is isomorphic to C5 x C5. Type G 2 ; This is the dihedral group of order 12 and its Sylow 2-subgoups are not cyclic. Now let W be & general finite Weyl group. That is, W is a product of the groups listed above. Then W has a cyclic Sylow /-group if and only if at most one component of the product has a cyclic Sylow /-group and all the other components have trivial Sylow /-groups. This is equivalent to the condition that I2 does not divide \W\. REFERENCES [1] S. Ariki and A. Mathas, The representation type of Hecke algebras of type B, to appear in Adv. Math., m a t h . R T / 0 1 0 6 1 8 5 . [2] S. Ariki and A. Mathas, The Hecke algebras with a finite number of indecomposable modules: a summary of results, preprint. [3] V.M. Bondarenko and J.A. Drozd, The representation type of finite groups, J. Soviet Math., 20 (1982), 2515-2528. [4] N. Bourbaki, Groupes et algebres de Lie, (1968), Hermann. [5] ft. Dipper and G. James, Representations of Hecke algebras of type Bn, J- Algebra, 146 (1992), 454-481. [6] K. Erdmann and D.K. Nakano, Representation type of Hecke algebras of type A, Trans. A.M.S., 354 (2002), 275-285. [7] M. Geek, Brauer trees of Hecke algebras, Comm. Alg., 20 (1992), 2937-2973. [8] D. Higman, Indecomposable representations at characteristic p, Duke Math.J., 2 1 (1954), 377-381. [9] J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics 29 (1990), Cambridge University Press. [10] H. Miyachi, Uno's conjecture for the exceptional Hecke algebras, in preparation.

UNO'S CONJECTURE

9

[11] K. Uno, On representations of non-semisimple specialized Hecke algebras, J. Alge bra, 1 4 9 (1992), 287-312. RIMS, K Y O T O UNIVERSITY, K Y O T O 606-8502, JAPAN

E-mail address: a r i k i a k u r i m s . k y o t o - u . a c . j p

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QUIVER VARIETIES, A F F I N E LIE A L G E B R A S , ALGEBRAS OF B P S STATES, A N D SEMICANONICAL BASIS IGOR FRENKEL,

ANTON MALKIN, AND MAXIM VYBORNOV

ABSTRACT. We suggest a (conjectural) construction of a basis in the plus part n-|- of the affine Lie algebra of type ADE indexed by irre ducible components of certain quiver varieties. This construction is closely related to a string-theoretic construction of a Lie algebra of BPS states. We then study the new combinatorial questions about the (classical) root systems naturally arising from our constructions and Lusztig's semicanonical basis.

1. INTRODUCTION

1.1. Since the appearance of the first manifestation of the connection be tween theory of representations of quivers and the structure theory of Lie algebras [Gab72], several authors discovered constructions of Lie algebras arising from the quiver theory. C. M. Ringel constructed the plus part of the simple Lie algebra of type ADE, [Rin90], and a (related) construction of the plus part of any Kac-Moody Lie algebra is implicit in Lusztig's construction of (quantized) enveloping algebras of Kac-Moody Lie algebras in terms of functions on a class of remarkable affine varieties A^, [Lus91, Lus92]. Inspired by Ringel and Lusztig, the present authors came up with a construction of the plus part n + of the affine Lie algebra of type ADE in terms of indecomposable representations of quivers [FMV01]. A similar construction is suggested by string theorists [FMOO], who use the stable representations of (double) quivers rather than indecomposable representations of oriented quivers (as do Ringel and the present authors). The concept of stability has been very useful in representation theory: H. Nakajima discovered that modules over Kac-Moody Lie algebras may be described using functions on the varieties Av/Gv where A^ are the stable points of Lusztig's varieties Ay [Nak94, Nak98]. 1.2. One of the main goals of this paper is to study the relationship be tween the Lusztig's construction of n+, the construction by the present authors [FMV01], and the "stable" construction suggested by physicists. Using the notion of semistable diagonal in A^ suggested in [HM98], we make li

12

IGOR FRENKEL,

ANTON MALKIN, AND MAXIM VYBORNOV

a conjecture (Conjecture 4.2.5) directly relating the Lusztig's construction and the "stable" construction. Moreover, we (conjecturally) obtain a basis in n+ parameterized by irreducible components of algebraic varieties. One can look at this conjectural basis as "semicanonical" basis for n + . 1.3. In section 5 we use our methods to obtain some simple results de scribing new aspects (arising from quiver constructions) of combinatorics of the root systems and Weyl groups, and ask many more questions than we can answer at the moment. One of the results of the Lusztig construction of the (quantized) universal enveloping algebras in terms of functions on the varieties Ay is the appear ance of the semicanonical basis in the non-quantized enveloping algebras of simply laced Kac-Moody Lie algebras [LusOO]. Semicanonical basis is indexed by the irreducible components of Ay. It was expected that this basis would coincide with the specialization of the Lusztig's canonical basis to q = 1, but a counterexample was found by M. Kashiwara and Y. Saito [KS97]. The relationship between these two bases is quite mysterious at the moment, even though they share their combinatorial properties. Let n + C U(n+) be the plus part of a simple Lie algebra of type ADE, and U(n+) be its universal enveloping algebra. For every positive root a G R+ we have a canonically (up to a sign) defined element .E* G na C E/(n+), where n a is the one dimensional root subspace of n+ corresponding to a. We study the decomposition of E^ with respect to the semicanonical basis. More precisely, let us consider Ay, dim V — a. Then the irreducible compo nents of Ay may be indexed by decompositions of a into the sum of positive roots. Let a = a\ + 012 H h aj, on S R+ be such a decomposition and let eai +tt2H i-ai be the element of the semicanonical basis corresponding to this decomposition (and the corresponding irreducible component of Ay). Then ^a

=

/

c

j

tt=Ql + «2 H

a i + a2H

h a i e c * i + a.7, H

haj 1

Y&l

where cai + a2 +...-\-ai € Z is the coefficient of eai + a2 _| y.ai. In other words, with the help of the semicanonical basis we assign an integer cai + a2 +—+a, to every decomposition a = a\ + a-i H + on of the root a into a sum of positive roots. It would be very interesting give a purely combinatorial description of these numbers without appealing to the semicanonical basis. We manage to obtain such a description in the An case in terms of the sign character of the Weyl group W = Sn+i. 1.4. Acknowledgment. We are grateful to Naihuan Jing who presented us with the opportunity to give this talk at the CBMS conference he or ganized at NCSU in June 2001. We tried to preserve a bit of the informal style of a talk in this paper. We are grateful to B. Fiol, J. Humphreys, G. Lusztig, M. Marifio, and C. Ringel for useful conversations. M.V. is grate ful to MSRI and Max-Planck-Institut fur Mathematik, where parts of this

QUIVER VARIETIES, BPS STATES, SEMICANONICAL BASIS

13

work were done, for their hospitality. The research of I. F. was supported in part by NSF. The research of A. M. and M. V. was supported by NSF Postdoctoral Research Fellowships. 2.

PRELIMINARIES

We always assume that our ground field is the field of complex numbers C. Our notation and conventions are mostly lifted from [Lus91, LusOO, FMV01]. 2.1. Quivers. 2.1.1. To a graph Q, with no edges joining a vertex with itself, we associate a pair of sets: I (vertices), and H (oriented edges), and two maps from H to I: (2.1.1.1) a map H —>•I denoted h —> h' (initial vertex), (2.1.1.2) a map H —>I denoted h -> h" (terminal vertex), (2.1.1.3) an involution h —> h on H which maps an oriented edge to the same edge with the opposite orientation. An orientation of (I, H) is a choice of a subset fl C H such that fiUfl = H and fi n Q. = 0. Abusing terminology we will call both (I, H) and (I, fi) quivers. 2.1.2. Let V be the category of /-graded vector spaces V = ©ie/V*. We define: Ev = 0 H o m ( ^ , 7 t » ) , Ev,n = 0Hom(V r v ) Vi l »). hen For an element x G ~Ey we denote by Xh, h G H its component in Hom(Vh,r,Vh"). A pair (V,x), x 6 E y (resp. x € E ^ n ) is called a rep resentation of (I,H) (resp. {I,fl)). We will also sometimes call (V,x) a module over (/, H), and moreover denote such a module by V if it is clear what x we consider. Let Z[7] be the free abelian group generated by the set I. The dimension of (V, x) with x G Ey or x G Ey,n is an element of Z[J] defined as follows: d i m ( V » = dim V = ^ d i m c ( ^ ) i G Z[J]. The algebraic group Gy = Yliei C-^(^i)

acts on

Ey in a natural way.

2.1.3. We say that x G Ey;n is indecomposable if (V, x) is indecomposable as a representation of (1,0). The subset of indecomposable elements in Ey,n is denoted by E 1 ^ .

14

IGOR FRENKEL, ANTON MALKIN, AND MAXIM VYBORNOV

2.1.4. We say that x € E y is nilpotent if there exists an N > 2 such that for any sequence hi, hi, ■ ■ ■, /ijv 6 H, such that h'x = h%, h'2 = h%,..., h^^-^ = h'jx, the composition XhxXh2 ■■■XhN = 0. The subset of nilpotent ele ments in E y is denoted by E™1. 2.1.5. Following [Lus91] we consider the moment map attached to the Gy-action on E y . The z-component m^ of this map is given by m X

i( ) =

e h X

X]

( ) hXh,

heH:h"=i

where e : H —>• C* is a function such that e(h) + e(h) = 0 for all h € H. Following Lusztig we introduce the subvariety Ay of Ey as follows Definition. Ay is the closed subvariety of all nilpotent elements x € Ey such that rrii(x) = 0 for all i € I. 2.2. Convolution product. 2.2.1.

Let C(V) =

MGv(Av)

be the set of constructible C-valued functions on Ay which are constant on Gy-orbits in Ay. Let £Q(V) (resp. £Z(V)) be the set of all / G £{V) with rational (resp. integral) values. 2.2.2. Let V,V, V" e V be such that d i m F ' + dimV" = dimV. Let / ' e C(V), f" e £(V"). We lift the convolution construction from [Lus91, 12.10]. Consider the diagram Ay, x A y . < - ^ — F ' - ^ - » F " - ^ - > Ay, where F " is the variety of all pairs (x, W) where x € Ay and W C V is an a;-stable subspace, dim W = dimK". F ' is the variety of all quadruples {x, W, R", R') where (x, W) e F", and R', R" are V-isomorphisms R' : V ~ V/W, R" : V" ~ W. We have p\(x, W, R", R') = (x1, x") where XhR'h, = R'h„x'h, and Xh,R^, = Rl„xl for all he H. We have p2(x, W, R", R') = (x, W), p3(x, W) = x. Let h e £(AV, x Ay») be given by h(x',x") = f'(x')f"(x"). Then there is a unique function / 3 6 £(F") such that p\fx = p2fz- Then by definition / ' * / " = (P3)l(/3).

There exists an analogous construction for the oriented quiver [Lus91, 10.19]. We will denote the corresponding convolution product by *Q for an orientation Q.

QUIVER VARIETIES, BPS STATES, SEMICANONICAL BASIS

15

2.2.3. If dim V = i e I, then Ay is a point, and we denote by Ei(Ay) = 1 the function which is identically 1 on this point. Let £o be the associative algebra with *-product generated by {Z?j}jej. (The associative algebra of functions on E ^ n with *n-product generated by {Ei}i^j will be denoted by £o,Q-) One can consider Co as a Z + [/]-graded Lie algebra over Q, with the following Lie bracket: [/, 9] = f * 9 - 9 * /■ We denote by n* the Lie subalgebra of Co generated by {.Ej} ie /. 2.3. Geometric realization of the enveloping algebra. 2.3.1. Let (I, H) be a quiver. For i,j e I define i ■ j = - Card{/i G H I h' = i, h" = j}, if i ^ j and i-j = 2 if i=j. Let U+ be the C-algebra defined by generators e,, i & I and the Serre relations:

(-l)^eA=0 p,0 p+q=-i-j+l

^

*

for any i ^ j in / . U+ = U(n) is the enveloping algebra of the Lie algebra n defined by generators e*, i € / and the Serre relations: (ade i )- i - J + 1 (e,) = 0 for any i / j in / . Let (V, x) be a representation of (I,H). Let C/^mV- denote the subspace of U+ generated by e^e^ . . . ej„ for sequences ii, 12,... ,i„ in which i appears dim V* times for any i £ I. Let t/g" be the subring of f + generated by the elements e?/p\ for all i£l,p£ Z> 0 . Then f/^dimy = t # f~l U+mV, see [LusOO, 1.1]. 2.3.2. In [Lus91, 12.12] Lusztig defines a C-linear map tpv '■ U^imV —>• C(V), such that (2.3.2.1) ip = © d i r n ^ i p v '■ U+ ~ Co is an isomorphism of algebras, (2.3.2.2) ip(ei) = E%, for any i £ I, (2.3.2.3) ipv(UldimV) C CZ(V). The isomorphism ^ also restricts to an isomorphism of Lie algebras ip : n — ► n* such that ip(ei) = Ei for any i £ I. In [LusOO] Lusztig defines a version of the map tpv '■ ^dimV ~* ^(^)> which restricts to a map Vv : ^z~dimV ~~* ^ z ( ^ ) 2.4. Lie algebra based on the Euler cocycle: classical A D E case. Let the graph underlying the quiver (I, H) be the Dynkin diagram of type ADE. In this case we can identify Z[I] with the root lattice of type ADE. The elements i £ I are considered simple roots, and we have the root system R C Z[I] and its positive part R+ C Z>o[i].

16

IGOR FRENKEL, ANTON MALKIN, AND MAXIM VYBORNOV

Let us fix a cocylcle e : Z[I] x Z[I] —> Z /2 Z. In the simply laced Dynkin diagram case, the Lie algebra n = n* defined above is isomorphic to the Lie algebra ne spanned by the elements ea, a E R+ with the bracket defined by e as in [FMV01, 4.2.1.a]. Now let us define the functions E^ e £(V), a G R+, d i m F = a as follows: (2.4.0.1)

E*a = V(fia).

Notice that if a = i € I, then Ay is a point and E* = E{. 2.5. Lie algebra based on the Euler cocycle: affine A D E case. Let the graph underlying the quiver (I, H) be the extended Dynkin diagram of type ADE. In this case we can identify Z[7] with the affine root lattice of type ADE. The elements i € / are considered simple roots, and we have the affine root system R C Z[J] and its positive part R+ C Z>o[I]. Let us choose an extending vertex p £ I, and let I' = I — {p}. Again, let us fix a cocylcle e : Z[7] x Z[7] —» Z/2 Z. Let n a be the root space of the algebra n corresponding to a € R+. Let 5 be the indivisible imaginary root. The Lie algebra n is isomorphic to the Lie algebra n£ spanned by the elements ea £ n^, a € R™, and ak(n) € n^s, n > 1, where we identify nenS = C\I'} = the vector space spanned by /', and Qfc = k € I' C Z[I'] are simple roots. The bracket is defined by e as in [FMV01, 5.2.1.a]. In particular,

{

e(a, P)ea+p e(a,/3)a(n) 0

ifa + /3ei^.e, if a + (3 = n5, ifa + /3<£R+.

Now let us define the functions 75* € £-{V), dim V = a € R™ as follows: (2.4.0.3)

El = ip(ea),

and the functions E£(n) € C{V), dim V = nS as follows: (2.4.0.4)

Et(n)=tKak(n)).

Again, if a = i G / , then Ay is a point and E* = Ei. 3. STABILITY AND SIMPLE LIE ALGEBRAS O F TYPE

ADE

3.1. Stability after King and Rudakov. 3.1.1. Stability after King. Let us consider the abelian category of nilpotent representations of a quiver [I,H) and its Grothendieck group KQ. A character on KQ is an additive function 6 : KQ -4 IR. Definition. [King94] A point x & E y is called 6-stable (resp. 6-semistable) if 0(V) = 0, and for any x-stable nonzero proper subspace V C V we have 6{V) > 0 (resp. fl(V') > 0).

QUIVER VARIETIES, BPS STATES, SEMICANONICAL BASIS

17

Definition. Two O-semistable points x',x" £ Ey are 5-equivalent (notac tion: x' ~ x") if the orbit closures Gy ■ x' n Gy ■ x" intersect in the set of 9-semistable points E y . 3.1.2. Stability after Rudakov. One can also define stable points using a "slope" stability condition: // = c/r where c and r are additive functions K0 -> R and r(V) > 0 for any V e V, see [Rud97, 3]. Definition. [Rud97] A point x € E y is called //-stable (resp. //-semistable,) if for any x-stable nonzero proper subspace V C V we have fi{V) < fi(V) (resp. n(V) < n(V)). If we fix such slope stability condition // and V 6 V we can define a character 0^ : KQ - > K a s follows: (3.1.2.1)

6ll(V')

= -c(V') +

^r(V').

According to [Rud97, Proposition 3.4] a point x G E y is //-stable if and only if it is 0^-stable. 3.1.3. Let us fix an orientation 0 C H until the end of this section. Fol lowing [King94] we construct a character 9 = 6y,n associated to 0. Let V, V € V. We define ©v.n(V') = ^(dimVfc. dim Vh'„ - dim V^, dim Vh»). 3.2. Stability Lemma, In this section we will consider quivers of finite ADE type, i.e. the underlying non-oriented graph Q is a Dynkin graph of (finite) ADE type. A fact similar to the lemma below was independently conjectured by M. Reineke [Rei02, Conjecture 7.1] L e m m a . Let x € Ay, dim V = a €. R+. Then x is Qy^-stable ifxeV$du.

if and only

Proof. Since in the Dynkin quiver case every root is a Schur root [King94], a generic point in Ey,n is Gy^-stable according to [King94, Scho92]. Since E y n is an open dense Gy-orbit in E y ^ , any x € E y n is 0y,n-stable. Let us prove the "only if" part. If x e Ay — E y n , then we have x = y+z, where y € E y ^ , z € E ^ ^ . Following [Lus91, 14] we see that there exists a decomposition: (V,y) = ®;=1(V»,y) such that Hom(V p , Vp ) = 0 whenever p' < p. There is a direct sum decomposition: Ey,n=

0 l
E $

where

E $ = Hom f c e n (V£, v £ ) .

18

IGOR FRENKEL, ANTON MALKIN, AND MAXIM VYBORNOV

Now we need the following claim: there exists y' € © p > p / E ^ such that y + y' =: x' e E ' v n . It is not very hard to prove this claim using the methods of [Lus90, 4], Following [Lus91, 14] we denote V^ = @p>qVp. Then we have the 2-invariant filtration: V = Vw D Vw D---D V{u) = 0. Denote V(9) = @p
o = v ( 0 ) c V ( 1 ) c - c v w = v. The vector space V^ is a submodule in (V, x'). Since x' is 0v,n-stable we have @v,Q(V(q)) > 0. However V^ is a quotient module in (V,a;). Therefore, (V, x) is unstable. □ 3.3. Stable construction. 3.3.1. Let us consider the cocycle en associated to our fixed orientation 0 , see [FK81], [FMV01, 1.3.4]. By construction, the functions E„, a € R+ on Ay defined as in 2.4 with the cocycle e = en form a basis of the Lie algebra n*, and the *-bracket is given by: (3.3.1.1)

[K,^)

= ha'0)Er^

^

+ peR+,

3.3.2. It is clear that the affine variety Ev,n C Av is one of the irreducible components of Av- We can define a constructible function Ea, a e R+ on Ev,n, dim V = a as follows, see [FMV01, 4.3.1.b]: E

(3.3.2.1)

° =\ n • tu 10 otherwise . According to (FMV01, 4.3.4], the space spanned by the functions Ea, a S R+ is a Lie algebra (which we will denote here by nn) with the bracket [/,s]n = / *Q 9 - 9 *Q / • The map n€ ->• u a given by ea -> Ea is an isomorphism. In other words, the bracket can be explicitly described as follows: (oo0o\

fe(a,P)Ea+p, if a+ f3eR+, < 10 ifa + pfR+. 3.3.3. Let us take the character 0v,n, and denote the identity function on Ay, dim V = a, by Ea, Ea(Av) = 1, where A^ is the set of points in Av stable with respect to 0v,n- Notice that by Lemma 3.2, Ay = E™^, and so the function Ea is constant on a single Gv-orbit in A^(3-3.2.2)

P i [Ea,E0\n=

I P

Lemma. Up to a sign:

QUIVER VARIETIES, BPS STATES, SEMICANONICAL BASIS

19

(3.3.3.1) E*\Ev^ =Ea, (3.3.3.2) JS£|A.v = Ea. Proof. Follows immediately from definitions, the proof of [Lus91, Theorem 12.13], and the formulas 3.3.1.1, 3.3.2.2. D 3.3.4. Let us denote the space spanned by the functions Ea, a € R+ by nstable. Since Av = E ^ Q , by lemma 3.2, the restrictions from Ay to Ey,n to Ay provide us with the based isomorphisms of vector spaces (cf. Lemma 3.3.3) «*

■—

> «fi

restriction

—

•.

^stable

restriction

defined by E£ <-> Ea >-> Ea for a 6 R+. It is clear that the first restriction is an isomorphism of Lie algebras, and the second restriction equips the space nstable of functions on Av constant on Gy-orbits (or equivalently, functions on Ay/Gy = pty) with the structure of a Lie algebra isomorphic to n* ~ nn ~ n e . 4. STABILITY, AFFINE L I E ALGEBRAS OF TYPE ADE,

AND A

CONJECTURAL CONSTRUCTION OF THE ALGEBRAS OF B P S STATES

4.1. Physics. 4.1.1. One of the current models of string theory defines £>-branes as ob jects in the derived category Db{Coh{Z)) of the coherent sheaves on the algebraic (Calabi-Yau) variety Z. Physicists consider the moduli space M.^{n) (where Q = (Ci; C2, • •• , C/v) are the Fayet-Iliopulos terms) of semistable £>-branes of charge n, cf. [HM98, DFROO, FMOO]. In the semiclassical approximation, the space of BPS states is given by the cohomology of the moduli space: KEFS =

H*(Mc(n)).

4.1.2. Dictionary. We will consider the L>-branes on the resolution C 2 / T of a simple singularity C 2 fT. Let Q be the extended Dynkin diagram corre sponding to r via the McKay correspondence. We suggest a mathematical model of the physical situation in this case with which we will work. In the next subsection 4.1.3 we will justify our model. For now, we provide the physics/mathematics dictionary: the charge lattice : the positive part Z>o[^] of the affine root lattice associated to Q; places in the the charge lattice occupied by the single particle BPS states : positive roots R+ C Z>o[/]; a D-brane of charge a € Z[I] on C 2 /T : a representation x € Ay, dim V = a of the quiver (I,H) (with the underlying non-oriented graph Q);

20

IGOR FRENKEL,

ANTON MALKIN, AND MAXIM VYBORNOV

Fayet-Iliopulos terms £ : an additive function c which defines Rudakov's slope stability together with r(V) = J^i^i dim V*; ^-stable D-branes : stable elements of Ay with respect to a stability condition; moduli spaces of D-branes of charge a G R+ : Av/Gy = pt a , dimV = a e R^; moduli spaces of D-branes of charge S = indivisible imaginary root : Ay/Gy = £ = £1, where £ is the exceptional fiber of the resolution of simple singularity; moduli spaces of D-branes of charge m5, m > 1 : Ay d/ (S — equivalence) = £ = £ m , where Ayd is the semistable diagonal (see 4.2.4.1 for the defi nition, cf. [HM98]), and where £ is the exceptional fiber of the resolution of simple singularity; BPS states : certain constructible functions on our moduli spaces, see 4.2.8 for details. 4.1.3. Justification. If we consider the D-branes on C /T then due to M. Kapranov and E. Vasserot [KVOO]: Db(Coh(C2 / r » ~

Db(Rep(Q))

where Q is the extended Dynkin diagram associated to the T C SL(2, C) via the McKay correspondence, and Rep(Q) is the category of finite-dimensional double representations of Q in the sense of [KVOO, 3.4]. We are only interested in D-branes of charge a G R+ which are 0complexes on the right hand side, i.e. double representations of Q rather than complexes of such representations. Such representations are identi fied with elements x € Ey, dimV = a satisfying rrii{x) = 0 for a l i i G / (see section 2.1.5 for the definition of mi). The notion of stability be comes the usual GIT stability adapted to the quiver situation by King and Rudakov (see section 3). We choose a "non-degenerate" stability condition (cf. 4.2.6). If dim V = a G i?+, then the moduli space is a point: A4<;(a) = pt a . If dim V = 5, where 5 is the indivisible imaginary root, then the moduli space is the resolution of a simple singularity: M<;{6) =

cVr.

In the two cases above one may replace the (middle) cohomology of the moduli spaces with the constructible functions on the exceptional fiber see [Nak94, 10.16]. Following this logic, we suggest the interpretation of BPS states as in the dictionary above and in 4.2.8 below if dim V = m5, m > 1 where the situation is more complicated and a clear connection between physical and mathematical results was not available before, as far as we know.

QUIVER VARIETIES, BPS STATES, SEMICANONICAL BASIS

21

4.1.4. Physics implies that the space of BPS states on C 2 /T of all possible charges should form a Lie algebra isomorphic to the plus part of the affine Lie algebra corresponding to Y via the McKay correspondence [FMOO]. The main purpose of this section is to offer a conjectural, but mathematically rigorous, validation of this claim in 4.2.8. 4.2. The conjecture., In this section we consider quivers of affine ADE type, i.e. the underlying non-oriented graph Q is an extended Dynkin graph of ADE type. 4.2.1. Let us fix an extending vertex p of the graph Q. For every a = d i m F let us fix the Nakajima's character: 0(V) = —1, if dim V = k € I' =1- {p}, and 6(V) = £ < 6 / , d i m c ( ^ ) if dimV = p e I (cf. [Nak98]). (Semi)stable points in this section are considered with respect to this sta bility condition. If Vp = 0 we can take a King's character Qv,n (see 3.1.3) associated to any orientation that "flows to the extending vertex" i.e. the extending vertex is a sink and one can get to the extending vertex from any other point in the quiver going along the oriented edges. In the DE case there is only one such orientation (once the extending vertex is fixed). In the An case there are n such orientations. 4.2.2.

For dim V = a e R% we have (cf. [CB01]):

(4.2.2.1)

Asv/Gv

= pta .

Let us denote by Ea the function identically 1 on the point pt a , £ a ( p t a ) = 1. 4.2.3. For dim V = 6 = indivisible imaginary root we have ([Kron89], [Nak94]): (4.2.3.1)

Ay/Gv = £,

where £ = £ i is the exceptional fiber of the resolution of simple singularity £ --» C 2 IT -» C 2 / r . Here V is the finite subgroup of 51/(2, C) corresponding to the diagram Q via the McKay correspondence. It is well known that £ is a configuration of lines P 1 which are its irre ducible components and which may be indexed by the vertices k & I' = I—{p}. We will denote the A:th irreducible component of £ by Yk, k £ I'. Let us denote the characteristic function of Yk by Ek(l), k € / ' , Ek(l)(Yk) = 1, £ * ( ! ) ( £ - Y f c ) = 0.

22

IGOR FRENKEL,

ANTON MALKIN, AND MAXIM VYBORNOV

4.2.4. Finally, for dim V = m6, m > 1 we introduce the semistable diag onal Af/d C Av as follows (cf. [HM98]): x is 5-equivalent to the direct sum of m isomorphic representations in K\. Here Af = Ay,, dim V = S. Formally: (4.2.4.1) A ^ s d = {x £ A y | X ~ Xi © X2 © • • • © Xm, Xi ~ X2 ^ • • • ~ X m , Xi £

A$s}

It is clear that: A^ d /(5'-equivalence) = £ where £ = £ m is the same exceptional variety as above. Let us denote the characteristic function of the fcth irreducible component Yk of £ = £ m by Ek{m), k € I', Ek(m)(Yk) = 1, Ek(m)(£ - Yk) = 0. 4.2.5. We have defined the functions E*a on Ay, dim V = a £ R+, and the functions J5^(m), k £ I' = I — {p} on Ay, dim V = TO5, m > 1 in section 2.5. We need one more definition (cf. 2.4.0.2): Et(m) :=

e(ak,f3k)[E*ak\Ki,EZk\A,V2],

where dim Vi = ak = k e I' C R™, dim V2 = /3k = (m<* - afc) € i?^_e (here we treat elements of I' as simple roots), and the bracket is in the sense of 2.2.3. Since the functions E^,Ek(m) are constant on GV-orbits, we can con sider them as functions on Ay/Gv- The equalities in the following conjec ture are understood in this sense. Conjecture. (4.2.5.1) Up to a sign, £ £ | A J , = Ea, dimV = a £ R+. (4.2.5.2) Up to a sign, 2%(l)|A(, = Ek{\), dim V = S. (4.2.5.3) Let d i m F = m5, m > 1. We conjecture that if x',x" £ Assd are two elements in the same S-equivalence class and Ek(m)(x') ^ 0, E*k{m){x") ^ 0, then E*k(m)(x') = E*k(m){x"). Thus we can consider 2S£(m)|A.»d as a function on Assd/S-equivalence by setting for an S-equivalence class X: fr*{ \(v\

_ J Ek(m)(x)> 10,

*/ there exists x £ X with Ek(m){x) otherwise.

(4.2.5.4) Up to a sign, EkK(m)\A..
^ 0,

dim V = m8.

The conjecture is verified in the A\ case. 4.2.6. Remark. We would like to explain here that there is nothing special about the Nakajima's choice of character. If dim V = m5 is an imaginary root, then the space of all characters defining stability may be naturally identified with a (classical) Cartan subalgebra f) = C-vector space spanned by i £ V. A character 8 is called non-degenerate if it does not lie on a wall of a Weyl chamber. We expect the conjecture to be true for any

QUIVER VARIETIES, BPS STATES, SEMICANONICAL BASIS

23

non-degenerate character in the same Weyl chamber as the Nakajima's character (this is the fundamental chamber). Moreover, we could take a non-degenerate character in any other Weyl chamber, but that would change our choice of simple roots which we identify with the set I' = I— {p}. One could perhaps reformulate our conjecture in this case modifying the functions E% (m) using the action of the Weyl group. 4.2.7. Remark. The functions Ek (m) parameterize the set Irr £ = Irr £ m of the irreducible components of the exceptional variety £. The components of £ are a basis in the space H2(C2 /T), see [Nak94]. 4.2.8. Denote the Lie algebra spanned by the functions E^ on Ay, dim V = a G R+, and the functions E^(m) k e I' = I — {p} on Ay, dim V = mS, m> 1 by n*(Q), cf. section 2.5. Let us denote the vector space spanned by the functions Ea, a G J?+, and the functions Ek(m) on £ = £ m = As^f/(S - equivalence) by n s t a 6 / e (Q). This is the vector space of constructible functions on moduli spaces of (semi)stable points in A, which are linear combinations of characteristic functions of irreducible components. The space nstable(Q) is our model of the space of BPS states at the orbifold C /T, with Q corresponding to T via the McKay correspondence. The conjecture, if true, would imply that the restriction from A to As (or Assd) gives us the based isomorphism of vector spaces n*(Q) —^—> n stoWe (Q) such that E1* •-»■ Ea and E^(m) i-» Ek(m) i-» Ek(m) for a G R+, k e I', m > 1. This map equips the space of BPS states n s t a W e (Q) with the structure of a Lie algebra isomorphic to the plus part of the affine Lie algebra corresponding to Q. On the other hand the above isomorphism provides the plus part of the affine Lie algebra corresponding to Q, with a basis {Ea, a € i?|jf; Ek(m), k G I',m € Z>o} indexed by irreducible components of algebraic varieties p t a = Asa/Ga, and £ m = A£$/(£ - equivalence), m € Z > 0 . 5. REMARKS ON SEMICANONICAL BASIS FOR SIMPLE L I E ALGEBRAS

5.1. Semicanonical basis. This subsection is lifted from [LusOO, 2.4-5]. Recall the setup of 2.3. Let V = ©jg/Vi be a /-graded vector space. If Y € Irr Ay is an irreducible component of Ay and / £ £z(V) then there is a unique c G Z such that / _ 1 ( c ) ^ ^ contains an open dense subset of Y. Note that / >-> c is a linear function py'■£-z{V) —>■ Z. L e m m a . Let Y G Irr Ay. There exists f = ey G ipv(U%v) Py(f) = 1 and py(f) = 0 for any Y' € Irr Ay - {Y}.

such that

24

IGOR FRENKEL, ANTON MALKIN, AND MAXIM VYBORNOV

Lusztig proves that the functions ey,Y € Irr Ay form a basis in tpy(Uy), and therefore, the collection of ey, Y € Irr Ay for all possible dimensions of V forms a basis in the algebra ip{U+) = Co isomorphic to U+. This basis is called semicanonical. In the remainder of this section we consider quivers of finite ADE type. 5.1.1. Since our quiver is of finite ADE type we know [Lus91, 14.2] that the irreducible components of Ay are the closures of conormal bundles of the various Gy-orbits in E y n , where O is some orientation. Such Gyorbits are indexed by the decomposition of (V,x), x £ E ^ n into a direct sum of indecomposable submodules (V,x) = (Vi,a:i) © • • • © (Yi,xi) with Xi € E y f t . (The decompositions are considered up to the order of the summands.) Thus, there is a one-to one correspondence between Irr Ay and the decompositions dim V = dim V\ + • • • + dim Vj of dim V € Z[J] into a sum of positive roots dim Vj € R+. (Recall that E y ^ ^ 0 if and only if dim V e R+.) Thus the elements of the semicanonical basis may be indexed by decompositions of dim V into a sum of positive roots. 5.2. Open problem. Recall that we have defined the functions £* on Ay, dimV" = a € R+ in 2.4. By construction these functions are defined canonically up to a sign which depends on the choice of the cocycle e. Since -E* G Vv(fz"dim y)> w e c a n decompose E* with respect to the semicanonical basis:

(5.2.0.1)

E*=

5Z

c

YeY-

YelrrAv

The problem is to calculate the integer coefficients cy for all irreducible components Y of Ay. These coefficients are independent of any choices up to simultaneous multiplication of all of them by —1. Suppose that an irre ducible component Y € Irr Ay, dim V = a corresponds to a decomposition (5.2.0.2)

a = ai +

\-at

of a into a sum of positive roots c*i,..., on . Then the formula 5.2.0.1 may be regarded as an assignment {a = a\-i httj}t->cy of the integer cy to every decomposition 5.2.0.2. It would be very interesting to describe this assignment in terms of combinatorics of the root system without appealing to the semicanonical basis. We obtain such a description in the An case using the Weyl group W = Sn+i. We also succeed in calculating cy when Y = Ev,ci a r e irreducible components "arising from orientations", and we calculate the examples of D4 a'nd D5. 5.3. Irreducible components arising from orientations.

QUIVER VARIETIES, BPS STATES, SEMICANONICAL BASIS

5.3.1.

25

Let us define the subvariety Oy Q Ay as follows:

Ov = U ®v,ti n where the union is taken over all orientations SI of our quiver. Since each Y = Ev,n is an irreducible component of Ay (we will call such Y irreducible components arising from orientations), we have IrrOy C IrrAy. We will split the sum 5.2.0.1 into two parts:

El = E'a + E'l where

(5.3.1.1)

E'a=

Cyey

^2

and

E

'a =

yelrrOv

Cyey

]C yglrrAv-IrrOv

5.3.2. If a root a € R+ is not simple, represent it as a sum of simple roots akl, ak2, • • •, akh , k e l , where h = bta (5.3.2.1)

a = akl + ak2 H

\-akh

in such a way that akl + Q.k2 + ■ ■ ■ + akj is a root for 1 < j < h. It follows from our construction and definitions that up to a sign (5.3.2.2)

E*a =

[...[Elki,E*akJ...Elkh}

Now we can calculate the coefficients of E'a (cf. section 3 for a similar discussion). Proposition. Up to a sign, E

'<*=

5Z Y = Ev,n€lrrOv

(Y[en(aki,akj))eY, i<3

where en is the Frenkel-Kac cocycle corresponding to SI, [FMV01, 1.3.4]. This presentation does not depend on the decomposition 5.3.2.1. Proof. Let us fix some orientation SI, and let us assume that Y = Ey,nAccording to Lusztig [Lus92, 12.13] we have a homomorphism of algebras Co -^ £o,n induced by the restriction on functions from Ay to ~Ev,a (see 2.2.3 for the definitions of £ 0 and £o,n)- Then (5.3.2.4) E*a\Y = E'a\Y = {...[Ekl,Ek2}a...Ekh}a

=

l[eQ(aki,akj)Ea, i<j

where Ea is defined by 3.3.2.1, and the last equality follows from [FMV01, 4.3.4.b]. Now it is clear that pY{E*a) = pY(E*a\Y). Then cY = pY{E*a) = pY(E*a\Y)

= J\en(aki,akj)pY(EaiQ) i<j

=

Y[eci(aki,aki). i<j

Notice that due to the formula 5.3.2.4 we know explicitly E£\o i-e-> the value El{x) at any point x 6 O C A. □

26

IGOR FRENKEL, ANTON MALKIN, AND MAXIM VYBORNOV

5.3.3. Let Q be the Dynkin graph of type An. In this case it is clear that for V € V, dim V = a € R+ any irreducible component of Ay is of the form Ev ( (j for some orientation 0 . Thus Proposition 5.3.2 gives a complete answer to our question 5.2 in this case. 5.4. Conjugacy classes of the Weyl group. 5.4.1. Recall the isomorphism ip : n —» n* (see 2.3.2), and let us take ea € ne ~ n such that ip(ia) = £*, a € R+, as in 2.4. Let us fix Y € Irr Ay and let a — Pi H h $ be the corresponding decomposition of a into the sum of positive roots. Now consider the element epx + ■ ■ ■ + epl € ne ~ n, and consider its adjoint orbit. In such a way we obtain a map from Irr Ay to the set TV of nilpotent orbits of the Lie algebra corresponding to the Dynkin diagram Q. Moreover, following Kazhdan-Lusztig [KL88] and Spaltenstein [Spal88], we can construct a map from Af to the set Cl(W) of conjugacy classes of the Weyl group W. Thus we obtain a map cl : Irr Ay —► Cl(W) for any V S V. We will use this map in the following subsection. 5.4.2. Example:

An.

Proposition. Up to a sign, E*a=

Y,

sgn(cir)ey,

YelrrAv

where sgn(cl Y) is the value of the one-dimensional sign character ofW = Sn+i on the conjugacy class c\Y. Proof. The proof is straightforward and is left to the reader.

□

5.4.3. Open question. We have the map cl : Irr Ay —> Cl(W) for all ADE quivers, see 5.4.1. Looking at the An case (Proposition 5.4.2), one may ask if in the DE case the coefficients cy are also governed by characters of the Weyl group. Unfortunately, the authors' naive attempt to replicate the An result fails already in the D4 case (however the explicit calculation for the maximal root of D\ is given below). 5.5. Example: D4. 5.5.1.

