Linear and Projective Representations of Symmetric Groups (Cambridge Tracts in Mathematics)

CAMBRIDGE TRACTS IN MATHEMATICS General Editors b. bollobas, w. fulton, a. katok, f. kirwan, p. sarnak, b. simon

163 Linear and Projective Representations of Symmetric Groups

Cambridge Tracts in Mathematics All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://publishing.cambridge.org/stm/mathematics/ctm/ 142. Harmonic Maps between Rienmannian Polyhedra. By J. Eells and B. Fuglede 143. Analysis on Fractals. By J. Kigami 144. Torsors and Rational Points. By A. Skorobogatov 145. Isoperimetric Inequalities. By I. Chavel 146. Restricted Orbit Equivalence for Actions of Discrete Amenable Groups. By J. W. Kammeyer and D. J. Rudolph 147. Floer Homology Groups in Yang–Mills Theory. By S. K. Donaldson 148. Graph Directed Markov Systems. By D. Mauldin and M. Urbanski 149. Cohomology of Vector Bundles and Syzygies. By J. Weyman 150. Harmonic Maps, Conservation Laws and Moving Frames. By F. Hélein 151. Frobenius Manifolds and Moduli Spaces for Singularities. By C. Hertling 152. Permutation Group Algorithms. By A. Seress 153. Abelian Varieties, Theta Functions and the Fourier Transform. By Alexander Polishchuk 156. Harmonic Mappings in the Plane. By Peter Duren 157. Affine Hecke Algebras and Orthogonal Polynomials. By I. G. MacDonald 158. Quasi-Frobenius Rings. By W. K. Nicholson and M. F. Yousif 159. The Geometry of Total Curvature on Complete Open Surfaces. By Katsuhiro Shiohama, Takashi Shioya and Minoru Tanaka 160. Approximation by Algebraic Numbers. By Yann Bugeaud 161. Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings. By R. R. Colby and K. R. Fuller

Linear and Projective Representations of Symmetric Groups ALEXANDER KLESHCHEV University of Oregon

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Contents

Preface

page ix

PART I: LINEAR REPRESENTATIONS

1

1

Notation and generalities

3

2 2.1 2.2 2.3

Symmetric groups I Gelfand–Zetlin bases Description of weights Formulas of Young and Murnaghan–Nakayama

7 7 12 17

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

Degenerate affine Hecke algebra The algebras Basis Theorem The center of
n Parabolic subalgebras Mackey Theorem Some (anti) automorphisms Duality Intertwining elements

24 25 26 27 28 29 31 31 34

4 4.1 4.2 4.3 4.4

First results on n -modules Formal characters Central characters Kato’s Theorem Covering modules

35 36 37 38 40

5 5.1 5.2

Crystal operators Multiplicity-free socles Operators e˜a and f˜a

43 44 47 v

vi

Contents

5.3 5.4 5.5

Independence of irreducible characters Labels for irreducibles Alternative descriptions of
a

49 51 51

6 6.1 6.2 6.3

Character calculations Some irreducible induced modules Calculations for small rank Higher crystal operators

54 54 57 60

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Integral representations and cyclotomic Hecke algebras Integral representations Some Lie theoretic notation Degenerate cyclotomic Hecke algebras The ∗-operation Basis Theorem for cyclotomic Hecke algebras Cyclotomic Mackey Theorem Duality for cyclotomic algebras Presentation for degenerate cyclotomic Hecke algebras

64 65 66 68 69 70 73 74 80

8 8.1 8.2 8.3 8.4 8.5 8.6

Functors ei
and fi
New notation for blocks Definitions Divided powers Functions i Alternative descriptions of i More on endomorphism algebras

82 83 83 87 90 92 99

9 9.1 9.2 9.3 9.4 9.5 9.6

Construction of U+ and irreducible modules Grothendieck groups Hopf algebra structure Contravariant form Chevalley relations Identification of K∗ , K∗ , and K Blocks

103 104 106 109 112 115 117

10 10.1 10.2 10.3

Identification of the crystal Final properties of B Crystals Identification of B and B

120 120 123 126

11 Symmetric groups II 11.1 Description of the crystal graph 11.2 Main results on Sn

131 131 136

Contents PART II: PROJECTIVE REPRESENTATIONS

vii 149

12 Generalities on superalgebra 12.1 Superalgebras and supermodules 12.2 Schur’s Lemma and Wedderburn’s Theorem

151 151 157

13 Sergeev superalgebras 13.1 Twisted group algebras 13.2 Sergeev superalgebras

165 166 168

14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8

Affine Sergeev superalgebras The superalgebras Basis Theorem for n The center of n Parabolic subalgebras of n Mackey Theorem for n Some (anti) automorphisms of n Duality for n -supermodules Intertwining elements for n

174 174 175 176 177 177 178 179 179

15

15.5 15.6

Integral representations and cyclotomic Sergeev algebras Integral representations of n Some Lie theoretic notation Cyclotomic Sergeev superalgebras Basis Theorem for cyclotomic Sergeev superalgebras Cyclotomic Mackey Theorem Duality for cyclotomic superalgebras

185 187 188

16 16.1 16.2 16.3 16.4

First results on n -modules Formal characters of n -modules Central characters and blocks Kato’s Theorem for n Covering modules for n

191 191 193 194 197

17 17.1 17.2 17.3 17.4

Crystal operators for n Multiplicity-free socles Operators e˜i and f˜i

200 200 203 204 205

15.1 15.2 15.3 15.4

Independence of irreducible characters Labels for irreducibles

181 181 183 184

Contents

viii 18 18.1 18.2 18.3

Character calculations for n Some irreducible induced supermodules Calculations for small rank Higher crystal operators

206 206 208 216

19 19.1 19.2 19.3 19.4 19.5 19.6 19.7

Operators ei
and fi
i-induction and i-restriction Operators ei and fi Divided powers Alternative descriptions of
i The ∗-operation Functions i Alternative descriptions of i

219 219 221 225 228 229 229 230

20 20.1 20.2 20.3 20.4 20.5 20.6

Construction of U+ and irreducible modules Grothendieck groups revisited Hopf algebra structure Shapovalov form Chevalley relations Identification of K∗ , K∗  and K Blocks of cyclotomic Sergeev superalgebras

238 238 239 241 244 246 247

21

Identification of the crystal

248

22 22.1 22.2 22.3

Double covers Description of the crystal graph Representations of Sergeev superalgebras Spin representations of Sn

250 250 255 259

References

270

Index

275

Preface

The subject of this book is representation theory of symmetric groups. We explain a new approach to this theory based on the recent work of Lascoux, Leclerc, Thibon, Ariki, Grojnowski, Brundan, Kleshchev, and others. We are mainly interested in modular representation theory, although everything works in arbitrary characteristic, and in case of characteristic 0 our approach is somewhat similar to the theory of Okounkov and Vershik [OV], described in Chapter 2 of this book. The methods developed here are quite general and they apply to a number of related objects: finite and affine Iwahori–Hecke algebras of type A, cyclotomic Hecke algebras, spin-symmetric groups, Sergeev algebras, Hecke–Clifford superalgebras, affine and cyclotomic Hecke–Clifford superalgebras,    . We concentrate on symmetric and spin-symmetric groups though. We now outline some of the ideas which lead to the new approach. Let us concentrate on the modular case, as this is where things get really interesting. So let F be a field of characteristic p > 0, and Sn be the symmetric group. Irreducible FSn -modules were classified by James. His approach is as follows (see [J] for details). Let S  be the Specht module corresponding to a partition  of n (the Specht construction works over any field and even over ). This module has a canonical Sn -invariant bilinear form. The form is non-zero if and only if  is p-regular, that is no non-zero part of  is repeated p or more times. In this case the radical of the form, call it Q , is a maximal proper submodule. Thus D = S  /Q is an irreducible FSn -module. Finally, D   is a p-regular partition of n is a complete set of irreducible FSn -modules. The main unsolved problem in modular representation theory of symmetric groups is to find (Brauer) characters and in particular dimensions of irreducible modules. These will be known if any of the following two equivalent problems

ix

Preface

x

can be solved (see [K6 ] for the proof of equivalence). Denote by n (resp. p n) the set of all partitions (resp. p-regular partitions) of n. Decomposition numbers problem. Find the composition multiplicities S   D

(0.1)

for any  ∈ n and ∈ p n. Branching problem. Determine the composition multiplicities of the restriction S

resSnn−1 D  D

(0.2)

for any  ∈ p n and ∈ p n − 1. Motivated by the second problem, the author [K1 ] obtained certain partial branching rules, including an explicit description of the socle of the restriction S resSnn−1 D , which is equivalent to the description of the spaces
 S HomFSn−1 D  resSnn−1 D   ∈ p n ∈ p n − 1  (0.3) Another result from [K1 ] describes the spaces
 S HomFSn−1 S  resSnn−1 D   ∈ p n ∈ p n − 1 

(0.4)

It turns out that the spaces in (0.3) and (0.4) are at most 1-dimensional, which gives two different generalizations of the multiplicity freeness of the branching rule in characteristic 0. The solution is in terms of delicate combinatorial notions of a normal node and a good node of a Young diagram (see Chapter 11): the space (0.3) (resp. (0.4)) is non-trivial if and only if is obtained from  by removing a good node (resp. normal node). As observed in [K2 ], it follows from this description that all irreducible modules appearing in the S socle of resSnn−1 D belong to different blocks, the fact sometimes referred to as the strong multiplicity freeness of the branching rule. A number of further results on the modular branching problem was established in [K5 , K4 , BK1 ]. For example, in [K4 ] we have described the multiplicity (0.2) when is obtained from  by removing a node. It turns out that this multiplicity is non-zero if and only if the node is normal, but it can be arbitrarily large. Many applications of the branching rules were obtained, see for example [K2 , FK, BO1 , BK4 ]. But the most important consequence was that they led to a discovery of deep connections between modular representation theory and the theory of crystal bases. The link was first made by Lascoux, Leclerc, and Thibon [LLT]. The nodes of Young diagrams come with residues, which are elements of /p, and we obtain a structure of a directed colored graph

Preface

xi

on the set of all p-regular partitions: by definition, there is an arrow from to  of color i ∈ /p if and only if is obtained from  by removing a good node of residue i. Lascoux, Leclerc, and Thibon made the startling combinatorial observation that this “branching graph” coincides with the crystal graph of the basic representation of the affine Kac–Moody algebra  = 1 Ap−1 , determined explicitly by Misra and Miwa [MiMi]. It turned out later that the same observation applies to the branching graph for the associated complex Iwahori–Hecke algebras at a primitive  + 1th root of unity and the crystal 1 graph of the basic representation of the affine Kac–Moody algebra  = A , see [B]. In this latter case, Lascoux, Leclerc, and Thibon conjectured moreover that the coefficients of the canonical basis of the basic representation coincide with the decomposition numbers (0.1) for the Iwahori–Hecke algebras. This conjecture was proved by Ariki [A1 ] (see also Grojnowski [G1 ]). More generally, Ariki established a similar result connecting the canonical basis of an arbitrary integrable highest weight module of  to the representation theory of the corresponding cyclotomic Hecke algebra, as defined in [Ch, AK, BM]. Note that Ariki’s work is concerned with the cyclotomic Hecke algebras over the ground field , but Ariki and Mathas [A2 , AM] were later able to extend the classification of the irreducible modules, but not the result on decomposition numbers, to arbitrary fields. For further developments related to the LLT conjecture, see [LT1 , VV, Sch, A3 ]. Subsequently, Grojnowski and Vazirani [G2 , G3 , GV, V2 ] have developed a powerful new approach to (among other things) the classification of the irreducible modules of the cyclotomic Hecke algebras. The approach is valid over an arbitrary ground field, and is entirely independent of the “Specht module theory” that plays an important role in Ariki’s work. Branching rules are built in from the outset, resulting in an explanation and generalization of the link between modular branching rules and crystal graphs. The methods are purely algebraic, exploiting affine Hecke algebras in the spirit of [BZ, Z1 ] and others. On the other hand, results on decomposition numbers do not follow, since that ultimately depends on the geometric work of Kazhdan, Lusztig, and Ginzburg. In this book, we explain Grojnowski’s approach to the theory in the case of degenerate affine Hecke algebra. In particular, we obtain an algebraic construction purely in terms of the representation theory of degenerate affine 1 Hecke algebras of the positive part U+ of the enveloping algebra of  = Ap−1 , as well as of Kashiwara’s highest weight crystals B and B for each dominant weight . These emerge as the modular branching graphs of the appropriate algebras. As a consequence, a parametrization of irreducible FSn modules, classification of blocks (“Nakayama’s conjecture”), and some of