We index the vertices of the Dynkin diagram as follows:

The simple roots are denoted by a0, ati, ct2, 0:3. Let a = 2ao+ai+a2+ct3 be the maximal root. Indexing the irreducible components of Ay, dim V =

QUIVER VARIETIES, BPS STATES, SEMICANONICAL BASIS

27

o by t h e decompositions of o we obtain by a straighforward computation using t h e properties of t h e semicanonical basis: Ea = E'a + Ea ~ ^a ~ e(cto+oti) + (ao+0!2+o!3) ~~ e (ao+a2)+(ao+cu+a3) — e

(ao+a3) + (oco+cti+a2) ~ e ao+ai+(ao + («2+a3)

— e

ao + a!2 + (ao+ai+a3)

— e

ao+a3 + (ao+ai+a2)

e

+ ^ a0 + (ao+ai+a2+a3)-

5.6. E x a m p l e : £) 5 .

5.6.1.

We index the vertices of the Dynkin diagram as follows:

So

3

4

2

T h e simple roots are denoted by 00,01,02,0:3,0:4. Let a = 2ao + 01 + 02 + 2«3 + 04 be the maximal root. There are 55 root partitions of a, e.g. a = 0 0 + 0 3 + ( 0 0 + 0 1 + 0 2 + 03 + 04). Indexing t h e irreducible compo nents of A y , dim V = a by the decompositions of o we obtain by a tedious but straightforward computation using the properties of t h e semicanonical basis:

K +

e

( 2 CXO + Oil + C*2 + «3) + ( a 3 + OCi)

+

e

'

e

ai +(c*o + a2) + (a0 + a 3 ) + (a 3 + a 4 )

+

e

a i + a 2 +(a 0 + a 3 ) + a 0 +(0:3 + a 4 )

+

e

(2 ao + c*i + 0:2 + a3)+oi3 + a 4

+

e

a2 +(ao + ai)+a3 + a 4 +(ao + 0-3)

+

e

a1 + (ao + a 2 ) + (ao + 0:3)4-03 + a 4

+

e

( a 0 + «3)+ao + a 2 + a3 + a 4

+

e

(a2 + a0 + a 3 ) + (ai + ao + 03 + 0:4)

+

e

( a i + a2 + ao+a3) + (ao-l-a3)+a4

~

e

(»2 + ao + ai) + (ao + ot3) + (oc3 + a 4 )

+

e

(a2 + a o + a i ) + a o + a3+(a3 + a 4 )

(c<2 + »o + ai)+oco +2 i»3 + a 4

—

e

( a 0 + ai + a 2 ) + (ao + a 3 )+a 4

'

e

e

a2 +(ao + ai) + (ao + ci3)-i-(a3 + a4)

+

e

~

e

(ao + a2) + (ao + a i + a3) + (a3 + a 4 )

~

e

" 2 +(ao + «i + a-3)+ao + (a3 + a 4 )

+

~

e

(ao + ai) + (a2 + ao + a3) + (a 3 + a 4 )

"■" e (a2 + ao + a3) + (ao + «i -I- a3 + a 4 )

~

e

a o + ai +(a 2 + ao + a3) + (a3 + a 4 )

+

e

a i +(02 -t- ao +a3) + (ao + a3 + at)

~

e

(ao + a2)+a 3 +(ao + ai + a 3 )+a 4

—

e

a o + 0 2 + a3+(ao + ai+Qf3)+a4

e

'" (ao + a 2 )+a3 +(a 0 + ai + a3 + a 4 ) +

e

ao + az + ot3 +(ao + ai + a 3 + a 4 )

(ao +ai + a3) + (ao + d2 +

a3+a4)

a2 +(ao + ai + a3) + (ao + oc3 + a 4 )

28

IGOR FRENKEL, ANTON MALKIN, AND MAXIM VYBORNOV

i ea.3 +(«o +oci)+(ao + ai 4-"3 +a*) + ea3 4- ao 4- a i 4-(oo + a 2 t « 3 + o 4 ) "r ^ e a o +(0:3 4- a4)+(ao + « i + 02 + 03) — ^ e a o 4- ci!3 4-(ao + a i 4 012 + t»3 4 04) — ^ e (2 a 0 4- ai 4- a 2 a 3 4- 04)403) — ^ e Qi 4-(c*o 4- 02)4(00 4- 03 4- 04)-fa 3 1 ^ e ( a o 4- 03 4- 04)4-(ao 4- Qi 4- 02)403

~ e Q3 4-(«o 4- oi)4-(oo + 02 +03)4-0:4 — e a 3 4- ao 4- a i +(«o + <*2 + 03)4-04 — ^ e (oo 4- 03 4- 04)4-(ao + a i + a 2 4 0:3) + ^ e a o 4- 03 4- 04 4-(ao 4- a i 4- 02 4- 0:3) — ^ea2 4-(<*o 4- oi) + (a 0 4- 03 4 04)4113 — ^ e (a<j 4- 03 + a 4 ) + a 0 4- oc\ 4- 02 4- 03

5.6.2. While in the D5 case the coefficients in the decomposition for the maximal root are ± 1 , ± 2 , in the DQ case the coefficients are ± 1 , ± 2 , ± 4 . We expect that in the Dn case all the coefficients are the powers of 2. 5.7. More open questions. The questions of decomposing "canonically" defined basis elements of affine Lie algebra (such as Ea,E£(n), see 2.5) with respect to semicanonical basis make sense for any affine Lie algebra of type ADE. It would be very interesting to develop a purely combinatorial approach to these questions in the affine case as well. REFERENCES [CB01] [DFR00] [FM00] [FK81] [FMV01] [Gab72] [HM98] [KVOO] [KS97] [KL88] [King94] [Kron89] [Lus90] [Lus91] [Lus92]

W. Crawley-Boevey, Geometry of the moment map for representations of quiv ers, Compositio Math. 126 (2001), no. 3, 257-293. M. Douglas, B. Fiol, and C. Romelsberger, Stability and BPS branes, preprint 2000, hep-th/0002037. B. Fiol and M. Marino, BPS states and algebras from quivers, J. High Energy Phys. 2000, no. 7, Paper 31, 40 pp. I. Frenkel and V. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980/81), no. 1, 23-66. I. Frenkel, A. Malkin, and M. Vybornov, Affine Lie Algebras and Tame Quiv ers, Selects Math. (N.S.) 7 (2001), no. 1, 1-56. P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 7 1 103; correction, ibid. 6 (1972), 309. J. Harvey and G. Moore, On the algebras of BPS states, Comm. Math. Phys. 197 (1998), no. 3, 489-519. M. Kapranov and E. Vasserot Kleinian singularities, derived categories and Hall algebras, Math. Ann. 316 (2000), no. 3, 565-576. M. Kashiwara and Y. Saito, Geometric Construction of Crystal bases, Duke Math. J. 89 (1997), no. 1, 9-36. D. Kazhdan and G. Lusztig, Fixed point varieties on affine flag manifolds, Israel J. Math. 62 (1988), no. 2, 129-168. A. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford ( 2 ) , 4 5 , (1994), 515-530. P. Kronheimer, The construction of ALE spaces as hyper-Kdhler quotients, J. Differential Geom. 29 (1989), no. 3, 665-683. G. Lusztig, Canonocal bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498. G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365-421. G. Lusztig, Affine quivers and canonical bases, Inst. Hautes Etudes Sci. Publ. Math. (1992), no. 76, 111-163.

QUIVER VARIETIES, BPS STATES, SEMICANONICAL BASIS

29

[LusOO]

G. Lusztig, Semicanonical bases arising from, enveloping algebras, Adv. Math. 151 (2000), no. 2, 129-139. [Nak94] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), no. 2, 365-416. [Nak98] H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515-560. [Rei02] M. Reineke, The Harder-Narasimhan system in quantum groups and cohomol°gy of quiver moduli, preprint 2002, math.QA/0204059. [Rin90] C. Ringel, Hall poynomials for the representation-finite hereditary algebras, Adv. Math. 8 4 (1990), no. 2, 137-178. [Rud97] A. Rudakov, Stability for an abelian category J. Algebra 197 (1997), no. 1, 231-245. [Scho92] A. Schofield, General representations of quivers, Proc. London Math. Soc. (3) 6 5 (1992), 46-64. [Spal88] N. Spaltenstein, Polynomials over local fields, nilpotent orbits and conjugacy classes in Weyl groups Orbites unipotentes et representations, I. Asterisque No. 168 (1988), 10-11, 191-217. DEPARTMENT OF MATHEMATICS, YALE UNIVERSITY, 10 HILLHOUSE AVE, N E W HAVEN,

CT 06520 E-mail address: frenkel8math.yale.edu DEPARTMENT OF MATHEMATICS, MIT,

77 MASSACHUSETTS AVENUE, CAMBRIDGE,

MA

02139 E-mail address: malkinflmath.mit.edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MASSACHUSETTS, LEDERLE GRADU ATE RESEARCH T O W E R , AMHERST, MA

01003

E-mail address: vybornovflmath.umass.edu

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D I V I D E D DIFFERENCES OF T Y P E D A N D T H E G R A S S M A N N I A N OF COMPLEX S T R U C T U R E S HAIBAO DUAN AND PIOTR PRAGACZ

Dedicated to Professor Tatsuo Suwa on his 60th birthday CONTENTS

1. Introduction 2. Preliminaries, Notation, and Conventions 3. Combinatorics of Divided Differences of Type D 4. A Group-Theoretic Approach to Schubert Calculus for CSn 5. Schubert Cycles of Complex Structures on R 2 n References

31 34 35 45 52 58

1. INTRODUCTION

Divided differences are discrete analogues of derivations. They were introduced by Newton in his famous interpolation formula in "Principia Mathematica" (1686). Their importance in geometry was shown in the early 1970's by Bernstein-Gelfand-Gelfand [BGG] and Demazure [De] in the context of Schubert calculus for generalized flag varieties associated with semisimple algebraic groups. More recently, simple divided differ ences, interpreted as correspondences in flag bundles, were used by Fulton in his study of the classes of degeneracy loci. Divided differences admit still another interpretation as Gysin maps in the cohomology of flag bun dles associated with semisimple algebraic groups (cf., e.g., [P2]). We refer to the lecture notes [FP] for a systematic discussion of these issues. The case of SL(n) has been recently developed extensively by Lascoux and Schutzenberger (cf., e.g., [LSc]), and serves nowadays as an important and useful tool for multivariate polynomials (cf. [L2]). The Grassmannian of complex structures parametrizes orthogonal auto morphisms of the Euclidean space R 2 n whose square is the minus identity. Equivalently, it parametrizes minimal geodesies from the identity to the mi nus identity in the orthogonal group 5 0 ( 2 n , R ) [Mi]. This space is usually denoted by CSn. It played a significant role in several important achieve ments in topology: 31

32

HAIBAO DUAN AND PIOTR PRAGACZ

• in the investigation of orthonormal vector fields on spheres by Hurewicz and Adams; • in the study of the existence of complex structures on even dimen sional spheres by Borel and Serre; • in the Bott's discovery of the eight-periodicity of homotopy groups of the stable real orthogonal groups. Also, CSn serves as the classifying space of all complex bundles whose real reduction is trivial, by a result of the first author [Dul]. The goal of the present paper is to develop in a systematic way a Schubert calculus for CSn. We hope that it will be useful also for topologists. The space CSn has two connected components, each isomorphic to the homogeneous space SO(2n,C)/U(n)

or

SO(2n,C)/P,

where P is the maximal parabolic corresponding to omitting the "right end root". This space is a connected component of the Grassmannian of all isotropic subspaces of C2n w.r.t. to the orthogonal form induced by the scalar product, and as such, it is also known as the orthogonal Grassmannian. With the help of the group-theoretic description, we can use the char acteristic map of Borel [Bo], and - via the theory of Bernstein-GelfandGelfand [BGG] and Demazure [De] - divided differences of type D to study the intersection theory on the space in question. In order to make the work with the characteristic map efficient, one needs a proper family of "invari ant" polynomials that are well suited to divided differences, and also to geometry/topology at the same time. A result of the second author [Pi] identified Schubert classes in the ho mogeneous space SO(2n,C)/U(n) with suitable Schur P-polynomials. In loc. cit. this identification used a geometric argument, namely an isomor phism 50(271, C)/C/(n) ~ S O ( 2 n - 1, C)/U(n- 1), and an identification of the Schubert classes for the latter Grassmannian with Schur P-functions. (This last identification was based on comparison of the Pieri-type formulas from [HB] and [Mo].) In the present paper we revisit the identification for the Schubert classes for SO(2n, C)/U(n) with Schur P-functions via a direct group-theoretic argument based on the calculus of divided differences of type D. More pre cisely, we give a group-theoretic proof of a Pieri-type formula that is based on some vanishing results for operators composed of divided differences of type D and simple reflections from the Weyl group of type D. These last results form the most technical part of the present work. Our proof of the Pieri-type formula follows a strategy for deriving similar formulas for var ious homogeneous spaces worked out by Ratajski and the second autor in a series of papers summarized in [P2]. This particular proof was promised in [PR2] - a paper that is now under revision. The proof uses esentially an

DIVIDED DIFFERENCES OF TYPE D

33

iteration of the Leibniz-type formula for a simple divided difference applied to the product of two functions. Combining the Pieri formula with a combinatorial lemma of Schur [S] for the projective characters of the symmetric groups, we get a formula for the degree of Schubert varieties in SO(2n, C)/U(n). (Occasionally, we discuss some alternative derivations of the lemma of Schur with the help of a specialization result from [DP].) We remark that there exists now a refinement of Schur P-functions that seems to be even better adapted for some aspects of geometry. These are the so called P-functions of [PR1], which are modeled on Schur P-functions. In [LP], Lascoux and the second author worked out a connection of orthogonal divided differences to P-functions using vertex operators. This has led to orthogonal Schubert polynomials that are useful in various cohomological computations (cf. a recent work of Kresch and Tamvakis [KT,T2], and Buch [BKT]). After presenting the Schubert varieties in a group-theoretic way, we also describe them via Schubert-type conditions relative to some flag of linear subspaces, and finally we study them in terms of complex structures. To this end, we are guided by the Mahowald-Vassiljev-type formula ([DV,V]): Hi(CSn) = ©jf=0i?i_fc(fc_i)(Gfe(CTl)), where Gfc(Cn) is the Grassmannian of all complex fc-planes through zero inCn. We end the paper by illustrating how the Schubert calculus developed here can be used to solve problems about enumeration of complex struc tures which satisfy some natural conditions of "partial overlaping" with a certain number of complex structures in general position in R 2 n . One of the applications leads to an interesting algebraic conjecture about homomorphisms between the cohomology ring of CS+ and that of the Grassmannian Gfe(C).

Acknowledgements. This work was partially supported by grant No.5031 of French-Polish cooperation C.N.R.S. - P.A.N. (the "French connection" of this grant being A. Lascoux whose comments clarified our Remark 4.11). While revising the paper and preparing its final version, the authors were supported by Polish KBN grant No.2 P03A 024 23. The present paper is a revised version of the Max-Planck-Institut fur Mathematik in Bonn Preprint 2001-107. The second author is grateful to the MPIM for excellent working conditions during the preparation of the first draft of this paper.

HAIBAO DUAN AND PIOTR PRAGACZ

34

2. PRELIMINARIES, NOTATION, AND CONVENTIONS

We start with some algebraic preliminaries on even orthogonal groups. We fix a positive integer n. Suppose that H — SO(2n, C) is the orthogonal group (of type Dn) over the field of complex numbers. Let us use the following notation: B - a fixed Borel subgroup of H, T C B - a fixed maximal torus, 1Z - the root system of H associated with T, £ - a set of simple roots of % associated with B, W - the Weyl group of (H, T). In a standard Bourbaki [Bu] realization, we have: K = {±ei ± e 3 : l < i < j < n } c R

n

K=iReu

=

£ = {ei - e 2 ) . . . ,e n _i - en,en-i +en}, 1 W = SnY Z ? - . A typical element of W can be written as a pair (r, e), where r e Sn and e = ( e i , . . . , en) is a sequence of elements of Z2 = {—1,1} such that # { i : €i = —1} is even. Multiplication in W is given by ( ¥ ) - ( r ' , f ' ) = (Tor',{), where "o" denotes the composition of permutations and Si = eT'(i) ' e'iThe following lemma can be easily verified (and is pretty well-known). For w € W, let l(w) denote the length of w taken w.r.t. to the above S. L e m m a 2 . 1 . For any w € W, l(w) is n

y^ # { j : j > i and w(j) < w(i)}+ ^ J 2#{q : q> p and w(q) > w(p)} . We will use the "barred-permutation" notation, indicating by a bar a place in the permutation w = [w(l),... ,w(n)] where e, = — 1. The following lemma, that is easy to prove from Lemma 2.1, gives us the lengths of some barred permutations important for this paper. L e m m a 2.2. Let y\ < ■ ■ ■ < yn-k and Zk > ■ ■ ■ > z\ be sequences of integers that are complementary in { 1 , . . . , n } . Assume that k is even. Then in W we have k

l([yi,y2,---

,yn-k,Zk,zk-i,---

,zi\)

=

^T(n-ZJ).

The barred permutations of this type form the poset, denoted W*, of the minimal length left coset representatives of Sn in W. Our terminology and all unexplained notation concerning partitions will follow [Ma]. We set p(k) := (k,k — l,... , 1), a "triangular partition" of length k.

DIVIDED DIFFERENCES OF TYPE D

35

Given a strict partition a = (oti > • • • > ati > 0) C p(n — 1), we set a+ := (on + 1, a2 + 1, • • • , « ; + 1) if Z is even, and a+ := (ai + l , a 2 + 1,. ■ ■ ,ai + 1,1) if I is odd. Note that a + is of even length. Given a strict partition p, = (/xi > [ii > • ■ - > Hk > 0) C p(n) of even length fc, we associate with it the following element w^ of W*. We set W/x : = bi,2/2,--- ,y„_fc,n + l - / t i f e , n + l - / i f c _ i , . . . , n + l - Mi] • Note that /(wM) = |JU| — k by Lemma 2.2. (Recall that the symbol |//| denotes the sum of the parts of /i.) Setting for a strict partition a C p(n -1), A := a+, we have Z(u>^) = |a|. Finally, we adopt a convention that all homology or cohomology groups in the present paper are taken with integer coefficients. 3. COMBINATORICS OF DIVIDED DIFFERENCES OF T Y P E D

We define simple divided differences of type Dn which are operators 9j : Z[X] -* Z[X], i = 1,... ,n, of degree - 1 acting on the ring of polynomials Z[X] where X is a fixed set of indeterminates X = {xi,X2,.-- ,xn}. To this end, we denote by Sj, 1 < i < n — 1, the transposition [ 1 , . . . ,i-l,i

+ l,i,i + 2,... ,n] € Sn C W,

acting on X by interchanging Xi and xi+\. Moreover, let sn = [ 1 , . . . , n - 2 , n , n - 1] be the reflection which transposes xn-\ with xn and changes the signs of both the variables. The remaining variables are invariant under the action of these transpositions. This action is extended multiplicatively to the ring Z[X]. Note that sn commutes with Si, i ^ n — 2, and Sn—2 * Sn • S n _ 2

==

Sn ' Sn—2 ' Sn -

Simple divided differences of type D are defined as follows:

di(f) = {f -Sif)/{xi-xi+i),

i= 1,... , n - 1;

9n(f) = (/ - s n / ) / ( x n _ i + xn). The algebra generated by the divided diferences di, 1 < i < n — 1 that goes back to Newton, has been recently extensively studied by Lascoux and Schiitzenberger [LSc]. Here we investigate the effect of adding one extra divided difference dn (which changes the picture in a drastic way). For every / , g G Z[X] and any i, we have: di(f-g) (a Leibniz-type formula).

= f-(dig)

+ (dif)-(si9)

(1)

36

HAIBAO DUAN A N D P I O T R P R A G A C Z

For a given a = (an, an-\,... erating function:

,02, a\) G {—1,0,1}™, we define the gen n

Ea^Hil+CLiXi).

(2)

i=\

In particular, for a = ( 1 , . . . , 1), the resulting generating function, denoted by E, is the generating function for the elementary symmetric polynomials ei(X) = ei(xi,... ,xn), i = 1 , . . . ,n. Lemma 3.1. (a) We have Si(Ea) = Ea>, where 1 _ («n. • • • ,aj + 2,ai,
i < n, i = n.

,n-l,

di(Ea) = d-Ea4f

ai=ai+1+d

(d = - 2 , - 1 , 0 , 1 , 2 ) ,

where a! = ( a n , . . . , 0 , 0 , . . . , a,\) is the sequence a with a^+i, a* replaced by zeros. (c) dn(Ea) = (an + a n _ i ) ■ ■E(o,o)a„_2,...,a1)In particular if A is a composition of some s- and d-operations, then for every a, A(E&) = (scalar) ■ E&', where a' is uniquely determined if this scalar is not zero. Proof. We prove e.g. (c). We have, with a' = (0,0, o„_2, • • • , a\), On\&&)

0)(i (1 + + anxn) - (1 - an-ixn){l

=

-

anxn-i)

— Xn—1 ~r Xn

• &&'

(q n _! + a n )(a: ra -i +x r a ) —

— Xn—\ -f- Xn

• £>a! — [0-n + dri-l)

■ &a> ,

as desired. D We now recall the following fact from [BGG] and [De]. For any w £ W and any reduced decomposition w = s^ ■ ■ ■ Sj, one can define dw = d^ o • • • o dii - an operator on Z[X] of degree —l(w). In fact, since divided differences satisfy braid relations, dw does not depend on the chosen reduced decomposition of w. Suppose a strict partition [i with even 1(/J.) is given. Let us use the following coordinates for boxes in the Ferrers' diagram £>M of fi:

37

DIVIDED D I F F E R E N C E S O F T Y P E D

We associate with /x a certain distinguished reduced decomposition of Wp e W. To this end, let us modify the diagram D^ in the following way. Remove one box from each row of D^ : from rows with even numbers remove the box in the n-th column, and from rows with odd numbers remove the box in the (n — l)-st column. We display the removed boxes in the picture using the symbol x and o

a

denote the so obtained set of boxes by £>M. For example, .0(8,7,4,2) is:

Assume now, that a subset D c D^ is given. A box belonging to D will be called a D-box and a box from the difference D^ \ D will be called a ~ D-box. Definition 3.2. Read DM row by row from left to right and from top to bot tom. Every D-box (resp. ~ D-box) in the i-th column gives us Sj (resp. di). Then d® is the composition of the resulting Si's and di's (the composition written from right to left). Definition 3.3. Read DM. Every D-box in the i-th column gives us s,. ~D-boxes give no contribution. Then, rr> is the word obtained by writing the resulting s^ 's from right to left. ( In other words, one obtains ro by erasing all the di's from d®.) For example, for n = 9 and /J = (8, 7,4,2), 9 8 7 6 5 4 3 2 1

d^ = ds o SQ o dy o dg o 83 o d4 o ss o 8Q o S7 o ss o 02 o d^ o S4 o 9 5 OSQ O

r&

S7 o Sg

= S6 ■ S5 ■ S7 ■ SS ■ S4 ■ S6 ■ S7 ■ Sg . o

One can easily prove that for D = D^, we have ro € R{wn) - the set of reduced decompositions of w^. This is our distinguished reduced decomposition of wM. For example, for n — 9 and fi = (8,7,4,2), 9 8 7 6 5 4 3 2 1

HAIBAO DUAN A N D P I O T R P R A G A C Z

38

w^ = sg • SQ ■ si ■ s9 ■ S3 • s4 ■ s5 ■ se ■ s7 ■ Sg • s2 ■ S3 ■ s 4 • S5 • s 6 - s7 ■ sg .

Let now / i b e a strict partition with even length. We will examine subsets D C D^ for which dj?(E) = 0 (we say: " D causes the vanishing"). In many computations in this section, we will apply compositions of the operators of boxes of £>p to the generating functions Ea. With the following example we illustrate how such operators act. Example 3.4. Let n = 9. We apply the operators of boxes from left to right to Ea, where a = (ag, as, a7, a$, a$, a4, a3, a 2 ,01), and obtain E^; "• " denotes a D-box and empty boxes are ~D-boxes. We give 2 examples of the action of the operators associated with a row in D^: 9 8 7 6 5 4 3 2 1

l*lxl*l*l*l*l*l*l

a

1

' =

(-a&,a7,ae,as,a4,a3:a2,0,0),

9 8 7 6 5 4 3 2 1

lxl*l*l*l

1*1*1

1*1

a'=

(as,a7,a6,0,a4,a3,0,ai,0).

Lemma 3.5. The following configurations of ~D-boxes (noted by "o") in Dfj, give the vanishing: (1) (A three ~D-box configuration) o

f of-

• • f o

(2) (.4 three ~D-box configuration) o 9

O

X

x

?■

? o

DIVIDED DIFFERENCES OF TYPE D

39

(3) (A three ~D-box configuration)

(Above, "?" can be o,«, or x; and the skew directions are all parallel to the antidiagonal.) Proof. Direct calculation using Lemma 3.1. □ Let us fix an element w = \yx,y2, ■ ■ ■ ,yn-k,Zk,Zk-i, ■ ■ ■ ,zi] S W*. Recall that k is even. We treat a given reduced decomposition w = SJ! ■ Si2 ■ ■ ■ Sit as a sequence of simple transposition operations, which pro duces the element in question from the identity permutation: [2/1,2/2,-•• ,2/„-fc,zjfe,Zfc-i,... ,zi] = (•••([1,2,...

,n]-sil)---)-sil.

In the following, the simple transpositions involved will be called the "sj h operations" (h = 1 , . . . , /). Proposition 3.6. We have the following two possibilities for the action of Sih -operations on the z 's: (1) If ih — n then this operation is: [... ,z,z'} where z — zv-\

-+ [...

,z',z],

and z' = zp for some even p.

(2) If ih < n, then this operation is: [... ,z,x,...]

-> [...

,x,z,...]

where x ^ Zj for j = 1,... , k. Proof. We must transpose each pair (zj,yi), for Zj < yi, at least once, because the y's preced the (barred) z's in w. Also, we must transpose each pair (zi,Zj), for i < j , at least once, because the (barred) z's appear in w in an descending order. In sum, we need at least

J2*{(zi'Vi) s

:z

o
:i<

J}

ih-operations to reach the sequence w. But by Lemma 2.1 this last number is equal to l{w). This means that the mentioned transpositions exchaust the family of all s, h -operations under consideration. As a consequence, no Sihoperation, for ih < n, interchanges two (bar-free) z's (on their way towards

40

HAIBAO DUAN AND PIOTR PRAGACZ

the end of the permutation). Moreover, we see that exactly fc/2 Sihoperations with ih = n appear. This implies immediately both assertions of the proposition. □ Now we assume that A is another strict partition, that D C D^, and that TD € R(w\). Suppose that a D-box appears in the i-th column where i < n. We define the mark of this box to be p, if the corresponding Si h -operation acts on the i-th and (i + l)-st places as follows: [. . . , Zp, X, . . . \ —> [. . . , X, Zp, . . ■ J ,

where x =^ Zj , j = 1 , . . . , k. AD — box in the n-th column has mark p — 1 if the corresponding (sih = sn,)-operation acts via [■ • • i zp—l)

z

p\

~~? [■ - ■ ? z p i Zp— i j .

In particular, boxes in the n-th column have only odd marks. In the fol lowing lemma, we collect some simple properties of marks. L e m m a 3.7. (1) (Connectedness) The D-boxes with a fixed mark in one row form a connected set. (The boxes marked by x are treated as empty boxes). (2) (Separation) In a fixed row, the two sets of D-boxes equipped with different marks are disconnected (i.e. there is at least one ~D-box between them; a box marked by x is not treated as as ~D-box). (3) The sequence of boxes with odd mark p is of the form: (tn,n),(tn-2,n-2),...

,(tZp,zp),

where p < tn < tn_2 <■■■
,(tZp,zp),

where p < tn-i < tn-2 < ■■■
,(i,j)

form an entire ribbon. The operation transforms them to (i + 2,n),(t + 2 , n - 2 ) , . . . ,(i + 2,j). Let i be even and suppose that the boxes (i,n-

l),(i,n-2),...

,(i,j)

DIVIDED DIFFERENCES OF TYPE D

41

form an entire ribbon. The operation transforms them to (i + 2 , n - l ) > ( t + 2 , n - 2 ) , . . . , ( t + 2 , j ) . (We assume that the {i + l)-st and (i + 2)-nd row are empty before this operation.) For example, the ribbons

can be pushed down to

• ("Breaking a ribbon") Let j < n — 2. The operation transforms a final segment (i,j),{i,3 - I ) , - - . of a ribbon to empty boxes (i + l,j),(i

+ l,j-l),...

,(i,h) ,(i +

l,h),

provided (i + 1, j + 1) is a ~D-box, or it is x and (i + 1, j + 2) is a ~£>-box. The box (i,j) (before the operation) is called the breaking box. For example, for a breaking box a, b a ~D-box, or b = x and C a ~£>-box: ? ? ? 1 1 1 1 ? c fa can be broken at 0 and transformed to ?

?

?

Using the braid relations in W one easily shows that if ro £ R(w\), then, after breaking a ribbon in D, we get D' such that rr,/ g R(w\). In the case of the push down operation, it is clear that we get D' with ro — ru1 ■ Note that any configuration of boxes D C Da such that m e R(w\) can be o

o

obtained from D\ C Du by a sequence of operations of the above described two types. Definition 3.9. ( "Maximal deformation" of D\ C D^)

42

HAIBAO DUAN AND PIOTR PRAGACZ

• Pick the lowest ribbon. Push it down as many times as possible. Then choose the leftmost breaking box on this ribbon (if it exists) and break the ribbon. o

• Pick a ribbon and suppose that lower ribbons in D\ have been already deformed. Push down this ribbon as many times as possible. Let o be the leftmost breaking box on the ribbon. Break the ribbon at a as many times as possible. Then choose the next leftmost breaking box b and break the ribbon at b as many times as possible etc. For some examples of maximal deformations of diagrams, see Example 4.7. Proposition 3.10. Let D c D^ be such that rD € R(wx). o

Ifdj?(E)

^ 0

o

then D is the maximal deformation of D\ C D^ . Proof. The proof is by descending induction on the mark of a ribbon. Pick the ribbon with mark p. Assume that the ribbons with marks p+1,... , l(\) have been already maximally deformed. Suppose that we have either a possibility of pushing down of a ribbon or breaking a ribbon. In the case of the former operation we will refer to boxes of the three involved rows; in the case of the latter operation, we will refer to the two involved rows. We note that • any box directly to the right or directly below of the rightmost box of a row of the (deformed) ribbon is a ~D-box; • any box directly to the left or directly above of the leftmost box of a row of the (deformed) ribbon is a ~D-box; • ~ D-boxes in the n-th or (n — l)-st column cannot be supplied by marks smaller than p. This implies that if we not perform the operations in a maximal way, then we will either obtain a configuration o o

?

•

•

■ ? o

or, we will get one of the following two possibilities: • x x b c a x • • • • •

x a x b x c • • • • •

where O, b and C are ~ D-boxes. By Lemma 3.5 all these three config urations of ~ D-boxes cause the vanishing. The obtained contradiction o

o

means that the maximal deformation of D\ C D^ is necessary to avoid the vanishing. □ As a corollary of this proposition, we get the following two results bound ing the size of /z w.r.t. A, if one wants to avoid the vanishing. We will need

DIVIDED DIFFERENCES OF TYPE D

43

some additional notation. Given a strict partition \i of even length I, we de note by /j," the strict partition (/ii — 1 , . . . ,/x; — 1). Note that (/i~) = fi. o

We also set l(fi) := l(fi~) ■ Setting for a strict partition a C p(n — 1), A := a+, we have /(A) = 1(a) = #£> a = #JDA. o

o

Proposition 3.11. //, for the maximal deformation D of Dx C D^, one has d$(E) + 0, then l(fi) < l(\) + 1. o

o

Proof. Suppose that /(//) > Z(A) + 2. Pick the highest, say i-th, row which contains a ~Z)-box in the n-th or n — 1-st column. This is the highest row from which some ribbon has been pushed down in the maximal deformation, or, if there was no pushing down, it is the row with number /(A) + 1. Since o

o

Kt1) > Kty + 2 and by the nature of maximal deformation, we see that the (i + l)-st row contains also a ~D-box in its n — 1-st or n-th row respectively. After breaking some higher ribbons we get the following configuration of ~Z)-boxes:

The boxes marked by O exist and they are ~£>-boxes (because \i is a strict partition). By Lemma 3.5 we get the vanishing - contradiction. □ o

o

Proposition 3.12. Assume that l(fi) < Z(A) + 1. If for the maximal deforo

o

motion D of Dx C D^ one has df?(E) ^ 0, then D^- \Dx- is a horizontal strip. Proof. Suppose that Aj < /ij + i for some i. We can assume that for some 3
but \j >

fj,j+i.

After the maximal deformation, we get in the consecutive rows with num bers i + 1, i,... , j + 1 the following configuration of ~D-boxes:

where a displays a ~D-box. We get the vanishing by Lemma 3.5 - contra diction. □

44

HAIBAO DUAN AND PIOTR PRAGACZ

The maximal deformation is obtained by breaking each row such that Aj = Hi+i, at one breaking point. o

o

Among the connected components of D^ \ Dx we have those which do not meet the n-th column: they are ordinary horizontal strips [Ma]. Those which meet the n-th component are of the form:

(3)

Note the following particular case of (3): x • • • • • • • o x o • • ■ o| x

(4)

(By "|" we visualize the end of a row.) After the maximal deformation an ordinary horizontal strip becomes

In turn, configuration (3) which is not of the form (4), is deformed to:

DIVIDED DIFFERENCES OF TYPE D

45

Of course, configuration (4) does not change under the maximal defor mation. Proposition 3.13. Suppose that A c n C p(n) are strict partitions such o

that Dp- \ D\-

o

is a horizontal strip (in particular, l(p) < l(X) + 1). Let o

o

Then djf(E) = 2m, where m

D be the maximal deformation of D^ \ D\. o

is the number of connected components of D^ \ D. Proof. Different connected components of Dy, \ D lie in separate rows and separate columns. Let us number these components from top to bottom. o

Pick a connected component of D^ \ D. The part of d® associated with the boxes in the rows preceding the rows of the component, transform E into 2m Ea, where m! is the number of components preceding the given one. If the first row of its appearance has odd (resp. even) number, then a = ( 1 , 1 , . . . , 1, * , . . . , *) (resp. a = (—1,1,... , 1, * , . . . , *) ) and the car dinality of displayed ± l ' s is the length of the first row of /i supporting the component, the count including the box x. In turn, the operators of rows supporting the component transform E& to 2E&* for some a'. The multiplic ity 2 comes from the highest leftmost box of the component; the operators of all remaining boxes give the multiplicity 1. If such highest leftmost box lies in the h-th. column where h < n, then one gets the multiplicity 2 by applying dh to Eh where b = (... , —1,1,...), the displayed entries being in (h + l)-st and h-th places. If the component is of the form (4), then we get the multiplicity 2 by applying dn to Ec where c = ( 1 , 1 , . . . ) . This proves the proposition. □ We summarize the results of this section in the following theorem (observe that the connected components of D^ \ D are in one-to-one corre spondence with the connected components of D^- \ Dx-)'O

Theorem 3.14. Let A C f t C p(n) be strict partitions. Then for D c D^, o

one has d^{E) ^ 0 iff D^- \D\-

is a horizontal strip (in particular l(fi) <

O

Q

O

Z(A) + 1), and D is obtained by the maximal deformation of D\ C D^. In this case, d®(E) = 2m, where m is the number of connected components of D„-\DX-. 4. A G R O U P - T H E O R E T I C APPROACH TO SCHUBERT CALCULUS FOR

CSn We first introduce some notation. Recall that H = S0(2n,C) and B C H is a Borel subgroup of H. We denote by P - the maximal par abolic subgroup of H containing B and corresponding to the subset S of

HAIBAO DUAN AND PIOTR PRAGACZ

46

simple roots minus the right end root, by F - an "isotropic" orthogonal flag manifold H/B, and by G - the orthogonal Grassmannian H/P. Moreover, the Schubert variety Xw, w e W, is defined as the closure of the Schubert cell B~wB/B in H/B (B~ is the opposite Borel subgroup to B). We record the following well-known result: Lemma 4.1. Xw is a {closed) subvariety of H/B of (complex) codimension l(w). Let X = {xi,... , xn} be a sequence of variables. For brevity, we denote also by the symbol Xw the class of the variety Xw in H2l^w\F). Let a C p(n — 1) be a strict partition and put A := a+; one has Xwx e H2W(F). Since w\ € W*, it follows from [BGG] that Xwx belongs already to H2^(G) C i ? 2 | a | ( F ) . Let us denote this element in # 2 W ( G ) (as well as the representing it Schubert variety in G), by aa. There exists a surjective ring homomorphism c : Z[1/2][X] -» H*(F) (called the Borel characteristic map) such that for a homogeneous / g Z[X] one has

c(/)=

E

9w{f)Xw.

(5)

l(w)=degf

(The original Borel's definition [Bo] of the characteristic map was different; the present description comes from [BGG] and [De].) Note (cf. e.g. [Bo]) that the ring H*(G) can be identified algebraically as H*(G) = Z[Xf«/(ei(X2),i = l,...,n-l;x1x2---xn), 2 where X = (x\,... ,a£). We have also another identification of H* (G) stemming from [Dul] and [PI]: Lemma 4.2. Let S be the tautological rank n subbundle on G. The Chern classes Ci(S) are all divisible by 2, and one has the identification o~i = ^Ci(S*) for i = l , . . . ,n — l. Moreover, H*(G) = Z[
1 n — 1, the relations Ri are given by Ri = U2 - 2(Tj_ 1 CT i+ i + 20i-2<Ji+2

+ 2(-l)i"1<71CT2i-l + ( - 1 ) V 2 < .

It turns out that after restriction to Z[X] 5n Z[l/2], the map c goes onto H*(G), and we have the following fact. Let, from now on, er = er(X) denote the r-th elementary symmetric function in X = {xi,... , xn}. Lemma 4.3. For every r — 1,... ,n— 1, one has c(e r ) = 2ar. Proof. We have <9„_ r ---9 n _ 2 d„(e r ) = 2.

DIVIDED DIFFERENCES OF TYPE D

47

Any other divided difference operator of degree r applied to er gives 0. This implies the assertion. □ For a strict partition a C p(n — 1), we choose a homogeneous fa £ Z[1/2][X] such that c(fa) = aa. Then, for w € W* with l(w) = 1(a), one has dw(fa) 7^ 0 iff w = w\ and dWx (fa) = 1 for A = a+. We want to find the coefficients dp in the expansion c(fa ■ er) = ^2 dPaP •

(6)

Proposition 4.4. In the above notation, setting p, :— f3+, one has

o

where the sum is over all D C D^ such that rp € R(w\)

(here, A = a+).