xii

Preface

the modular branching rules mentioned above follow from the special case  = 0 of the main results. Part II of the book deals with representation theory of Schur’s double covers of symmetric groups. This is equivalent to studying projective (or spin) representations of symmetric groups. The spin theory in characteristic p was developed in [BK3 ] (cf. [BMO, ABO1 , ABO2 , MoY2 ])–it is parallel to the theory of linear representations described above, but with the role of 1 the Kac–Moody algebra  = Ap−1 played by the twisted Kac–Moody algebra 2  = Ap−1 . We note that the modular irreducible spin representations of Sn were first classified in [BK2 ], following a conjecture in [LT2 ], using a more classical approach via “Specht modules”. However, that approach did not allow us to obtain any branching rules. We hope that having both linear and spin theory under one cover will be useful for the reader. The two theories are actually very similar, if developed from the new point of view adopted in this book. In fact, a glance at the contents shows that many sections of Part II are exactly parallel to the corresponding sections of Part I. Let us now describe the contents of the book in more detail. We note that each chapter has its own introduction where the results of the chapter are motivated and described, sometimes informally. Chapter 1 contains notation and some basic preliminary results. Chapter 2 is a presentation of the beautiful theory of Okounkov and Vershik in characteristic 0. It is not directly related to the rest of the book. However, it is a nice introduction to some ideas employed further on, and might be a good place to start for less-advanced readers. Also, it is perhaps the shortest way to some key results of the classical representation theory of symmetric groups in characteristic 0, such as classification of irreducible representations, Young’s formulas, and the Murnaghan–Nakayama formula. Degenerate affine Hecke algebras
n are introduced in Chapter 3. Basis Theorem for
n is proved and the center of
n is described. We introduce parabolic subalgebras
⊂
n and the corresponding induction and restriction functors indn and resn . “Mackey Theorem” is a result describing a filtration of resn indn by certain induced modules. An important result relating induction and duality is proved in Section 3.7. In Chapter 4, we introduce the formal characters of finite dimensional
n -modules, discuss central characters and blocks, and then study in detail a remarkable irreducible
n -module, called Kato module, as well as its “covering modules”. The functors ea and their versions a and am , which are affine analogues of Robinson’s a-restriction functors for symmetric groups, are studied in

Preface

xiii

Chapter 5. The (affine version of the) strong multiplicity freeness of the branching rule is established in Corollary 5.1.7. This allows us to define the crystal operators e˜ a and f˜a in Section 5.2 as the socle and the head of a certain direct summand of restriction and induction, respectively. We then prove that the formal characters of irreducible modules are linearly independent and use the crystal operators f˜a to label the irreducible
n -modules. Section 5.5 contains some further branching results. In Chapter 6 we establish a sufficient condition for the irreducibility of the module, induced from an irreducible module over a parabolic subalgebra (this condition is far from necessary–see for example [LNT] for results on the subtle problem of finding necessary and sufficient condition). Then we calculate characters of some irreducible
n -modules for n ≤ 4. These calculations provide the reader with concrete examples to play with, but also are important for the theory, as they will imply that the operators ei on the Grothendieck group satisfy Serre relations. Finally, some higher crystal operators are introduced in Section 6.3–these play only a technical role. Integral representations and (degenerate) cyclotomic Hecke algebras are treated in Chapter 7. We explain why it is enough to study integral representations and reveal their relations with cyclotomic Hecke algebras
n . We introduce Lie theoretic notation related to the Kac–Moody algebra  of type 1 Ap−1 , which will be used until the end of Part I. The main results of the chapter are Basis Theorem for cyclotomic Hecke algebras, Cyclotomic Mackey Theorem, and the fact that -duality commutes with induction for cyclotomic Hecke algebras. In Chapter 8 we study the cyclotomic analogues ei of the functors ei and introduce the “dual” functors fi and related notions. We also introduce the divided powers functors which generalize ei and fi . The main goal of the Chapter is to get the fi -analogues of the results on ei obtained in Chapter 5, that is to get the results on induction similar to our previous results on restriction. This turns out to be quite a bit harder. Key Chapter 9 begins by defining a Hopf algebra structure on the Grothendieck group K of integral
n -modules for all n ≥ 0, operations coming from induction and restriction. We prove in Theorem 9.5.3 that the dual algebra K∗ can be identified with the universal enveloping algebra of the positive part of the Kac–Moody algebra . We also obtain a natural action of K∗ on the Grothendieck group K of finite dimensional
n -modules for all n ≥ 0–under this action the Chevalley generators ei act as operators ei , which come from representation theory of
n . This action is extended with the operators fi and hi to give the action of the full Kac–Moody algebra , and the module K is identified with the irreducible high weight module

xiv

Preface

V. Finally, the weight spaces of V are interpreted in terms of blocks of
n -modules, see Section 9.6. In Chapter 10 we identify the crystal graph of V as the socle branching graph of the cyclotomic Hecke algebras
n . The consequences for symmetric groups are deduced in Chapter 11, where we specialize to the case  = 0 and use the explicit description of the corresponding crystal graph to label the irreducible modules by p-regular partitions, to describe the blocks, and to deduce some of the branching rules. We also get useful results on formal characters. Part II starts with Chapter 12, which is a review on superalgebras and their representations. This will be needed for the spin theory, as it turns out extremely convenient to consider the twisted group algebra n of Sn as a superalgebra and then work in the category of n -supermodules. In fact, it is even more convenient to work with the Sergeev superalgebra n , which is defined in Chapter 13. We also prove in this Chapter that n is “almost Morita equivalent” to n . Chapters 14–22 are all parallel to the corresponding chapters of Part I, as indicated in the beginning of each of them, so we do not review them here in detail. I have benefited greatly from collaboration with Jon Brundan. I was greatly influenced by the work of Jantzen and Seitz; Lascoux, Leclerc, and Thibon; Ariki; and, especially, Grojnowski. The idea to write this book appeared when I lectured on the topic at the University of Wisconsin-Madison during my sabbatical leave from Oregon. I am very grateful to Georgia Benkart, Ken Ono, and especially Arun Ram for their hospitality. I should also thank the editors–without their repeated (patient) emails I would never have finished. Most of all, I am indebted to my family (without whom this book might have been written faster, but that’s not the point...).