Proof. We have dp = dWit(fa ■ e r ), and dW[i = d^. The integer dp = &u(fa ' e r) is computed by a consecutive application of the Leibniz-type formula (1): we apply only the d^s (and the identity operators) to fa, and both the Si's and dj's to the factor er. We get d

0=

^2drD(fa)-d^(er), o

o

the sum over all D c D^. The summand corresponding to a subset D C is not zero only if # D = d e g / a and #(£>M \D) = r. By the choice of dro(fa) = 0 if TD £ R(w\), and equals 1 if ro G R(w\), and thus we the desired equality.

D^ fa, get D

Remark 4.5. This use of an iterated Leibniz-type formula to compute the multiplicities dp stems from a series of papers of Ratajski and the second author (cf [P2]). It was also known to Kostant and Kumar - see [KK]. Combining this proposition with Theorem 3.14, and taking into account Lemma 4.3, we get a group-theoretic proof of the following result (that is refered to as a "Pieri-type formula"): Theorem 4.6. Let a C p(n — 1) be a strict partition. r < n — 1,

Then for any 1 <

aa-ar = J22m" °P> P where the sum is over all strict partitions [3 C p(n—l) such that Dp\Da is a horizontal strip of length r and mp is the number of connected components of Dp \ Da minus 1. (Cf. also [PI, Theorem 6.17'].) Example 4.7. Let n = 8. We examine the product 05,3 • 04: •

•

•

• • •

•

•

times

0 0 0 0

HAIBAO DUAN AND PIOTR PRAGACZ

48

On the LHS we depict the /3 's; on the RHS we display the unique D C D^ such that dj?(E) ^ 0: and

and

and

and

and

and x

Thus we get:

"5,3 • er4 = o-7>5 + 4er7i4ii + 2
DIVIDED DIFFERENCES OF TYPE D

49

So by (i), deg(*a) = aa ■ o-S^-W-W . We invoke Schur P-functions P\ = P\(X) of [S] whose definition reads: (1) For a nonnegative integer i, Pi := ^sx, where the sum is over all hook partitions A of i, and sx denotes the corresponding Schur S-function (cf., e.g., [Ma]). (2) For integers i > j > 0, J2 (-V9Pj+«Pi-,

P(id) := PiPj + 2

+ (-l)* +, ><+* ■

(3) For a strict partition A = (Ai,... , Afc) written with an even k (byputting Afe = 0 if necessary), P

:

{\i,...,\k)

=

Pi P

[ {\P<Xq)]l
,

where Pf denotes the Pfaffian. See [S], [Ma], [HH], and [PI] for more on Schur P-functions. Sometimes it is more handy to work with Schur Qfunctions defined by Qx = Q\(X) = 2l^P\ for a strict partition A. Comparing the Pieri-type formula for P-functions [Ma,III.8.15] (extracted in [PI] from [Mo]) with Theorem 4.6, we get that deg
p«("-l)/2-|"l

or the coefficient of P a in pn(n-l)/2-|a|

= p\a\

_

Here a is the partition whose part complement the parts of a in { 1 , . . . , n — We define for a partition 7 = (71,72, • • •),

7!

t^ii+v

where 7! = 71172! •••. Proposition 4.8. One has deg(cra) = g1 for 7 = a. Remark 4.9. Certain special cases of this formula were obtained by Hiller [Hi]. Some related computations were performed by Tamvakis [Ta2] in the context of heights of homogeneous spaces in arithmetic intersection theory. The proposition follows from the following lemma due essentially to Schur [S]. Lemma 4.10. One has the sum over strict partitions 7 of k.

50

H A I B A O DUAN AND P I O T R P R A G A C Z

Proof. We give here a proof using a specialization result from [DP] and the following formula (8). Let Pi(X) = x\ + ■ ■ ■ + x\ be the power sum. For a partition p, = (/xi,/i 2 ,.. •) we set p^X) = Y\.iVm{X) and zM = Yli>iimifni\, where m^ = # { j : p,j = i}. Moreover, by an odd partition we understand the one whose all parts are odd. Let Y = {t/i,... ,yn} be another set of indeterminates. Then we have [Ma, III.8.13], [HH, Cor.7.15]:

PX(X)QX(Y) = J2 2l^z-%(X)Plt(Y).

£ A strict

(8)

/z odd

We use the following specialization. We set pi(Y) = 1/2 and piiY) = 0 for i > 2. Using

Qi{Y) = J2

z-^^Pv(Y)

v odd

(Ma, p.260], [HH, (7.9)), we see that under this specialization, we have Qi(Y) = \/i\. The following equality was proved in [DP]: via this special ization, for a strict partition A, Qx(Y)=gx/\\\l.

(9)

Therefore the specialization under consideration transforms equation (8) into the assertion of the lemma. □ Remark 4.11. As a matter of fact the key in the original Schur's calcu lation [S] (see also [Ma, p.267] and in more detail [HH]) is the proof of the following equality: for a strict partition 7 = (71 > • • • > 7/ > 0),

(10)

where 7W is the strict partition obtained from 7 by subtracting 1 from the i-th part 7$ 0/7. The original argument rests on the expansion into partial fractions of the function

(2t i ) n f t + 7 i ) ( ^ ~ 7 i ~ 1 )

(t + 7 i - l ) ( * - 7 < ) '

Here is another way of obtaining (10) for those who prefer Lagrange interpolation to expansion into partial fractions. Suppose that 7 1 , . . . ,7/ are I indeterminates. We start with the equation: (71 + • ■ ■ + jOjJdi

=^(-1)^7,

-

ij)

n

( 7 i -7,)n(7 P +7i),

(11)

51

DIVIDED DIFFERENCES OF TYPE D

which, as Lascoux points out, is exactly the content of the Lagrange inter polation for 71 H + 7; (cf.[L2]). (Equation (11) is easy to prove, e.g. by showing that its RHS is skew-symmetric.) Letting Qx be a function in the 7 's given by the expression for Q\(Y) in (9), we rewrite (11) as 1

(71 + -"- + 7<)Q7 = I> 1 L ) i "" 1 ^r,-iQ7 l ,...,-»-i,7i + i,..,7. ■

(12)

We now record the following identity for general Q-functions that follows rather easily from their definition by induction: l

l

2__,\^) i=l

Qii-lQ-yx,...

,7i-ii7»+i.--- .71

=

/ ;V7i J=l

Tj-li-.7i ■

\*-^)

Comparing (12) and (13), we get 1

(71 + • • • + lOQ-y = X ! £ 7 1 , - ,7,-1

7< '

3=1

which gives (10). R e m a r k 4.12. Here is still another derivation of the lemma for the reader knowing Hall-Littlewood functions: combine [Ma, Ex.III.8.1 p.259; formula III.7.1, p.246; and Ex.III.8.12 p.266]. Up to now, we have considered Schubert varieties as purely grouptheoretic objects. We end this section by recalling their interpretation in terms of Schubert-type conditions. This description is a recollection from [LSe] and [PI], and we will need it in the next section. Let U be a 2n-dimensional vector space endowed with a nondegenerate orthogonal form £ : U x U -» C. Consider Z = {L c U : L is maximal isotropic subspace in U} . This subvariety is canonically embedded in the Grassmannian Gn(U). This last variety has Schubert (sub)varieties which are denned w.r.t. to flag U1cU2C---cU2n

=U

(where dim(£/j) = i), in the following way: Given a sequence 1 < ix < ■ ■ ■ < in 5: 2n, we set «(»!, ...,in)

= {L€ Gn(U) : dim(L

nUir)>pVp=l,...,n}.

One has d i m i l ( i i , . . . ,in) =i\-\ \- in - n(n + l ) / 2 . It is known that Z has two connected components which are isomorphic to G = H/P. Let vi,... , vn,wi,... , wn be a basis of U such that

HAIBAO DUAN AND PIOTR PRAGACZ

52

£(vi,Vj) = £(wi,Wj) = 0,£(vi,Wj) = £(wj,Vi) = Sij. Let V* be the vector space spanned by the first i vectors of the above basis. Then the Schubert varieties in Gn(U) (determined by the flag V\ C • • ■ C V-in = V) which give rise to the Schubert varieties in G (in the sense of [LSe] and [PI]) are in dexed by the sequences (i\,... , in) where ip ^ 2 n + l — iq for p, q = 1 , . . . , n, and if k denotes the largest number such that i\, < n, then n — k is even. Let us denote by fl[z'i,... ,ik] the Schubert variety in G determined (via restriction to G) by this Schubert variety in Gn(U), that is: n[h,...

,ifc] :={LeG:dim(LnVip)

> p V p = 1 , . . . ,fc}.

(Instead refering to the flag V\ C ••• C Vn, we will also say that this Schubert variety is defined w.r.t. the ordered basis {i>i,... , vn}.) One has dim Cl[ii,... , ik) = h H

+ ik + n(n - k) - n(n + l ) / 2 .

The Schubert classes in H* (G) determined by these Schubert varieties are related in following way to the Schubert classes aa considered earlier in this section. For a strict partition a = (ai,... , afc) C p(n — 1), one has aa = f2[n — ot\,... ,n — a.k\ if n —fc is even, and aa = fi[n— a.\,... ,n — ak,n} if n — k is odd. The corresponding Schubert variety aa in G can be defined in the fol lowing way w.r.t. the above flag V\ C • • • C Vn; it is: {L € G : dim(L n Vn-ap) >iVp=l,...,k

and c o d i n g (L n V„) is even}.

5. SCHUBERT CYCLES OP COMPLEX STRUCTURES ON

R2n

We will adopt the following convention. Let R 2 n be the real Euclidean 2n-space with the standard orthonormal basis { e i , . . . , e2n}The 2n-dimensional complex Euclidean space C 2 n will be considered as the complexification of R 2 "; C 2 n = R 2 n ® R C. Note the following simple facts: (a) The set {ei 1 , . . . , ein ® 1} is an orthonormal basis for C 2 n . (b) If L, K C R 2 " are two linear subspaces satisfying diniR.(L C\K) >i, then their complexifications LC,KC C C2n satisfy d i m c ( i C n Kc) > i. (c) Corresponding to an orthogonal decomposition L = L\ © Li of a subspace L c R 2 n , one has the orthogonal decomposition Lc = Lf © Lf of L c c C 2 n . (d) An R-linear endomorphism of a subspace L c R 2ra induces a C-linear endomorphism of the subspace Lc C C 2 n . Let V be an oriented even dimensional real Euclidean space and Iso(V), the group of orientation preserving isometries of V. Consider CS(V) = {Ae Iso(V) : A2 = -Idv} It is the space of complex structures on V.

.

DIVIDED DIFFERENCES OF TYPE D

53

If V = R 2 n , one has the identification Iso(R2n) = SO(2n, R), the special orthogonal group of order 2n, and CS(R2n)

= {Ae

SO(2n, R) | A2 = -I2n} ■

Note that if A S CS(Rn) then A is a skew-symmetric matrix. Let us abbre viate C5(R 2 n ) by CSn as is common. The space CSn has two connected components which are distinguished by the Pfafnan function Pf : CSn -» {±1} . We write CSn = CS+ U CS~ with Pf(CS±) = ±1.( The symbol U denotes the disjoint union.) Both manifolds CS^ are isometric to SO(2n, C)/U(ri). Our goal in this section is to interpret Schubert varieties in SO(2n, C ) / U(n) in terms of complex structures. We will give an interpretation, in terms of Schubert varieties, of the following Mahowald-Vassiljev-type for mula Hp(CSn) = ©fc=o-ffp-fc(fc-i)(Gfc(Cn)), n where Gfc(C ) is the Grassmannian of all complex fc-planes through zero in C " (see [DV] and also [V]). We will now define Schubert varieties of complex structures. Let us fix a complex structure Jo 6 CSn. By convention, we will denote by CS+ the connected component of CSn that contains Jo- We will work here with the component CS£, leaving to the reader details concerning the other compo nent CS~. The results about the component CS~ will be summarized in Proposition 5.5. Let R2™ = Lx © L2 ® ■ ■ ■ © L„ , where dimR. Li = 2, be an invariant subspace decomposition of the orthog onal operator J0 : R 2 " -» R 2 ". This yields a flag in R2n F1cF2C---cFn=IL2n,

(14) 2n

where i^ = L\ © L2 © ■ ■ • © Li. Furnishing R with the complex structure J 0 , we get an re-dimensional complex space C™ = (R 2 n , Jo)- Since each Fi is an invariant subspace w.r.t. Jo, the flag (14) gives rise to a complex flag ^ c i y 2 c - c w » = cB, (15) where dime W* = i. Consider the Grassmannian Gj(C n ) of all complex Z-planes through zero in C n . For a sequence I < ji < ■ ■ ■ < ji < n, one defines a Schubert variety fiO'i. ...,Ji)

= {L€ Gt(Cn) : dim(L n Wip) > p Vp = 1,...,/} .

One has dime n(j'i,. - - ,ji) =ji + ---+jiFor an even I, we define CS+(ji,... ,ji) as

1(1 + l ) / 2 .

{A e CS+ : 3L € fiO'i. • • • Ji) s-t. A(LR) = L R & A|L R = J 0 | L R } ,

54

HAIBAO DUAN AND PIOTR PRAGACZ

where L^ is the orthogonal complement of the real reduction L R of L. This is a closed subvariety in CS£- One has dime CS+(ju

... ,ji) = J! + • ■ • + ji - 1 .

One verifies easily that the class of the variety CS+(ji,- ■ ■ ,ji) is indepen dent of the choice of Jo. Indeed, a path in CS£ joining Jo to another J € CS+ yields a one-parameter family of varieties from CS+{j\,... , ji) attached to Jo, to that attached to J. Define the complex structure Jn in the initial basis { e i , . . . , e2«} by Jn = Ji © Ji © • • • © Ji

n times,

where

Example 5.1. Suppose that J0 = J„. For a sequence 1 < i\ < ■■ ■ < ik < n, we put J(ii,... ,ik) = eiJi ©e 2 Ji © •■•©£nJi where e/, = —1 if h = ip for 1 < p < k Then J ( H , . . . ,ik) € CS^ iff k is even. We now want to identify the variety CS£(ji,- ■ ■ ,ji) with a suitable Schubert variety fl\i\, ■ ■. ,ik] in G. To this end, we first describe an imbed ding i: CSn —> G n (C 2 "). Every A £ CSn has two eigenvalues ±i (here i is the pure imaginary complex number) with equal multiplicities n. Thus, as an endomorphism of C 2 n , (cf. (d)), A has the eigensubspace decomposition C 2 n = L{A, +) © L(A, - ) where dimL(A,+)

= dimL{A,-)

= n,

with A(v) = iv for all v e L(A, +) and A(v) = ~iv for all v e L(A, - ) . The embedding i : CSn -> Gn(C2n) defined by A -> L(A, +) has as its image the Grassmannian of all isotropic subspaces of C2™ w.r.t. the or thogonal form induced by the scalar product. Let us fix the complex structure Jo to be Jn. By using a simple linear algebra, one shows that w.r.t. the flag (15), associated with this complex structure, the following identification takes place. Proposition 5.2. Let I be even. isomorphism, of varieties: CS+{ji,...

,ji)

where t\ < ■ ■ ■ < tn-i

and

Then the embedding i restricts to an

Sl[n + 1-tn-i,-■■

,n +

l-ti],

is the complement of ji < ■ ■ ■ < ji in { 1 , . . . , n}.

Remark 5.3. The equality in this proposition is a refinement of a result announced by Dynnikov and Veselov in [DV], where some variant of Morse theory was used.

DIVIDED DIFFERENCES OF TYPE D

55

As a consequence of Proposition 4.8, we get: Corollary 5.4. The degree of C5+ (j\,... where 7 = (ji - 1 , . . . , j i - 1). Since, for even I, Q[n + 1 — tn-i,... {A:3Ke

fi(j'i.

,ji) is equal to the number g1,

, n + 1 — t\] is equal to

• • • >3i) s-t. A{KK) = KR & A\K& = J0\K^}

then, rewriting it for even n — k, il[ii,...

,

, ik] is equal to

{A:3K€ to(n + 1 - s„_fc,... , n + 1 - s t ) s.t. j4(tfR)=tfR& J 4|Jrjfc = .7o|tfk}. By taking L = if-1, we can present this fl[ii,...

,ik] as

{i4 : 3L € « ( » ! , . . . , ik) s.t. A(L R ) = LR & A|L R = J 0 | L R } .

(16)

This last identification (16) seems to be the most handy for applications. In the following proposition, keeping the notation from this section, we collect properties of Schubert varieties in CS~. For an odd I, we define CS'iji,... ,ji) as {A € CS-

: 3L e n(j'i. • ■ • .3i) s.t. ^ R ) = LR k A\L& = ^ O | L R } ,

where LR is the orthogonal complement of the real reduction LR of L. Proposition 5.5. (i) CS-(ji,... ,ji) is a closed subvariety in CS~ of di mension ji + ■ • • + ji — I. (ii) CS~{j\,... ,ji) can be identified with the restriction to CS~, properly embedded in Gn(C2n), of the Schubert variety fi(n + 1 -tn-i,...

,n + l-t1,n

+ ji,...

,n + ji)

in this last Grassmannian. (iii) CS~(ji,... ,ji) is also identified with {A € CS~ : 3L £ n{ix,...

,ik) s.t. A(LR)

= L R and A\LR

= J0|£R} .

(iv) The degree of CS~ (ji,. ■ ■ , ji) is equal to g~*', where 7 = (ji — l,... !)• Example 5.6. We describe the Schubert varieties in CS^ which are divi sors. We have different description according to the parity of n. If n is odd, then the divisor o\ = fl[n — 1] is {A : 3L C W n _i s.t. dime L = 1, A(LR) = LR and A\LR =

J0\LR}.

If n is even, then the divisor a\ = Cl[n — l,n] is {A : 3L C Wn s.t. dime L = 2, A(LR) = LR and A\LR =

J0\LR}.

,j\ —

HAIBAO DUAN AND PIOTR PRAGACZ

56

We will end this paper with some applications. The identification made in (16) allows us to solve enumerative problems about the number of general complex structures satisfying some constrains. To this end, we need the following definition. Definition 5.7. Let A andB be two orthogonal operators o n R 2 n . A linear subspace L C R 2 " is said to be a common k-space of A and B iff A(L) = B{L) = L, A\L = B\L and d i m R L = k. We will work in CS+. Let n and 2 < k < n be even integers. Let 1 < ii < • • • < ifc < n be a sequence of integers. Set d = dimfi[ii,... , ik]. Suppose that a list {-Bj}, 0 < i < d, of general complex structures on R 2 n is given. Then the number of complex structures A s CS+ s.t. A has a common 2fc-space from fi(ii,... , ik) with B0, and A has a common 4-space with any other Bi from the list, is equal to degfi[ii,... , ik]As a particular case, we have: Proposition 5.8. Let n and 2 < k < n be even integers. Suppose that a list {Bi} of general complex structures on R 2 n is given, where 0 < i < 1 + (n - k)(n - k - l ) / 2 . Then the number of complex structures A € CS+ having a common fixed 2k-space ( W ^ ) R with Bo and a common 4-space with any other Bi is given by gin-k-l.n-k-2,...

,2,1) =

[(„

_

fc)(f|

_

fc

_

1)/2]!

" JJ ' | _ ^ i .

Indeed, this is a restatement of the formula about the degree of il[l,2,...,k-l,k]

=

CS+(l,2,...,n-k),

which is simplified in this case of a triangular partition, cf. [DP]. Example 5.9. For n = 8 and k = 4, a list of 7 complex structures {BQ, Bi, B2, B^, B4, Bs,Be} is given. There exist exactly 2 complex struc tures A € CS£ s.t. A has a common fixed 8-space with Bo and at least a common 4-space with every Bi, where 1 < i < 6. If n = 10 and k = 4, a list of 16 structures {Bo, B\,... , 2?is} is given. There exist exactly 286 complex structures A G CS^ s.t. A has a common fixed 8-space with Bo and at least a common 4-space with every Bi, where 1 < i < 15.

DIVIDED DIFFERENCES OF TYPE D

57

Let now n and 2 < k < n be odd integers. Let 1 < ii < ■ ■ ■ < ik < n be a sequence of integers. Put d = dimfi[ii,... , i*]. Suppose that a list {Bi}, 0 < i < d, of general complex structures on R 2 n is given. Then the number of complex structures A € CS£ s.t. A has a common 2fc-space from tt(ii,... ,ik) with Bo, and ^4 has a common 2-space from ( W „ _ I ) R with any other Bi from the list, is equal to degCl[ii,... , ik]. We leave it to the reader to deduce from it a result analogous to the one in the last proposition. We will give now another example of enumerating complex structures satisfying some constraints and state some conjecture. For a complex structure BQ € CSn, we have an n-dimensional complex space C n = (R 2 n , B0). Note that for all L € Gk(Cn), both L R and £ R are invariant subspaces of Bo- We have then an embedding a : Gk(Cn) -> CSn defined by a(L) = {B0\LK)®(-B0\L&). Without loss of generality, we can assume that the image of a lies in CS£Let R be the canonical complex A;-bundle over Gk(Cn) and R1- its orthog onal complement in the trivial complex n-bundle. Let S be the canonical complex n-bundle over C5+. From the definition of a we have a*S =

R®R~I,

where R1- denotes the complex conjugation of R-1. We infer that the pullback of the total Chern class of S, a*(l + Ci(S) + ■ ■ - + cn(S)), is equal to (l + Cl(R) + --- + ck(R))(l - ci(R) + c2(R) -■■■ +

(-l^c^R))-1.

From this we get that the induced homomorphism a*

:H*(CSn)->H*{Gk{Cn))

satisfies a*(^ci(S)) = —a*(o"i) = ci(R). That is, the embedding a pre serves the classes of hyperplane sections, or, the Kahler classes of both varieties. As a consequence, we get results summarized in the following proposi tion. Proposition 5.10. (i) Let n be an even integer. Suppose that 2 < k < n is another integer. Let {Bi}, 0 < i < k(n — k), be a list of general complex structures on R 2 n . Then the number of complex structures A £ CS£ s.t. A and BQ have a common 2k-space, A and (—Bo) have a common 2(n~k)space, and A and each Bi, i > 1 have at least a common 4-space, is equal to the degree ofGk(Cn). (ii) Let now n be an odd integer. Suppose that 2 < k < n is another integer. Let {Bi}, 0 < i < k(n— k), be a list of general complex structures on R 2ra . Then the number of complex structures A G CS£ s.t. A and BQ have a common 2k-space, A and (—BQ) have a common 2(n — k)-space, and A and

58

HAIBAO DUAN AND PIOTR PRAGACZ

each Bi, i>\ have at least a common 2-space in (Wn-i)R> is equal to the degree o/G f c (C"). (Recall that degGfe(C

}

~ (n-*)!(n-Jfe + l ) ! - - . ( n - l ) ! '

a result which goes back to Schubert (1886).) It is well known that the Grassmannian Gfc(Cn) is an approximation space for the classifying space BU{k) of all complex fc-bundles. On the other hand, the space CSn serves as the classifying space for all complex n-bundles with a trivial real reduction [Dul]. Thus a homotopy classifica tion of continuous maps Gk(Cn) —» CSn may suggest possible interesting operators between these two vector bundle theories. Let (5 : Gfc(Cn) —> CSn be a continuous map. Since o\ = -^ci(S) € H*{CS±) = Z and a(R) e H*(Gk(Cn)) = Z are the only generators in dimension 2, then the induced map /?* : H*{CS±) -> H*(Gk(Cn)) satisfies /3*(\c1(S))

= -p*(o-1) =

m-c1(R)

for some m G Z. We finish this paper by stating the following conjecture. Conjecture 5.11 If m ^ 0, then the map (3* : H*(CS±) is given by p*(x)=mpa*(x), forxe

->

H*(Gk(Cn))

H2P(CS±).

We refer the reader to [Du2] and [Ho] for some background related to this conjecture. REFERENCES [BGG] I.N. Bernstein, I.M. Gelfand and S.I. Gelfand, Schubert cells and cohomology of the spaces G/P, Russian Math. Surveys 28:3 (1973), 1-26. [Bo] A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogenes des groupes de Lie compacts, Annals of Math. 5 7 (1953), 115-207. [Bu] N . Bourbaki, Groupes et Algebres de Lie, Chapters 4,5, and 6, Herrmann, Paris, 1968. [BKT] A.S. Buch, A. Kresch and H. Tamvakis, Gromov- Witten invariants on Grassmannians, preprint (April 2002). [DP] C. De Concini and P. Pragacz, On the class of Brill-Noether loci for Prym varieties, Math. Ann. 3 0 2 (1995), 687-697. [De] M. Demazure, Desingularisation des varietis des Schubert generalisees, Ann. E.N.S.7 (1974), 53-88. [Dul] H. Duan, The secondary Chern characteristic classes, Proc. A.M.S. 128(8) (2000), 2465-2471. [Du2] H. Duan, Self-maps of the Grassmannian of complex structures, to appear in Compositio Math.

DIVIDED DIFFERENCES OF TYPE D [DV]

59

I.A. Dynnikov and A.P. Veselov, Integrable gradient flows and Morse Theory, Algebra i Analiz 8 (1996), 78-103 (Russian); English translation: St. Petersburgh M a t h . J., 8 (1997), 429-446. [FP] W . Fulton and P. Pragacz, Schubert varieties and degeneracy loci, Springer LNM 1689 (1998). [GH] P. Griffith and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. [HP] W.V.D. Hodge and D. Pedoe, Methods of Algebraic Geometry, Cambridge Uni versity Press, 1954. [Hi] H. Hiller, Combinatorics and intersection of Schubert varieties, Comment. Math. Helvetici 5 7 (1982), 41-59. [HB] H. Hiller and B. Boe, Fieri formula for 5 , 0 2 n + i / { / „ and Sp„/Un, Adv. in Math. 6 2 (1986), 49-67. [Ho] M. Hoffman, Endomorphisms of the cohomology of complex Grassmannians, Trans. A.M.S. 281 (1984), 745-760. [HH] P.N. Hoffman and J.F. Humphreys, Projective representations of the symmetric groups, Oxford University Press, 1992. [KK] B . Kostant and S. Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G*, Adv. in Math. 6 2 (1986), 187-237. [KT] A. Kresch and H. Tamvakis, Quantum cohomology of orthogonal Grassmannians, Preprint (2001). [LSe] V. Lakshmibai and C.S. Seshadri, Geometry of G/P II, Proc. Indian Acad. Sci. A 8 7 (1978), 1-54. [LI] A. Lascoux, Notes on interpolation, Nankai University, Tjanjin (1996). [L2] A. Lascoux, Notes for the CBMS Conference on Algebraic Combinatorics, N.C.S.U., June 2001. [LP] A. Lascoux and P. Pragacz, Orthogonal divided differences and Schubert poly nomials, P-functions, and vertex operators, Michigan Math. J., Fulton's volume 4 8 (2000), 417-441. [LSc] A. Lascoux and M.-P. Schiitzenberger, Decompositions dans I'algebre des differ ences divisees, Discrete Math. 99 (1992), 165-179. [Ma] I.G. Macdonald, Symmetric functions and Hall polynomials, (Second edition), Clarendon Press, Oxford, 1995. [Mi] J. Milnor, Morse Theory, Annals of Mathematics Studies 5 1 , Princeton, 1963. [Mo] A. Morris, A note on the multiplication of Hall functions, J. London Math. Soc. 3 9 (1964), 481-488. [PI] P. Pragacz, Algebro-geometric applications of Schur S- and Q-polynomials, Seminaire d'Algebre Dubreil-Malliavin 1988-89, (M.-P. Malliavin, ed), Lecture Notes in Math., Vol. 1478, Springer-Verlag, Berlin and New York, 1991, 130191. [P2] P. Pragacz, Symmetric polynomials and divided differences in formulas of inter section theory, in "Parameter Spaces", Banach Center Publications 3 6 (1996), 125-177. [PR1] P. Pragacz and J. Ratajski, Formulas of Lagrangian and orthogonal degeneracy loci; Q-polynomial approach, Compositio Math. 107 (1997), 11-87. [PR2] P. Pragacz and J. Ratajski, A Pieri-type theorem for even orthogonal Grassman nians, MPIM-Preprint 1996-83. [S] I. Schur, Uber die Darstellung der symmetrischen und alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 1 3 9 (1911), 155-250. [Tl] H. Tamvakis, Height formulas for homogeneous spaces, Michigan Math. J., Ful ton's volume 48 (2000), 593-610.

60

HAIBAO DUAN AND PIOTR PRAGACZ

[T2] [V]

H. Tamvakis, Quantum, cohomology of Lagrangian and orthogonal Grassmannians, "Arbeitstagung 2001", Bonn, MPIM-Preprint 2001-50. V.A. Vassiljev, A geometric realization of homology of classical groups Algebra & Analysis 3(4) (1991), 113-120 (Russian); English translation: St. Petersburg Math. J. 3 (1992), 809-815.

INSTITUTE OF MATHEMATICS, CHINESE ACADEMY OF SCIENCES, BEIJING 100080, P . R . OF CHINA

E-mail address: dhbOmath.ac.cn INSTITUTE OF MATHEMATICS, POLISH ACADEMY OF SCIENCES, SNIADECKICH 8, P L 00-956 WARSZAWA 10, P . O . B o x 2 1 , P O L A N D

E-mail address: pragaczOimpan.gov.pl

T A B L E A U X STATISTICS FOR TWO PART M A C D O N A L D POLYNOMIALS L. LAPOINTE AND J. MORSE ABSTRACT. The Macdonald polynomials expanded in terms of a mod ified Schur function basis have coefficients called the q, t-Kostka poly nomials. We define operators to build standard tableaux and show that they are equivalent to creation operators that recursively build the Macdonald polynomials indexed by two part partitions. We un cover a new basis for these particular Macdonald polynomials and in doing so are able to give an explicit description of their associ ated q, i-Kostka coefficients by assigning a statistic in q and t to each standard tableau.

1. INTRODUCTION

The Macdonald polynomials, J\[X;q,t], are a two parameter family of polynomials in N variables, forming a basis for the space of symmetric functions. The polynomials, expanded in a modified Schur function basis { 5 A [ X ' ] } A , have coefficients called the q,t-Kostka polynomials, K\^(q,t). Macdonald conjectured that Kx^q, t) is a polynomial in q and t with pos itive integer coefficients. This conjecture has been proven [3] based on the representation-theoretic interpretation [2] of the q, i-Kostka polynomials. However, the proof does not reveal a combinatorial interpretation for these coefficients. In this article, we consider the case that J\[X;q, t] is indexed by partitions with no more than 2 parts. We uncover a new basis for the these particular Macdonald polynomials and in doing so are able to give an explicit description of their associated q, i-Kostka coefficients by assigning a statistic in q and t to each standard tableau. Our results appeared on the web a few years ago x. The recent general ization [6, 7] of the methods introduced here have prompted us to submit this article for publication. Prior to this work, a rigged configuration inter pretation for the coefficients associated to Macdonald polynomials indexed by partition with no more than 2 parts was given in [1]. In addition, simi lar tableaux statistics in the case of partitions whose first part is not larger than 2 were discovered [13] at the same time as ours. 1991 Mathematics Subject Classification. 1 math.CO/9812001 61

Primary 05E05.

62

L. L A P O I N T E A N D J . M O R S E

This paper can be seen as a study of the case k = 2 of the A;-Schur functions, sx '[X;t], associated to a filtration of the symmetric function space introduced in [7]. There is strong computational evidence to support conjectures asserting that the fc-Schur functions obey fc-generalizations of many Schur function positivity properties. The results of this paper prove, for k = 2, that the fc-Schur functions expand positively in terms of Schur functions and that the Macdonald polynomials indexed by partitions whose first part does not exceed 2 can be expanded positively in terms of ft-Schur functions. Conjectures generalizing this phenomenon are given in [7]. The connection between the work in this paper and that of the fc-Schur functions in [7] can be made using the following notational correspondance:

F-xHx[X-q,t\

<—>

qn^Hx,[X;l/q}

Ft-1Bi0)Ft^B2t-D

t-+l/q

Ft-lU^lb^qn^s^[X;l/q},

(1.1)

where on an arbitrary symmetric function P[-X"], we define Ft P[X] = P[X(l - t)} and Ff1 P[X] = P[X/(1 - t)}, and where A is the partition A = (2 a , l 2fe ,e). Note that from Property 5, F t _ 1 uie) is, for any v, equal up to a constant to a 2-Schur function. Also, letting t —>■ 1/q has the effect of transforming the 2-Schur functions from a basis of the linear span of Macdonald polynomials indexed by partitions whose first part is not larger than 2, to a basis of the linear span of Macdonald polynomials indexed by partitions whith no more than 2 parts. The paper is divided as follows: section two covers basic definitions used in symmetric function theory. We present in the third section, the creation operator B2 =tB^ +Bj, q~D~x, showing several properties that include the action of B% on the Hall-Littlewood polynomials H\[X;q,t] and an expansion of J\[X\ q, t] in terms of products of B\ ' and i?2 f° r ^W — 2Further, a new basis for these Macdonald polynomials having coefficients in the parameters q and t with positive integer coefficients is uncovered. The fourth section begins with basic definitions used in tableaux theory and then introduces operators on tableaux which correspond, under a morphism F, to the operators of section 3. Finally, a statistic on standard tableaux is presented in the fifth section. 2.

DEFINITIONS

Partitions are sequences of integers A = (Ai, A2,...) such that Xi > A2 > • • • > 0. The order of A is |A| = Ai + A2 + . . . , the number of non-zero parts in A is denoted ^(A) and n(A) refers to X^(* — 1)^»- The dominance order

TABLEAUX STATISTICS FOR 2 PART MACDONALD POLYNOMIALS

63

on partitions is defined for two partitions with ]A| = \(i\, by A < \i when Ai H + Aj < //! -| 1- [ii for alH. A partition A may be associated to a Young diagram with \ lattice squares in the ith row, from the bottom to top. The Young diagram of A = (5,4,2,1) is

zEEfc].

(2.i)

For each square s = (i,j) in the diagram of A, let £'(s),£(s),a(s) and a'(s) be respectively the number of squares in the diagram of A to the south, north, east and west of the square s. The transposition of a Young dia gram associated to A with respect to the main diagonal gives the conjugate partition A'. For example, the conjugate of (2.1) is

±Eb,

(2.2)

which gives A' = (4,3,2,2,1). We shall use A-rings, needing only the formal ring of symmetric functions Sym to act on the ring of rational functions in X\,..., zjv, q, t, with coeffi cients in R. The ring Sym is generated by power sums \I/j, i = 1,2,3 The action of \]>j on a rational function J2a caua/ J2B dpvp is by definition

with ca,dp € R and ua, vp monomials in x\,..., XN, q, t. Since any sym metric function is uniquely expressed in terms of the power sums, (2.3) extends to an action of Sym on rational functions. In particular, a sym metric function f(xi,..., arjy) can be denoted f[xi-\ harjv]- We shall use the elements X := xi + - ■ - + xN, Xtc> := X(t-l)/(q-l) and X* := X(t-1). The Macdonald polynomials can now be defined using a scalar product ( , )qtt defined on Sym Q[q,t) by e W

<* A [X], *„[*]>,,* = 5XliZx ft

i _ T

A,

-|X7,

(2.4)

where we associate to a parition A with mi (A) parts equal to i the number zx = lmim1\2m2m2\---

(2.5)

64

L. LAPOINTE AND J. MORSE

Macdonald polynomials J\[X;q,t]

are thus uniquely specified [11] by

(i)(JA,^>,,t=0, (ii) Jx[X;q,t]

ifA^/i,

(2.6) (2-7)

= ^vxlt(q,t)Sll[X],

(hi) Vxx{q, t) = H(l - g»Wt<« + 1 ), (2.8) sex where 5^[X] is the usual Schur function and vx^(q,t) G Q[g,*]. The expansion coefficients of the Macdonald polynomials when expanded in terms of the basis {S^lX*]}^. Jx[X;q,t] = ^K^iq^S^X*],

(2.9)

are studied here in the case that £(A) < 2. 3.

ALGEBRAIC SIDE

A Macdonald polynomial indexed by any partition can be constructed by repeated application of creation operators Bk defined [4, 8] such that BkJx[X;q,t]

= Jx+1H[X;q,t],

l(X) < k,

(3.1)

or more specifically, when k = 2, B2 Jm,n[X;q,t] = Jm+i,n+i[X',q,t]. If V is the Q[g,t]-linear span of [j\[X;q,t]}e(x)<2, that

(3.2)

because it is known [11]

Jx[X;q,t] = J2^(Q>t)SAxt9},

(3-3)

for some c^\(q,t) G Qfa, *], V must also be the Q[q, i]-linear span of {S\[Xtq]}i(X)<2. The action of B2 on V can thus be defined by its ac tion on {S\[Xtq]}t( 2, introduced in [5] as R 9 2

fY*«l - dpt & - *? m+1 )5 ro+ l[X*«] (1 qm+2)Sm+2[Xtq] a e t n tc n+1 ' J~ {\-tq )Sn[X >} (I - q )Sn+1[Xtq] •

m n[A

(3.4) It will be convenient to split the operator B2 into a sum of two operators, B2 = tB^+Bi1)q'D-\ (3-5) where D is the operator such that Df[X] = df[X] on any homogeneous function of degree d and where B2 and B2 are defined on V by -qm+lSm+1[Xtq} (1 qm+2)Sm+2[X^} (3.6) B^ S^nlX**] = det n -q Sn[X^] (1 qn+1)Sn+1[Xtq)

TABLEAUX STATISTICS FOR 2 PART MACDONALD POLYNOMIALS

and R(1) Q rytgl _„m+n+l J 0 + Sm+i[Xtq] j * 2 -5m,„[A \-q aet ^ ^

(l - g m + 2 )S m + 2 [X* g ] (1 - g "+ 1 )5„ + 1 [X t »]

65

(3.7)

These expressions, obtained by expanding (3.4), provide that (3.5) holds onV. We now introduce a deformation of the Hall-Littlewood polynomials: Hx[X;q,t]

== ^ y W f f ^ l / g . O ^ f X * ]

(3.8)

which is a basis for the ring of symmetric functions since the H\[X;q,tys are linearly independent; i.e., the i i T ^ l / g , 0) matrix is triangular with respect to the partial ordering and K\\(l/q,0) = 1. In fact, these poly nomials are specializations of the Macdonald polynomials. More precisely, let Jx[X;q,t]W (resp Jx[X;q,i\^tq^) denote the {S\[X*]}-expansion (resp {S\[Xtq]}-expansion) of the Macdonald polynomial, J\[X;q,t\. Then P r o p o s i t i o n 1. H\[X; q,t] is obtained by taking the coefficient of the max imal t-power in J\[X;q,t]^ (or J\\X\q,t}^tq^). Equivalently, Hx[X;q,t]

= Jx[X;q,t]W

:=^K^q,t)

= Jx[X;q,t]M

S^X*]

:=£>A(g,i)

S„[Xtfl] ■

(3.9)

Proof. Given the first identity, the second follows from (3.3) and the relation: Sx[Xt\ =

(3.10)

'£ivllX{q)Sp[X«>].

for some v^xiq) € Q[
n

/x=q

t"<*)

Since JsT^/^l/g, 1/i) is a polynomial in l/q,l/t rewritten K»x(q,t)

F»(A)

(3.11)

^K^x(l/q,l/t)

(eg. [4],[8]), (3.11) can be

=gnJ^A(l/?,0).

(3.12)

Therefore, the first identity in (3.9) is equivalent to (3.8). Substituting the known [11] relation, Jm[X; q, t] = (q; q)mSm[Xtq], (q;q)m = (l-q)(l-q2)---(l-qm), into Hm[X;q,t] A = (ra), yields

=

Jm[X;q,t}^

m > 0;

(q;q)o = l,

D where (3.13)

, that is into Proposition 1 in the case t°

66

L. L A P O I N T E A N D J. M O R S E

Corollary 2. Hn[X;q,t]

= Jm[X;q,t]

= {q-q)mSm[Xt(i].

(3.14)

With H\ now characterized as a specialization of J\, we have Lemma 3. V is the Q[q,t]-linear span of {H\[X; <7>*]}«A\<2 • Proof. (3.9) gives that the Q[q,£]-linear span of {H\[X;q,t}}e(>,<2 is in t(! cluded in the Q[<7,£]-linear span of {S\[X }} <2 which is equal to V. The lemma then follows because {H\[X;q,t]}e( . is a linearly indepen dent set. □ We now establish several properties of the operators B2 ' and B2 ' that will later enable us to associate them to tableaux operators. First, by Lemma 3, we we define the action of B2 on V by finding its action on Hm,n{X\q,t\. Property 4. The action of B2 ' on Hm,n[X;q,i\ B2 Hm>n[X;q,t]

is given by

= Hm+i,n+\[X\q,t\

.

(3.15)

Proof. Definitions (3.6) and (3.7) show that the action of B2 and B2 on Sm,n[Xtq] gives coefficients involving only the parameter q when expanded in terms of the {Sx[Xqt}},(x)<2 basis. This implies that the successive ac tion of B2 and B2 on Jm[X;q,t) = (q;q)mSm[Xtq} produces coefficients involving only q when expanded in the {S\[Xtg]}((x)<2 basis. Therefore, from Jm+^[X;q,t}

= (B2fjm[X;q,t}

= (tB<0) + ^ V ^

- 1

the coefficient of the maximal t-power in Jm+e^\X;q,t} Jm+eAX;q,t}W\te

) '

Jm[X;q,t], (3.16)

is given by

= ( f i f yjm[X;q,t].

(3.17)

The assertion follows by replacing left side of this expression with yield[X;q,t} by (3.9), and Jm[X;q,t] = Hm[X;q,t}.

Hm+t^ D

Property 5. On the space V, we have the q-commutation relation B[2l)B{2a)=qB{2a)B^. Proof.

(3.18)

The action defined in (3.6) and (3.7), and the Pieri rule n tci

t

1

Sm[X )Sn{X < ) = YJSm+n-iAXtq\,

where

m>n,

(3.19)

TABLEAUX STATISTICS FOR 2 PART MACDONALD POLYNOMIALS

67

yield s f ) B | 1 ) 5 T O , n [ x t « ] = g"+™+2(g™+i _ g ») ^

J2(qr

m+n+3 r

-

-)

q

r=01=0

Sm+n+4-(te[Xtq]

x +

q2m+2n+5-e(qm+l

_ gn) ^ ( ^ + 1 _ 1)

[X*9] 71+1

n+m+1

+ q

(l

n+1

- q

n+1

)(q

- g

ro+2

)

£ f=o

[Xtg] +

^m+n+3^ _ ^n+l^n+2 _

^

X 5 m+2 ,„ + 2[X*«]

(3.20)

and n

B^B{°]Smtn[Xtq}

= qn+m+\qn

T

_ g m + J ) J^ ^2(qm+n+*-r

-

r q

)

r=0 1=0 X +

Sm+n+4—ltt gn+m+3(qn

_ gm+lj

^ ( 1

qi+l)

-

t=0

71+1

+

? 3ra + n +5^ n +l

_ JJ Y^{qm+2

-

qn+1)

X <S,?7»+n+4-£,*[-X' q) + g 2 m + n + 4 ^ _ g n + l j ^ n + 2 _ -Q

x 5m+2,n+2[Xt9].

(3.21)

First exchange the order of summation in the right side of the top line in (3.20) and (3.21), and then use ST(qr - qm+n+3-r)

=

(1

9

H9

1

)

(3 22)

Next, send £ -> .£ — 1 in the right side of the second terms in (3.20) and (3.21). By fixing i, we have that qB^ B^ Smtn[X**] = Bk°} B{21] S^X^}

68

L. L A P O I N T E A N D J. M O R S E

if /(l-g"-<+1Kg<-g'"+3)

+ r+n+4-e{qe

_1)

+ {(f+±

_

1}\

=(/ 1 -^ 1 l ( f ) -^ ) + ( t f '-l) + ^(^-l)), (3.23) which holds by algebraic manipulation.

□

Property 6. Let e € {0,1}. B^ 0) and B ^ are ( S f +Bi21))mHe[X;q,t} Proof.

SMC/I

ifeai

= H2m+e[X;q,t}.

(3.24)

Definitions (3.6) and (3.7) give that (B^

+ B^)Sk[X^}

= (1 - qk+1)(\ - qk+2)Sk+2[X^),

(3.25)

which, using Hk[X;q, t] = (q; q)kSk[Xtg], implies that (B™ +Bi1))Hk[X;q,t]

= Hk+2[X;q,t].

(3.26)

We apply this identity m times, starting with k = e, to complete the proof.

□ With relation (3.5), this property shows that any Macdonald polynomial indexed e d lby a partition with no more than 2 parts can be built using B2 and B2 since Hm[X;q,t] = Jm[X\q,t). Definition 7. Let v = (vi,... define

,vk) with all Vi £ {0,1}. For e € {0,1} we

UU = B
(3.27)

Proposition 8. For e,t>, 6 {0,1} we have J2m+i+eti[X;q,t}=

Y,

^-OM'+^V-M'I/W,

(3.28)

v=(v1,...,vm+t)

where d — 2m+2£+e, \v\i = v\-\ Proof.

\-ve andn(v)e = U2+2t>3H

\-(£—l)ve.

From Property 6, we have that J2m+([X;q,t}

=

^ V=

U&, (vi,...,Vm)

(3.29)

TABLEAUX STATISTICS FOR 2 PART MACDONALD POLYNOMIALS

69

where Vi e {0,1}, proving (3.28) for £ = 0. Proceeding by induction, we assume that (3.28) holds for every I. We thus have, acting with B2, that v' — (0,v)

x ^+i-KU+i[/W +

V^

g(l-d)(|«"|t+i-l)+2(n(«")/+i-|«"|<+i

+ l)-d-l

v"=(l,u) x

f M-l-l«"l«+lf/w.

(3.30) Combining the two sums, we obtain B2J2m+l+e,t

=

g( 1 -( d+2 ))l 0 l«+i+ 2 "W'+»t<+l-l B l« + i^ e ),

Y, v={vi,...,vm+i+i)

(3.31) which completes the induction argument since B2J2m+e+e,t — J2m+e+i+e,e+i-

a Example:

We have

+ tq~M°lo + tq-3uQA

J4i2[X; q, t] = M°l0 + Ml,

+ tq-*Uglo + « 9 - 6 < i + «-8tfffio + ?- 8 ^ ( S,i • (3-32) Corollary 9. The maximal t-power in (3.28) is

H2m+l+^[X-q,t\

= YjU^

,

(3.33)

V

summing over all v = (0e, v) where v = (vi,...,

vm) such that v\ G {0,1}.

4. TABLEAUX SIDE

4.1. Definition and background. Let A* be the free monoid generated by the alphabet . 4 = { 1 , 2 , 3 , . . . } and Q[A*} be the free algebra of A. The elements of A* are called words. The degree of a word w is denoted |to| and its image in the ring of polynomials Z[A] is called the evaluation, denoted ev{w). For example, w = 131332 has degree 6 and evaluation (2,1,3). A word w of degree n is said to be standard iff ev(w) = ( 1 , 1 , . . . , 1). A tableau T will be the pair (A, w), where A is a partition and w is a word, such that |A| = \w\. We say that A is the shape of T. A Young diagram as sociated to Afilledwith the letters of w from left to right and top to bottom is a planar representation of T. For example, T = ((4,2,2,1), 114356234) corresponds to m

H

•

(4-1)

70

L. LAPOINTE AND J. MORSE

A semi-standard tableau is a tableau such that the entries in every row are nondecreasing and such that the entries in every column are increasing. In this case, we do not specify A in the pair (A, to), since it can be extracted from w. For instance, T = 67 445 11123 has the representation 6 7 4 4 5 1 1 1 3|3|

(4.2)

Notice that a semi-standard tableau is a tableau. Finally, a standard tableau T is a semi-standard tableau of evaluation ( 1 , 1 , . . . , 1). For ex ample, T = 7 46 1235 or

r

fflel

(4.3)

|l[2|3|5|

is a standard tableau. As with the Ferrers diagrams, we define T* to be the transpose of a tableau T. For example, with T as given in (4.1), we have _4 3 2 5 14 6 3 11

T^IL

•

(4-4)

Notice that the transpositon of a semi-standard tableau may not be a semistandard tableau while the transposition of a standard tableau must be a standard tableau. Words can be associated to numbers called the charge and cocharge where charge(w) = n(ev(w)p) — cocharge(w),

(4-5)

for ev(w)p the partition obtained by reordering ev(w). The cocharge of a standard word w is defined by the following algorithm; 1. Label the letter 1 in w by c\ = 0 2. If the letter i + 1 appears at the left of the letter i in w, then Cj+i = Ci + 1. Otherwise Cj+i = Cj. 3. cocharge(tu) = C\ + ■ ■ ■ + cn. For instance, cocharge(413265) = 0 + 0 + 1 + 2 + 2 + 3 = 8. Recall that semistandard tableaux and standard tableaux are simply words, and therefore have an associated charge and cocharge. Lascoux and Schiitzenberger defined [9] an action of the symmetric group on Z[A*} that sends a word of evaluation (evi,..., eu», euj+i,...) to a word of evaluation (ei>i,..., evi+i, evi,...) under an elementary transposition, a*. Their action induces the usual action of the symmetric group on Z[A]. For our purposes, we define this action only on words such that (e^,evi+i)€{(l,2),(2,l)}: aab <4 abb,

aba <4 bba,

baa <4 bab,

(4-6)

TABLEAUX STATISTICS FOR 2 PART MACDONALD POLYNOMIALS

71

where a and b stand for i and i + 1 respectively. This action sends a semistandard tableau to a semi-standard tableau while preserving its shape. For example, 04(215345) = 215344 and cr3 jjrg]= ][ 5T5. 1112131 111214 We will use several simple linear operations on words, r^ is a translation of a to a + k for every letter a in an alphabet. For instance, T2(231567) = 453789. The restriction of a word w to the alphabet a, b, c,..., is denoted w {a,b,c,...}- i-e-> w {3,4} = 3343 for w = 12334223. If w is such that wtatby = ab, r( ab _ >cd ) sends ab to cd. For example, r(23_>46) (121543) = 141546. We further define an operator Ra to remove all letters a in w, and finally, Ai+i,n+i is an operator on a tableau T, that adds a horizontal 2-strip of the boxes n + 1 in all the possible ways to T. For instance (4.7)

There exists a shape and cocharge preserving standardization [9] of semistandard tableaux, denoted by VS, that we will use only on semi-standard tableaux with evaluation (evi,..., evk), where evi G {1,2}. In the case of a semi-standard tableau T, it is defined as 1. If evi = 2 then T —> rir( 11 _ >0 i)^'- Proceed to step 2. 2. If ev(T) = ( 1 , 1 , . . . , 1) then the standardization is complete. Other wise, T —> o\ ■ ■ ■ Oi-\T for the smallest i such that evi = 2. Proceed to step 1. For example, T = 45 235 124 undergoes the following standardization pro cess: (4.8)

Note that VS^ will denote the standardization of only the first n letters of a semi-standard tableau of degree N >n which sends such a tableau to a semi-standard tableau of evaluation (l n , evn+\,..., ev^). 4.2. Tableaux operators. The primary goal of this section is to present tableaux operators that can be related to B^ and B\ using the linear morphism on semi-standard tableaux, F :T

-+ g cocharge < T >S shape(T) [X*].

(4.9)

It is known [10] that the Hall-Littlewood polynomials Qx{X-t] = Y,K^,t)Slx[X%

(4.10)

are equivalently expressed as a sum over semi-standard tableaux T such that Qx[X;t]=

J2 T;ev(T)=X

^"^Sshapeml**].

(4.11)

72

L. L A P O I N T E AND J. M O R S E

The substitution oiK^l/q, yields

0) = K^\i{Q, l/q) [11] in expression (3.8) thus

HX{X; q,t] = X y ( V ) * M ' v ( 0 , l/q)Sfl,{Xt] =

E

^^^^shapelT)^*],

(4-12)

T;ev(T) = \'

using cocharge(T) = n(ev(T)) — charge(T). Since standardization VS pre serves cocharge and shape, we have Hx[X-,q,t]=

^CharSe(VS(T))5shaPe(vS(r))[^-

Y,

( 4 - 13 )

T;ev(T) = X'

This given, we can equivalently express H\ [X; q, t] as a sum of semi-standard tableaux under the action of F ■ More precisely, Definition 10. For all partitions A where £{\) < 2, let Mx =

^

VS(T).

(4.14)

T;ev(T)=\'

Therefore, (4.13) gives that F(Mx) =

Hx[X;q,t}.

(4.15)

In the spirit of section 3, we define two linear operators on standard tableaux. Definition 11. On any standard tableau T such that \T\ = n, let B 2 0) : T ->• VS{An+lyn+iT)

= T i ^ n ^ o i ) ^ ' ' ' o-nAn+i
(4.16)

and # > -:T ^

(B^T*)*.

(4.17)

Example: Given a standard tableau, T = Fl , ,, we add all possible hori zontal 2-strips containing the letter 5 and then standardize:

(4.18) Note that B 2 0) and B 2 J ) send T to a sum of standard tableaux. We now undertake the task of proving that these operators satisfy prop erties analogous to Properties 4, 5 and 6 as this will imply that sequences (0) ?(D of B»<°> and Bs(i) correspond, under F, to similar sequences of B 2 and B\ 2 2 (see Theorem 26). First we prove the analog of Property 4.

TABLEAUX STATISTICS F O R 2 P A R T M A C D O N A L D P O L Y N O M I A L S

73

Property 12. With A a partition such that £(X) < 2, B 2 0) H A = M A+1 2.

(4.19)

Proof. Since -<4n+iin+i commutes with VS^n\ using the action of J 4 n + 1 ] „ + 1 on a semi-standard tableau of degree n = |A| and (4.14),

^i,.+i

VS
Y

J2

T;ev(T)=X'

VSM(T),

(4.20)

T;ev(T) = (\',2)

where A' is a vector of length n. Thus by the definition of u^ '

4 0)

Y

VSW{T)=

T;ev(T)=W

J2

r1r{11^01)a1---anVS^(T).

T;ev(T) = (\',2)

(4.21) and further by the recursion VS^n+2^ — 0) ( n

4

Y

T;ev{T) = X>

VS - '>(T)=

TI?"(II->OI) (7 I

' ' " °~n we have

Y

T;ev(T) = {\> ,2)

VS(n+2)(T).

(4.22)

Because the symmetric group action (4.6) preserves charge and shape, for /?(//) any permutation of the vector ^, we have

vs

( 4 - 23 )

VS{n+2){T).

(4.24)

Property 12 follows now from definition (4.14). The analog of Property 6 in the tableaux world is stated:

□

Y

vs(T)= Y

T;ev(T)=p

^>

T;ev(t)=/3(li)

which then implies B 0)

2

v (n)

Y

S {T)=

J2

T;ev(T)=\'

f;ev(T)

=

(\+l2)'

Property 13. For e S {0,1}, we have that M2m+e = (M{20)+Mi21))m^eProof.

(4.25)

From Definition 10, H

»=

E

VS(T) = Y,T,

T;eu(T) = ( l » )

(4.26)

T

and Hn+i,i=

E

VS(T)= Y,

r;ev(T)=(2,l")

V

(4-27)

T ' ; T { ' 1 2 } = 12

where T and T' are standard tableaux of degree n and n + 2 respectively. Property 12 gives that B 2 H„ = Hn+i^, implying R(°>IHT 2

-

uj.n =

T

Y r

r

" {'i.2}=

'■

12

(4-28)

74

L. L A P O I N T E A N D J. M O R S E

On the other hand, since M21]T = (B^T*)* and Mn = H L we have B^Hn = (l^Mn)* =

^ r

7

T'.

'; {i.3}=

(4.29)

21

H„ + 2 is the sum of all standard tableaux of order n + 2, of which each tableaux contain the subword 12 or 21. Therefore we have Wn+2 = (B20)+M21))Mn,

(4.30)

from which (4.25) follows.

□

It remains to show that W>2 and B 2 satisfy a relation similar to the g-commutation relation of B2 ' and B2 '. The proof of this last important property (stated in Property 25) requires a lengthy development of lemmas and identities. We need two linear operators on standard tableaux, defined by: Definition 14. On any standard tableau T with T{\t2y = 12 and |T| = n, .(0)

B2

:T->.R„-i
(4.31)

and on any standard tableau T such that 7{i,2} = 21 and \T\ = n, B ^ : T -»• ( B 2 ° V ) ' . -(1)

Example: IL

Acting with B 2 t GEL (1) DEL

(4.32)

[5]

on i l l , we go through the following steps: 111314 [

(2) DEL (3) DEL (4) \3\

(5) m

(6) pq

t m

2.61 —> 111 —f 2_J\ —f 2_ J\ —f 1IL — ' 1 I L ~* 1 I L ~~' 2. —» , IL |J . IUUI 11 j 2 [ 51 |i|iUi I1I2U1 11131 -11 hUUi iU|r,| |T[4l I I^I I (4.33) For any sum S = Y^k T(fc) with T(fc) ^ T(fc'' for all k ^ k', we shall say

that T e S if and only if T = T

w

for some k. Then, the operators B 2

* ^~*

and B 2

(0)

(1)

can be seen as the inverse of operations B 2 and B 2 . That is,

P r o p e r t y 15. For T{ € B(2]Ti and T2 € Bi21)T2, we have . (o) . (1) B 2 T{ = 71 and B 2 T2' = T2.

(4.34)

Proof. These identities follow directly from the definition of the elemen tary operations comprising the operators and the fact that RaT' = T for any tableau T € Aa aT. D .(i)

It is useful to alternatively define the action of B 2 . This is accomplished by introducing a new permutation aa, defined by its action on a and b — o + l; aab <4 bab,

aba <4 abb,

baa «4 bba

(4.35)

T A B L E A U X STATISTICS F O R 2 P A R T M A C D O N A L D P O L Y N O M I A L S

75

N o t e t h a t aa is such t h a t on a word w with ev{w{a^y) € {(1,2), ( 2 , 1 ) } , we have aawR = (aaw)R, where t h e superscript R is t h e operation sending a word W = W\- ■ -wn to t h e word wR = wn---W\. .(i)

P r o p e r t y 1 6 . The action o / B 2 on any standard tableau T where 7(1,2} = 21 and \T\ = n, can equivalently be expressed as B2

: T-> Rn-i
Example:

■ • ■ air^o^nyT-iT.

(4.36)

[5]

This action of B 2

o n 2. in consists of t h e following steps: 1113 rsi

2^1 —> i l l — f 2.11 - * J.I1 - 4 i l l "4 4.11 —> 1-1-1lUUl 11 [2I3 [ 111213 I 1112131 |l|2|3| 1112131 ' ' ''

(4-37)

N o t e t h a t we avoid t h e transposition steps shown in Example (4.33), a n d t h a t each t a b l e a u here is t h e transposition of t h e corresponding tableau from Example (4.33). Proof. Definition 14 gives t h a t for any s t a n d a r d tableau T such t h a t 7{i,2} = 21 a n d \T\ = n, we must show (cr n -2---0'i?-(oi->.ii)r-iT t )

= a„-2 •••CTir(io_m)T_i7',

(4.38)

where we have used t h a t Rn-\ commutes with transposition. If we let T b e t h e semi-standard tableau f = 7"(on-ii)' r —IT - *, we get t h a t T* = J*(io->n)T_i7". Expression (4.38) is then verified since we can obtain (
for any semi-standard T with

€ {(1,2), ( 2 , 1 ) } .

ev(T{ab=a+1y) (4.39)

T h i s identity can b e proven by observing t h a t under such conditions, we have (T{atb})R

= T*a6}.

For example, T = f l l ^ ^ is such t h a t T{3<4} =

433 a n d Tj3 4 , = 334. T h e only possible planar distribution of a a n d b t h a t could cause this t o fail are s a m e as Tt

by.

" or

b a

, since T^^y would b e t h e

T h e first case never holds a n d t h e second occurs only if

we have b o t h another a and another b, which we d o not. Thus, using T*a,b} = (TR){a,b)

= Via*})*

=

T

{a,6}'

W e h a V e

*»!?„,» = ^T^b}

=

R

(o-aT{a,b}) = {
-> (aio-i+1w1,aiai+1W2),

(4.40)

76

L. L A P O I N T E AND J. M O R S E

and consider 6 pairs of words with a and b = a + 1; Ci(a) = (aabb,baab), Ci{a) = (baab,bbaa),

C?2xa) = (abab,abab), Cs(a) = (baba,baba),

Cs{a) = {abba, aabb) C§{a) = (bbaa,abba). (4.41)

We insert letters into such pairs by defining an insertion operator on words, Pka'wi •■■wn = wi--- Wk-iawk ■ • -w„

(4.42)

with the understanding that I^'w = aw and In+iw = wa- ll a c * s o n pairs of words by I^'{w\,W2) = (1^ wi,Ij^ xi>2)Computer experimentation using ACE revealed that applying Ej to any pair Cj (?) with the letter (i + 2) inserted recovered a pair of the same type where i -» i + 1 with the extra letter i occuring in the same position of both elements of the pair. More exactly, Lemma 17. Let 1 < k, k! < 5, 1 < j,j' < 6 and i > 1. For any k,j and i, Vi(l%+2)Ci{i))=I$Cj.{i

+ l),

(4.43)

for some k' and j ' . Example:

We have E 4 ( 4 6 ) C 3 ( 4 ) ) = E 4 (46554,46455) = (5-4(7546554, cr4a-546455) = (46565,46565) = 7 1 (4) C 5 (5).

(4.44)

Proof. This is a lemma having 30 possible configurations which are easily verified with a computer. □ Proposition 18. Let 71 be any standard tableau of degree n such that 7i{i,2,3,4} is 4312 or 3124 and let 7^ = TJ2<_>3:, i.e. 7i is obtained by per muting 2 and 3 in 71. If /.(1).(0)

.(0).(l)

x

B : (71, r 2 ) -> (B 2 B 2 71,B 2 B 2 T2) ,

(4.45)

B(7!,r2H(f,f),

(4.46)

then

for some standard tableau T■ pf] Example: Given 71 = I s ]

, we have the pair,

^ ■ ^ ^ ( i b i ^ ' i u . ) =(Hna-fflm) ■

<4"7»

TABLEAUX STATISTICS F O R 2 P A R T M A C D O N A L D P O L Y N O M I A L S

77

.(0)

Proof. B 2 reduces the degree of 71 to n—2, giving by Definition 14 and Property 16, .(l).(O)

B 2 B 2 71 = itre_3(Tn_4 • • • o-ir( 10 -ni)T_iiJ n _ 1 (T n _2 • • • ci^'(oi->n)''"-i7i =Rn-3Rn-.2

(^n-4^ n -3) ' ' ' (^lO'2)?'(10-)-ll)'r-lO'2Cri^(0H-ll)''"-l7I , (4.48)

where we have considered the relations T-\Rn-i

= Rn-2T-\

and T-\Oi =

.(i)

CTJ_ 1 T_ 1 .

Similarly, acting first with B 2 >

• (o). (i) B 2 B 2 Ti = i?„_ 3 cr n _ 4 - ••<7ir(oi_».ii)T_ifln_i<7„_2 •••crir(io-ni)r_i7 2 = i ? n _ 3 i 2 n _ 2 ( o ' n - 4 ^ n - 3 ) " • " (c r lCT2)r( 0 1 _>ii)T_ 1 cf 2 a-i7'( 1 o_>i 1 )T_ 1 72 .

(4.49) r

on

We act first with r(1O-m)T_icr20'i'''(oi->-ii)' -i 71 in the case that 71 {^2,3,4} is 3124, obtaining a word with subword 1122 and letters 3 , . . . , n—2 occurring exactly once. 7-2, defined by permuting 2 and 3 in 7i, thus contains the subword 2134 which is sent to 2112 under J"(oi_>n)T_i<72<7i7'(1o-ni)T_i, while the remaining letters occur exactly as they do in 7"(io->ii)T_i0'2 o"i^(oi-Mi)T-i7i- Consequently, 1(7-1,75) = -R T ,-3*n-2S n _ 4 ---Si (/J- 2 ) ---/£ ) _ 5 j£ ) _ 4 (1122,2112)) , (4.50) for some k\,..., fc„_4. Since Si acts only on the letters 1,2 and 3, we may now use Lemma 17, where i = j = 1, to determine that the action of S i results in B (7i,T 2 ) = J R n _ 3 i?„- 2 S„_ 4 • • • S 2 ( / ^ - 2 ) • • • /£ ) _ 5 /( 1 ) C i (2)) ,

(4.51)

for some j and r. Lemma 17, applied repeatedly in this manner, gives B (71, T2) = Rn-aRn-2

(4X)42) • • •

fc4)^("

- 3)) ,

(4.52)

for some r\,... ,r n _4 and some j ' . This is to say that the action of B on such a pair is equivalent to acting with Rn^zRn-2 on a pair of tableaux that are identical in all letters except n—3 and n—2. Further, since Rn-zRn-2 removes these letters, we have proved the identity in the case 3124. A sequence of similar arguments may be used in the case that 7i {1,2,3,4} ls 4312, and we get B(Ti,T 2 ) = i ? n _ 3 f i „ _ 2 S n _ 4 - - - S 1 ( 4 ; - 2 ) - - - 7 ^ / ^ ( 2 1 1 2 , 2 2 1 1 ) ) . (4.53) Again, successive applications of Lemma 17, beginning with the case i = 1 and j = 4, prove the identity. □

78

L. LAPOINTE AND J. MORSE

L e m m a 1 9 . For T ' a standard tableau of degree n > 4, we have that T

e B^ 0) B< 0)

^ T T;|T|=n-4

r

€ »20)B^

£ T < = ^ ^ , 2 , 3 , 4 } € {4312,3124} , T;|T|=n-4

T' e B^B<0)

T £ T;|T|=n-4

T € B ^ B ^

]T T;|n=n-4

Proof.

T

^

^ , 2 , 3 , 4 } € {1234,4123,3412} ,

^

T

€

i4213'2134).

^ ^

^I.W}

€

{4321,3214,2413}.

WA}

(4,54)

We begin by simultaneously proving the first two cases of (=>)

and the others of (=>) follow by transposition. Notice first t h a t B 2 T = £r„T",

where T('12} is 12 and B^T

= £ r „ T",

where

^",2}

is

21

(see the proof of P r o p e r t y 13). T h e following action of J3 2 begins with j 4 n + i ) T , + i adding a horizontal 2 strip t o T " resulting in tableaux t h a t are all semi-standard and containing the subword 12 (or 21). We act next with the succession of cr„, o - n _ 1 , . . . , cr3 implying t h a t the tableaux remain semistandard a n d t h u s must each contain, in t h e first case, the subword 1233, 3123 or 3312, a n d in t h e second, 3213 or 2133. T h e remaining operations, aside from T\ , act exclusively on these subwords as follows; T'i?'(ii_>oi)OriO-2l233 = 1 2 3 4 , n?-(ii->oi)^iC23123 = 4 1 2 3 , T]T(ii->oi)0'i0'23312 = 3412 ,

(4.55)

for the first case and Ti'"(ii-j.oi)0'i0'23213 = 4312, Tir(n-yoi)<7iO"22133 = 3124 , for the second, ), we are given the four defined V, which is e

(4.56)

thus proving (=>) for the two first cases. To prove (<= a standard tableau T' with subword T!123i\ in one of disjoint sets; call this set 5 £ l i £ 2 . Property 13 gives t h a t Mn, for n = \T'\, is such t h a t V £ B 2 El) B 2 ?2) ]HI n _4 =

B2ei'B2 ]C{|T|=n-4} 7~> f ° r s o m e ^i € {Oi ! } • B u t since we have just proven t h a t for such T', TL 234} is contained in t h e set Silt?2, we see t h a t e, = e, and the l e m m a is proven. D P r o p o s i t i o n 2 0 . On any standard tableau T, we have B20)B21)T= ( l ^ B ^ r ) where 2 «-* 3 denotes a permutation

2 0 3

,

(4.57)

of the letters 2 and 3 in each tableaux.

TABLEAUX STATISTICS FOR 2 PART MACDONALD POLYNOMIALS

Proof. Lemma contain contain

79

Suppose there exists V £B£0)B£1}T such that T ' 2 " 3 g B ^ B ^ T . 19 gives that every element in B 2 B 2 T , in particular T', must either the subword 4312 or 3124. This implies that T' ** must either 4213 or 2134 and thus by the same lemma we have that T'2°3 GB ^ B ^ T "

(4.58)

. (i) . (0)

for some T" ^ T . Observe now that B 2 B 2 T ' = T, implies by Proposi tion 18 that B 2 B 2 T'2^3 must also be T. Expression (4.58) then yields B 2 B 2 (B^B^T")

= T + other terms

(4.59)

which by Property 15 gives that T" = T and we reach a contradiction. We thus have that B ^ B ^ T C ( B ^ B ^ T ) manner that B ^ B ^ T 2 ( B ^ B ^ T )

. We can also show in the same , which proves the proposition.

We now define four pairs of words on consecutive numbers; a,b = a + l , c = b + 1 and d = c + 1. D\(a) = (bacd,cabd), D2(a) = (dbac,dcab), Dz(a) = (acdb,abdc),

D4(a) = (cdba,bdca).

(4.60)

These pairs appear as the only distinct subwords in certain pairs of semistandard tableaux. More precisely, such a pair of semi-standard tableaux, called (Ti, T2)Dj{a), satisfies ^ = T 2 6 ~ c and (T1{aAc,d}, T2{ab^) = D^a). For example, I I l U L i 4"6j8l

I

(4-61)

vtmnn] \MM\J D3(5)

is such a pair. One should note that {T\,T2)D (a) ls a P au " of tableaux of the same shape since in any such semi-standard tableaux, b and c never occur in the same row or column. With fij defined such that on pairs of words fij : (wi,W2)

-> (o-iO-i+lO-i+20-i+3Wi,0-iO-i+1O-i+20-i+3W2)

;

(4-62)

Lemma 21. Let 1 < k2,k'2,k2' < 5, 1 < ki,k[,k'{ < 6, 1 < j,j',j" and i > 0. For any such ki,k2,j and i, we have r ^ u ^ d ^ I ^ D ^ )

e {i^I^D.ii

+ 1),

<4

I^I^Drii)} (4.63)

and

r^^nj^i^DM

G {igr1)7™JDJ«(*+.i), j g r ^ ' z v w } , (4.64)

/or some fc^,fc",k'2,k2,j' and j " .

80

L. LAPOINTE AND J. MORSE

Example:

Starting with ifh^D^)

= (798965,698975), we get

ft4(798965,698975) = (597865,596875).

(4.65)

Under r^5^-i5), we recover I{'l[9'D4(5) and under r^b-^-si), If'I^9'D3(5). Proof. For each i there are 60 cases that have been verified using a computer. □ Lemma 22. Let T and T' be standard tableaux of type (T, T')u-(t) for some i,j. If standard tableaux 1~ € B 2 7~ and 7~' € Bj 7 1 are standard tableaux of the same shape for e S {0,1}, then T and T' is a pair of type {T,t')Djl{i+i) or {T,T')D.,(i+2), for some f. Proof. We start with the case e = 0 and split the action of B2 into a sequence of operations beginning with .<4n+iin+i. As such, we consider a semi-standard tableau T obtained by adding an arbitrary horizontal 2strip to T- We denote by T', the semi-standard tableau of the same shape that is obtained by adding this horizontal 2-strip to 7~' and thus T and T are a pair of type (T, T')j>(,). Next in the sequence of operations defining J8 2 is o~i+4 • • • o~n which, acting only on the letters i + 4 , . . . , n, must preserve the similarity in T and T". If i ^ 1, since acting with <7i_i amounts to applying either r^ii_).ii_i') or r(ij_>j_ij), acting on both elements with ai^iGiai+\o-1+20-1+3 using Lemma 21, gives a pair of semistandard tableaux of type (T,T')D.,^i+1^ or {T,T')o.,(i)There remains to act with Tirn-^oiCi • • •o'i-2> which leads to pairs of standard tableaux or of type (T,T')D.,(%+2) (T, T')D -/(t+i)- In t n e c a s e where i = 1, acting on both elements with Tir(11^.Oi)cri<J20'3Cr4, gives, from Lemma 21, pairs of standard tableaux of type (T, T')D /(3) o r (7~,7~')D;(2)> finally proving the lemma for e = 0. To prove the lemma in the case B 2 , we observe that, for 71 and 7-2 standard tableaux, (7I,72)DI(O)

= > (7j,7f).D2(o)

(7I,72)D2(O)

=> (T2>7l)Di{a),

(Tl,T2)D3(a)

==> (7j,7^)D 4 (a)

(71, T2)D4(a)

==> O2 ,T?)D3{a) ■

(4.66)

B 2 T = (B2 7"*)' thus implies that the proof in this case is exactly the proof for B 2 with every pair reversed plus an additional reversal of the pairs at the end, accounting for the last transposition in B 2 . □ Lemma 23. IfT and T' are a pair of standard tableaux of type (7~, 7~')zx,(i) then F(T) = qF(T').

(4.67)

Proof. We have already noted that (T, T')DJ (%) ls a P a i r of tableaux with the same shape. Further, the definition of cocharge gives cocharge(T) =

TABLEAUX STATISTICS FOR 2 PART MACDONALD POLYNOMIALS

81

cocharge(T') + 1; for example, for D\{a) = (bacd,cabd), we have cocharge(T) = cx + ■ • • + ca + (ca + 1) + (c0 + 1) + (ca + 1) +Ca+4 H

\-Cn

cocharge(T') = cx + • • • + ca + (ca) + (c a + 1) + (c a + 1) +c a +4 H

H c„ ,

(4.68)

giving cocharge(T) = cocharge(T') + 1 as claimed. □ We now finally have all the ingredients needed to prove the commutation relation. Definition 24. We define, for v = (i>i,..., Vk) with Vi G {0,1}, U<e> = B ^ 1 ' • • • M(2Vk)Be, Property 25. For any v = (vi,... {0,1} and k, k' > 0, we have

e G {0,1} .

,Vk) and v = (v\,...,

/"(Uilo,*) = < ? F ( u i l M )

(4.69)

Vfc'), with Vi,Vi G

e e {0,1} -

(4.70)

Proof. We begin by showing that this identity holds in the case that v is empty. Since v[% = B ^ B ^ £ r T and U ^ i 0 = B ^ B ^ E r 7 " - e a c h tableau 7i GU^, c can be paired with some T J ' G U Q ' J ^ such that T{ = T^**Z by Proposition 20 and such that 7i{1>2,3,4} € {4213,2134} by Lemma 19. This implies that 7i and T[ are of type {TI,TI)DJ{I) where j = 1 or 2 which, using Lemma 23, proves the identity for ■; = (). We now proceed to the case when v = (0) or (1) by acting with B2 on the pairs obtained when v = (). These pairs (T±,T{)D (1) a r e thus sent to a pair of sums of standard tableaux that, by Lemma 22, can be paired by types (72,7^)^.(2) for some j = 1, 2,3 or 4 and some i = 3 or 4. For any v, we repeat this process and obtain that each T G U„ ■[ 0 a c a n be paired with some standard tableau T ' € U ^ i ^ where this pair is of type (T,T')Dj(i), and 1 < i < n — 3. Lemma 23 then proves the property.

for 1 < j < 4 □

Given Properties 12, 13 and 25, analogous to those proven in section 3, we can finally prove the main result of this section. Theorem 26. Let e,Vi € {0,1}. For any v = (v\,... ,i>fc) we have F(UW) = U^ .

(4.71)

Proof. Recall that the action of 2?2 and Bi, ' + B\ \ determined in Properties 4 and 6, led to Corollary 9. That is, to

H2m+l+t>l{X^A

= YJU^, V

(4.72)

82

L. LAPOINTE AND J. MORSE

where v = (0e, v) for some v = (v\,..., vm). Observe that we have proved equivalent actions for the operators B 2 and B 2 + B 2 in Properties 12 and 13, giving

M2m+e+i,t = ^2lJie\

(4.73)

V

where v is as before. We thus have, since F ( H m ^ ) = Hmj[X] q, t], H2m+e+eAX;<7,

*] = £ F ( U £ > ) = £ U ^ V

.

(4.74)

V

We convert the expression such that we are summing only over dominant vectors Vd = (0m+i~k,lk) for some fc by using the following implication of the ^-commutation relations proven in Properties 5 and 25: UicK =
H2m+i+£AX;q,t] = E C f ^ ^ ) = E C t ? ) ^ > Vd

(4-75)

Vd

where d™je(q) = Ylp(v)=vd Q1^- F° r 2m+2£+e = n, the number of possible Vd is L n /2j + 1 , exactly the number of partitions of n of length < 2. We thus have, from (4.75), that U$ and r ( U ^ ) are both bases for V, the Q[q,t]linear span of {H\[X;q,<]}^(A)<2We see from expression (4.75) again, that the transition matrices from {H\[X;q,t]}^x)<2 to {UyJ}Vd and from are identical. Since these are invertible matrices, we have that UvJ = F(VvJ), which can be extended to Uv = F(Vy) using Properties 5 and 25. □ Vd

5. A STATISTIC FOR MACDONALD POLYNOMIALS IN 2 PARTS

It is now clear from the previous theorem and (3.28) that for d = 2m + 21 + e, we have

J2m+t+eAX;q,t}=

J2

^-^W'+^V-W'FCUW),

(5.1)

v=(vi,...,vm+e)

where \v\t and n(v)g are as denned in Proposition 8. To provide an expres sion for J\[X;q,t] with coefficients that are determined by statistics, we associate to any standard tableau a vector G {0, l}k called a "domino" vec tor. The domino vector is determined by the succession of of operators B 2 and B 2 ' that build the associated standard tableau. Since any standard tableau T such that T{ij2y = 12 (or 21) can be obtained by acting with B 2 (or B 2 ) o n a predecessor T', such a succession is determined recursively using Property 15.