PART I Linear representations

1 Notation and generalities

Throughout the book: + is the set of non-negative integers and F is an algebraically closed field of characteristic p ≥ 0. Throughout Part I I =  · 1 ⊂ F

(1.1)

If p = char F > 0 then I is identified with 0 1     p − 1 , and if p = 0 then I = . If  is an associative F -algebra we denote by -mod the category of all finite dimensional left -modules and by -proj ⊂ -mod the full subcategory of all projective -modules. We write K-mod K-proj for the corresponding Grothendieck groups. The embedding -proj ⊂ -mod induces the natural Cartan map   K-proj → K-mod Note that in general  does not have to be injective. Let M ∈ -mod. The socle of M, written soc M, is the largest completely reducible submodule of M, and the head of M, written hd M, is the largest completely reducible quotient module of M. If V is an irreducible -module, we write M  V for the multiplicity of V as a composition factor of M. For algebras  , an -module M, and a -module N , we write M
N for the outer tensor product, that is the tensor product of vector spaces M ⊗ N considered as an  ⊗ -module in the usual way.  If is a subalgebra of , and M is a -module we write ind M or ind  for the induced module  ⊗ M. We may consider ind as a functor from the category of -modules to the category of -modules. This functor is left adjoint to the restriction functor res (or res ) going in the other direction. If  is free as a right -module the induction functor is exact. We denote by  the center of . By a central character of A we mean a (unital) algebra homomorphism    → F . For a central character  the 3

4

Notation and generalities

corresponding block is the full subcategory A-mod of -mod consisting of all modules M ∈ -mod such that z − zk M = 0 for k 0. We have a decomposition  -mod -mod = 

as  runs over all central characters. If  is finite dimensional, then two non-isomorphic irreducible -modules L and M belong to the same block if and only if there exists a chain L  L0  L1      Lm  M of irreducible -modules with either Ext1 Li  Li+1  = 0 or Ext 1 Li+1  Li  = 0 for each i. Let be a subalgebra of an F -algebra  and  be the centralizer of in . If V is an -module and W is a -module then Hom W res V  is naturally a -module with respect to the action cf w = cfw for w ∈ W  f ∈ Hom W res V  c ∈ . Lemma 1.0.1 Let ⊆  be semisimple finite dimensional F -algebras. If V is irreducible over  and W is irreducible over then Hom W res V  is irreducible over . Proof By Wedderburn–Artin, we may assume that  = EndV . Decompose res V = W ⊕k ⊕X, where W is not a composition factor of X. Then the algebra End W ⊕k , naturally contained in , acts on the space Hom W res V  as the full endomorphism algebra. For any n ≥ 0, let  = 1  2      be a partition of n, that is a nonincreasing sequence of non-negative integers summing to n. If p > 0, the partition  is called p-regular if for any k > 0 we have  j  j = k < p By definition, every partition is 0-regular. Let n (resp. p n) denote the set of all (resp. all p-regular) partitions of n. Thus n = 0 n. Set    = n and p = p n n≥0

n≥0

We identify a partition  with its Young diagram  = r s ∈ >0 × >0  s ≤ r  Elements r s ∈ >0 × >0 are called nodes. We label the nodes of  with residues, which are elements of I. By definition, the residue of the node r s

Notation and generalities

5

is s − r mod p if p is positive, and simply s − r if p = 0. The residue of the node A is denoted res A. Define the residue content of  to be the tuple cont = i i∈I 

(1.2)

where for each i ∈ I, i is the number of nodes of residue i contained in the diagram . Let i ∈ I be some fixed residue. A node A = r s ∈  is called i-removable (resp. i-addable) for  if res A = i and A =  \ A (resp. A =  ∪ A ) is a Young diagram of a partition. A node is called removable (resp. addable) if it is i-removable (resp. i-addable) for some i. Thus, for example, a removable node is always of the form m m  with m > m+1 . Throughout the book, Sn denotes the symmetric group on n letters. The permutations act on numbers 1     n on the left so that for the product we have, for example, 1 22 3 = 1 2 3. Sn also acts on n-tuples of objects by place permutations on the right: a1  a2      an  · w = aw1      awn  or on the left: w · a1  a2      an  = aw−1 1      aw−1 n  The length function on Sn in the sense of Coxeter groups is denoted by . The number w can be characterized as the number of inversions in the permutation w. Finally we recall one classical result. Let n = F x1      xn be the polynomial algebra in n indeterminates,

n = F x1      xn Sn be the ring of symmetric polynomials, and + n ⊂ n be the symmetric polynomials without free term. The following fact is well known over . That it holds over , and hence over F , is proved in [St]. Theorem 1.0.2 n is a free module of rank n! over n . Moreover we can take the set a

B = x1 1    xnan  0 ≤ ai < i for all 1 ≤ i ≤ n

Notation and generalities

6

as a basis. In particular, the cosets of elements of B form a basis of the algebra n /n + n. A slightly more general result easily follows: Corollary 1.0.3 Let r ≤ n. Then nSr is a free module of rank n!/r! over n . Moreover we can take the set a

r+1    xnan  0 ≤ ai < i for all r + 1 ≤ i ≤ n

B = xr+1

as a basis. Proof It suffices to prove that elements of B generate nSr as a module over

n . Let f ∈ nSr . In view of Theorem 1.0.2, we can write  a f = fa x1 1    xnan  (1.3) where the summation is over all n-tuples a = a1      an  with 0 ≤ ai < i, and fa ∈ n . Using Theorem 1.0.2 with n = r we can also see that n is a free a nSr -module on basis x1 1    xrar  0 ≤ ai < i for all 1 ≤ i ≤ r . Now note that ar+1 the polynomials fa xr+1    xnan are in nSr , so, since f is also in nSr , it follows that fa = 0 in (1.3) unless a1 = · · · = ar = 0. This completes the proof.

2 Symmetric groups I

In order to illustrate the theory we are trying to develop let us start from an “easy” special case, namely the case of complex representations of the symmetric group Sn . We explain the beautiful elementary approach of Okounkov and Vershik [OV] (see also [DG]). The idea of this approach is not new: to study all symmetric groups at once. However, it is rather amazing that in this way the whole theory can be built quickly from scratch using only the classical Maschke and Wedderburn–Artin Theorems. We will obtain the following well-known results: labeling the irreducible Sn -modules by partitions of n, construction of Young’s orthogonal bases in irreducible modules, explicit description of matrices of simple transpositions with respect to these bases, and the Murnaghan–Nakayama formula for irreducible characters.

2.1 Gelfand–Zetlin bases Define the kth Jucys–Murphy elements (JM-element for short) Lk ∈ FSn as follows:  m k (2.1) Lk = 1≤m
These elements were introduced in [Ju], [Mu1 ]. Note that L1 = 0 and Lk commutes with Sk−1 . As Lk ∈ FSk , it follows that the JM-elements commute. Here and below, if m < n, the default embedding of Sm into Sn is with respect to the first m letters. A copy of Sm embedded with respect to the last m letters is denoted by Sm . Denote by n the center of the group algebra FSn . Also let

nm = FSn+m Sn 7

8

Symmetric groups I

be the centralizer of FSn in FSn+m . It is clear that nm is spanned by the class sums corresponding to the Sn -conjugacy classes in Sn+m . These conjugacy classes can be thought of as cycle shapes with “fixed positions” for n + 1 n + 2     n + m – we call them marked cycle shapes. For example, the symbol ∗ ∗ ∗ ∗ ∗∗ ∗∗∗12 ∗ 13 14 ∗15