TABLEAUX STATISTICS FOR 2 PART MACDONALD POLYNOMIALS

83

Theorem 27. The Macdonald polynomials indexed by partitions with no more than 2 parts are given by Jlm+i+eJ = 2 ^ Stat(T)S' s h a pe(T)[X*],

(5.2)

\r\=d where d = 2m + 21 + e and Stat(T) = qcochars
^5 ^

with the domino vector, dv(T) = ( d v i , . . . , dv m + ^), obtained recursively by (0,dv(B 2 T))

<MT>= (i,dv(6^V)) 0 Proof.

»/T{i,2} = 12

(5-4)

,/r{1,2} = 2i iff

has degree < 1

The theorem follows directly from (5.1) and Property 15.

□

Example: The statistic associated to a standard tableau T = I I ? ] in the Macdonald polynomial J§^[X\q,t] vector of T.

is determined by finding the domino

(0, dv (if & = PL ) ) = (0,1, dv f Bj15 Sel = SI ) ) \

11[a 161

1113151 /

\

I X 13 IS I

|l|2| /

= (0,l,0,dv('B2 0 ) | = g A ) = (0,1,0,1). (5.5) This gives that |dv(T)| 2 = 1 and n(dv(7~))2 = 1. The cocharge of T = 48 357 126 is 0 + 0 + 1 + 2 + 2 + 2 + 3 + 4 = 14, and we have Stat(T) = q^q-^H2-1 = g9*. (5.6) Acknowledgments. The authors would like to thank Alain Lascoux for his significant contribution to this work. This work was carried out at Marnela-Vallee and the authors are grateful for the resources provided there. The author, J. Morse, expresses thanks to Adriano Garsia for the NSF support that made her visit to Marne-la-Vallee possible. L. Lapointe was supported through an NSERC postdoctoral fellowship (Canada). The algebraic com binatorics environment, ACE, Maple library [12] has been used extensively. We thank Glenn Tesler for allowing us to use his tableau WFgK. package. REFERENCES [1] S. Fishel, Statistics for special Kostka ■polynomials, Proc. Amer. Math. Soc. 123 (1995), 2961-2969. [2] A.M. Garsia and M. Haiman, A graded representation module for Macdonald's polynomials, Proc. Natl. Acad. Sci. USA V 9 0 (1993) 3607-3610. [3] M. Haiman, Hilbert schemes, polygraphs, and the Macdonald positivity conjecture, J. Amer. Math Soc. 14 (2001), 941-1006.

84

L. LAPOINTE AND J. MORSE

[4] A. Kirillov and M. Noumi, Affine Heche algebras and raising operators for Mac donald polynomials, Duke Math. J. 9 3 (1998), 1-39. [5] L. Lapointe, A. Lascoux and J. Morse, Determinantal expressions for Macdonald polynomials, IMRN 18 (1998), 957-978. [6] L. Lapointe and J. Morse, Schur function identities, their t-analogs, and k-Schur irreducibility, Adv. in Math., to appear. [7] L. Lapointe and J. Morse, Schur function analogs for a filtration of the symmetric function space, JCT-A to appear. [8] L. Lapointe and L. Vinet, A short proof of the integrality of the Macdonald (q,t)Kostka coefficients, Duke Math. J. 91 (1998), 205-214. [9] A. Lascoux and M.-P. Schiitzenberger, Le monoide plaxique, Quaderni della Ricerca scientifica 109 (1981), 129-156. [10] A. Lascoux and M.-P. Schiitzenberger, Sur une conjecture de H.O. Foulkes, C.R. Acad. Sc. Paris. 294 (1978), 323-324. [11] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Clarendon Press, Oxford, 1995. [12] S. Veigneau, ACE, an Algebraic Combinatorics Environment for the com puter algebra system MAPLE, Version 3.0, Universite de Marne-la-Vallee, 1998, http://phalanstere. univ-mlv.fr/~ace/. [13] M. A. Zabrocki, A Macdonald vertex operator and standard tableaux statistics for the two-column (q,t)-Kostka coefficients, Electron. J. Combinat. 5, R45 (1998), 46pp. INSTITUTO DE MATEMATICA Y FLSICA, TJNIVERSIDAD DE TALCA, CASILLA 7 4 7 , TALCA, CHILE

E-mail address:

lapointe(Dinst-mat.utalca.cl

UNIVERSITY OF MIAMI, DEPARTMENT OF MATHEMATICS, CORAL GABLES, FL

USA E-mail address: morsejamath.miami.edu

33124,

A CRYSTAL TO R I G G E D C O N F I G U R A T I O N BIJECTION FOR N O N E X C E P T I O N A L A F F I N E ALGEBRAS MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO ABSTRACT. Kerov, Kirillov, and Reshetikhin defined a bijection be tween highest weight vectors in the crystal graph of a tensor power of the vector representation, and combinatorial objects called rigged configurations, for type An . We define an analogous bijection for all nonexceptional affine types, thereby proving (in this special case) the fermionic formulas conjectured by Hatayama, Kuniba, Takagi, Tsuboi, Yamada, and the first author.

1. INTRODUCTION

The fermionic formula, denoted by M, is a certain polynomial expressed as a sum of products of ^-binomial coefficients. It originates in the Bethe Ansatz analysis of solvable lattice models in two dimensional statistical mechanics. The prototypical example is given by the Kostka polynomial K\fj,(q) € Z>o[g], which is indexed by a pair of partitions X,/J,. According to Lascoux and Schiitzenberger [12],

£ *C{T)-

KXM)=

T€T(\,n)

Here T(A, fx) is the set of semistandard tableaux of shape A and weight /x, and c(T) is the charge of the tableau T. We consider the case that fi is a single column (1 L ). Kirillov and Reshetikhin [8] gave a fermionic formula for the Kostka polynomial: (1-1)

KXi{1,)(q)

q(>)M(\,(lL);q-1)

=

1991 Mathematics Subject Classification. 81R50 05E15 05A30 11B65. 85

Primary 17B37 82B23 05A19; Secondary

86

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

where

M(A,(1*);,) = £,-«•»» II \P'a)+$a) > {m}

l
m

i

t>l

cc({m}) = ^

C

*b 2 ^ min(i, j ) m | a ) m j 6 ) ,

^2, l
^

a)

*ij>l

= L<5 a l - ^ l<6
vy _J_ -p-i

=

is

(q)p+m/(q)p(q)m 2

m

Ca6^min(ii,j)mjb), j>l

the g-binomial coefficient, s

(q)m a

=

(1 — 7)(1 — q ) ■ • ■ (1 — g ), the sum Y^im} * taken over {m\ ' e Z>o | 1 < a < n,i > 1}, satisfying pf > 0 for 1 < a < n,i > 1 and 2 D 1 «m, = A a+ i 4- A a+2 + ••• + A n+ i for 1 < a < n. Here n is an integer not less than the length of A minus 1, and (Ca&)i
RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

87

The purpose of the paper is to construct the analogue of the KKR bijection and thereby prove the X = M conjecture, for all nonexceptional affine Lie algebras, in the case of the simplest crystal bases. For An', this case corresponds to the Kostka polynomial K^{\^){q) discussed above. 2. QUANTUM AFFINE ALGEBRAS AND CRYSTALS

2.1. Affine algebras. We adopt the notation of [1]. Let g be a KacMoody Lie algebra of nonexceptional affine type X^', that is, one of the types A^\n > 1), Bnl\n > 3), d 1 } ( n > 2), D^\n > 4), A%>(n > 1), A^\n > 1), A{2n-i(n > 2), D^lx{n > 2). Note that A^1 is the same diagram as A\^ but with the opposite labeling. The Dynkin diagram of g = X^ is depicted in Table 1 (Table Aff 1-3 in [4]). Its nodes are labeled by the set / = { 0 , 1 , 2 . . . , n } . Let ai, hi, Aj (i € I) be the simple roots, simple coroots, and fundamen tal weights of g. Let 5 and c denote the generator of imaginary roots and an( the canonical central element, respectively. Recall that S = J2teiai0ii ^ c = ^2i€i^hi, where the Kac labels a, are the unique set of relatively prime positive integers giving the linear dependency of the columns of the Cartan matrix A (that is, A(ao,.. ■, a n )* = 0). Explicitly, a0-\

+ an

if g = A\

a0 + a-i + 2a 2 H h 2a„ a0 + 2a1 + --- + 2an-1 + an m-,\

x

(2.1)

0 = <

\a0 + ai+2a2

+

if g = JB„ if g = C^]

h 2a„_ 2 + a n _ i + an

if g = Dnl) r2\

2a0 + 2ai + • • • + 2an-i + an a0 + 2al + --- + 2an-1+2a„

if Q = A\^ ifg = ^ ) t (2)

ao + ot\ + 2a 2 H h 2a„_ 1 + an .ao + aiH han_i+an

if 0 = A\^_x ifg = D ^ 1 .

The dual Kac label af is the label m for the affine Dynkin diagram obtained by "reversing the arrows" of the Dynkin diagram of g, or equivalently, the coefficients giving the linear dependency of the rows of the Cartan matrix A. Note that a^ = 2 for g = A\j} and a^ = 1 otherwise. Let (-|-) be the normalized invariant form on P [4]. It satisfies (2.2)

(oulotj) =

av &i

^Aij

for i,j £ I. In particular (2.3) if aa is a long root.

(aa\aa) =

a

^ o

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MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

TABLE 1. Dynkin diagrams for X^'. The labeling of the nodes (by elements of I) is specified under or the right side of the nodes. The numbers tj (resp. iV) defined in (2.4) are attached above the nodes for r = 1 (resp. r > 1) if and only if t4 ?£1 (resp. tY ^ 1).

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

89

For i G I let a. ti = m a x ( - ^ , a o ) ,

(2.4)

aY = max(—,a0).

tf

The values U are given in Table 1. We shall only use £V and U for « G 7* = I\{0}. For a G 7* we have
1

if

T-

= l,

ta = a% if

r>l.

We consider two finite-dimensional subalgebras of g: g, whose Dynkin diagram is obtained from that of g by removing the 0 vertex, and QQ, the subalgebra of Xjv fixed by the automorphism a given in [4, Section 8.3]. TABLE 2 v(l) 7V XN XJV

0 0 flo

A

4( 2 ) -^2n

,(2)t ^2n

C« S„

Bn Bn

Cn

Bn Bn

cn

Let 0 (resp. 0Q) have weight lattice P (resp. P), with simple roots and fundamental weights aa,Aa (resp. aa,Aa) for a G /*. Note that 0 = 0Q f° r 0 ^_42„>. For 0 = A%>, g = Cn and g„ = B„. P is endowed with the bilinear form (-|-)', normalized by (2.5)

(aa\aa)'

= 2r/a%

if aa is a long root of 0Q.

(2)

For ^ , the unique simple root a\ of 0g = B\ is considered to be short. Note that aa,Aa and (-|-) may be identified with aa,Aa and (-|-)' if 0 ^ -

Define the Z-linear map t : P —>• P by (2.6)

t(A a ) = e a A a

for a G /*,

where ea is defined by (2.7)

ea = / 2 l f Q = A^ a n d 1 1 otherwise.

a =

n

In particular t(aa) = eaaa for a G I*. We have (2.8)

(t(ab)|t(a 6 ))' = ao(afc|aj,)

for all b G I*.

If 0 = ^2n both sides of (2.8) are equal to 8 if b = n and 4 otherwise. Especially for 0 = A2 (n = 1), we have (ai\ai)' = 2 and ( a i | a i ) — 4. In the rest of the paper we shall write (-|-) in place of (-|-)'.

90

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

2.2. Simple subalgebras. For later use, specific realizations are given for the simple roots and fundamental weights of the simple Lie algebras of types Bn, Cn, and Dn, which appear as the subalgebras g and gg- of g. In each case the sublattice of P given by the weights appearing in tensor products of the vector representation, is identified with Z n . Let {EJ | 1 < i < n} be the standard basis of Z n . The simple Lie algebra Bn. (2.9)

aa = ea - ea+x

for 1 < a < n

Aa = ei + ■ ■ • + ea

for 1 < a < n

An = 2 ^ 1 + --- + £ n)A € Z™ is .Bn-dominant if and only if (2.10)

Aa - A a + i > 0 A„>0.

for 1 < a < n

The simple Lie algebra Cn. (2.11)

aa = ea-

e a +i

for 1 < a < n

an = 2e n Aa = ei + ■ ■ ■ + ea

for 1 < a < n.

A £ Z™ is C7„-dominant if and only if it is I? n -dominant (2.10). The simple Lie algebra Dn. (2.12)

aa = ea - ea+i

for 1 < a < n

A.a = ei + ■ ■ ■ + ea An-i = -(ei H

for 1 < a < n — 2

he„_! - e„)

An = ^(ei + " ' + £ n - i + e " ) A e Z n is Z?„-dominant if and only if (2.13)

Aa - Aa+i > 0 A n -i + An > 0.

for 1 < a < n

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

91

2.3. Crystals. Let g' be the derived subalgebra of g. Denote the cor responding quantized universal enveloping algebras of g D g' D g by Uq(g) D U'(g) D 17,(5). In [2] it is conjectured that there is a family of finite-dimensional ir reducible f/g(g)-modules {W\ \ a € I*,i e Z>o} which, unlike most finite-dimensional £/^(g)-modules, have crystal bases Ba'%. This family is conjecturally characterized in several different ways: 1. Its characters form the unique solutions of a system of quadratic re lations (the Q-system) [9]. 2. Every crystal graph of an irreducible integrable finite-dimensional C/q(g)-module, is a tensor product of the Ba,t. 3. For A G P let V(X) be the extremal weight module denned in [3, Section 3] and B(X) its crystal base, with unique vector u\ £ 5(A) of weight A. Then the affinization of Ba'% (in the sense of [5]) is isomorphic to the connected component of u\ in S(A), for the weight A = iha (except when fl = A^ and o = n, in which case A = 2iAa). In light of point (2) above, we consider the category of crystal graphs given by tensor products of the crystals Ba'%. We introduce notation for tensor products of Ba'%. Let /J, — {Li)aei',i£l>0 be a matrix of nonnegative integers, almost all zero. Define (2.14)

flW

=

(g)

(B°'')® L i o) .

(a,i)€7*xZ > 0

In type An this is the tensor product of modules, which, when restricted to An, are irreducible modules indexed by rectangular partitions. The set of classically restricted paths (or classical highest weight vectors) in B^ of weight A e P = © i G j . Z>oAj is by definition (2.15) 7>(A,/x) = {b€ B^

| wt(6) = A and e^ undefined for all i € / * } .

Here ei is given by the crystal graph. For b, b' e B°"'1 we have b' = ei(b) if there is an arrow b' —1-> b in the crystal graph; if no such arrow exists then 2,(6) is undefined. Similarly, b' = fi(b) if there is an arrow b —> b' in the crystal graph; if no such arrow exists then fi{b) is undefined. If Bx and Bi are crystals, then for b\ 62 £ B\ B2 the action of e* is defined as ei{bi®b2) = < ■ _ - , ^&i®ei&2

, else,

where £,(&) = max{/e j e^ is defined} and . BW = B®L w h e r e B = _gi,i. We shall write B^

92

M A S A T O O K A D O , A N N E SCHILLING, AND M A R K S H I M O Z O N O

TABLE 3. Crystals 5 1,1

The crystal graphs B1'1 are listed in Table 3. In each case (other than A)?') the elements of B = B1,1 consist of {k, k 1 < k < n} and possibly elements 0 and <j>.

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

93

Remark 2.2. By glancing at Table 3, one may check that the following are equivalent for b = bL bi-\ ® • • • ® &i G B®L and A G P . (1) 6 is a classically restricted path of weight A € P . (2) A — wt(&j[,) G P , foi_i ® ■ • • ® &i is a classically restricted path of weight A - wt(6i,), and if &L = 0 G B then An > 0 (where A is viewed as an element of Z n ). The weight function wt : B —> Z n is given by wt(fc) = £fc

for 1 < k < n

wt(/c) = —efc

for 1 < k < n

wt(0) = wt(4>) = 0. The weight function wt : B®L -> Z™ is denned by wt(6 L ® • • • 6X) = S , = i w t ( ^ ' ) - So if A = wt(p) where p G B® L , then Afc is the multiplicity of k in p minus the multiplicity of k in p. 2.4. One-dimensional sums. The energy function D : fiW -> Z gives the grading on B ^ . In the case B^) = B®L it takes a simple form. Due to the existence of the universal .R-matrix and the fact that W\ is irreducible, by [5] there is a unique (up to global additive constant) function H : B1'1 B 1 , 1 -> Z called the local energy function, such that (2.16) H(ei(b®b'))=:H(b®b')+

'-1 < 1 0

if i = 0 and e0(&<8>&') = b®e0b' if i = 0 and e0(b®b') = e0b®b' otherwise.

Let $ G B1'1 be the unique element such that
#(1®1)=0.

Then L-l

, , ( 2 - 18 )

E(bL®---®bl)=LH{b1®$)

+ V(L-j) ,=1

Jffo+iOfy),

£>(&£ ® ■ • • ® &i) = E(bL ® • ■ • ® 6i) - E ( l • • • 1). Define the one-dimensional sum X(X,fi;q) (2-19)

X(A, M ;<7)=

G Z[q,q~l) by
£ 6€P(A,/i)

Since B W is completely reducible as a £/?(g)-crystal, one has

£

e w t (V ( 6 ) = £

XA*(A,M;
94

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

where xA is the character of the irreducible £/g(g)-module of highest weight A. It can be shown that X(X,/j,;q) € Z>o[<7-1]. For convenience we define (2.20)

H=-H,

X(\,fi;q) = X(X,fi-q-1).

D = -D,

3. RIGGED CONFIGURATIONS AND THE BIJECTION

3.1. T h e fermionic formula, g ^ A£n . This subsection reviews defi nitions of [1, 2]. Let Q be a Kac-Moody algebra of nonexceptional affine type that is not of the form A^ ■ Fix A G P and a matrix n = {L\ ) of nonnegative integers as in subsection 2.4. Let v = (m\a') be another such matrix. Say that v is a A-configuration if /

(3.1)

a)

\ iL a) K

X

^ i m l a a = t J2 i ~ «~ ael* J€Z> 0

■

I ael* \i6Z>0

/

Say that a configuration v is //-admissible if (3.2)

p\a) > 0

for all a € I* and i e Z > 0 )

where (3.3)

p0 V be//

Write C(A, /i) for the set of /x-admissible A-configurations. Define (3.4)

cc(u) = - ^jT

Y^

(a 0 |a6)min(ti,j,t a fc)m] a) m[. 6) .

The fermionic formula is defined by

(3.5)

M(A,wg)= Y: ^ n n K^r^" • I/€C(A,M)

a6/'i£Z>o

L

*

9a

The X = M conjecture of [1, 2] states that (3.6)

X(\,v;q) = M(\,v,q).

3.2. Rigged configurations, g ^ A£n. The fermionic formula M(X,n) can be interpreted using combinatorial objects called rigged configurations. These objects are a direct combinatorialization of the fermionic formula M(X,fi;q). Our goal is to prove (3.6) under Assumption 2.1 by defining a statistic-preserving bijection from rigged configurations to paths. For this purpose it is convenient to use an indexing slightly differing from that used above.

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

95

For a € I*, define

(3.7)

if a = n and g = C« if a = n and g = Bit' otherwise.

\ 1

va is half the square length of aa for untwisted aftine types and is equal to 1 for twisted types. A quasipartition A of type a e I* is a finite multiset taken from the set u a Z>o. Denote by rrii(X) the number of times i e u a Z>o occurs in A. The diagram of such a quasipartition has, for each i G u a Z>o, mj(A) rows consisting of i boxes, where each box has width va. Set (3.8)

U = {(a,i)

\a&r,ievaZ>0}.

Denote by {v*, J') a pair where u' = {v^}aeim is a sequence of quasipartitions with i/(°) of type a and J* = {■^°'^}(a,i)e'H is a double sequence of partitions. For (a,i) G H, define

'

(3-9)

y:

Then a rigged configuration is a pair (v*,Jm) subject to the restriction (3.1) and the requirement that j( a , t ) be a quasipartition contained in a mf'(u') x P>a\u*) rectangle. The set of rigged configurations for fixed A and /i is denoted by RC(A, /J,). Then (3.5) is equivalent to

F(AlM)=

« ee( "'' J " )

E (I/«,J«)€RC(A,AI)

where cc(u',J*) = cc(i/) + \J'\ and \Jm\ = E( a ,i) e -H f al j ( a , i ) l sponding to v* under (3.9).

for v

corre

3.3. A\,^ rigged configurations. In this subsection let g = A(22 • As this case is not considered in [1] we shall only give the definition in terms of rigged configurations, although it is easy to express the result as a sum of a product of g-binomials (see [13, Section 7.6]). The important feature is that the riggings of odd-sized parts of v^n\ must have the form x/2 where x is an odd integer. So let fi and A be as in subsection 3.1. Given a matrix v = (m\a'), let P^'(f*) and m\a'(u') be defined as before. Call v* /x-admissible if P\ (y') > 0 for all a £ I* and i € Z > 0 , together with the extra condition that (3.10)

Pin\v')

> 1

if » is odd and m\n)(v*) a J

> 0.

A rigging J" consists of quasipartitions j( > ) for a e I* and i € Z>oFor a ^ n or t even, j( a ' J ) is an ordinary partition satisfying the usual properties. For a = n and i odd, j( n , t ) is a quasipartition contained in a

96

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

rectangle with P>n'{v*) columns and m^ (i/*) rows, but it has cells of width 1/2 and each part size must be of the form x/2 for x an odd integer. This defines the set RC{X,p) for g = A^ . Then F(X,fx) is defined as before where \J'\ is the sum of the areas of all the quasipartitions Ja,%. This definition is compatible with the virtual crystal realization which embeds paths (and rigged configurations) of type A^ into those of type A2n-\ [13]. 3.4. T h e bijection from R C s t o p a t h s . We now describe the general form of the bijection $ : RC(A,/i) —► V(X, /x) under Assumption 2.1. Let H = (L^) be such that B^ = B®L, that is, L\a) = L5alSn. Let p. be such that B<« = £ ® ( ^ - i ) . Let (v*, J') e RC(A, /j,). We shall define a map rk : RC(A, / J ) - > B which associates to (v*, J') an element of B called its rank. Denote by RC(,(A, /x) the elements of RC(A, p) of rank b. We shall define a bijection 8 : RCb(A,/i) —> RC(A — wt(b),p,). The disjoint union of these bijections then defines a bijection <5 : RC(A, p,) —> \JbeB RC(A — wt(6), p). The bijection <& is defined recursively as follows. For b G B let ^ ( A , p) be the set of paths in B^) = B®L that have b as leftmost tensor factor. For L = 0 the bijection $ sends the empty rigged configuration (the only ele ment of the set KC(X,p)) to the empty path (the only element of V(X,p)). Otherwise assume that <E> has been defined for f?®(L_1) and define it for B®L by the commutative diagram RC6(A,/i) (3.11)

—*->.

Vb{\p)

s[

|

R C ( A - w t (&),£) —^—>• P(A-wt(&),/2) where the right hand vertical map removes the leftmost tensor factor b. In short, (3.12)

$(!/•, J*) = rk(i/', J*) <8>

$(5{v',J')).

R e m a r k 3 . 1 . For <3? to be well-defined, by Remark 2.2 it must be shown that if b = rk(i/*, J*), then p = X — wt(fc) is dominant, and if 6 = 0 then A„ > 0 . We also require the bijection $ : RC(A,/i) -> V(X,p) given by $ = $ocomp where comp : RC(X,p) —► RC(A,/x) with comp(^*, J*) = (u*,J*) is the function which complements the riggings, meaning that J' is obtained from J* by complementing all partitions j(a>*) in the mf1' x P>a'(v') rec tangle. T h e o r e m 3.2. $ : RC(A,//) —> V(X,p) is a bijection such that (3.13)

cc{v', J') =7){§(v',

J*))

for

all(u',J')eRC(X,p).

RIGGED CONFIGURATION BIJECTION F O R AFFINE ALGEBRAS

97

For type A„ a generalization of this theorem for all fj, was proven in [10]. For other types Theorem 3.2 is proved in section 5. 4.

T H E BIJECTION FOR EACH ROOT SYSTEM

In this section the maps rk and S are denned in a case-by-case manner. For each g, an explicit formula is given for the vacancy numbers P>a'{v') (see (3.9)), obtained by writing (3.3) in terms of the function Qi (see (4.1)) using the data for the simple Lie algebras given in section 2.2. Then for (i/*, J') € RC(A, /i), an algorithm is given which defines b — rk(i/*, J*), the new smaller rigged configuration {v*,J') = 5{um, J') such that (i>*, J') £ RC(/?, ji) (where p = X — wt(&)), and the new vacancy numbers in terms of the old. For a quasipartition T with boxes of width v and i € vZ>o, define

(4.1)

Qi(r) = J2mm(Tj,i), 3

the area of T in the first i quasicolumns. The quasipartition j( a , J ) is called singular (with respect to the configu ration v') if it has a part of size P>a' (v*). If A is a statement then x(^) = 1 if A is true and x(A) = 0 if A is false. We also use the Kronecker delta notation Sa (o V) = Qi{v{a~l))

~ 1Qi{v{a)) + QMa+1))

Pln-2\v')

= Qiiv^-Q)

- 2Q i ( I >~ 2 >) + Qi(^n-V)

i f ^ V )

= Qi{v(n-2))

~ 2Q i (*>- 1 ) )

+ LSa,x for 1 < a < n — 2 + Qi(v{n))

i> ( n ) (i/') = Q j (i/(»- 2 ))-2Q i (i/W) Constraints. a X

|i/°)| = L~Y^ b ^

(4.3)

1^-1)1=

( *

forl
6=1 n—\ L

_^A

6

+ An)

6=1

\»w\=\{L-ib>*) Z

6=1

98

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

Algorithm 5. Set £^ = 0 and repeat the following process for a = 1, 2 , . . . , n — 2 or until stopped. Find the minimal index i > £( a_1 ) such that j( a '*) is singular. If no such i exists, set b = a and stop. Otherwise set £(a) = i and continue with a + 1. If the process has not stopped at a = n — 2 continue as follows. Find the minimal indices i,j > £^n~2'> such that j(n_1>«) and j("^) are singular. If neither i nor j exist, set b = n — 1 and stop. If i exists, but not j , set g(n-i) _ ^ £ _ n a n ( j s t; 0 p jf j exists, but not i, set ^ n ' = j , b = n and stop. If both i and j exist, set ^(™-1) = •£, £ £(a+1> where ^ n _ 1 ) = m a x ^ " - 1 ) , ^ ) such that j(aa' [v*) in j( a '*)). If no such i exists, set b = a + 1 and stop. If the process did not stop, set b = 1. Set all yet undefined ^°) and fca) to oo. New RC.

(4.4)

1 -1 (a) a , ml (£>')=mj V )+{ 1 -1 0

if i = £^ - 1 if i = £^ if i = J(») - i and 1 < a < n - 2 if i =Z
The partition j( a>1 ) is obtained from j( a '*) by removing a part of size P / a ) ( 0 for i = ^ and i = £ia\ adding a part of size P}a)(i>') for i = £(a) — 1 and z = ^ a ' — 1, and leaving it unchanged otherwise. Change in vacancy numbers. (4.5)

P}a)(P)

= P}a)(v')-x(£{a~1}

- X ^ - D < i) + 2x{£{a) < t) - X ' " r "f"l n) <' 't) for 1 < a < n - 2 i ( n - 2 ) (j/') = p / n - 2 ) ( i / « ) - x ( ^>» - 3 ) < i) + 2x(^ ( n _ 2 ) < i) - x ( ^ ( n _ 1 ) < t _ x (£(n-3) < ^ + 2x(^ re - 2 > < ») - x ( / n ) < t) p.("-D(^)_ x (^-2) < j) _ x( /(« _ ( " _ 1 ) < i) >- 2 ) _ < i)t + 2x(^

= P i ( " V ) - x ( * ( n _ 2 ) < t) - X(£(n~2)
RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

99

4.2. Bijection algorithm for type Bn ■ Vacancy numbers. (4.6) P[a\u')

= Qi^—V)

- 2Qi(vM) + Qi{v[a+l))

P<>n-l\v')

= Qi{^n-V)

- 2Q i (/>- 1 >) + 2Qi{vV>)

P«

( B

V)

+ W„,ifor i € Z> 0 1< a 0

= 2Qi(i/<"-1)) - 4Qi(vM)

for i € 2\- > o -

Constraints. a

| i » | =L-^2\b (4.7)

forl
6=1

6=1

Algorithm <5. Call a partition quasi-singular if it is not singular and has a part of size Pi (y*) — 1. Set £(°1 = 0 and repeat the following process for a = 1,2,... ,n — I or until stopped. Find the minimal index i > £( a_1 ) such that j( a - 1 ) is singular. If no such i exists, set b = a and stop. Otherwise set €^ = i and continue. If the process has not yet stopped, continue as follows. For brevity let us denote by (S) and (Q) the following conditions: (S) i > £(n~V and ./("'*) is singular. (Q) i = ^ n _ 1 ) - \ and •/("■*> is singular; or i > 6n~^ singular.

and J ^

is quasi-

Find the minimal index i > £^n~^ — A such that (S) or (Q) holds (note that (S) and (Q) are mutually excluding). If no such i exists, set b = n and stop. If (S) holds set #n> = i and t^ = i - \. Say that case (S) holds. If (Q) holds set £(n~> = i and find the minimal index j > i such that (S) holds. If no such j exists, set b = 0 and stop. Say that case (Q) holds. Otherwise set £("■*> — j and say that case (Q,S) holds. If the process has not yet stopped continue in the following fashion for a = n— 1, n — 2 , . . . , l o r until stopped. Find the minimal index i > £(a+1) such that j(a'*) is singular (if £^ = i then j(a-*) actually needs to have two parts of size P^{v')). If no such i exists, set b = a + 1 and stop. Otherwise set £^ = i and continue. If the process did not stop for a > 1 set 6 = 1 . Set all undefined £ and 1^ for 1 < a < n to oo.

100

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

N e w RC.

(4.8)

mW(P') = mS B) (0 +

1 -1 1 -1 0

if i = £(<*) - va ifi = ^°) if i = l^ - va ifi = £(a) otherwise.

Note that if two or more conditions hold, all of the changes should be performed. For 1 < a < n the partition j( a ' 1 ) is obtained from j( a -') by removing a part of size P$a)(i>u) for i = £^ and i = Ha\ adding a part of size P^a)(i>') for i = £^ — 1 and i = 1^ — 1 and leaving it unchanged otherwise. If case (S) occurred j("' s ) is obtained from j(n-*) by removing a part of size Pln\v') for i = £(n\ adding a part of size P^n)(i>') for i = 1^ - 1, and leaving it unchanged otherwise. If case (Q) holds remove the largest part in j("- i ) for % = &^ and add a part of size P\n){v') for i = ^ n > - \. If case (Q,S) holds, then apply (S') for t = l^ and (Q') for t = 1^ where (S') obtain ./("•*) from ./("•*) by removing the largest part for i = t and adding a part of size P>n'(v') for i = t — | , leaving all other j(n>1) unchanged; (Q') obtain ■/("•*> from j( n '*) by removing the largest part for i = £ and adding a part of size P*>n) {£>') - 1 if t < ^ " " ^ and of size P^n) (v') if t = £(™_1) for i = £ — | , leaving all other j(n>1) unchanged. Change in vacancy numbers. (4.9)

pW(P) = /f1 V ) - x(^ (a_1) < 0 + 2X(^(a) < i) - x(^(a+1) < i) - x(£(a_1) < *) + 2x(^(a) < i) - x{(-{a+l) < i) for 1 < a < n — 1 and

i>(i/«) = ^ ( " V ) - x^"" 15 - \ < i) - x(e{n~1}
RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

101

Vacancy numbers. P t ( a V ) = Qi(v(a~l)) P^n\v') (4.10)

= Ql(v(n-V)

~ 2Qi(^ (a) ) + Qi(*> + 1 ) ) + L6a,i for i € Z> 0 1 < a < n - Qi(i/(")) for i € 2Z> 0 .

Constraints.

L~Y,Xb

» l

(4.11)

for 1 < a < n.

6=1

Algorithm 5. Set ^°) = 0 and repeat the following process for a = 1,2,... , n or until stopped. Find the minimal index i > £(° -1 ) such that j(a'*) is singular. If no such i exists, set b = a and stop. Otherwise set £(a) = i and continue. If the process has not stopped continue as follows for a = n — l,n — 2 , . . . ,l_or until stopped. Set #n> = £
=

^(o+l)

set

e(a)

=

£(a)

a n d

reset

g(a)

=

£(a) _ j _ g a y

cage

a+1

a J

1 -1 1 -1 0

if i = £^ -- 1 if » = *(«> if i = l^ --1 ifi = # a > otherwise.

(g)

holdg

Otherwise find the minimal index i > £( 1 such that j( ' ) is singular. If no such i exists, set b = a + 1. Otherwise set l^a^ = i and continue. If the process does not stop for a > 1 set b = 1. Set all undefined I^ and £^ for 1 < a < n to oo. New R C .

(4.12)

m<
If two or more conditions hold then all changes should be performed. If a = n or case (S) holds for 1 < a < n the partition j(°'') is obtained from j( a - 1 ) by removing a part of size P^'iy*) for i = l^a\ adding a part of size P>a'(i>') for i = 1^—2, and leaving it unchanged otherwise. Otherwise j(a.») i s obtained from j( a ' 1 ) by removing a part of size P>a'(vm) for i = £^ and i = Ha\ adding a part of size P^a\v') for i = £^ - 1 and i = #a> - 1, and leaving it unchanged otherwise. Change in vacancy numbers. (4.13) P(a\v°)

= ^ ( a V ) - x ( ^ ( a - 1 ) < i) + 2X(£{a) < 0 ~ X(^ (a+1) < ») _ x (/(a-D < ^ + 2 x ( | W < j) - x ^ + D < i)

102

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

for 1 < a < n — 1 and

pW>{v») = i > ( n V ) - x ( * ( n _ 1 ) < i) ~ X{i(n-l) < i) +X(l(n)

Recall here that g = Cn and

Vacancy numbers. The vacancy numbers are the same as for type C„ (4.10) with the only exception that now i e Z>o even for a = n. Constraints. The constraints are the same as for type d1}

(4.11).

Algorithm S. Set £^> = 0 and repeat the following process for a = 1,2,... ,n or until stopped. Find the minimal index i > ^( a_1 ) such that j(a,i) j s s m g U i a r . jf n o s u c r i i exists, set b = a and stop. Otherwise set £(a) = % and continue. If £(") = 1 set b — and stop. Otherwise say case (S) holds for a = n and continue. If the process has not stopped, set fln) = &^ and reset £^ = fcn) - 1. Continue as follows for a = n — l,ra — 2 , . . . ,1 or until stopped. If £^ = /( a + 1 ) set (^ = £^a) and reset £^ = £^ - 1 . Say case (S) holds. Otherwise find the minimal index i > ^ a + 1 ) such that j(a>J) is singular. If no such i exists, set b = a 4- 1. Otherwise set £^ = i and continue. If the process does not stop for a > 1 set b = T. Set all undefined £^ and £^ for 1 < a < n to oo. N e w R C . The configuration changes in the same way as for type C„ (4.12). If case (S) holds for 1 < a < n the partition j(a>J) is obtained from J(<M) by removing a part of size P± (v') for i = £^a\ adding a part of size P4 (£*) for i = Z'°) — 2, and leaving it unchanged otherwise. Otherwise J(<M) i s obtained from j(a>») by removing a part of size P>a\v*) for i = £^ and i = Ha\ adding a part of size P / a ) ( £ ' ) for i = £^ - 1 and i = £^ - 1, and leaving it unchanged otherwise. Change in vacancy numbers. The change in the vacancy numbers is the same as for type C„ ' (4.13). 4.5. Bijection algorithm for type .A^n-i-

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

103

Vacancy numbers. (4.14)

i f 2 V ) = Qi{v{a-l)) - 2Qi{u^) + Q,(i/<°+1>) + LSaA for 1 < a < n — 1 P^-X\v»)

= Qi^"" 2 )) - 2Qi(v)

i\ ( n ) (i/') = Q i (i/("- 1 ))-2Q i (i/W). Constraints. a

|i/(°)|=L-^A6

forl
b=l

(4-15)

n

(n)

l" l = ^ - X > ) . Z

6=1

0

Algorithm <5. Set ^( ) = 0 and repeat the following process for a = 1,2,... , n or until stopped. Find the minimal index i > £( a_1 ) such that j(a>») j s singular. If no such i exists, set b = a and stop. Otherwise set ^(a) = i and continue. If the process has not stopped set I^> = l^ and continue as follows for a — n— l , n — 2 , . . . ,1 or until stopped. Find the minimal index i > £( a+1 ) such that j(a'*) is singular (if i = lSa> then there need to be two parts of size Pt(v') in j ( a ' ^ ) . If no such i exists, set b = a + 1 and stop. If the process did not stop, set b = 1. Set all yet undefined ^ and 1^ to oo. New R C .

(4.16)

'l -1 m| o) (i? , ) = m j o ) ( i / ' ) + < 1 -1 v0

ifi = ^ ° ) - l ifi = ^°) if i = #°> - 1 and 1 < a < n - 1 if i = £^ and 1 < a
The partition j(a>8) is obtained from j(°>*) by removing a part of size P>a'{v') for i = ^(a) when 1 < a < n and i = £(°) when 1 < a < n, adding a part of size P^a\i>') for i = £^ - 1 when 1 < a < n and i = £^ - 1 when 1 < a < n, and leaving it unchanged otherwise. Change in vacancy numbers. (4.17) /> (a) 0>') = Pla){V)-X(l(a-l)

-x{i{a~l)

<») + 2 x(^ (a) < 0 - x(^ (a+1) < 0

104

M A S A T O O K A D O , A N N E SCHILLING, A N D M A R K S H I M O Z O N O

for 1 < a < n — 1 and P$n\v')

= i ^ O * ) - ^ " - 1 ) < {) + 2X{l{n) < i) - x{i(n~l)

< *')■

(2)

4.6. Bijection algorithm for type £>„+!• Vacancy numbers. (4.18) Pt{a\u') i>

(n

= Qi(iy(a-V) - 2 Q i ( i » ) + Qi(^a+V) {n 1}

V ) = 2Qi(v - )

- 2Qi(^

(n)

+ W a ,ifor 1 < a < n - 1

).