(2.2)

corresponds to the S11 -conjugacy class in S15 , which consists of all permutations whose cycle presentation is obtained by inserting the numbers 1 through 11 instead of asterisks. Proposition 2.1.1 [O1 ] The algebra nm is generated by Sm , n , and Ln+1      Ln+m . Proof It is clear that Sm , n , and Ln+1      Ln+m are contained in nm , so they generate a subalgebra  ⊆ nm . Conversely, let us filter nm so that the ith filtered component inm is the span of the class sums which correspond to the marked cycle shapes moving at most i elements. For example the class 11 sum corresponding to (2.2) belongs to 12 114 , but not 114 . We prove by i induction on i = 0 1    that nm ⊆ . For i = 0 and 1, we have inm = F · 1 ⊆ . We explain the inductive step on example. Let z ∈ 12 114 be the class sum corresponding to the marked cycle shape from (2.2). Let c ∈ 11 denote the sum of all elements of S11 whose cycle shape is ∗ ∗ ∗ ∗ ∗∗ ∗∗∗ Also, let x = 12 13L12 13 14L14 − 12 14 − 13 14 ∈  (Note that L12 is the class sum corresponding to ∗ 12, and L14 − 12 14 − 13 14 is the class sum corresponding to ∗ 14.) Then xc is equal to z modulo lower layers of our filtration. From now on until the end of Chapter 2 we assume that F = . The following key multiplicity-freeness result is well known – it is a special case of the branching rule, which describes the restriction of an irreducible Sn -module to Sn−1 . However, usually the branching rule is proved after some theory has been developed and irreducible modules have been studied. In the approach explained here the multiplicity-freeness result is proved from scratch and then used to develop a theory.

2.1 Gelfand–Zetlin bases

9

Theorem 2.1.2 Let V be an irreducible Sn -module. Then the restriction resSn−1 V is multiplicity free. Proof It follows from Proposition 2.1.1 that the centralizer of Sn−1 in Sn is commutative. So the theorem comes from Lemmas 1.0.1. We now define the branching graph  whose vertices are isomorphism classes of irreducible Sn -modules for all n ≥ 0 (by agreement S0 = ); we have a directed edge W → V from (an isoclass of) an irreducible Sn module W to (an isoclass of) an irreducible Sn+1 -module V if and only if W appears as a composition factor of resSn V ; there are no other edges. Our main goal is to find an explicit combinatorial description of the branching graph. This will give us a labeling of the irreducible Sn -modules for all n. This will also yield the branching rule. To achieve this goal we will actually do more. Let V be an irreducible Sn -module. Pick an Sn -invariant inner product · · on V (it is unique up to a scalar). Theorem 2.1.2 implies that the decomposition  resSn−1 V = W W →V

is canonical. Decomposing each W on restriction to Sn−2 , and continuing inductively all the way to S0 , we get a canonical decomposition  VT resS0 V = T

into irreducible S0 -modules, that is 1-dimensional subspaces VT , where T runs over all paths W0 → W1 → · · · → Wn = V in . Note that Sk · VT = Wk

0 ≤ k ≤ n

(2.3)

Choosing a vector vT ∈ VT , we get a basis vT of V called Gelfand–Zetlin basis (or GZ-basis). Vectors of GZ-basis are defined uniquely up to scalars. Moreover, if   V → V  is an isomorphism of irreducible modules then  moves a GZ-basis of V to a GZ-basis of V  . Note also, for example using (2.3), that a GZ-basis is orthogonal with respect to · ·. Now decompose the algebra Sn according to the Wedderburn–Artin Theorem  Sn = End V  (2.4) V

Symmetric groups I

10

where the sum is over the representatives of the isoclasses of irreducible Sn -modules. This decomposition is canonical. Let us pick a GZ-basis in each V . Then we also identify  Mdim V  (2.5) Sn = V

Define the GZ-subalgebra n ⊆ Sn as the subalgebra which consists of all elements of Sn , which are diagonal with respect to a GZ-basis in every irreducible Sn -module. In terms of the decomposition (2.5), n consists of all diagonal matrices. In particular: Lemma 2.1.3 n is a maximal commutative subalgebra of Sn . Also, n is a semisimple algebra. We give two more descriptions of the GZ-subalgebra. Lemma 2.1.4 (i) n is generated by the subalgebras 0  1      n ⊆ Sn . (ii) n is generated by the JM-elements L1  L2      Ln . Proof (i) Let eV ∈ n be the central idempotent of Sn , which acts as identity on V and as zero on any irreducible Sn -module V   V (in terms of (2.4); eV is the identity endomorphism in the V -component and zero endomorphism in other components). If T = W0 → W1 → · · · → Wn = V is a path in  then eW0 eW1    eWn ∈ 0 1    n acts as the projection to VT along ⊕S =T VS and as zero on any irreducible Sn -module V   V . So the subalgebra generated by 1  2      n contains n . As this subalgebra is commutative and n is a maximal commutative subalgebra of Sn , the two must coincide. (ii) Note that Lk is the sum of all transpositions in Sk minus the sum of all transpositions in Sk−1 , that is Lk is a difference of a central element in Sk and a central element in Sk−1 . So by (i), the JM-elements do belong to n . To prove that they generate n , proceed by induction on n, the inductive base being trivial. By (i), n is generated by n−1 and n . In view of the inductive assumption, it suffices to prove that n−1 and Ln generate n . But this follows from the obvious embedding n ⊆ n−11 and Proposition 2.1.1, as n−1 ⊆ n−1 . Now, we will try to have the GZ-subalgebra play a role of a Cartan subalgebra in Lie Theory. As n is semisimple we can decompose every

2.1 Gelfand–Zetlin bases

11

irreducible Sn -module V as a direct sum of simultaneous eigenspaces for the elements L1      Ln . If  = 1      n  ∈ n and V is the simultaneous eigenspace for the L1      Ln corresponding to the eigenvalues 1      n , respectively, then we say that  is a weight of V and V is the -weight space of V . It follows from definitions that vectors of a GZ-basis are weight vectors. Also, since in terms of (2.5), n consists of all diagonal matrices, each weight space is 1-dimensional. Thus the weight spaces are precisely the spans of the elements of a GZ-basis. It also follows that if  is a weight of an irreducible module V , then it is not a weight of irreducible V   V . Thus, via GZ-bases, we get a one-to-one correspondence between all possible weights (for all symmetric groups) and all paths in . The weight corresponding to a path T will be denoted T and a path corresponding to a weight  will be denoted T . We will also write v for vT . A path T ends at a vertex V if and only if the corresponding weight T is a weight of V . It is clear now that in order to understand  it suffices to describe the sets Wn =  ∈ n   is a weight of a Sn -module

n ≥ 0

(2.6)

and the equivalence relation  ≈ ⇔  and are weights of the same irreducible Sn -module (2.7) on Wn. Indeed, note that =



Wn/ ≈

n≥0

and for equivalence classes  and  of  ∈ Wn − 1 and ∈ Wn we have  →  if and only if  = 1      n−1  for some  ≈ . Remark 2.1.5 For the reader who is spoiled by knowing what the final answer should be: yes, the elements of the set Wn/ ≈ will be labeled by the partitions  of n, and the elements of the set Wn will be labeled by the standard -tableaux for all such , with two tableaux being equivalent if and only if they have the same shape . To be more precise, if t is a standard -tableaux, then the corresponding weight  is obtained as follows: i is the residue of the box in  which is occupied by i in the -tableaux t (1 ≤ i ≤ n). The following notation will be convenient: if  = 1      n  ∈ Wn, we write V for an irreducible Sn -module, which has  as its weight.

12

Symmetric groups I

The weight  determines V uniquely up to isomorphism, but V  V  if and only if  ≈ Now, (2.3) can now be restated as follows: Sk · v = V1      k 

0 ≤ k ≤ n

(2.8)

2.2 Description of weights We write si = i i+1 ∈ Sn , 1 ≤ i < n, for basic transpositions. Note important relations si Li = Li+1 si − 1

si Lj = Lj si

j = i i + 1

(2.9)

The second relation immediately implies: Lemma 2.2.1 Let  = 1      n  ∈ Wn, and 1 ≤ k < n. Then sk v is a linear combination of vectors v such that i = i for i = k k + 1. If the role of Cartan subalgebra is played by n , then the role of sl2 subalgebras will be played by i = Li  Li+1  si . In view of (2.9), every i is a quotient of the rank one degenerate affine Hecke algebra (of which we will see much more later in this book):
2 = s x y  xy = yx s2 = 1 sx = ys − 1 We now construct some
2 -modules. Fix a pair of numbers a b ∈ . If b = a + 1, let La b = v with the action of the generators xv = av yv = bv sv = v Clearly, the relations are satisfied, so we have a well-defined action of
2 . Similarly, if b = a − 1, we have La b = v with xv = av yv = bv sv = −v Finally, assume that a = b ± 1. Let La b be 2-dimensional with the action of the generators x y s, given, respectively, by the matrices       0 1 b 1 a −1  (2.10)   1 0 0 a 0 b Note that, if a = b, then x and y do not act on La b semisimply, while, if a = b b ± 1, then we can simultaneously diagonalize x and y, so that the matrices of x y s are       b − a−1 1 − b − a−2 b 0 a 0    (2.11) 0 a 0 b 1 a − b−1

2.2 Description of weights

13

To achieve this, change basis from v1  v2 to v1  v2 − b − a−1 v1 . If instead we change to v1  1 − b − a−2 −1/2 v2 − b − a−1 v1   the matrix of s becomes orthogonal:  b − a−1 1 − b − a−2



 1 − b − a−2  a − b−1

(2.12)

(2.13)

Proposition 2.2.2 (i) Every irreducible
2 -module is isomorphic to some La b. (ii) If a = b ± 1, then La b  Lb a, and there are no other isomorphic pairs among La b  a b ∈  . Proof (i) Let V be an irreducible
2 -module. There exists v ∈ V which is a simultaneous eigenvector for x and y. So xv = av, yv = bv for some a b ∈ . If sv is proportional to v, then V = v, and we must have sv = ±v, as s2 = 1. This immediately leads to b = a ± 1 and V = La b. If sv is not proportional to v, then v sv must be a basis of V , which leads to the formulas (2.10), but these formulas determine an irreducible module only if a = b ± 1. (ii) That no other pairs are isomorphic is clear, because if La b and Lc d are isomorphic, then their restrictions to x y are isomorphic. Finally, if a = b b ± 1, it is easy to write down an explicit isomorphism between La b and Lb a using the formulas (2.11). Corollary 2.2.3 Let  ∈ Wn, V = V, 1 ≤ i < n, and = si  = 1      i−1  i+1  i  i+2      n  Then: (i) i = i+1 . (ii) If i+1 = i ± 1 then si v = ±v and is not a weight of V . (iii) Let i+1 = i ± 1. Then is a weight of V . Moreover, the vector w = si − i+1 − i −1 v is a non-zero vector of weight , the elements si  Li  Li+1 leave X = spanv  w invariant, and act in the basis v  w of X with matrices (2.11), respectively. Proof By (2.8), Si+1 · v  V1      i+1 . Consider M = HomSi−1 V1      i−1  V1      i+1 

14

Symmetric groups I

as a module over i−12 =  i  i−1 , see Proposition 2.1.1. This module is irreducible by Lemma 1.0.1. By Schur’s Lemma, i−1 acts on M with scalars, so M is irreducible even as a i -module. Note that the i -module M is isomorphic to the Bi -submodule N = i · v ⊆ V . Inflating along the surjection
2 → i , N becomes an irreducible
2 -module, with i and i+1 appearing as eigenvalues of x and y, respectively. Hence N  Li  i+1 , see Proposition 2.2.2. Now the result follows from the classification of irreducible
2 -modules obtained above, noting for (i) that x and y do not act semisimply of La a, so this case is impossible. Corollary 2.2.4 Let  = 1      n  ∈ n . If i = i+2 = i+1 ± 1 for some i, then  ∈ Wn. Proof Otherwise, Corollary 2.2.3(ii) gives si v = ±v and si+1 v = ∓v , which contradicts the braid relation si si+1 si = si+1 si si+1 . Lemma 2.2.5 Let  ∈ Wn. Then: (i) 1 = 0. (ii) i − 1 i + 1 ∩ 1      i−1 = ∅ for all 1 < i ≤ n. (iii) If i = j = a for some i < j then a − 1 a + 1 ⊆ i+1      j−1  Proof (i) is clear as L1 = 0. If (ii) fails, apply Corollary 2.2.3(iii) repeatedly to swap i with i−1 , then with i−2 , etc., all the way to the second position. Now, if i = 0, we get a weight which starts with two 0s, which contradicts Corollary 2.2.3(i). Otherwise, again by Corollary 2.2.3(iii), we can move i to the first position, which contradicts (i). If (iii) fails, let us pick i j with the minimal j − i for which this happens. By Corollaries 2.2.3(i), (iii) and 2.2.4, we have  =     a a ± 1     a ± 1 a     By the choice of j − i, there must be both a ± 1 + 1 and a ± 1 − 1 between the two entries a ± 1 in . So there is an entry equal to a in between, which again contradicts the minimality of j − i. Recall partition notation introduced in Chapter 1. Define the Young graph  as a directed graph with the set  of all partitions as its set of vertices; moreover, for   ∈  we have  →  if and only if  = A for some