Constraints. The constraints are the same as for type C^] (4.11). Algorithm <5. Call a partition quasi-singular if it is not singular and has a part of size P^a'(v*) — 1. Set £^ = 0 and repeat the following process for o = 1, 2 , . . . ,n — 1 or until stopped. Find the minimal index % > £(a~l) such that j(a>*) is singular. If no such i exists, set b = a and stop. Otherwise set £^ = i and continue. If the process has not yet stopped, continue as follows. Consider the following conditions: (S) j( n '') is singular and i > 1; (P) j( n , ! ) is singular and i = 1; (Q) j( n '') is quasi-singular. Find the minimal index i > ^(™_1) such that one of the mutually exclusive conditions (S), (P) or (Q) holds. If no such i exists, set b = n and stop. If (P) holds set e^ = i, b = <> / and stop. If (S) holds set £^ = i - 1, #") = i, say case (S) holds for o = n and continue. If (Q) holds set £^ = i. Find the minimal j > i such that (S) holds. If no such j exists, set b = 0 and stop. Else set £(") = j , say case (Q,S) holds and continue. If the process has not stopped continue in the following fashion for a — n - 1, n - 2 , . . . , 1 or until stopped. If ^ a ) = fia+V set £^ = 6^ and reset £(a) = ((a) _ l_ g a y case (S) holds for a. Otherwise find the minimal index i > £(a+l) such that j( a ' 1 ) is singular. If no such i exists, set b — a + 1 and stop. Otherwise set v-a> = i and continue. If the process did not stop for a > 1 set b — 1. Set all undefined £^ and £^ for 1 < a < n to oo. New R C . The new configuration v* is given by (4.12). If case (S) holds for 1 < a < n the partition j(a>^ is obtained from a j( >») by removing a part of size P\ {v') for i = £("\ adding a part of size P>a'{p') for i = £(a) — 2, and leaving it unchanged otherwise. If (Q) or (Q,S) holds for a = n, then J^n'l> is obtained from j( n '*) by removing a part of size P$n\v*) - 1 (resp. Pt(n)(V)) for i = £^ (resp. i = £<")), adding a part

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

105

of size P>n){v') (resp. P/ n ) (£*) - 1 ) for i = e^ - 1 (resp. i = ttn) - 1 ) , and leaving it unchanged otherwise. Otherwise J(a>J) is obtained from J^a^ by removing a part of size P>a'(v*) for i = t^ and i = g(a\ adding a part of size PJ;a'(i>') for i = l^ - 1 and i = l^ — 1, and leaving it unchanged otherwise. Change in vacancy numbers. (4.19) Pla\u')

= Pla\v*)-x{Z{a~l)
for 1 < a < n — 1 and P/ n ) (i>') = p/ n ) (z/*)-2 X (^ ( n - 1 ) < i) + 2*(* < i) -2X{I(n~l)

<»)■

4.7. Bijection algorithm for type A\n . Vacancy numbers. The vacancy numbers are given by the same formula as for type C i (4.10) with the only exception that in this case i £ Z>o for all a € r . Algorithm 6. If a = n and i is odd, then j( n , t ) is never singular. For i odd, call f(° _1 ) such that j(a'^ is singular. If no such i exists, set b = a and stop. Otherwise set £^ = i and continue. If the process has not yet stopped, continue as follows. Consider the conditions (S) i is even and j(n'^ is singular; (Q) i is odd and j("' J ) is quasi-singular. Find the minimal index i > l^n~^ such that one of the mutually exclusive conditions (S) or (Q) holds. If no such i exists, set & = n and stop. If (S) holds set l^ = i — 1,1^ = i, say case (S) holds for a = n and continue. If (Q) holds set £^ = i. Find the minimal j > i such that (S) holds for j . If no such j exists, set b = 0 and stop. Else set 1^ = j , say case (Q,S) holds and continue. If the process has not stopped continue in the following fashion for a = n - 1, n - 2 , . . . , 1 or until stopped. If (.^ = £^a+1'> set l^ = £ ^ and reset £(<*)= (W-1. Say case (S) holds for a. Otherwise find the minimal index i > £(a+l) such that j(°' 1 ) is singular. If no such i exists, set 6 = a + 1 and stop. Otherwise set £^ = i and continue. If the process did not stop for a > 1 set 6 = 1 . Set all undefined &■) and £^ for 1 < a < n to oo.

106

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

N e w R C . The new configuration v* is given by (4.12). If case (S) holds for 1 < a < n the partition j(a>') is obtained from a 1 j( < ) by removing a part of size P} (u*) for i = Ba\ adding a part of size P>a'(v') for i = i^ — 2, and leaving it unchanged otherwise. If (Q) or (Q,S) holds for a = n, then J("'0 is obtained from ■/("'*> by removing a part of size P>n'(v') — 1/2 for i = £^> (and a part of size P^n\v') for i = /<") < oo), adding a part of size ^ ( n ) (£*) for i = £^ - 1 (and a part of size P>n'(v') — 1/2 for i = £^ — 1 < oo), and leaving it unchanged otherwise. Otherwise j( a '*) is obtained from j(°' 1 ) by removing a part of size P^ (v*) for i = £^ and i = £<-a\ adding a part of size PJ;a){i>') for i = £^ - 1 and i = £(a) — \7 and leaving it unchanged otherwise. C h a n g e in vacancy numbers. The vacancy numbers P>a (v') change as in (4.13). 5. P R O O F OF THEOREM

3.2

In the following subsections Theorem 3.2 is proved case-by-case for the various root systems. The following notation is used. Let (u',J') € RC(A,/*), b = rk(i/ , ) Jm) e B, p = A - wt(fo), and (u',J') = S(v',Jm). There are three things that must be verified: (I) p is dominant and b can be appended to any path in V(p, p) to give an element of V(X,p). (II) {y9, J*) e RC(p,jJ,) where B^ = S ® ^ " 1 ) . (Ill) The conditions of Lemma 5.1 are satisfied. Parts (I) and (II) show that S is well-defined. The proof that 5 has an inverse, is omitted as it is very similar to the proof of well-definedness. Part (III) suffices to prove that 3> preserves statistics. Ebr ( I / - , J') € RC(A, p), define A{cc{v',J')) = cc(v*,J*)-cc(5'(v*,J')) and A 2 (cc(z/', J ' ) ) = A(cc(i/ , ,J*)) - A(cc(5'(u\ J*))) where 5' = comp o S o comp. Lemma 5.1. To prove that (3.13) holds, it suffices to show that it holds for L = 1, and that for L > 2 with 3>{i>*, J*) = &L ■ ■ ■ ® b\, we have (5.1)

A(cc(v-,J'))

= %a[V-X(bL a o

= 4>),

and H(bL ® 6i_i) = %(a[1) - d ^ ) - X (6i = )+ x ( 6 1 _ 1 =
(5.2)

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

107

Proof. If L = 0, RC(A,/i) and "P(A,/x) are both empty unless A = 0, in which case RC(A,/z) (resp. V(X,fi)) is the singleton set containing the empty rigged configuration (resp. the empty path). Both of these objects have statistic zero. The case L = 1 is given by hypothesis. For L > 2, by the definition (2.18) and (2.20) of D, (5.3) L-l

~D(bL <8> • ■ • ® h) - Z?(&L_i ® • • • <8> &i) = #(&i ® &*) + ^

#(&j+i ® &?)•

Therefore by induction on L it suffices to prove that A(cc(^*, •/*)) is given by the right hand side of (5.3). By induction and again "taking the differ ence" it suffices to prove that A2(cc(S,J'))

= H(bL ® &L_i).

But this follows from (5.1) and (5.2).

□

We also need several preliminary lemmas on the convexity and nonnegativity of the vacancy numbers P>a'(v'). Lemma 5.2. For large i, we have Pz{a)(!/•) = Aa - A a + 1

for 1 < a < n

(2A„

for B{n1], D{?+1

IA„_I+A„

forD(nl).

Proof. This follows from the formulas for the vacancy numbers (4.2), (4.6), (4.10), (4.14), (4.18), the constraints (4.3), (4.7), (4.11), (4.15), and the fact that for large i, Qi(v{a)) = | i » | . □ Direct calculations show that (5.4)

Type £ # ) -P£\{v') { l

m r \v')

+

2PfXV')-Pll\{y') )

- 2mf {vm) + m^ + 1 ) (j/ # ) + LSaASiA

™ i " ~ 3 V ) - 2 m f " 2 ) ( i / ' ) + m| n _ 1 ) (i/*) + m f V ) m^n~ (v") — 2m\n'(i'm)

for 1 < a < n - 3 for a = n - 2 for a = n — l , n .

108

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

(5.5)

Type B™

- ^ 1 (o + 2;f V ) - ^ > 9 ) ' ™ | a _ 1 V ) - 2m| o ) (^) + m f + 1 V ) + L<J0,i*i,i 2

1)

for 1 < a < n - 2

mr )K)-2m|"- (^) +2(2mj n) (i/') + m("+\(vm) + m j ^ i / ' ) ) 2mf~1]{u') - 4raj B ) (/) *

'

for

for a = n - 1 a = n.

(5.6)

Type C «

- ^ L ( " ' ) + 2 ^ ( ° ) (" , )- p i+i.(o 'm<0_1 V ) - 2m
Types 4 2 J and 4 f -J> ( !>(^)

+ 2 ^ ( 0 - ^ ( 0

_ (mj° -1) (i/") - 2roj°V) + ™i°+1 V ) + M,,i*i,i ~\m\n~l)(iy*)-m[n)(iy')

for 1 < a < n - 1 for a = n.

(5.8)

Type 4 1 i -/>(!)1(0 + 2 ^ ( 0 ) ( 0 - ^ ) 1 K ) 2

n 1)

n

= i m<"- V ) - 2m| - (i/«) + 2 m j V )

U - V ) - 2m<*V)

for 1 < a < n — 1 for a = n — 1 for a = n.

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

109

(5.9)

-P£\(y') =

+

2pM{y')-P&{y')

( m j ° _ 1 ) K ) - 2 m j ° V ) + mj a+1) (i/*) + M , , ^ \2mjn"1)(i/*)-2mjn)K)

for 1 < a < n - 1 for a = n.

In particular these equations imply the convexity condition (5.10)

J f V ) > \{P£\a{v')

+*&>•))

if m<(° V ) = 0.

L e m m a 5.3. Let v* be a configuration in C(X,p,). The following are equiv alent: 1. Pla\v*) 2. P^\v')

> 0 for all i e vaZ>0, a € I*; > 0 for all i £ vaZ>0, a € I* such that m ^ V * ) > 0.

Proof. This follows immediately from Lemma 5.2 and the convexity condi tion (5.10). (See also [11, Lemma 10]). □ 5.1. Proof for t y p e

D^.

Proof of (I) for D„ . Here it suffices to show that p satisfies (2.13). Sup pose not. lib — k with 1 < k < n then (a) Afc = Xk+i if 1 < fc < n — 2 (b) A n _i = |A n | if k = n - 1 (c) A„_i = - A n if k = n. In case (a) we have P± {vm) = 0 for large i by Lemma 5.2. Let £ be the largest part in v^k\ By convexity this implies P> \v°) = 0 for all i > £. Equation (5.4) in turn yields m\ ~ {v') = 0 for all i > £ so that 1 < £(k~l) < I. But this is a contradiction since there is a singular string of length £ in (vm, Jm)W since P\k){v%) = 0 and mf\v*) > 0 so that we would have rk(j/*, J') > k. In case (b) let us first assume that A„_i = A„. Then for large i, p / " - 1 ^ * ) = 0 and by convexity p/" - 1 ^!/*) = 0 for i > £ where £ is the largest part in i/( n _ 1 ). By (5.4) we have m\n~ {v°) = 0 for i > t. Hence 1 < ^(™-2) < £ which yields a contradiction since there is a singular string of length £ in (um, J')(™ -1 ) so that rk(z/% J") ^ n - 1. If A„_i = —A„ the same argument goes through with n — 1 replaced by n. The case (c) is analogous to the second part of case (b). Now suppose b = k for some 1 < k < n. We show again that p not dominant will yield a contradiction. If p is not dominant one of the following

110

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

has to be true: (d) Afc = Afc_i if 2 < k < n - 1 (e) Xn = Xn-i if k = n. Case (e) is analogous to case (b). In case (d) some caution is in order. By lemma 5.2 and convexity (5.10) we have P^ ~ ( O = Q ior i > £ where £ is the largest part in iAfc"- 1 ). By (5.4) it follows that m f V " ) = 0 for i > I. Hence | W < L Since P^k~l\v') = 0 and m f - 1 ) ( 0 > 0 there is a singular string of length £ in {y*, J*)( fc_1 ). Hence £(fc_1) < ^ unless ^(fc-i) _ £ a n ( j m l - '(z/#) = 1. We will show that the latter case cannot occur. Equation (5.4) with a = k — 1 and i = £ implies that P\S\ =0 2) k and mf~ {v') = 0 since by assumption ^ ~^ = £^ = #*> = £ and hence m f \v*) > 2 (or m^ _ 1 ) (i/*) > 1 and m^n)(i/*) > 1 for k = n - 1). However this implies that m ^ (^*) = 0 since otherwise ^ fe ~ 1 ) < t - 1 and not £ since there is a singular string of length £ — 1 in (J/*, J*)( f c - 1 '. Now by induction oni = £ — 1,£ — 2 , . . . ,1 it follows from (5.4) at a = k — 1 that P> ~ {vm) = m\ (vm) = rn\ (y°) = 0. However, this means in 2) particular that mf ~ (^*) = 0 for all 1 > £ which contradicts ^ ' ^ = L D Proof of (II) for D^. Denote by J^(i/*,J*) the biggest part in J ^ * ) . To prove admissibility of (£*, J*) we need to show for alii > 1,1 < a < n that (5.11)

0<j££{v\J')
Fix a > 1. Only one string of size £( a ' and one string of size l^ change in the transformation (v*, J*) ( o ) -> (£*, J ' ) ( a ) . Hence j M ^ ' . i ' ) = ^(a)(£')

0 < j££{vm,J')

for

< J ^ V , J')

* = £{a) - 1 and i = £{a) - 1

else.

Hence by (4.5) the inequality (5.11) can only be violated when £(a~1'> 0. Hence (5.11) is fulfilled for these i. It remains to prove that P>a'(i>') > 0 for all i such that rnf\v') = 0 and £(-a~1'> *) > 0. Therefore the only remaining case for

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

111

which (5.11) might be violated occurs when For 1 < a < n - 2: m{i\{y')

= 0, P}l\(vm)

= 0, 6a~l) < £ (resp. £(a+1> < £)

and £ finite where £ = £{a) (resp. t = ^ (a) ) For a = n— l,n: mf\{v*)

= 0, P£\(vm)

= 0, 6n~V < £

and £ finite where £ = £{a). We show that these conditions cannot be met simultaneously. Let p < £ be maximal such that mj, (i/*) > 0; if no such p exists set p = 0. By (5.10) P^\(u') = 0 is only possible if i * / " ^ * ) = 0 for all p < i < £. By (5.4) we find that rnf~^\v') = 0 (resp. mf+l\v') = 0) for p 1 (resp. £<° ) > 1). Hence assume that p > 0. Since Pp (^") = 0 and rrip{v') > 0 there is a singular string of length p in (i/*, ,/")(") and therefore £^a' = p (resp. £^ = p). However, this contradicts p < £. This concludes the proof that (vm,J') is well-defined. □ Proof of (III) for D{nl). Here 6" = 1, If (6 ®j/) = 0 if b < b',H(b ®V) = l if b n,n ® n or 6 > b' where b ^ 1, b' ^ 1, and ff(l ® 1) = 2. If L = 1 then the path is 1, the rigged configuration is empty, and both sides of (3.13) are zero. Here (5.1) and (5.2) are given by = a{1)

(5.12)

A(cc(v',J'))

(5.13)

H(bL ® bL-i) = x(^ ( 1 ) = 1) + X(i{l) = 1)

where £^ and £^ are determined by the algorithm 5. -(a)

Let £(a) and £ be the length of the selected strings defined by the algorithm 6 on (v9,Jm) = 5(v*, J'). To check (5.13) note that if £^ = 1 it follows from (4.5) that £^ > £<-a+1) for 1 < a < n - 2. Hence \ibL n— 1 then 2 a > £(a~l) and bL-X < bL. If bL < T then both sides of (5.13) are 1. If &L = 1 and £^ = 1 then £^ = co and hence &£_i = 1. In this case both sides of (5.13) are 2. If &£, = 1 and £^ > 1 then there is a singular string in (&', J')^ so that &L_I > 1. In this case both sides of (5.13) are 1. If £(!) > 1 then £^ < £^ for 1 < a < n - 2 and the cases can be checked in a similar fashion as before.

112

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

To prove (5.12), by (3.4) and (4.4) we have 1 ( '*) = o 2 E

cc £

" E min(i,j)(a a |a 6 )

i,j>l a,6=1

x (m^

- SjiW

+ <^(6)_i -x(f>
2)(5jm

- ^w_i)J •

Applying the data for Dn and using (4.5), a tedious but straightforward calculation yields

Acc

n

(o = E E (^(,V) - ^(l

x (™! a) - *i,/(") - X(a < n - 2)*M-(.,)) + E ™ ^ For A|J*| we obtain from the algorithm 5'

A J,

i i = EE(^ (a) ^)-^ (0) (^)) a=l i>l

x ( m ^ - <5Mim!

a

i >

we

obtain (5.12).

D

5.2. Proof for type Bn1] . Proof of (I) for Bn1]. Let us assume that either p is not dominant, or that 6 = 0 (so that p = A) and A„ = 0. For b = k with 1 < k < n the proof that this cannot happen is the same as for type Dn ■ Now assume that b = n and A„ = 0. Then P^(v') = 0 for i > (, where I is the largest n part in z/ ) by Lemma 5.2 and (5.10). By (5.5) with a = n we find that rn i [y*) = 0 for i > ^, so that ^n~l~> < £. But there is a singular string of length I in {v*, J*)( n ) which contradicts £^ = oo. Next assume that 6 = 0 and Xn = 0. By the same arguments as in the previous case £^n~1> < £. But there is a singular string of length £ in (y*, J*)( n ) since m^'{v') > 0 and Pj;n'(v') = 0. Since (Q) must hold for 6 = 0, there must be a singular string at ^(™_1) - | o r a quasisingular string at ^( n _ 1 ) < i < £. But then (S) holds for I which contradicts 6 = 0. The case 6 = k with 1 < k < n is the same as for type Dn '. □ Proof of (II) for Bnl). Denote by j££(v',Jm) the biggest part in J ^ . To prove admissibihty of (v*, J') we need to show for alH > 1,1 < a < n

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

113

that

o<j£iV,.n
(5.14)

Up to small alterations, the proof of (5.14) for 1 < a < n is the same as for type Dn '. Let us assume that a = n. Only one string of size £^ and one string of size £^ change in the transformation (v',J*)^ —> (y',J9)(n\ Hence for the different cases:

for i = tW - 1

0<J<&Hu',J')<J<£g(v

, J*)else for i = £{n) - 1/2

, J*)else o<J'' for i = £{n) (Q,s) J^(i>-,J-) = pt\i>' n)

for t = B

- 1/2 - 1/2, £(") =

- 1 for » = £{n) - 1/2, £^

fr-V

< I*""1)

, J*)else. Let us first assume that (S) holds: By the definition of £^ and £^ there is no singular string at £^n~l~> — | and no singular or quasisingular string of length £(n~l"> 0, we have j}&$(v',Jm) < P^\P') - 2 for ^n~l) 0- By lemma 5.3, (5.14) can only be violated if ■m(?-i(v') = 0,

Pi-\(V)

* ( n - 1 ) <£-l,

= 0 or 1,

£ finite.

(«), ; The case P^_1(i'm) = 0 is the same as before. Hence assume thatP;_ 1 K) = n 1. If mt_x{v*) = 0, then by (5.5) and (5.10) P^ \v') = 1 for p 0. By (5.5) we also have m ^ n _ 1 ) K ) = 0 for p 0 there is a (quasi)singular string of length p in (i>*, J')^ which contradicts p < £. If m{^x(u*) > 0, then P^\(i>') > 2 since otherwise >(n)

*■

2

*

2

there would be a (quasi)singular string of length £ — \ in {y*, Jm)(n\ By convexity (5.10) and (5.5) this implies P;_s(u') — 0 and mt_3.{v*) > t

2

2

0. Since ^ (n ~ 1 ) < € - 1, (Q) would hold for £ - § which contradicts our assumptions.

114

M A S A T O O K A D O , A N N E SCHILLING, A N D M A R K S H I M O Z O N O

One more problem might occur when / ( n _ 1 ) = £^ = £ and me_i(v')

0, Jmax^V, J') = PiZlK)-

>

B u t in this

case there is a singular string

2

of length £ - \ in (f*, •/•)("> which contradicts ■£(") = 1 Now assume that (Q) holds: In this case £(n~1') = £("■) = oo. By similar arguments as before (5.14) can only be violated if (n_1) £ { n ) £{n) lfinite

"£.i("") = °> ^ ( » ' " ) = o, * 1

or

2

2

-l<

<

= t>

2

(!/•) = 0, P.W (I/*) = 1, ^ n - x ) < £{n) < £{n) = £,

m^\ 1

*

2

*•

£ finite.

2

n

Since (Q) holds, we must have P} '(i/') > 1. Hence by convexity (5.10) it follows that P}n'(vm) = 1 for p 0. Equation (5.5) implies that m\n~1)(u') = 0 for p < i < £ so that £ ( " - 1 ' < p. But there is a (quasi)singular string of length p in {v*, J*)( n ) which contradicts p < £. Finally assume that (Q,S) holds: For i < £^ the same arguments hold as for case (Q). Since by definition there are no singular strings of length £(n~x) 0, the only problem occurs when „>)(„•) *

2

= 0

,

P , ^ (!/') = 0, *

£{n) + l<£,

£ finite.

2

If p < £ is maximal such that mp (c') > 0, then by (5.5) and (5.10) p/ n) (z/") = 0 for p < % < L Since m^{v*) > 0 and P^v') > 0 we must have £^ < p. But then there is a singular string of length p in {v*, J')^ which contradicts £^ = £. O Proof of (III) for B^]. Here b* = T. Note that ~H{b ®b') = 0iib 6') = 2 if 6 b' = T 1, and H(b ®b') = l otherwise. If L = 1 then the path is 1, the rigged configuration is empty, and both sides of (3.13) are zero. Here (5.1) and (5.2) are given by (5.15) (5.16)

A(ccK,J,))=a[1) H(bL ® &£_!) = X(i{1) = 1) + X{i{l) = 1).

where £^ and £^ are determined by the algorithm 5. -(a)

Let f(a) and £ be the length of the selected strings defined by the algorithm 6 on {t>*, Jm) = 5(um, J*). To check (5.16) note that if £^ = 1 it follows that £^ > £( a+1 ) for 1 < a < n - 1 . Hence if bL < n then 6 L _i < 6L and both sides of (5.16) yield 1. If 6^ = 0 then bt-\ < &L by (4.9) and both sides of (5.13) are 1. If n < &£, < T, then 6/,_i < &L by (4.9) and both sides

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

115

of (5.16) are 1. If bL = 1 and £^ = 1, then 6 L _i = 1 by (4.9). Hence both sides of (5.16) yield 2. Finally, if &L — 1 and tS1^ > 1, then there exists a singular string of length £^ — 1 in (v*, J*) so that &L-I ^ 1. Hence both sides of (5.16) are 1. If £^ > 1 then £^ < £^ for 1 < a < n - 2 and the cases can be checked in a similar fashion as before. To prove (5.15), by (3.4) and (4.8) we have 1 cc(i>m) =- ^2 i,j>l

" E

min(tbi,taj)(aa\<Xb)

a,b=l

x Imf' -5.

eM

+8. <(.) - ~ 5i

IW + 5 Ha)

Applying the data for Bn and using (4.9), a tedious but straightforward calculation yields

a=l»>l

+ ^

m

(D-

x

i>l

^(«) = £("-D _ I ) + X(£(«) = #»-U) ^

(n)

' (n)

+ X(i =oo)x(^
AJ

i *i = E E O^"') - p ^ * ) ) H a ) - ^ y « - *«.<,*•>) a=li>l

+ x(*(") = *(»-D - I ) -

X (£W

= /("- 1 )) - x (£(") = oo)X{(-{n) < oo),

where the last three terms come from the fact that for n-th rigged partition singular strings can be transformed into quasisingular strings and vice versa. Hence altogether, using ^2i>1m\ — a\', we obtain (5.15). □ 5.3. Proof for type

c£].

Proof of (I) for C„ . If b = k with 1 < k < n the proof that p is dominant is analogous to type D„'. For b = n a problem occurs if A„ = 0. In this case P>n'(v*) = 0 for i > £ where £ is the largest part in v^ by Lemma 5.2 and (5.10). By (5.6) this implies m^ n_1) (i/*) = 0 for i > £. Hence £(n-i) < £_ B u t there is a singular string of length £ in (um, J*)(") which contradicts f.^> = oo. If k = n a problem occurs if An — A n _i. In this case J^ (n_1) (j/*) = 0 for i > £ where £ is the largest part in i / " - 1 ) . By (5.6), m- (^") = 0 for i > £. Hence £^ < £. But there is a singular string of size

116

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

£ in (um, J ^ " - 1 ) (this also works if t^-V l(n-l)

=

= l^

= £) which contradicts

0Q_

D

Proof of (II) for C{n]. We show that (vm,Jm) € RC(p,/I). We use the same notation and set-up as in type Dn . Then •4°«)(«J". J') = pia)("')

for i = £^ - 1 and i = £{a) - 1 or i = £{a~> - 2 if Ba) = ^ a + 1 )

< J&2{S,Jm)

0 < J££iy\n

else.

The proof that 0 < J^{v*,J') < P^a\vm) for 1 < a < n is the same as Q+1 a+l usual if ^°> ^ Z( ). If ^°) = H \ by (4.13) the only problem occurs if mf}2(»/•) = 0,

P\%(!/•)=

0,

/? (a ~ x) < £ - 1,

£ = 5°) = £{a) + 1 finite.

We show that these conditions cannot be met simultaneously. Let p < £ — 1 be maximal such that mpa' {v') > 0; if no such p exists set p = 0. By (5.10), P^}2(vm) = 0 is only possible if P}a)(v*) = 0 for p < i < £ - 1. By (5.6) this requires rnf~ ' (v') = 0 for p p. Finally for a = n the only problem occurs if m{^2(u')

= 0,

p£\{vm)=0,

£{n~1)<£-l,

£ = £^

finite.

By convexity (5.10), P^ (v') = 0 for p 0. Then by (5.6) we also have m\n~ {v') = 0 for p = £ > p. □ Proof o/ (7/Jj /or c£ x ) . Herefcl= T, H(b®b') = 0 if b < b' and H(b®b') = 1 otherwise. If L — 1 then the path is 1, the rigged configuration is empty, and both sides of (3.13) are zero. Here (5.1) and (5.2) are given by (5.17)

A(cc(v,,J'))

= a{1)

(5.18)

H(bL®bL-1)

= X(t{1) = l)

where £^ is determined by the algorithm 5. Note that there is no contri bution from B-1) in (5.18) since l^ > 1. -(a)

Let £^ and £ be the length of the selected strings defined by the algorithm S on {u*,J') = 8{v',J'). Note that if £^ = 1 then (4.13) implies that DL_I < bi, so that both sides of (5.18) are 1. If lM > 1 then 1(a) < g(a) for 1 < a < n so that bL-X > bL and both sides of (5.18) are 0.

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS -(n)

For bL = n,£

117

< £
and hence £^ = 1 which contradicts our assumptions. If £

< £^

then

-(o)

also £ < £^ which implies that &L_I > &£. Hence both sides of (5.18) are 0. To prove (5.17), by (3.4) and (4.12) we have 1 " cc

(^*) = 2 E

E

min

(*6^taj)(aa\ab)

x ( m | a ) - x(a < n)(8ilW

- SiiiM_1)

- 6 jw + S{ i w _ )

x (m* 6) - x(6 < n)(6jtim

- SjiW_1)

- 5

m

+5 m

).

Applying the data for Cn and using (4.13), a tedious but straightforward calculation yields

a=l»>1 a=l»>1

»>1

For A|J*| we obtain from the algorithm 5'

A j

I ' \ = E E ( p i:^*) - p£i(**)) H a ) - x(« < «)*,,*.> - sVaitIM). a=l»>1

Hence altogether, using $^ i > 1 mf ) = oSi\ we obtain (5.17).

D

5.4. Proof for type A^. The proofs of (I) and (II) are analogous to the previous cases. In particular, the proof of (II) is very similar to that for type CP. Proof of (III) for A%>. Here 6» = 4>, ~H(b ® b') = 0 if b < b', H(b ®b') = 2 if b > b' or b b' — <j> ® <j> and H(b b') = 0 otherwise. If L = 1 then the path is 1 or . In the former case, the rigged config uration is empty, and both sides of (3.13) are zero. In the other case both sides of (3.13) are 1. Here (5.1) and (5.2) are given by (5.19)

A(cc(i/*, J')) = 2a[l) - X(£{n) = 1)

(5.20)

H(bL ® &£_!) = 2X{£{1) = 1) - X(e{n) = 1) + x{i{n) = i)

where £W is determined by the algorithm J. Note that there is no contri bution from IV in (5.20) since I™ > 1.

118

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

Equation (5.20) can be checked in a similar fashion as to the other cases. To prove (5.19), applying the data for Bn and using (4.13), a tedious but straightforward calculation yields

a=li>l

x (m
- 6iJW)

- X(i(n)

= 1) + 2 X > < ( 1 ) i>\

For A|J*| we obtain from the algorithm 8'

A| J'\ = 2J2 E ( ^ ° V ) - P,{a\>')) H a ) - X(a < n)SitiM - Si>iw) . a=l t > l

Hence altogether, using 2 i > i m !

a

i

>

we

° b t a m (5.19).

□

5.5. Proof for type A ^ ; ^ . Proof of (I) for A^-i- The proof that p is dominant for b = k with 1 < k < n is analogous to the other types. For b = k with 1 < k < n, p is not dominant if Afc = \k-i- In this case P> '{v') = 0 for i > £ where i is the largest part of u^-V by Lemma 5.2 and (5.10). By (5.8), mf'(z/*) = 0 for i > I so that £(fe> < I. But since P ^ V * ) = 0 and mf_1) > 0, there is a singular string of length £ in (v*, J # )( f c _ 1 ). Hence £(fc-1) < £ (which contradicts ^ f c _ 1 ) = oo since k = rk(v',J')) unless £(k~l) = | W = £ a n ( j mf~l){y') = 1. Since m f V * ) > 2 for 1 < k < n and m f V ' ) > 1, (5.8) for a = /c — 1 and i = £ implies that m[ ~ (vm) = 0 and i^_7 C^*) = 0. Hence £(fc-2) < £ and mf."^ 1 ^*) = 0 since otherwise £^~^ < £ - 1. By induction on j = £ — 1, £ — 2 , . . . , 1 (5.8) for a = k — 1 implies that m ! ~~ ( O = 0 and P>_i {v') = 0 which in turn requires m^_^ (^*) = 0 since else ^ " ^ ^ w n i c n contradicts our assumptions. D Proof of (II) for A^n-i- To prove that (y*, J") is admissible, one finds sim ilarly to the proof of type D„ ' that the only problem occurs if 771^ = 0,

P^\(v*)

= 0, £ finite,

6a~l) < £ (resp. £{a+l) < £ for 1 < a < n), £ = 6a) (resp. £ = £(a) for 1 < a < n).

Analogous to the case Di 1 } it can be shown that these conditions cannot hold simultaneously. □

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

119

Proof of (III) for j ^ n - i - Here &» = T, # ( T ® 1) = 2, 77(6 6') = 0 if b < b' and H(b ® 6') = 1 otherwise. If L = 1 then the path is 1, the rigged configuration is empty, and both sides of (3.13) are zero. Here (5.1) and (5.2) are given by = a[1)

(5.21)

A(cc(u',J'))

(5.22)

H(bL ® 6 L _!) = X (^ ( 1 ) = 1) + x(t{l)

= 1)

where t^ is determined by the algorithm 5. The proof that (5.22) holds is very similar to the previous cases. To prove (5.21) we apply the data for Cn to (3.4). Using (4.16) and (4.17) a tedious but straightforward calculation yields

a ==il j»>i >l X

For A|J"j we obtain from the algorithm 8'

a=l%>1

Hence altogether, using ^ i > i m !

= a

i >

we

obtain (5.21).

D

5.6. Proof for type D ^ j . Proof of (I) for D^+i- The proof proceeds as before except in the cases b = n and b = 0 (there is nothing to prove for b = n\v%) = 0 for i > £ where £ is the largest part in i/™) by Lemma 5.2 and (5.10) and the admissibility of v*. By (5.9) it follows that m i ~~ \v') = 0 for z > £ so that ^ n _ 1 ) < I. But there is a singular string of length t in (um, J')W since P^n\v') = 0 which contradicts £^ = oo. To prove that 6 = 0 cannot occur if A„ = 0 we find as for the case k = n that £(™_1) < £ and that there is a singular string of length £ in (v*, J')^ since P\ {v*) = 0. For the 6 = 0 case (Q) must hold so that there must be a quasisingular string of length £^ < £ in {v*, J')^n\ But observe that there is a singular string of length £ > £^ in (i/*, J ' ) W which contradicts e~M = oo. D Proof of (II) for D^. Next we need to show that {v',J') 6 RC(p,/2). The case that (£>*, J ' ) ( a ' is admissible for 1 < a < n works as usual.

120

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

Consider a = n. First note that there is no problem in case (Q,S) setting the new string of length I^> - 1 to be quasisingular since the string of length IW - 1 is not singular by definition so that P^\_1(v') > 0and also PlM-ilP') > ° b y (4.19). The only problem occurs if m^\{ym)

= 0,

P£\{V*)

= 0 or 1,

£{n~l) < £,

1 = £ (n) finite.

Note that P^'iy') is always even so that P^[ = 1 is impossible. The proof that these conditions cannot hold simultaneously works as usual. □ Proof of (III) for D^]+1. Here 6" = and ~H{®) = 2,H(b®) = H ( 0 ® &)_ = 1 if b £ 4>, H{b ® b') = 0 if b,b' ^4>,b< b' and b + V if b = 0, and H{b ® b') = 2 if 6 > b' or b = V = 0. If L = 1 then the path is either 1 or <j>. In the former case the rigged configuration is empty, and both sides of (3.13) are zero. In the latter case it is also not hard to check that both sides of (3.13) are 1. Here (5.1) and (5.2) are given by (5.23)

A(cc(i/*, J')) = 2a[ x) - X(£{n) = 1)

(5.24)

H(bL ® 6 L _j) = 2X(£W = 1) - X(t{n)

= 1) + x(* ( n ) = 1)

where £^ and £^ are determined by the algorithm 5. To obtain (5.24) we used the fact that by definition 1^ > 1. Here £^ is defined by the algorithm 5 on (i>*, J') = 6'(u\ J " ) . It can be checked directly that (5.24) holds. For example, if £^> = 1 it follows that #°) > i^+V for 1 < a < n. Hence if bL < n then bL-i < bL and both sides of (5.24) yield 2. If bi = <j> then both sides of (5.24) are 1 for &L-I 7^ 4> and 2 if bi,-i = - If &L — 0 then both sides of (5.24) are 2 if &£,_i < 0. Note that &L_I = <j> or &£_i > n is not possible. Finally, if =(o)

bij>n then ^ > f w and &L-I < &£. Note that 6^,-1 = 1 which implies i^ > 1. Both sides of (5.24) yield 2 in this case. If ^ (1) > 1 then 1^ < £^ for 1 < a < n and the cases can be checked in a similar fashion as before. To prove (5.23), from (3.4) and (4.12) we obtain cc i>

1

™

( ')=2 5Z ]C ™in(i,j)(aa\ab) i,j>l a,b=l X (ml*

- (Jj^(a) - Sii(a)

x (m]- - (Jj^(6) - 5jiw

+ 0~ite(*)-l + <5j/(a)_l)

+ SjiiW_1

+ 5^(6)_i).

RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

121

Expanding out and using (4.19) a tedious but straightforward calculation yields

a = l" ii>>l l

+ 2 Y^m\1]

- x(?{n) =

OO)X(* (B)

< oo).

t>l

For A | J"| we obtain from the algorithm 8'

a=l»>1

+ x(^ ( n ) = oo)x(l < ^ ( n ) < oo) where the last term comes from the fact that in case (Q) a quasisin gular string is changed into a singular string. Hence altogether, using £ » > i m i X ) = a i X ) a n d t h e f a c t t h a t Z{n) = oo if .£(") = 1 by the algorithm 8, we obtain (5.23). □ 5.7. Proof for type A ^ . Proof of (I) for A$}. The only case that proceeds differently than before is b = 0. Suppose A„ = 0. Let £ be the longest part of i/(nh As in the proof of the D^lx case, P$n)(v') = 0 where £ > ^n~1\ If £ is odd then this is a contradiction of the admissibility of v'\ see (3.10). If £ is even then j( n ' f ) is singular and £^ < £ (as £^ is odd and £ is the longest part), contradicting 6 = 0. □ Proof of (II) for A^n . The admissibility of (v',J') for 1 < a < n is as before. Let a = n. We first observe that in all cases, (5.25)

£{1) < £{2) <•■■<

£ ( n ) < £(n) < £{n~l)

< ■ ■ ■ < £{1),

with £^> odd and £^ even (when they are finite). We also note that by (4.13), 1

(5.26)

/f V) >/f'V) - 1

with equality if and only if ^ " - ^ n\ Let us verify (3.10) for (£', J'). Let i be odd such that m\n\i>') > 0. Suppose first that m j " V ) > 0. By (3.10) for {u*,J'), P^\v*) > 0. By (5.26) we may assume that P{n)(i/') = 1 and ^ n _ 1 ) m) > 0 we are in case (Q, S) with i = £ ( " ) - 1 . In case (Q, S) £^ < #">, so £^ < i < Hn\ Now i ^ £^ since

122

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO

rnf\um) = 0. So ^ ') = Pi+l(v') = 0. Since i - 1 is even, if m\"\(u') > 0 then j(n,t-i) w o u i d have been singular with i ^ < i — 1 < fin\ contradicting t h e choice of Hn\ So m(?\(v*) = 0. Applying (5.10) again, P£\(V') = 0. Since (v*,J') was admissible and i — 2 is odd, by (3.10) it follows that m l-2( I / *) = 0- Continuing in this manner, a contradiction is reached since Now suppose i is even. It must be checked that P}n'(i>') > 0. The only problem is if P^n\v') = 0 and i^-V 0 then S would have chosen the singular partition j ( n , t ) . So m\n'(um) = 0. By (5.10) it follows that P i +{(i / ) = 0. Arguing as above but with the index increasing from i, a contradiction is reached since PJ£,\(v°) > 0. D Proof of (III) for A^} ■ One has $ = 1, #(&'2 b[) = 0 if 62 < b[ (except for ff(0 0) = 1), and ~H(b'2 ® &i) = 1 for b'2 > b'v If L = 1 then the path is 1, the rigged configuration is empty, and both sides of (3.13) are zero. Here (5.1) and (5.2) are given by = a{1)

(5.27)

A(cc(v',J'))

(5.28)

H ( & L ® & L - I ) = X(*(1) = 1 )

where £W and £^ are determined by the algorithm S. The term x(& = 1) disappears since the definition of the algorithm forces 1^ > 2. The proof of (5.27) is very similar to that in the Z?„+j case. Straightforward computations yield A c c K ) = X > „ | a 6 ) ( X ( t > t{b)) + x(i > t{b)))(rnf\v')

- *<,«.) ~ *i,*->)

+ X (^ ( 1 ) < oo) + x(i ( 1 ) < oo) - ^x(* ( n ) < oo) + \x(t{n)

< oo)

and

£ ( / f V ) - ifV))K W (0 - JM(B) - 5iJM) = $ > a | a 6 ) ( x ( i > ^ ( 6 ) ) ; + X(* > ^ ( b ) ) ) K ( a ) ( ^ ) - Si
- ^ ^ ^ ' ^ ^ ^

<°°) + X(t(1) < 0 0 ) .