2.2 Description of weights

15

removable node A for . A path in  ending in  will be referred to as an -path. Thus an -path T can be thought of as a sequence of nodes T1      Tn of  such that Tn is removable for , Tn−1 is removable for Tn , etc. If, for all 1 ≤ i ≤ n, we substitute the number i for the box Ti , we get an -tableau, that is an array of integers 1     n of shape  and such that the numbers increase from top to bottom along the columns and from left to right along the rows. In this way we get a one-to-one correspondence between the -paths in  and the -tableaux. We will not distinguish between the two. Thus, if T is an -tableaux, Ti is the box occupied by i in T . To each -tableau T , we associate the n-tuple T = res T1      res Tn  ∈ n  Example 2.2.6 If  = 4 2 1, an example of an -tableau is given by 1 2 4 5 T= 3 7 6 In this case T = 0 1 −1 2 3 −2 0. Set W  n = T  T is an -tableau for some  ∈ n 

(2.14)

Note that the shape  of T can be recovered from the weight T : the amount of as among the i s is the amount of nodes on the ath diagonal of the Young diagram . So the n-tuples  ∈ W  n come from tableaux of the same shape if and only if  can be obtained from by a place permutation, in which case we write  ∼ . Lemma 2.2.7 The set W  n is precisely the set of all n-tuples  ∈ n which satisfy the properties (i)–(iii) of Lemma 2.2.5. In particular, Wn ⊆ W  n. Proof Easy combinatorial exercise. If  = 1      n  ∈ n , and i = i+1 ± 1, then a place permutation which swaps i and i+1 will be called an admissible transposition. If  = T for a tableau T , then an admissible transposition amounts to swapping i and i + 1 that do not lie on adjacent diagonals in T . It is clear that such a swap always transforms an -tableaux to an -tableaux. Let  = 1 ≥ 2 ≥ · · · ≥ k  ∈ n. We define the corresponding canonical -tableau T to be the -tableau obtained by filling in the numbers

16

Symmetric groups I

1 2     n from left to right along the rows, starting from the first row and going down. Lemma 2.2.8 Let  ∈ n. If T is an -tableau, then there is a series of admissible transpositions which moves T to T. Moreover, these transpositions si1  si2      si can be chosen in such a way that  = si1 si2    si . Proof Let A be the last node of the last row of . In T, A is occupied by n. In T , A is occupied by some number i. Note also that in T , i + 1 and i do not lie on adjacent diagonals. So we can apply an admissible transposition to swap i and i + 1, then to swap i + 1 and i + 2, etc. As a result, we get a new -tableau in which A is occupied by n. Next, forget about A, and take care of n − 1. Continuing this way we will get a chain of admissible transpositions which transform T to T. Finally, note that this chain yields a reduced word. Lemma 2.2.9 If  ∈ W  n and  ∼ for some ∈ Wn, then  ∈ Wn and  ≈ . Proof Let  = T . By Lemma 2.2.7, = S . As  ∼ , the tableaux S and T have the same shape. In view of Corollary 2.2.3(iii), it suffices to show that we can go from S to T by a chain of admissible transpositions. But this follows from Lemma 2.2.8. The following is the main result of this chapter. Theorem 2.2.10 We have Wn = W  n Moreover, T ≈ S ⇔ the tableaux T and S have the same shape ⇔ T ∼ S . In particular, the branching graph  is isomorphic to the Young graph  . Proof By Lemma 2.2.7, Wn ⊆ W  n. As the number of (isoclasses of) irreducible Sn -modules equals the number of conjugacy classes of Sn , which are labeled by partitions of n, we have Wn/ ≈  = n = W  n/ ∼ 

(2.15)

Now, let  ∈ W  n. In view of Lemma 2.2.9, the ∼-equivalence class of  either contains no elements of Wn or is a subset of ≈-equivalence class of Wn. In view of (2.15), this now implies Wn = W  n and ∼ is equivalent to ≈.

2.3 Formulas of Young and Murnaghan–Nakayama

17

Now to every irreducible Sn -module V we can associate a partition  ∈ n. Indeed, if  ∈ Wn is a weight of V then  = T for some tableaux T , and we associate to V the shape  of T . We will write V = V  . This notation is better than V for  ∈ Wn, because we have a one-to-one correspondence between the isoclasses of irreducible Sn -module and partitions of n. The weights of V  are precisely res T1      res Tn   T is an -tableau . Example 2.2.11 (i) If  = n, the only -tableau is 1 2 3 · · · n . So n V n is 1-dimensional, and its only weight is 0 1     n − 1. Similarly, V 1  is 1-dimensional with the only weight 0 −1     −n. It is clear from this n information that V n is the trivial and V 1  is the sign modules over Sn . (ii) Let  = n − 1 1. Then the -tableaux are Ti = 2 ≤ i ≤ n, and the corresponding weights are i = 0 1     i − 2 −1 i − 1     n − 2

1 2 i

···

n

for

2 ≤ i ≤ n

Note what we have done so far. We have started from a nested family of algebras S0 ⊂ S1 ⊂    , proved the multiplicity-freeness of the branching rule from scratch, defined the branching graph , and tried to learn enough facts about , so that we could identify it with some known graph. This have lead to a classification of irreducible Sn -modules for all n and a description of the branching rule at the same time. On the way we have obtained other useful results about irreducible modules. We are going to follow this scheme again and again in this book for various families of algebras, although to realize it we will need more sophisticated tools.