6i
RIGGED CONFIGURATION BIJECTION FOR AFFINE ALGEBRAS

123

Together these yield

Acc(0 = £ ( / f V ) - J?V))(mj 0 V) - 8ittW - <Ja(.>) a,i

+ ^ 1 ) - ^ ( " ) < o o ) + iX(^n)
w\ = £(*f V ) - ^(°)(*,))fo('V) - v-) - %<»>) + i X ( ^ ( n ) < oo) - ^ x ( * » < oo). This proves (5.27).

□

REFERENCES [1] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Z. Tsuboi, Paths, crystals, and fermionic formula, MathPhys odyssey, 2001, 205-272, Prog. Math. Phys., 23, Birkhuser Boston, Boston, MA, 2002. [2] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Remarks on fermionic formula, Contemporary Math. 248 (1999) 243-291. [3] M. Kashiwara, On level zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, 117-195. [4] V. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge University Press, 1990. [5] 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) (1992) 449484. [6] S-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992) 499-607. [7] S. V. Kerov, A. N. Kirillov and N. Yu. Reshetikhin, Combinatorics, the Bethe ansatz and representations of the symmetric group, Zap.Nauchn. Sem. (LOMI) 155 (1986) 50-64. (English translation: J. Sov. Math. 41 (1988) 916-924.) [8] A. N. Kirillov and N. Y. Reshetikhin, The Bethe Ansatz and the combinatorics of Young tableaux, J. Soviet Math. 4 1 (1988) 925-955. [9] A. N. Kirillov and N. Yu. Reshetikhin, Representations of Yangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras, J. Sov. Math. 52 (1990) 3156-3164. [10] A. N. Kirillov, A. Schilling and M. Shimozono, A bisection between LittlewoodRichardson tableaux and rigged configurations, Selecta Math. (N.S.) 8 (2002), no. 1, 67-135. [11] A. N. Kirillov and M. Shimozono, A generalization of the Kostka-Foulkes polyno mials, J. Algebraic Combin. 15 (2002), no. 1, 27-69. [12] A. Lascoux and M. P. Schiitzenberger, Sur une conjecture de H.O. Foulkes, CR Acad. Sci. Paris 2 8 6 A (1978) 323-324. [13] M. Okado, A. Schilling, and M. Shimozono, Virtual crystals and fermionic formulas of types D^^, A^n , and C„ , Representation Theory, to appear (math.QA/0105017).

124

MASATO OKADO, ANNE SCHILLING, AND MARK SHIMOZONO DEPARTMENT O F INFORMATICS AND MATHEMATICAL SCIENCE, GRADUATE SCHOOL OF

ENGINEERING SCIENCE, OSAKA UNIVERSITY, TOYONAKA, OSAKA 560-8531, JAPAN

E-mail address:

okadoOsigmath.es.osaka-u.ac.jp

DEPARTMENT O F MATHEMATICS, UNIVERSITY OF CALIFORNIA, O N E SHIELDS AVENUE,

DAVIS, C A 95616-8633, U.S.A. E-mail address: annefflmath.ucdavis.edu DEPARTMENT O F MATHEMATICS, 460 M C B R Y D E HALL, VIRGINIA T E C H , BLACKSBURG,

VA 24061-0123, U.S.A E-mail address: mshimoiBmath.vt.edu

LITTLEWOOD'S FORMULAS FOR C H A R A C T E R S OF ORTHOGONAL A N D S Y M P L E C T I C G R O U P S ALAIN LASCOUX

ABSTRACT. Littlewood gave expansions of products of the type f l l / ( l —
Given an infinite alphabet A = {a\, 0,2,03,... }, Littlewood [7] gave ex pansions of products of the type

Hil-OifiJlil-aiaj)*1

and

I](l -

i<j

a

^ IR1 ~ ^ ^

'

W

i<j

the case J7(l — a i ) _ 1 Ili<.j(l — ajOj) - 1 was due to Schur [10]. By functoriality of symmetric functions, Littlewood's formulas imply without further computations expansions of more complicated products like

11(1 - a i ) ±1 IJ(1 - h)*1 JJ(1 - aiaj)^ JJ(1 - &,&,)=" IJ(1 - a^-)*1 i<j

i<j

(2)

involving a second alphabet B = {61,62,-••}■ However, Littlewood's method is not limited to expand the above prod ucts, and it also allows to expand products of the type J-A (1 - xiai) ■ • • (1 - xna,i) -*-1 1 _ aiaj involving a finite alphabet X = {x\,... ,xn}, the alphabets A and X playing now a disymmetrical role. To obtain his formulas, Littlewood describes in fact the adjoint of the multiplication by certain series of symmetric functions (see the theorem below). We shall state his results in terms of A-rings. The case X = {ari, #2} has been considered by Ishikawa [2], Ishikawa and Wakayama [3], Bressoud [1], and the case X = {xi,X2,xz} by Jouhet and Zeng [4].

125

ALAIN LASCOUX

126

Before reinterpreting their formulas, let us recall some definitions, for which we refer to [8]. Given two alphabets A, B, the complete functions Sj(A-M) are defined by the generating series (with z an extra variable) az(A-M):=Y[(l-zb)/l[(l-za). b€B

(4) a€A

Given A = (Ai, A 2 ,... , Ar) € Nr, the Schur function S ' A ( A - B ) is 5A(A-B)=

SX^A-B) l
When A is weakly decreasing, it is a partition. In that case, following Frobenius, one uses another description of partitions, by decomposing their diagram into its diagonal hooks, and writing A = (a \ f3) ([8, p.3]). On the space &ym(A) of symmetric functions in A, there exists a scalar product (, )A, and Schur functions indexed by partitions constitute an orthormal basis with respect to it : (5 A (A),5 A (A)) A = 1 and (S A (A), S„(A)) A = 0 \±v . (5) Another way to state (5) is to assert the existence of a reproducing kernel A"(A,B):=CTI(AB)=

(1 - ab)'1

JJ a€A,6£B

which satisfies V/ e &ym , (K(A, B), / ( A ) ) A = /(B) ,

(6)

because the Cauchy formula states that tf(A,B)

= £;Sx(A)S\(B),

(7)

A

where t h e sum is over all partitions. Given any symmetric function / , the adjoint D(f; •) of the multiplication by / is defined by V/, g,h€

&ym , (D(f;g),

h) = (g, fh).

(8)

The formulas

(A2(A)) = ]TS M (A), all H with columns of even lengths

(9)

and ai(A + A2(A)) = ^2sx(A) 2

, all partitions A 2

(10)

are equivalent, because A (A + ' l ) = A -f- A (A). Combinatorially, consid ering the letter 1 to be bigger than all the letters in A, it is equivalent to enumerate all tableaux in A, or all tableaux in the alphabet A + 1 := AU{1}

LITTLEWOOD'S FORMULAS

127

with columns of even length (we write unions of alphabets with a '+'). In deed, the l's occupy the horizontal strip such that the inner shape /x has columns of even lengths. The computations of Littlewood in chapter XI of his book [7] can be rephrased in the A-rings paradigm as follows : T h e o r e m 1 (Littlewood). Given two alphabets A, X, then one has the identities a1(AX + A2(A)) = J2^WD{a1(A2);S„))(X),

(11)

(12)

* ! ( A X - A2(A)) = ^ S ^ ( A ) i ? ( f f i ( - A 2 ) ; 5 ^ ) ( X ) I

(13)

(
(14)

where the sums are over all partitions fi. Proof. All four generating series factorize : cr^AX + A2(A)) = <Ti(AX)cri(A2(A)). Since o-i(AX) is a reproducing kernel, one has for any H : (a 1 (AX + A 2 (A)), S„(A)) A =

(a1(AX),D(a1(\2);Sfi)(A))A

= D(a1(A2);SfM)(X)

.

and this proves the first identity. The same proof would work for cri(AX)/(A), / being any symmetric function. Indeed property (6) entails (/;S, 1 )(X) , and formulas (12), (13) and (14) are special cases of it.

(15) □

Let us mention that other expansions are possible. Indeed, 5 2 (A ± X) = 5 2 (A) ± S1(A)S1(X)

+ S 2 (±X),

A2(A ± X) = A2(A) ± A 1 (A)A 1 (X) + A 2 (±X) . It implies, for example, that CTI(AX

+ A2(A)) = o-i(-A 2 (X))
S„(A + X ) ,

(16)

^—^u

sum over all partitions v with columns of even lengths. To go further, one would decompose Sl/(A + X) in the basis of Schur functions in A, and use in general Littlewood-Richardson's rule to collect functions in X. But in the case where X is of small cardinality, the expansion can be written directly.

ALAIN LASCOUX

128

For example [1, Theorem IV], if X = x\ +x2, Equation (16) can be rewritten as a1((xi + x2)k + A2(A))=(l-x1x2)'J2

S„(A + xi+x2)

(17)

and is easy to expand in terms of the S^(A), interpreting it as the sum of all Young tableaux with even columns, on an alphabet with last two letters X\,

X2.

Let us show in more details how to explicitly calculate the coefficients in Littlewood's identities, for card(X) = n = 1,2,3, recovering the related results of [1, 2, 3, 4]. We shall treat only the case of <7i(XA + A2A), the other being similar. When n = 1, X = {xi}. Then D(SV; SM)(X) is non zero iff the diagram of n/v is an horizontal strip. Therefore, given any partition /x, there is one and only one partition v with even columns such that D(SU; 6'/x)(X) is non zero. Equation (11) specializes to o-i(anA + A2(A)) = J2xlddM

5M(A) ,

(18)

where odd(/x) is the number of columns of /x of odd length. Take now n = 2. A skew diagram n/v gives a non-zero function S^/^xi +x2) iff it is made of horizontal strips of height 1 and 2. Decompose the partition into blocks Bk, k = 1,2,... of columns of the same length k (denoting l^ its width). Iffcis odd, then the restriction of fi/v to this block must be an horizontal strip of length l^, giving a factor Stk{x\ + x2). If k is even, the restriction of njv is a 2 x i rectangle, with 0 < i < tk- The figure shows the restriction of a strip to two consecutive blocks :

• □ ■ .. .■■... am . a □...

■ m □ •■ □ •■ ■ a ■ •■ □ •■.. a □ ■■ a ■■..

D O •■ • t2*

D D •■■ HIT

n

- i

The sum over all possible i gives a factor Stk(l + x\x2). Moreover, there is no interference between the different blocks, and therefore, one gets that the coefficient of SM(A) in o\ ((xi +£2)A + A2(A)) is oo

I>(o-i(A2); S^ixx

+ x2) = Y[ Se2k (1 + Xlx2) S(2k+1 (Xl +x2) . fc=o

(19)

The case n = 3 is treated by Jouhet and Zeng [4]. The independent blocks are now the blocks Bk of columns of length 2fc +1, 2k.

LITTLEWOOD'S FORMULAS

129

There are two possible configurations for those p/v giving a non-zero contribution :

□ ... □ ■ ... ■■...■ n - n ■••■■■••■ ■

.

a ... DD - nn ••• n

.

, p = t2k+i, q = hk,

«>l-

v The sum over all t = 1 , . . . , p gives X1X2X3Sp-i(l

+X1X2X3)Sq(A2(x1

□ D - D D - D I - ■ D D - D D - D I ■ an-DD-DD-a

{ =

+X2

0

0

+X3)).

< j <

q

.

a a - D D ■•• D D •■• D p

q

The sum over all j = 0 , . . . , q is Sp(x! +x2+

x3) 5,(1 + A2(X)) .

For example, for fi = [7,7,4], the possible configurations, together with their factors, are : ■■■■ ■■■■ ■■■■ ■■■■ DDDIIII DDIIIII DBBBBBB ■■■■■■■ DDDIIII + DDIIIII + Dllllll + ■■■■■■■ S4n(X)S33(X) Si22(X) S33(X) 5433(X) 5 33 (X) 5 444 (X) 5 33 (X) ■■■■ ■■■■ ■■■■ ■■■■ DDDDDDD DDDDDDI DDDDDII DODDIII + DDODDDD + DDDDDDI + DDDDDII + DDDDIII . 5 4 (X)5 0 0 (X) 54(X)5U(X) 5 4 (X) 5 22 (X) 5 4 (X) 5 33 (X) In summary, the contribution of the block Bk is p

c(Bfc) = ^ 5 p _ i ( X ) 5 9 g ( X ) + 5 p X ) ( l + 5 n ( X ) + • • • + 5 n n (X))

(20)

and the coefficient of 5M(A) in ^ ( ( x i +x2 +x3)A + A2(A)) is the product of all the factors c(Bk) given by the decomposition of p, into blocks. This expression is equivalent to Formula (8) of Jouhet and Zeng [4]. Littlewood's expansions were motivated by the description of characters of orthogonal and symplectic groups. In turn, these characters allow to expand Littlewood's formulas (13) and (14).

130

ALAIN L A S C O U X

Following [7, p.240], using a second alphabet X, define the orthogonal Schur function
- a^)-1

- aiaj)]H[(l

%<j

i

= ^0A(X)SA(A) ,

j

(21)

X

that is, en (AX - S 2 (A)) = J2 °*(A) x

5

A(X) •

(22)

Formula (14) can now be rewritten : Ox = D{a1(-S2);Sx)

(23)

and the inverse formula is Sx = subs (s = 0 , D f a (S2); 5 A ))

(24)

which means that in that last case, one uses the operator D(ai{S2) ;•) as a formal operator on a vector space, the basis of which is indexed by partitions (and the action being obtained by identifying this basis with Schur functions). For example, Littlewood [7, p. 241] gives (writing in the expansion of <7i(—S2) and o-\{S2) only the terms which give a non-zero contribution) : 0332 =

D(SQ-S2

+S31-S33 ; S332) = S332 — £332/2 + £332/31 — $2

and the identity D(S0 +S2 +S22 +S222; £332) = £332 + $33 + £321 + £211 + £31 + "Sii implies, by formal change of S into 0 : £332 = 0332 + 033 + S321 + 0211 + 031 + Cll • Ishikawaand Wakayama [3] give the expansion of cri((xi+a;2)A—5 2 (A)), with X1X2 = 1- It means computing the specialization of orthogonal Schur functions for such X. In fact the complete functions (resp. power sums) of X are Tchebychef polynomials of the first and second kind in the variable (#i +x2)/2. Ishikawa and Wakayama characterize those partitions for which 0\ (X) ^ 0 in terms of their Frobenius decomposition, and obtains that in that case ± 0 A ( X ) is equal to a Tchebychef polynomial of the second kind. Using symmetrizing operators, one easily expands oi(—S 2 ) as : Cl( — S ) = 2 - ,±S , eil e 2 ,e 3> ... = Sooo... — £200... ~ "So40... + 5240... + " ' {25) = S0 - S2 + S31 - 5 3 3 + • • • •

(26)

In the first summation, Schur functions are indexed by vectors [ei, €2, £3, • ■ • ] such that €i € {0,2i}, and not by partitions, and the sign alternates with the parity of the number of non-zero components.

LITTLEWOOD'S FORMULAS

131

It is remarkable that the operator D(ai(—S2); •) can also be written as an enumeration of vectors :

Sx^Ox=D(a1(-S2),Sx)=

Yl ±^/[n,e2,...]

(27)

£l,<=2,

(recall that, for any two integral vectors A, v, £>/„ stands for the determi nant \SXi+j-i-Vj\). Indeed, the fact that D(SU;S\) = S\/v for two partitions, extends, by reordering the rows of the determinantal expression of S„ and the columns of S\/„, to the case where v is not a partition, i.e. is not decreasing. Thus 0332 can also be written C332 = £332/000 ~~ 5332/200 S3 S4 S5 5*2 £3 §4

So Si S2

—

—

£332/040 + £332/240 S3 So S$

Si S4 S& S3 S4 0 Si S2

— s2 0 s4

SQ

So 0 S2

5i 5"o S$

+

So 0

S4

0 0 52

By mutilinearity of determinants, one can write the above sum as a single determinant, due to Weyl [11, Theorem 7.9.A], that we display for the order 3: Sxi ~ £ \ i - 2

Ox, A2A3

£A2-I ~ £A2-I-2 £A3-2 -

£A3-2

£ \ : + l — £Aj + l - 4 £ \ I + 2 — £Ai+2-6 £A2

—

£A3-I -

£A2-4

£A2+I — £A2+I-6

£A3-I-4

£A3

_

(28)

£A3-6

For example, O332, again, is equal to £3 — £1 £4 — £0 £5 — 0 £2-£o £3-0 £4-0

£o-0

Si - 0 £ 2 - 0

We shall refer to [5] for determinantal expressions of orthogonal or symplectic characters. The case of "nearly rectangular shape" leads to an interesting combinatorics obtained by Krattenthaler [6]. Similarly, define the symplectic Schur function Spx by Spx =

D(a1(-A2);Sx),

(29)

but this time, it will be evaluated into the alphabet Y := {xi, l/a;i,X2,1/2:2,... ,xn, l/xn} instead of X. The identity

(30)

sum over all Cj € {0,2?}, with e0 = 0, extends, as before, to SPx = D(cr1(-A2);Sx)=

£

5

A/[o,ei,o,...] ,

(31)

ALAIN LASCOUX

132

and this sum can be written as a single determinant due to Weyl [11, Theorem 7.8.E] : SXi S\2-I

S\1

+

i -

^ 2

5 * 3 - 2 S\3-l

_

SAI + 1 - 2 -SAI+2 — S.\i+2-4 £A2-2

^2+1 -

- ^A3-l-2

^2+1-4

•

(32)

S\3 — S\3-4

Lovers of A-rings will prefer the expression Spx = SX(Y - 0, Y - 2, Y - 4 , . . . ) := |s\,+j-i(Y - 2j + 2)| ,

(33)

with, for any alphabet Y, any number A;,
:=az(Y-l

1) = (1 - z)k az(Y) .

Indeed, because

Si(Y -k)= SiOO - (* W i ( Y ) + Q W 2 ( Y ) - • • ■t-fcOO, ± 5, expanding each term of the determinant S\i+j-i(Y — 2j + 2) one obtains Weyl's determinant by eliminating in each column multiples of the columns on its left. In particular, when specialized to 1, the determinant spe cializes to a determinant of binomial coefficients 'In + Xt - i - j + 1 (34) Xi+j-i l
LITTLEWOOD'S FORMULAS

133

[9] R. Proctor, Interconnections between symplectic and orthogonal characters, Contemp. Math. 8 8 (1989) 145-162. [10] I. Schur, Gesammelte Abhandlungen, vol. Ill, p.456, Springer (1973). [11] H. Weyl, The classical groups, Princeton Univ. Press (1939). INSTITUT GASPARD M O N G E , CNRS, UNIVERSITE DE MARNE-LA-VALLEE, 77454 M A R N E LA-VALLEE C E D E X , F R A N C E

E-mail address:

Alain.Lascouxauniv-mlv.fr

This page is intentionally left blank

A g-ANALOG OF SCHUR'S Q-FUNCTIONS GEANINA TUDOSE AND MICHAEL ZABROCKI

ABSTRACT. We present a family of analogs of the Hall-Littlewood symmetric functions in the Q-function algebra. The change of basis coefficients between this family and Schur's Q-functions are g-analogs of numbers of marked shifted tableaux. These coefficients exhibit many parallel properties to the Kostka-Foulkes polynomials.

1. INTRODUCTION

The space of (J-functions, Y, is defined to be the algebra generated by the odd power sum elements {pi,Pz,Ph, ■ • • } as a subalgebra of the space of symmetric functions, A. V is associated to the representation theory of the spin group and is also related to the projective represenation theory of the symmetric and alternating groups. The fundamental basis for this space are Schur's Q-functions, Q ^ X ] , which are indexed by strict partitions A. These functions hold the place that the Schur S-functions, s^[X] for /i a partition, represent in the algebra of the symmetric functions. The space of symmetric functions contains an important basis, H^X; q], the Hall-Littlewood symmetric functions [6]. Through specializations of the parameter q these functions interpolate several well studied bases of the symmetric functions and generalize features of these bases. They have elegant properties and may be seen in many different contexts of combina torics, algebra, representation theory, geometry and mathematical physics. It is natural to ask the question of what the analog of the Hall-Littlewood symmetric functions in the Q-function algebra should be. In this paper we introduce a family of functions G\[X;q] € T that answers this question since we observe that this family shares many of the combinatorial and algebraic properties of the Hf,\X;q\ functions in the space of symmetric functions. We expect that these functions will also be interesting from the perspective of other fields as well. From the combinatorial standpoint, we note that the coefficient of s\[X) in the symmetric function H^X; q] is the well known Kostka-Foulkes poly nomial. This family of coefficients are known to be polynomials in the parameter q with non-negative integer coefficients and at q = 1 represent

135

136

GEANINA TUDOSE AND MICHAEL ZABROCKI

the number of column strict tableaux of shape A and content fi. The combi natorial tools of jeu de taquin and the plactic monoid were in part developed t o explain the connections between the Kostka-Foulkes polynomials and the column strict tableaux [15]. By comparison the coefficient of QA[-^] in the function G^X; q] is also a polynomial in q and we conjecture (and prove in certain cases) that it also has coefficients that are non-negative integers. At q = 1 we know that these coefficients are the number of marked shifted tableaux with shape A and content /j,. A version of the RSK-algorithm was developed by Sagan, Worley and others [5], [19], [26], [24], and used to develop the theory of marked shifted tableaux. We hope that this theory can be extended to help answer the question of a combinatorial interpretation for these coefficients. Our definition for the functions G\\X;q] is motivated by viewing the symmetric functions sM[X] and H^,[X\q] as compositions of operators. In the case of the Schur functions, the Bernstein operator S m e End(A) ([17] p. 96) has the property for m > fii, Sm(s,j,[X])

= S(m,Mi,...,MJ(M))[^]-

That is, this formula is a recursive defintion for the Schur functions of degree n + m as an algebraic relation that raises the degree of a symmetric function by m acting on a Schur function of degree n. For the Hall-Littlewood symmetric functions, the operator H m e End(A) with H m (# M [X;g]) =

H(m
for m > ii\ is due to Jing [8]. In [27], it was noticed that these operators (as *--> q

well as many others) are related by a simple algebraic g-twisting, S m = HTO (the defintion of ~q is stated precisely in equation (11) below). The Schur's Q-functions, QA[-X] may also be seen from this perspective ([7], [14], [17] p. 262-3). That is, there exists an operator Q m £ End(T) such that for m > Ai, Qm(Q\[X})

= <9(m,Ai,A2,...A<w)[^]-

Since the defintion of the q-twisting ~q extends to EndiT), a natural defin tion for an analog to of the Hall-Littlewood symmetric functions in Y is to define G m := QTO and then for m > Ai, we set G(m,A1,...,A,(A))[^;<7] : =

Gm{G\[X;q\).

This framework provides us only with a possible definition for the HallLittlewood analogs in V. It remains to show that these functions share properties similar to those of the Hall-Littlewood functions. In this case we find some striking similarities that say we have indeed found the cor rect analog. We remark that similar functions were defined in [9] that are deformations of Schur's Q-functions and defined because of Lie algebra considerations.

Q-HALL-LITTLEWOOD FUNCTIONS

137

This work is inspired by the results of the Hall-Littlewood functions and the desire to find analogous structure in the Q-function algebra. In addition, part of the motivation of defining these functions and identifying their properties is to find what features of the Hall-Littlewood symmetric functions are not unique to the symmetric function algebra and should hold in a more general setting. A goal of this reasearch is to possibly identify what the g-twisiting of equation (11) represents on a combinatorial, geometric or representation theoretical level and to show that the G\[X; q] are another example of a structure that seems to exist in a more general context. The remainder of this paper is divided into three sections and an ap pendix. The first section is simply an exposition of definitions and and notation related to the symmetric functions and Q-function algebra. We develop in some detail the perspective that bases of the symmetric func tions and Q-functions can be seen as compositions of operators that have simple algebraic definitions. In the next section we introduce the G\[X;q] functions and derive recurrences and some properties that are analogous to those that exist for the Hall-Littlewood symmetric functions. In a final sec tion we discuss a generalization of the functions G\[X;q] that are indexed by a sequence of strict partitions (fj,^, fx^2\ . . . , / i ^ ) and the motivation for this generalization. These functions correspond to a g-analog of the product QM(i) [X]Qtlm [X] ■ ■ ■ Q^k) [X] and the coefficients of these functions corre spond to analogs of the generalized or parabolic Kostka polynomials of [11], [13], [20], [22], [21] and [23]. Finally, in the appendix we include tables of transition coefficients be tween the G\\X; q] basis and the <3A[^] basis for degrees 3 through 9. These tables are evidence of a very strong conjecture that these coefficients are polynomials in q with non-negative integer coefficients and represent a q analog of the number of marked shifted tableaux. This suggests that the marked shifted tableaux should have a poset structure similar to the charge poset for the column strict tableaux. 2. NOTATION AND DEFINITIONS

2.1. Symmetric functions, partitions and columns strict tableaux. Consider Ax the ring of series of finite degree in the variables x\, X2, £ 3 , . . . which are invariant under all permutations of the variables. This ring is algebraically generated by the set of elements {pfcpf] = Y^,i xi\ c A x and hence Ax is isomorphic to the ring A = C\pi,p2,P3, ■ ■ ■] with deg(pk) = k. We will refer to both A and Ax as the ring of symmetric functions. A is a graded ring and a basis for the component of degree n is given by the monomialsp\ '■=p\1p\2 ■ ■ -p\e where A is a non-increasing sequence of non-negative integers such that the values sum to n. Such a sequence is called a partition of n (denoted A h n ) . The entries of A are called the parts of the partition. The number of parts that are of size i in A will be

138

GEANINA TUDOSE AND MICHAEL ZABROCKI

represented by mi{\) and the total number of non-zero parts is represented by £(A) := ^ mj(A) and the size by |Aj := Yk kmk(X) = ]T\ X%. A common statistic associated to partitions is n(A) := X^i(* ~ 1)-V The partial order on partitions, A < fj, if and only if Y,%=i ■*-» — !Ci=i M» for all 1 < k < i(X), is called the dominance order. We call the operators RijX = (Ai,... , Aj + 1 , . . . , \j - 1 , . . . , $($) for 1 X. We will consider three additional bases of A here. Following the notation of [17], we define the homogeneous (complete) symmetric functions as h\ := h\1hx2 ■ ■ ■ hxHX) where hn = Z ^ P A / ^ a n d z* = Yl^=l i m<(A) "ii(A)!. The elementary symmetric functions are e\ := ex1e\2 ■ ■ -e\tw where en = 52\\-n(~l)n~e^P^/z^^ y convention we set p0 = h0 = e0 = 1 and p_fe = /i_fc = e_fe = 0 for k > 0. The Schur functions are given by s\ = detlhxi+i-jl^ij^^y The sets {p\}\tn, {hx}x\-n, {ex}xt-n and {s;\}Ahn all form bases for the symmetric functions of degree n. We will consider the elements of A as functors on the space Ax. If E is an element of A* then we let Pk\E] = E with Xi replaced with x\, so that pk : t\.x —>• Ax. We then extend this relation algebraically, px[E] will represent the expression px1 [E]px2 [E\- ■ ■ pxew [E] ■ I n particular, if we take E = X = x\ + X2 + Xz H then Pk[X] = Yi xi a n d the map that sends Pk !-> Pk[X] is a ring isomorphism A —► Ax since Pk[X] = x\ + x% + x\ H . Identities which hold in the ring A specialize as well to the ring of sym metric polynomials in a finite number of variables. We will use the notation Xn to represent x\ + x-i + ■ • ■ + xn, and the map which sends A to AXn by Pk !->■ Pk {Xn) corresponds to the operation of specializing variables in a symmetric series from an infinite set variables to a symmetric polynomial in a finite set of variables. For notational purposes, capital letters used as variables the end of the alphabet will be used to represent a series of variables (e.g. X = J2ixi ov Y = YiVi)j while capital letters indexed by a number will represent a polynomial sum of variables (e.g. Xn = Yd=i xi or Zn = YJi=i zi)We will need to adjoin to each of the rings A, Ax and AXn a special element q (or many special elements, if necessary) which acts mucli like a variable in this ring, however q will specialize to values in the field, q has the special property that pk \qX] = qkpk [X] and hence is not an element of our base field since for c e C, we have that pk[cX] = cpk[X\. Notice that by definition we have in general Pk[aX + bY] = apk\X] + 6pfc[y] for a, b s C. This implies that /[—X] does not represent the symmet ric series f[X] with Xi replaced by — x^ since p/t[X]| _ = (—l)kPk[X], while pk[—X] = —pk\X]. To this end we introduce the notation f[eX] =

Q-HALL-LITTLEWOOD FUNCTIONS

139

/[gX]| __ . In the case ofthe power sums we have that pfc [eX] = (—l)kpk[X] <md hence f[eX} = f[X]\Xi^_Xi. Consider the series ft = Y^n>o ^™ which is not an element of the ring A, but lies in the completion of this ring. We will use the morphism pk M- pk [X] on this element as well and manipulations of this notation allow us to derive the following identities, which we will use repeatedly in our calculations.

(i)

n

[*] = I I n b : = E M * ] i

(2)

l

n>0

n[-x\ = H(i - Xi) = i/a[x] i

(3)

(4)

S1[X + Y] = il[X]il\Y]

Sl[-eX] = J ] ( l + Xi) = J2 en[X]. i

n>0

Define a generating function of operators S(z) = J2m S m 2 m where for an arbitrary symmetric function P[X], S(z)P[X] = P[X - l/z]fl[zX}. In this manner Um acts on any symmetric function raising the degree of the function by m and has the action S m P[X] = S(,z).P[X] . A composition ofthe operators S(ZJ) produces the expression

(5)

s(Zl)s(z2)---s(zk)i

= n{zkx]

J J (i-^-M). l
where Zk represents the sum z\ + z2 + • • • + zk. Since the coefficient of zx in Sl[ZkX] then it must be that the coefficient of zx in the right hand side of (5) is given by Y\i 2 • • • S^ t l = s^[X]. These operators are due to Bernstein [17] p.96. It follows that S m (s^[X]) = S(m>A)[-X] where (m, A) denotes (m, A i , . . . , A ^ ) ) and s(au.„iai)[X] = det\hai+i-j[X]\. R e m a r k 1: We follow [17] in the use of raising operators for our definitions, however we are being imprecise since our raising operators are not associa tive or commutative as defined. We will consider a symmetric function as a composition of operators (for example in equation 5) and the operators Rij serve to raise or lower the indexing integer of the operator in the i and j t h positions respectively. By acting on an arbitrary symmetric function using these operations and the relations in equations (1) (2) and (3) commutation relations ofthe operators follow very nicely. By expanding the left and right side of the

140

GEANINA T U D O S E A N D MICHAEL

ZABROCKI

following expression verifies that S(z)S(u)P[X]

=

--S(u)S(z)P[X].

By taking the coefficient of umzn in both sides of the equation, we find that S«S m = —STTJ—ISJJ+I which also implies that S m S m + i = 0. Many of the calculations of commutation relations for other operators are of a similar sort of manipulation. A Young diagram for a partition will be a collection of cells of the integer grid lying in the first quadrant. For a partition A, ^(A) = {(i, j) : 0 < j < £(X) and 0 < i < Xj+i}- The reason why we consider empty cells rather than say, points, is because we wish to consider fillings of these cells. A tableau is a map from the set Y(X) to N, this may be represented on a Young diagram by writing integers within the cells of a graphical representation of a Young diagram (see figure 1). The shape of the tableau is the partition A. We say that a tableau T is column strict if T(i,j) < T(i + l,j) and T(i,j) < T(i,j + 1) whenever the points (i + \,j) or (i, j + 1) are in Y(X). Let rrik(T) represent the number of points p in Y(X) such that T(p) = k. The vector (mi(T),m2(T),...) is the content of the tableau T. The Pieri rule describes a combinatorial method for computing the prod uct of /im[A"] and s^X] expanded in the Schur basis. We will use the no tation X/fj, e Hm to represent that |A| — \/j,\ = m and for 1 Xi+i. It may be easily shown that

(6)

hm[X]slt[X]=

J2 **[*]■

This gives a method for computing the expansion of the h^X] basis in terms of the Schur functions. Consider the coefficients K\M defined by the expression (7)

hM[X] = £

KXflsx[X}.

AHMI

Kxfj. are called the Kostka numbers and are equal to the number of column strict tableaux of shape A and content fx. 2.2. Kostka polynomials and Hall-Littlewood symmetric functions. Define the following symmetric functions

(8)

=

n

(I+(?-I)^-+(92-?)4+---)MX].

l
They will be referred to as Hall-Littlewood symmetric functions as they are transformations of the symmetric polynomials defined by Hall [6] (see [17] for a modern account where QfM(x; q) in their notation is iI M [X(l — q); q] in

Q-HALL-LITTLEWOOD

FUNCTIONS

141

ours). The coefficient of s\[X] in H^X; q] is known as the Koskta Foulkes polynomial Kxn(q)- That is, we have the expansion

(9)

Hlx[X-q)=Y,KXll{q)sx[X]. A

We will present some of the properties of the Kostka-Foulkes polynomi als and the Hall-Littlewood symmetric functions below. First, it will be important to establish some identities for manipulating these functions. Let H(;z) = Y,m H m-z m be defined as the operation H(z)P[X] = P[X (1 — q)/z]£l[zX}. Taking the coefficient of zm defines the operator H m = H(z) which has the effect of raising the degree of the symmetric function it is acting on by m. A composition of these operators has the expression (10)

H(2i)H(z 2 )..-H(z f c )l = n[ZfcA-]

* ~Zitz;

TT

.

X qz Zi

i
, it is clear that the coefficient of zx

Now since HR^X = j:Cl[ZkX]

in the right hand side of (10) is exactly the right hand side of (8) and hence H > J H A 2 • ■ -H^ ( A ) 1 = Hx[X;q]. This operator also satisfies the rela tions H m _ i H m = ^ H m H m _ i and H m _ i H n — g H m H „ _ i =
VqP[X] = VYP\qX + (1 - q)Y]

_ ,

where VY denotes that as an operation on symmetric functions in the Y variables only and Y = X represents setting the Y variables equal to the X variables after the operation is completed. This is a q-analog of the •

g

operator V and we remark that S(z) = H(z). This follows by calculating _,.., S(z); « P[X) = S^YY(z)P[qX

+ (1 - q)Y] Y=X

= P\qX + (1 - q)(Y - l/z)]Sl[zY] = P[X - (1 - q)/z]il[zX] =

y=x

H(z)P[X}.

This relationship between S(z) and H(z) is the motivation for our definition of the g-analog of Schur's Q-functions. The functions Hx[X;q] interpolate between the functions s\[X] = H\[X;0] and /i^pf] = H\[X; 1]. The Kostka-Foulkes polynomials are de fined as the g-polynomial coefficient of s\[X] in H^[X\ q] and hence we have

GEANINA TUDOSE AND MICHAEL ZABROCKI

142

the expansion analogous to (7). (12)

H„[X;q}=YtKxfi(q)s,[X}. AH | Ml

The coefficients K\^(q) are clearly polynomials in q, but it is surprising to find that the coefficients of the polynomials are non-negative integers. A defining recurrence can be derived Kx^(q) in terms of the KostkaFoulkes polynomials indexed by partitions of size |/x| —\i\ using the formula for HTO. This recurrence is often referred to as the 'Morris recurrence' for the Kostka-Foulkes polynomials [18]. The action of HTO on the Schur functions is given by

(is)

Hm(s„[x]) = ]T Y. q\m+i,x)[x]i>0 ii/Xe-Hi

It is not immediately obvious that at q = 1, the previous equation reduces to the Pieri rule and at q = 0 the formula is simply Sm(sA,[X]) = S(m,p) [X]Using (13) and equating coefficients of s\ on both sides of the equation HOT(-ftrM[X; q\) = J2\ K\ll(q)'Hm(sx\X]), we arrive at the Morris recurrence

(i4)

#«,(„>,„)(?)= ]r ( - i r v - - m

J2

s:aa>m

A:V<*(s) €«(<»„-„,)

K

^)>

where m > Hi and a^ is a with part as removed. The Kostka-Foulkes polynomials and the generating functions H^X-^q) have the following important properties which we simply list here so that we may draw a connection to analogous formulae. For a more detailed reference of these sorts of properties we refer the interested reader to the excellent survey article [1]. (1) the degree in q of Kx^q) is n(^i) — n(\). (2) KXll(0) =
(q;q)n where (q;q)n = II?=i( 1 ~ ^*)-

(6) If £ is kth root of unity, H^[X; (] factors into a product of symmetric functions. (7) Set K'^iq) := qW-^K^l/q), then K'^q) > K^(q) for A < v.

Q-HALL-LITTLEWOOD

4 3 2 1

6 5 2 1

7 5 3 4 6 1 1 2 3|

FUNCTIONS

143

Z DZ Z I Z

FIGURE 1. The diagram on the left represents a col umn strict tableau of shape (6,5,3,3) and content (4,3,3,3,2,2,2,1). The diagram on the right represents a shifted marked tableau of shape (7,5,4,1) and content (2, 5,5,3,2). This tableau has labels which are marked on the diagonal.

(8) K\+(a)tll+(a)(q) > K\i(l(q), where A + (a) represents the partition A with a part of size a inserted into it. (9) KXfl(q) = Y,wesn sign(w)Vq{w{\ + p) - (/x + p)) where Vq{a) is the coefficient of xa in rii a 1 analog of the Kostant partition function and p = (£(p) — 1, £(p) — 2 , . . . ,1,0). (10) H„[X; q}Hx[X; q] = £ 7 d^(q)Hu[X; q], for some coefficients d^(q) with the property that if the Littlewood-Richardson coefficient c\ = 0 then dux(q) = 0. These coefficients are a transformation of the Hall algebra structure coefficients. (11) For the scalar product (SA[X], s^Xj) = Sx^, the basis ^ [ ^ ( l — q);q] is orthogonal with respect to H\[X;q], that is (H\[X;q\, Hli[X{l-q);q])=Qil\^p.

2.3. Schur's Q-functions, strict partitions, and marked shifted tableaux. The Q-function algebra is a sub-algebra of the symmetric func tions r = C[pi,p3,p5,...]. A typical monomial in this algebra will be p\, where A is a partition and Aj is odd. A partition A is strict if Aj > Aj+i for all 1 < i < £(\) — 1 and a partition A is odd if A, is odd for 1 < i < £(X). We will use the notation A h s n (respectively A h 0 n) to denote that A is a partition of size n that is strict (respectively odd). Note that the number of strict partitions of size n and the number of odd partitions of size n is the same (proof: write out a generating function for each sequence). The analog of the homogeneous and elementary symmetric functions in T are the functions qx := qXlq\2 ■■■q\e(x), where qn = £ A h o n 2e^p\/z\. Define an algebra morphism 8 : A —> T by the action on the pn generators as 0(pn) = (1 - (-l)n)pn. That is 6(pn) = 2pn if n is odd and 6{pn) = 0 for n even. 0 has the property that 9(hn) = 0(en) = qn and may be represented in our notation as 9(pn[X]) = pn[(l — e)X]. Under this morphism, our Cauchy element may also be considered a generating function for the qn

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GEANINA T U D O S E A N D M I C H A E L ZABROCKI

elements since

(is)

m_e)x}

= J2in[x} = ll\±j:n>0

l

i

It follows that {p\}\\-0n, {q\}\\-0n, {qx}xt-sn are all bases for the subspace of Q-functions of degree n. Another fundamental basis for this space are the Schur's Q-functions <5A[^] = S(H\[X; —1]). These functions hold a similar place in the Q-function algebra that the Schur functions hold in A. In particular, {Qx[X}}xt-sn is a basis for the Q-functions of degree n. In analogy with the Schur functions, Qx\X] may also be defined with a raising operator formula by setting q — — 1 and applying the 9 homomorphism to equation (8). We arrive at the formula: (16)

Qx[X] = J ] \^Qx[X]

= 11(1 - 2 % + 2 i $ - • • • )qx[X],

where the operators now act as Rijq\[X] = qni:jx[X]. Furthermore, they have a formula as the coefficient in a generating function: (17)

Qx[X]=Q[(l-e)ZnX]

J]

TT7J7 3I

l
As with Schur functions and the Hall-Littlewood functions, the raising operator formula leads us to an operator definition. By setting Q(z)P[X] = P [X — i ] Q[(l — e)zX], it is easily shown that Q(zi)Q(z2)---Q(zn)l

= n[(l-e)ZnX]

J] l
l l ^ . ' ■>'

l

and hence if we set Q m P[X] = Q(z)P[X] then Qm(Qx[X}) = Q{m,x)[X] as long as m > Ai. The commutation relations for the Q m are (18)

(19)

Q m Q n = -QnQm

QmQ-m

for

771 ^

= 2 ( - l ) m - Q-mQm

-71,

if m + 0,

(20) Q ^ = 0 if m y£ 0 and Qj; = 1. These formulas allow us to straighten the QM[X] functions when they are not indexed by a strict partition. The Q-function algebra is endowed with a natural scalar product. If we set (px,Pii)r ~ 2e^$\itz\ f° r \ v ^~o n, then it may be shown that we also have (21)

{Qx[X],Qn[X])r

=

2?M5^.

A shifted Young diagram for a partition will again be a collection of cells lying in the first quadrant. For a strict partition A, let YS(X) = {(i,j) :

Q-HALL-LITTLEWOOD

FUNCTIONS

145

0 < j < £{X) and j < i < Xj+\ +j}- A marked shifted tableau T of shape A is a map from YS(X) to the set of marked integers {1' < 1 < 2' < 2 < . . . } that satisfy the following conditions

• T(i,j) < T(i + l,j) and T(i,j) < T(i,j + 1) • If T(i,j) = k for some integer k (i.e. has an unmarked label) then T{i,j + l)^k • If T(i,j) = k' for some marked label k! then T{i + l,j) ^ k'. We may represent these objects graphically with a Young diagram rep resenting A and the cells filled with the marked integer alphabet. If T is a marked shifted tableau, then we will set m^T) as the number of occur rences of i and i' in T. The sequence (rrii(T),m2(T), 771.3(T),...) is the content of T. The combinatorial definition of the marked shifted tableaux is defined so that it reflects the change of basis coefficients between the q^ and Q\ basis. The rule for computing the product of <J/m[X] and <5^[X] when expanded in the Schur Q-functions is the analog of the Pieri rule for the T space. If A//J G Tim then a(A///) will represent 1+ the number of 1 < j < £(\) such that Xj > \ij and jfj-i > A,. We may show that

(22)

qm[X]Qlt[X)= J2

2<^>-WWQA[X].

Denote by L\^ the number of marked shifted tableaux T of shape A and content /x (where A is a strict partition) such that T(i, i) is not a marked integer. We may expand the function q^{X] in terms of the Q-functions using (22) to show

(23)

3.

q^X] = Y^ L^QX[X}.

T H E Q-HALL-LITTLEWOOD BASIS

G\[X\q]

FOR THE ALGEBRA

V

In this section we define a new family of functions G\[X;q] which seems to play the same role as the Hall-Littlewood functions H\[X;q] in the Qfunctions algebra. These functions are introduced via a raising operator formula similar to (8). This definition permits an equivalent interpretation via a corresponding vertex operator G m whose properties are analogues to both the Hall-Littlewood vertex operator H m and Q m . Note: From here, unless otherwise stated, all partitions are considered strict.

146

GEANINA TUDOSE AND MICHAEL ZABROCKI

3.1. Raising operator formula. We define the following analog of the Hall-Littlewood functions in the subalgebra T

We call the functions G\ € Y c ^(l) the Q-Hall-Littlewood functions. In T ® C(q) this family can be expressed in the basis of Q-functions as

(25)

J2L^)Qx[X],

G^[X;q} = A

which can be viewed as a g-analog of (23). We call the coefficients L\^,{q) the Q-Kostka polynomials. We shall see that this family of polynomials shares many of the same properties with the classical Kostka-Foulkes poly nomials. Tables of these coefficients are given in an Appendix. It follows from (24) that Lx^q) have integer coefficients and L\fJj(q) = 0 if A < JJ,. This shows Proposition 1. The G\, A strict, form a Z-basis for T ®% Z[q]. The basis G\ interpolates between the Schur's Q-functions and the func tions <7M because G\[X\0) = Q\[X] and G A [ ^ ; 1 ] = q\[X] as is clear from (24). Since the coefficient of zx in fi[(l — e)ZnX] is q\[X] equation (24) implies

P.)

CjlXid _

n

(i^)(i±^)n|0-.Wr,

l<2<j
v

J i

/

\

JI

/

Define the operator G{z) acting on an arbitrary symmetric function P[X] as (27)

G(z)P[X] = P X

Sl[(l-e)zX].

The operator G(z) defines a family of operators as G(z) = J2mez

£*mzm

and hence GmP[X] = G(z)P[X] . If we consider a composition of these operators acting on the symmetric function 1 we obtain the following

GWo(,,...G(,,^JJiT^)(ii^) (28)

xn[(l-e)ZnX],

which together with relation (26) gives Gx[X;q] =

GXl...Gi{X)(l).

Q-HALL-LITTLEWOOD FUNCTIONS

147

Next we investigate some properties of this operator. First, Gm satisfies the following commutation relation. Proposition 2. For all r,s £ Z we have 0-—Q

)(GrGs+GsGr)+q,(Gr_iGs-)-i—G5+iGr_i+Gs_iGr+i—Gr+iGs_i) = 2(-ir(i-g)25r,_5.

Proof We will prove this relation in a few steps. Consider G(w) and G(z) the operator G defined above on the variable u and z respectively. We are looking at the composition of these two operators. Step 1. We may write G(«)G(z) =

G(z,u)F(z/u)F(-qz/u),

where G(z, u) is an operator symmetric in z and u defined by G(z,u)P[X] and F(t) := (29)

1-t

:=P X

1-q]

tt[(l - e)(z + u)X]

. This is easily seen from

G{u)G(z)P[X]

= P X

1-q

v,'l

1 - 9)

1-q

= G(u)P X

1 - + \z u

n[(l - e)zX] =

Jl[(l - e)(z + u)X}Cl [(1 - e)(q - 1)~] .

z Note that first two factors are exactly G(u,z) and f2 (1 — e)(q — 1)— is equal to F(z/u)F{-qz/u). Denote by a(t) := F(t) + F(t~l) = f^r€Z r r 2(-l) t . Step 2. Let qj- be the adjoint of multiplication by qi[X] in the algebra T under the scalar product < , >r- Consider Q±(t) its generating function, It is not difficult to see that P[X + t) = Q±(t)P[X]. In fact it suffices to show this for a suitable basis element in T, namely the power sum PA[X], where A is an odd partition. We show that q^p\\X] = p\\X + t] . We have

pil[x + t}\ti = (p^lx] + tfll)---(pM

(so)

+ t>Mt)

At the same time we also know that pj^p^ = 2~e^ ^pv /

(31)

VxVv

2 <")

P,r{

p

^-

if v W A = [i since 2e^z„

z e

Now because q{ = £ A h < j i 2 Wpx/zx, then q^p^X] = £ A a i / = / i ~;Pv A h 0 i and this is exactly the right hand side of (30).

where

GEANINA TUDOSE AND MICHAEL ZABROCKI

148

Also recall that Cl\(l — e)tX] = Q(t), where Q is the generating functions for qi. If we replace these in the expression of G(z, u) we obtain Q(z)Q(u)Q±(-u-l)QJ-(-z-1)Q±(qu-1)Q±{qz'1).

G(z,u) =

We know that Q(u)Q(z)a(u/z) = a(u/z) ([17] Chap III. 8, p. 263), and hence we have as well Q±(u)Q±(z)a(u/z) = a(u/z). Therefore G(z,u)a(u/z)

=

a(u/z).

Step 3. Consider the following expression. [G(u)G(z)(l

- qz/u)(l + quiz) + G{z)G(u)(l

=(1 + qz/u)(l + qu/z)G(u, =(1 + qz/u + qu/z +

- qu/z)(l + qz/u)}

z)a(u/z)

2

q )a(u/z),

and so G(u)G(z)(l

- qz/u + qu/z - q2) + G(u)G(z)(l = (1 + qz/u + qu/z + q2)a(u/z)

- qu/z + qz/u - q2)

= (1 -

q)2a(u/z).

since we have as a series ta(t) = —a(t). By taking the coefficient of urzs in the left hand side when expanded as series and equating with the corre sponding coefficient in the right hand side we obtain the desired relation.

a For q = 0 in the relation above we recover the commutation relations of the operator Q given in equations (18), (19) and (20). At q = 1, G m becomes multiplication by qm and hence is commutative. Formula (26) may be used to derive the action of the operator G m on the basis of Schur's Q-functions. Proposition 3. For m > 0, (32)

G m (Q A [X]) = £ y

£

2a^\-l)<m+i^Qll+{m+i)[X},

where a + (k) denotes the partition formed by adding a part of size k to the partition fx, and e(k, /u) + 1 represents which part k becomes in \i + (k) (Qii+(k)\X\ = 0 if fJ, contains a part of size k). For m < 0 a similar statement can be made using the commutation relations (18), (19) and (20).

Q-HALL-LITTLEWOOD FUNCTIONS

149

Proof From (26) the action of G m on a function P[X) € T can be written as GmP[X] =P[X - (1 - q)/z]Cl[(l - e)zX] = J2 q'z-'iqj-PXX

- l/z]n[(l

- e)zX]

i>0 4

■*

1 vTn,-

i>0 p x tl „-l-l since P[X + t}= J2i>0 qf l}-

Thus

GmQx[X]

^Y^qtQm+iiqiQxlX}), i>0

where qf- applied to Q\ is

qj-Qx\X}=

J2

2»W*>QM[X],

If m > 0, equation (32) follows from (18) and (20). If m < 0 we may need to use the commutation relation (19) for Q n to straighten the index to a strict partition. In general we have Q,m+i{Qp\X\) = Q^m+i^[X], where we prepend the part (m + i) (possibly negative) to the partition /i. □ Example 1. We compute G(3t2,i)[X;q] using the proposition above. We have Gl3,2,i)[X;q]=G3(G2(Qli)[X])) = G

3|E

2 a « 1 )^(-l)^ 2 + ^Q M + ( 2 + i ) [X])

E

V>o (.1)/^eni =G3(Q{2tl)[X}) + 2qG3(Q{3)[X})

=E

/

2"«2'1)^(-l)^3+^)Q,+(3+i)[X] +

E

»>0 (2,l)//i€«i

\i>0

(3)/«/€«< 1 1

= (9°2°Q(3,2,i) +9 2 Q( 4 ,2) +
+ « 3 2 1 Q (6) )

=
150

GEANINA TUDOSE AND MICHAEL ZABROCKI

polynomials. In what follows we will consider some of the other remaining properties. An important consequence of equation (32) is a Morris-like recurrence which expresses the
(33) Lat{nill)(q) = Yl ( - i r V * - " S:Q

2»W"(*)>£A/1(g),

£ s

»>"

A:A/a< )e"H((>3_n)

where n> H\ and a^s' is a with part as removed. Proof If n > Hi we have that (34)

GnG^X- q] = GM[X;

?]

= £

(q)Qa[X}. Ct

On the other hand GM[A"; q] = J2x ^jXti(q)Q\[^]

Gn \TL^(l)Q^x})

an

^ so

=J2L^)Gn(Qx[X]).

Using the action in (32) we have (35)

GnGlt[X;q] = '£iI<xMT,9i \

i>0

E

20(^(-l)£(»+<'")Qv+(„+i)[X].

v:\/veUi

For a = v + (n + i), equating the coefficients of Qa in (34) and (35) we get

£«,(»,,*) («) = X)E« <2a(Va " (n+<)) (- 1 ) e(n+1,a " ( " + ' ))L *M(9)A

i>0

By reindexing i := a a - n for a s - n > 0 we obtain the desired recur rence (33). D Example 2. Let n = 5 and L(6I2),(5,2,I)(Q) = 2q + Aq2. Using the recur rence we have one s such that as > 5, i.e. c\\ — 6. So i(8,2),(5,2,l)(?) = 9 6 " 5

E

2°^< 2 »L A ( 2 i l ) ( 9 )

A/(2)6Wi

= ?(2L(2i),(2i)(9) + 2L(3),(2i)(g)) = g(2 + 2 • 2q) = 2q + Aq2. As a consequence of the Morris-like recurrence we have the following Corollary 5. Let /i < A in dominance order. 1. Ifn > Ai then L(„,A),(n,/x)(g) = L\M)2. Lxx(q) = 1 and Lmx(q) = 2«*>-V(A)e ( 3. 2 M- W divides Lx„(q).

Q-HALL-LITTLEWOOD FUNCTIONS

151

Proof 1. There is only one term in the recurrence (33) in this case which is exactly L\^{q). 2. The first is a consequence of (1). For the second, we have that the only term on the right hand side is q^~Xl2LQx\-Xi)(X2,— ){Q) which by induction is qM-*i+*«>*,-))2 • 2i^~2 = 2<(*)~V (A) . This is the analog of Property 2.2 for the Kostka-Foulkes polynomials. 3. We use induction on £(fi) and the Morris recurrence to derive this property. If l(n) = 1 we know that G(m)[X;q] = Q(m)\X] and thus A can be only (m) and the assertion holds. For the induction step we use equation (33). We need to show that 2*0i)-*(«)+i divides La^n^{q). From the induction hypothesis we know that 2 ^ ) - ^ ) divides LX)l{q) for every such A on the left hand side of equation (33). The partitions A have £(X) € {£(<*), £(a) - 1}. If £(X) = £(a) - 1 we are done. If ^(A) = £(a) then A ^ a^s\ since a ^ has length one less than a, and thus a(X/a^) > 1. This implies that 2e^~t^+1 divides each ) and thus La>M(q). D 2-W^' )L^{q) Using the Morris recurrence we are also able to obtain a formula for the degree of L\^(q) similar to Property 2.2 for the Kostka-Foulkes polynomials. Proposition 6. If fx < X in dominance order, we have degqLXlJL{q) = n(fi) - n(X). Proof We prove this assertion by induction on £{fi). For £(fj) = 1, the equality is obvious. For the induction step we use the recurrence (33). Fix now an index s on the right hand side of the equation (33). Denote by fi\^l = (cti + cts — n,a2:... ,as-i,as+i,...). We claim that n(/xS) < n(A) for any other A such that X/a^ € H(a,-n)- We have that s-l n

(^(oj) = H(* - l)ai + Y^(s i=2

+i

~ l)as+i+i

i>0

while s-l n A

( ) = X ] ^ ~ l^Ui + ^ i=2

+

X ^ S + i ~ ViGs+i+l + C*+i)' i>0

where A = (on + e i , . . . , a 5 _i + e s _i, a s + 1 + es,...) and ]T\ e, = as - n. Moreover if A ^ jt;*; there exists at least one e, with i > 2 such that e* ^ 0.

152

G E A N I N A T U D O S E AND M I C H A E L Z A B R O C K I

Therefore n ( / / S ) < n(A). We thus have proved that among the polyno mials L\Aq) in the second sum of (33), the polynomial L (,) (q) has the M( 0 ) >^

highest degree, namely n(fi) — n ( / i S ) . Next we show that in the first sum, the highest degree is obtained for

s = l. That is to say degq (q^^L^^q))

> degQ

fq^^L^Jq)

hence oti-n

+ n(fj.) - n(^oj) >as-n

s-1

+ n(/x) - n ( ^ ) , s-1

a i + 5 ^ ( i - 1)* + 5 Z (7 - 2)a,- > a, + £ ( i - 2)a< + ^ ( j j=2

J>»+1

»=3

2)aj

j>s

which is 5-1

S-l

a i + a 2 + ^ ( « - l)ai > ^ ( i - 2) a* + (s - l ) a s i=3

t=3

= (a 3 + a s ) + (2a 4 + a,) + • • • + [(s - 3)a s _i + as] + 2as. The last inequality is true as (i — l)a< > (i — 2)a, + s for i = 3 , . . . , s — 1 and a i + 0J2 > 2a s . Thus we have degqLa^n^(q) — (aj — n) + n(^i) — ra(^i|0j). Finally we need to show this is in fact n((n, /*)) — n(a). That is a i - n + ^ ( i - l)/i» - ^ ( i - 2)ai = ^2 in - ]T](i - l)«i, i>2

j>3 a

j>0

j>2

n =

and by simplifying we obtain J2 i ~ S A'ti which is obviously true. Hence degq (La,(n,^))(<7)) = n((n, /z)) — n(a) and the proof is complete.

n The property that is most suggestive that these polynomials are analogs of the Kostka-Foulkes polynomials is Conjecture 7. The Q-Kostka polynomials L\^(q) have non-negative coef ficients. We will prove this conjecture for some particular cases. In general we believe that there should exist a similar combinatorial interpretation as for the Kostka-Foulkes polynomials. More precisely there should exist a statistic function d on the set of marked shifted tableaux, similar to the charge function on column strict tableaux, such that T

summed over marked shifted tableaux of shifted shape A and content /x with diagonal entries unmarked.

Q-HALL-LITTLEWOOD FUNCTIONS

153

The polynomials Lx^q) suggest that many of the tableaux of the same shape and content will have the same statistic. This suggests that the statis tic d might satisfy d(T) = d{S) for S and T differing by only marking or unmarking certain cells. This property does not seem to hold for larger sets of tableaux and for large enough examples (for example fj, = (5,4,3,2,1)) it is impossible that the statistic is completely independent of the markings. Another intriguing property of this statistic function d is that the values it takes are not too different than the charge function. It seems that in general we have that for given A and /x the set of {d(T) : T in the summation of L\l_i(q)} is a subset of {c(T) : T column strict tableaux of shape A and content ^ } , where c is the usual charge. This suggests that there should be relationship between these two statistics; however, we have so far not been able to establish what that link might be. Proposition 8. (1) For [i a two-row partition and A > /J. we have Lx^q) 2gn(ii)-n(X)

=

_

(2) If /J, has the property //j >

y . A*j» Conjecture 7 is true. j>i+l

Proof (1). Let us consider G,j,[X;q] and let JJL = (n,m). G(ntm)[X;q} = G„(Q ( m ) [X]) which by Proposition 3 is

G„(Q(m)[X]) = 5 > '

J2

We have that

^^'"H-ir^^QrHn+iilX]-

From this we deduce that i = 0 , 1 , . . . , m and /i = (m — i). Thus 771

Gn(Q(m)[X\)

= Q(„,m)[-X"] + 2_^ 2qlQ(n+i,m-i) [X] i=l

and the proof is complete. (2). In this case we prove it by induction on (.{p) and using the Morris recurrence (33). The case £(/J,) = 1 is clear as Lxn(q) = Sx^. For the induction step consider Z/a(«,lAi)( (n, /J.) in dominance order and n > |/x|. Thus the right hand side does not contain negative signs and by induction it is non-negative . Hence La^n^{q) has non-negative coefficients. D We also note that monotonicity properties, similar to Property 2.2 and 2.2, hold for the Q-Kostka polynomials. Conjecture 9. Let L'^(q) := qnM-n{X)Lx^q'1). L'xfiiq) > Z^^^L'xviq),

for n
We have in dominance order.

We can prove this fact by using induction and the recurrence (33) for the case ji\ — v\.

154

GEANINA TUDOSE AND MICHAEL ZABROCKI

Example 3. Let A = (6,2), /z = (4,3,1), v = (5,2,1). We have n(A) = 2, n{\i) = 5, and n(y) = 4. The U polynomials are L'Xli = q5-\4/q2

+ 4/ff3) = 4 + 4g,

and thus L'X(t(q) >

L'x„ = q4-2(2/q

+ 4/q2) = 4 + 2q,

23-3L'Xl/(q).

Another property of the Kostka-Foulkes polynomials case that seems to hold in our case refers to the growth of the polynomials L. For the KostkaFoulkes polynomials the conjecture belongs to Gupta (see [1] and references therein). Conjecture 10. / / r is an integer that is not a part in either partition A or fi, then L\+(r),n+{r){q) > L\n(q). The case where r > X\ (which also ensures that r > zzi) is obviously true since £<(r,A),(r,^)(<7) = ^A^(g) (see Corollary 5). Example 4. Let A = (5,3), /z = (4,3,1) and a = 2. We have i(5,3,2),(4,3,2,i)(«) - £(5.3),(4,3,i)fa) = 2q + 4q2 + 8q3 - (2q + 4q2) = 8q 3 .

3.3. Another expression for Lx^q). The polynomials Lxfi{q) have a similar interpretation to property 2.2 using an analog of the g-Kostant partition function. We follow the construction in [1]. In order to write equation (16) as (36)

^ [ x ] = n ( f ^ )

_ 1

Q A W

we will use linear maps from the group algebra Z[Z n ] to the algebra V. A basis of Z[Z n ] will consist of formal exponentials { e a } a g which satisfy relations eae/3 = ea+<3. In fact we identify the ring with the ring of Laurent polynomials in xi,... ,xn and set ea = xa. With this in mind we are viewing all our polynomials in Tn (or A„) as linear homomorphisms from Z[Zn] to T i.e. Q : ex -> Q(ex) = Qx If we now set £„ :=

TT I

q : ex -»• q{ex) = qx.

~r^~ ) > l

we

Xi X

l
have that £n = \ J TZ(a)ea Q €Z«

where 1Z(a) = £) t at2t and at counts the number of ways the vector a can be written as a sum of positive roots of type A i - i , t of which are distinct. The positive roots in the root lattice of An-i are {e, — ej}i
qx[X}=q(ex)=Q(^n^)eaeX)

Q-HALL-LITTLEWOOD FUNCTIONS

= £

155

n(a)Qx+a[X}.

If we consider the same argument for G\ [X; q] —

J|

I

l
- ^ ] Q\ [X] ~ ^

ij

'

we need to define the ^-analog of £n as

^.«(o:)e a , where 7?.q(a) = Ylit,kat,k%tQk and at,fc

and thus (n(q) = E a€Z"

counts the number of ways the vector a can be written as a sum of k positive roots, t of which are distinct. Hence

Gx\X;q} = J2 aezn

ng(a)Qx+a[X}.

This yields another expression for the Q-Kostka polynomials in terms of 7?.q(a) as

Lv(
E

±ynq{a).

The index of the sum reflects the straightening of a Q-function indexed by an integer sequence and it is a consequence of the commutation rela tions (19). It is possible to express the equation above using the action of the symmetric group on Schur's Q-functions, yielding an alternating sum similar to Property 2.2. Unfortunately the action of the symmetric group on Schur's Q-functions indexed by a general integer vector is not as elegant as for Schur functions. E x a m p l e 5. To compute G(3i2ti){X;q] we compute flq(a) for all of the relevant compositions a. There will be a finite number for which the func tion Q\+a[X] is non-zero and we list only those the expression below. Note that trailing Os of these composition are removed. G(3,2,i) [X; q] =Q(3,3,i) [X] + (2q + V ) Q ( 4 , 2 ) [X] + ( V + V ) Q ( 5 , 1 ) [X] + 2q2Q(5M[X]

+ ( V + 4<73)Q(6)[X] + 2g 3 Q ( 6 ,_ 1 , 1 ) [X].

The commutation relations say that Q (5,0,1) [-X1 = —Q(5,i)[-X] and Q(6,-i,i) [X] = —2Q(6)[X] and hence we have used this formula to show G( 3 ,2,i)[*; q] = Q(3,2,i)[*] + (29 + V)Q(4,2)[*] + (2
+VQ (6) [X]. Remark: Most of the properties of the Q-Kostka polynomials L\IJ/(q) are analogous to the Kostka-Foulkes polynomials, but a few properties do not seem to generalize.

GEANINA TUDOSE AND MICHAEL ZABROCKI

156

(1) The analog of Property 2.2 does not seem to hold since computations of G\\X; q] where q is set to a root of unity do not factor. (2) There does not seem to exist an elegant relationship between G\ [X;q] and its dual basis (Property 2.2). (3) A property similar to that of Property 2.2 does not seem to hold. We do not know if there is a relationship between G\[X;q] and a Hall-like algebra. (4) We do not know if an analog of the Macdonald symmetric functions should exist. A family of functions which mimic the formulas for the Macdonald symmetric functions in [27] may easily be defined, but the specializations of the variables indicate that the same sort of positivity and symmetry properties of the coefficients cannot hold through this definition. 4. GENERALIZED (PARABOLIC) Q - K O S T K A POLYNOMIALS

There exists in the literature a few generalizations of the Kostka-Foulkes polynomials that correspond to g-analogs of multiplicities of irreducibles in tensor products of irreducible representations (Littlewood-Richardson co efficients). In [23] formulas were introduced for realizing 'generalized' or 'parabolic' Kostka coefficients [22] as coefficients appearing in families of symmetric functions defined as compositions of operators. This construc tion may also be extended to the Q-function algebra providing a means of defining a generalization of the Q-Kostka polynomials that corresponds to a (7-analog of coefficients in products of Q^X]. Let fi* — (fj.^,/J,(2\ ... ,/j'fc') be a sequence of partitions and fi* the concatenation of all those partitions. Take r] = (£(fj,(-1^),£(fx(-2')),... , l(^)) and n = YliVii then set Rootsv = {(i,j) : 1 < i < rji + ■ ■ ■ + r\r < j < n for some r } . Now set

(37)

Htl.[X;q]=

]J

^-J—s^X}.

The parabolic Kostka polynomials are defined as the coefficients of the Schur basis in these symmetric functions. That is, the polynomials K\-til- (q) are defined by the coefficients in the expansion (38)

Hll.[X;q]=

£

Kx,^(q)Sx[X}.

Ah| M *|

The functions H^[X;q] and the parabolic Kostka coefficients have the following properties. • If /z* is a partition then it is conjectured that K\]IJ,'(q) has nonnegative integer coefficients (in certain cases this is known). • Hp.- [X; 0] = 5^» [X]. /J,* need not be a partition, but this is consistent with the definition of s\ in section 2.1.

Q-HALL-LITTLEWOOD FUNCTIONS

157

Hp* [X; 1] = s^u) [X]s^2) [X] ■ ■ ■ s^(fc) [X] and in this sense the coeffi cients K\;fM* (q) are g-analogs of the Littlewood-Richardson coefficients If (i* = ((71), (72), • • • , (lt(y))) where 7 is a partition, then H^ [X; q] = H^X-q). s • If / ? is a partition then H^ [X;q] = s^. [X] + X)A>^-. ^A;M* (l) *[-^1• There exists an operator H 7 such that H7(i7A,«[X;q']) = Hhtti(1)t...!liW)[X;q} (see [23]). In addition, analogs of most properties of the Hall-Littlewood functions and the Kostka-Foulkes polynomials also seem to hold (see for instance [22]). We should also mention that there is an analog of several other formulas for the Hall-Littlewood functions. It follows from the definition of the functions H^ [X; q] that (39)

Hll.[X;q]=Q[ZnX}

]\

{l-Zj/zi)

l
fl

\ ■

(i,j)£Rootsv

1 i'

*

For k > 0, if we define the operation, (40)

H(Zk)P[X]

= P[X - (1 - q)Z*]Q[ZX]

JJ

{1-Zj/zi),

l
where Z* = £ * = 1 j : , then

H(Z"1)H(Zr'2)---H(Z7"('")l = n[Z„X]

JJ l
(41)

*(i-*/*)

n

rr^M H

{i,j)€Roots„

3I

and therefore U(Zk)H^[X;q]I ^ = # ( 7 j M .)[X;g]. This construction exists in complete analogy within the Q-function al gebra. We will create a family of functions in T which are indexed by a sequence of strict partitions. Let fi* = (/Z 1 ),/^ 2 ),... ,fi^) where ^W j s a strict partition and set 77 = ((.(fi^),(.(^),... ,£((1^)). Define Rootsv as before and then define the function

(42)

Gll.[X;q]=

JJ (i,j)€Rootsn

I^f^M-MtJ

We may also view these elements of T as the result of a family of operators acting on 1. Consider the composition of the operators QxP[X]:=QXlQXa---QxkP[X]

= p[x-z;)n[(i-e)zkx] n l
\=£l£ +

Z /Zi

*

158

G E A N I N A T U D O S E AND M I C H A E L Z A B R O C K I '— Q

and then set G\ := QA , that is (43)

GxP[X] = P[X + (q-l)Zm[(l-e)ZkX}

]J

\^fi

i
As with the other operators of this sort, it is easily shown that a composition of G^i) acting on 1 is equivalent to the defining relation (42) of the functions G^ [X; q] and hence that (44)

G 7 G^[X;<7] = G(7i/l(i>

^))[X;q}.

The G^ [X; q] functions seem to share many of the same properties of the HV' \X\q\ and Gy[X;q] analogs. Define the polynomials L\^*{q) by the expansion

(45)

• G^»[X;0] = Q^,[X]. fi* need not be a strict partition, however if it is not then the straightening relations (18), (19) and (20) may be applied to reduce the expression. • G^ [X; 1] = Q^D \X]Q^2) [X]--- Q^m [X] and hence LA;M- (1) is equal to the coefficient ofQ\[X] in the product Qll(i)[X}Q^2)[X] ■■ -Q ik)[X}. • If IX* = ((71), (72),... , (7^(7))) where 7 is a partition, then G^. [X\ q] = GyiX-q}. • If fj,* is a strict partition then GM* [X; q] = Q^. [X] + J2\>n' L\w (q) Qx[X). Computing these coefficients suggests the following remarkable conjec ture and indicates that these coefficients are an important (/-analog of the structure coefficients of the QA[-^] functions in the same way that the K\-,n" (9) polynomials are ^-analogs of the Littlewood-Richardson coeffi cients. Conjecture 11. For a sequence of partitions fi*, if [L* is a partition then L\;^ (q) is a polynomial in q with non-negative integer coefficients. If this conjecture is true then the polynomials LA;M* (
Q-HALL-LITTLEWOOD FUNCTIONS

159

for example [20], [21] or [13]) which imply that the coefficients K\tfi*(q) are positive. By contrast, for the generalizations of the Q-Kostka polynomials we know that if /z^ has two equal parts for any i then G^ [X; q] = 0, hence this special case is not of interest in this setting. 5.

APPENDIX: TABLES OF

2ew~i^Lxfl(q)

FOR.

n = 4,5,6,7,8,9.

(3,1) (4) 1 q 0 1 (3,2) (4,1) (5) 1 2q q2 0

1

q

0

0

1

(3,2,1) (4,2) (5,1) (6) 1 2 q2 + q 2 q3 + q2 q" "4 0

1

2q

q2

0

0

1

q

0

0

0

1

(4,2,1) (4,3) q

(5,2) (6,1) (7) 2qz +q2q3 + q2 q4

o

l

2q

2q2

o o o

o o o

1

2q

0

1

0

0

(4,3,1) (5,2,1) (5,3) (6,2) (7,1) (8) 2q 2 q2 + q 2 q2 + 2 q3 q3 + 2 q4 q-l 5 1

2q2+q

2q3 + q2 q4

0

2q

2q2

q3

0

1

2q

q2

0

0

1

q

0

0

0

1

GEANINA TUDOSE AND MICHAEL ZABROCKI

160

p(4, 3, 2) 1

(5,3,1)

0

1

0

0

0

0 0

2q

1

0

0

0 0

(6,2,1)

q

0

0

0

(5,4)

(6,3)

(7,2)

(8,1)

(9)

2 q + 4 q2 2 q3 + q2 2 q2 + 4 q3 q2 + 2 q4 + 4 q3 4 q4 + q3 + 2 q5 2 q6 + 2 q5

1

0 0

0 0

0 0

ACKNOWLEDGMENT:

0 0

2

2q

+ q

2q q

2

3

2q

+ 2q

3

4

q

+ 2q

2

3

2q

2q

2

3

2q +q 2<J

q7 q5 q4

2

+ q

q4

2

q3 q2

2q

0

1

2q

0

0

1

2q

0 0

0 0

1 0

q 1

We thank Nantel Bergeron for many helpful sugges

tions on this research. REFERENCES [1] Desarmenien, J., Leclerc, B., and Thibon, J.-Y., "Hall-Littlewood Functions and Kostka-Foulkes Polynomials in Representation Theory," Seminaire Lotharingien de Combinatoire, B32c(1994), 38 pp. [2] Foulkes, H. O., "A survey of some combinatorial aspects of symmetric functions." in Permutations. (1974) Gauthier-Villars, Paris. [3] Garsia, A. M., "Orthogonality of Milne's polynomials and raising operators." Dis crete Math, 9 9 , 247-64. [4] Fulton, W., Young Tableaux. Cambridge University Press, Cambridge, 1997. [5] Haiman, M., "On Mixed Insertion, Symmetry, and Shifted Young Tableaux." J. Comb Theory (A), 50 (1989), 196-225. [6] Hall, P. "The algebra of partitions." Proc. ith Canadian Math. Congress, Banff, (1959). [7] Jing, N., "Vertex Operators, Symmetric Functions and the Spin Group r n . " J. Algebra 138(1991), 340-398. [8] Jing, N., "Vertex Operators and Hall-Littlewood Symmetric Functions." Advances in Math. 87(1991), 226-248. [9] Jing, N., "Vertex operators and generalized symmetric functions," in Proceedings of Quantum Topology, World Sci. Singapore (1993), 111-128. [10] Jozefiak, T., "Schur Q-functions and applications." Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras, (1991) 205-224, . [11] Kirillov, A., "Ubiquity of Kostka polynomials." Phys. and Comb., Proceedings of the Nagoya 1999 International Workshop on Physics and Combinatorics, Nagoya University, August 23-27, (1999), 85-200. [12] A. N. Kirillov and N. Yu. Reshetikhin, The Bethe Ansatz and the combinatorics of Young tableaux, J. Soviet Math. 4 1 (1988), 925-955. [13] Kirillov, A. N., Shimozono, M., "A generalization of the Kostka-Foulkes polynomi als," math.QA/9803062, to appear in J. Algebraic Combin. [14] Lascoux, A., Pragacz, P., "Orthogonal Divided Differences and Schubert Polynomi als, P-Functions and Vertex Operators." Mich. Math. J. 48(2000), 417-441. [15] Lascoux, A., Schiitzenberger M. P., "Sur une conjecture de H. O. Foulkes." C. R. Acad. Sci. Paris, 286A(1978) 323-324. [16] Lascoux, A. and Schiitzenberger, M. P., "Le Monoid plaxique." Noncommutative structures in Algebra and Geometric Combinatorics, CNR, Rome. 109(1981), 129156.

Q-HALL-LITTLEWOOD FUNCTIONS

161

Macdonald, I. G., Symmetric Functions and Hall Polynomials. Cambridge Univer sity Press, Cambridge, 1995. Morris, A. O., "The characters of the group Gl(n;q)", Math. Zeitschr. 8 1 (1963) 112-123. Sagan, B. E., "Shifted Tableaux, Schur Q-functions and a Conjecture of R. Stanley." J. Comb. Theory (A), 45(1987), 62-103. Schilling, A., Warnaar, S. O. "Inhomogeneous lattice paths, generalized Kostka polynomials and An-i supernomials." Commun. Math. Phys. 2 0 2 (1999) 359-401. M. Shimozono, "A cyclage poset structure for Littlewood-Richardson tableaux," European J. Combin. 22 (2001) 365-393. Shimozono, M., Weyman, J., "Graded characters of modules supported in the clo sure of a nilpotent conjugacy class." European J. Combin. 21 no. 2(2000) 257-288. Shimozono, M., Zabrocki, M., "Hall-Littlewood Vertex Operators and Generalized Kostka Polynomials." Adv. Math. 158(2001), 66-85. Stembridge, J. R., "Shifted Tableaux and the Projective Representations of Sym metric Groups." Advances in Math. 74(1989), 87-134. Stembridge, J. R., "On Schur's Q-functions and the Primitive Idempotents of a Commutative Hecke Algebra." J. Alg. Combinatorics 1(1992), 71-95. Worley, D. R., "A Theory of Shifted Young Tableaux." Thesis, MIT (1984). Zabrocki, M., "^-Analogs of symmetric function operators," Discrete Math. 256 (2002), 831-853. SCHOOL OF MATHEMATICS, UNIVERSITY OF MINNESOTA, MINNEAPOLIS, MINNESOTA,

55455 E-mail address: tudoseSmath.unui.edu DEPARTMENT OF MATHEMATICS AND STATISTICS, YORK UNIVERSITY, T O R O N T O , O N

TARIO, M 3 J 1 P 3

E-mail address: zabrockilBmathstat.yorku.ca

A L G E B R A I C COMBINATORICS AND QUANTUM GROUPS Algebraic combinatorics

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