2.3 Formulas of Young and Murnaghan–Nakayama Formulas of Young describe explicitly the matrices of simple transpositions si with respect to a nice choice of a GZ-basis. The formulas come more or less from (2.11) and (2.13). But we need to scale elements of a GZ-basis in a consistent way.  correIn order to do this, fix  ∈ n. Pick a basis vector vT ∈ VT sponding to the canonical -tableau. Let T be an arbitrary -tableau. Write T = w · T for w ∈ Sn . Define T to be w. Denote by T the projection to VT along ⊕S =T VS , and set vT = T wvT 

(2.16)

18

Symmetric groups I

By Lemma 2.2.8, there is a reduced decomposition w = si1    si with all simple transpositions being admissible. So Corollary 2.2.3(iii) implies  wvT = vT + cS v S  (2.17) S S<T

and vT = 0. Theorem 2.3.1 (Young’s seminormal form) Let  ∈ n, vT be the GZ-basis of V  defined in (2.16), and 1 ≤ i < n. Then the action of the simple transposition si ∈ Sn is given as follows: (i) If res Ti+1 = res Ti ± 1, then si vT = ±vT . (ii) Let  = res Ti+1 − res Ti −1 = ±1 and set S = si T . Then

if S > T, vT + vS  si vT = 2 −vT + 1 −  vS  if S < T. Proof If res Ti+1 = res Ti ± 1, the result follows from Corollary 2.2.3(ii). Otherwise si is an admissible transposition for T . We may assume that S > T. As weight spaces of V  are 1-dimensional, Corollary 2.2.3(iii) implies that vS equals si vT − vT up to a scalar multiple, and, using (2.17), we see that the scalar is 1. Corollary 2.3.2 Irreducible representations of Sn are defined over  and are self-dual. Theorem 2.3.3 (Young’s orthogonal form) Let  ∈ n. There exists a GZ-basis wT of V  such that the action of an arbitrary simple transposition si ∈ Sn is given by si wT = wT + 1 − 2 wsi T where  = res Ti+1 − res Ti −1 (note that when  = ±1, the coefficient of wsi T is zero, so this term should be omitted). Proof Let vT be the basis of Theorem 2.3.1, and set wT = vT / vT  vT  Let S = si T . We may assume that si is an admissible transposition and S > T. As si preserves · ·, the formulas 2.3.1(ii) imply of Theorem vS  vS  = 1 − 2 vT  vT , whence wS = vS / vT  vT  1 − 2 . Now, the result follows from (2.12) and (2.13).

2.3 Formulas of Young and Murnaghan–Nakayama

19

Example 2.3.4 Let  = n − 1 n. Using the notation of Example 2.2.11(ii) and writing vj for vTj , 2 ≤ j ≤ n, the formulas of Young’s orthogonal form become ⎧ v if j = i i + 1, ⎪ ⎪  ⎨ j 1 1 si vj = i vi + 1 − i2 vi+1  if j = i, (2.18)  ⎪ ⎪ ⎩ 1 1 1 − i2 vi − i vi+1  if j = i + 1. Let M be the natural permutation Sn -module with basis e1      en . It has   the irreducible submodule N = i ai ei ∈ M  i ai = 0 . Set vj =

1 jj − 1

e1 + · · · + ej−1 − j − 1ej 

2 ≤ j ≤ n

Then v2  v3      vn is a basis of N with respect to which the simple permutations act by formulas (2.18). Let  ∈ n and  ∈ n − k. Set V / = HomSn−k V   resSn−k V   It is clear from the branching rule that V / = 0 if and only if the Young diagram  is contained in the Young diagram , in which case we denote the complement by /. A set of nodes of this form will be called a skew shape. The number of nodes in / will be denoted /. The number of rows occupied by / − 1 will be denoted by L/. A skew shape is called a skew hook if it is connected and does not have two boxes on the same diagonal (equivalently, if the residues of the nodes of the shape form a segment of integers). We know that V / is an irreducible n−kk -module. On restriction to  Sk ⊂ n−kk it becomes a (not necessarily irreducible) Sk -module. Let  / be the character of this Sk -module. If  = ∅ we get the character   of V  . The results on GZ-bases and Young’s canonical forms can be easily generalized to skew shapes. For example, define an /-path to be any path which connects  with . We will not distinguish between /-paths and /-tableaux (defined in the obvious way). Then Theorem 2.3.3 implies: Proposition 2.3.5 (Young’s orthogonal form for skew shapes) Let / be a skew shape with / = k,  = n − k. There exists a basis wT  T is an /-tableau of V / such that the action of an arbitrary simple transposition si ∈ Sk is given by si wT = wT + 1 − 2 wsi T

Symmetric groups I

20

where  = res Ti+1 − res Ti −1 (note that when  = ±1, the coefficient of wsi T is zero, so this term should be omitted). Moreover, each vector wT is a simultaneous eigenvector for Ln−k+1      Ln ∈ n−kk with eigenvalues res T1      res Tk , respectively. The final main result of this section is: Theorem 2.3.6 Let / be a skew shape with / = k. Then


 −1L/  if / is a skew hook,  / 1 2     k = 0 otherwise. The following is a very effective way to evaluate an irreducible character on a given element. Corollary 2.3.7 (Murnaghan–Nakayama rule) Let / be a skew shape with / = k, and c be an element of Sk whose cycle shape corresponds to a partition  = 1 ≥ · · · ≥ l > 0 ∈ k. Then   / c = −1LH  H

where the sum is over all sequences H of partitions  = 0 ⊂ 1 ⊂ · · · ⊂ l =  such that i/i − 1 is a skew hook with i/i − 1 = i for all  1 ≤ i ≤ l, and LH = li=1 Li/i − 1. Proof By the branching rule for m < k we have resSm ×Sk−m V / = ⊕ V /
V /  where the sum is over all partitions  with  ⊂  ⊂ , such that / = m. More generally, resS

1

×···×Sl V

/

= ⊕H V 1/0
· · ·
V l/l−1 

where the summation is over all H as in the statement of the corollary. Now the result follows from Theorem 2.3.6. We proceed to prove Theorem 2.3.6. Fix a skew shape / with / = k.

2.3 Formulas of Young and Murnaghan–Nakayama

21

Lemma 2.3.8 Theorem 2.3.6 is true for  = ∅. Proof It is easy to see that L2 L3    Lk is the sum of all k-cycles in Sk . If v ∈ V  is a weight vector of weight , then L2 L3    Lk v = 2 3    n v, which is zero unless  is a hook, see Theorem 2.2.10. However, if  = k − b 1b  is a hook with L = b, then, again
by Theorem 2.2.10, we have 2    n = . Now the result follows from the fact −1b b!k − b − 1! and dim V  = k−1 b that there are k − 1! k-cycles in Sk . Lemma 2.3.9 In the notation of Proposition 2.3.5, Sk · wT = V / for any T . Proof As V / is irreducible over n−kk , we have n−kk · wT = V / . However, in view of (2.9), every element of n−kk can be written as gxz, where g ∈ Sk , x ∈ Ln−k+1      Ln , z ∈ n−k . As x and z act on wT by multiplication with scalars, the result follows.
 Lemma 2.3.10 If / is not connected, then  / 1 2     k = 0. Proof Let / =  ∪ , where  and  are skew shapes disconnected from each other, that is res C − res D > 1 for any C ∈  and D ∈ . Let c =  and d = . There exists an /-tableau T such that T1      Tc ∈  and Tc+1      Tk ∈ . By Proposition 2.3.5, the subspace of V / , spanned by vectors wT for all such tableaux T , is invariant with respect to Sc × Sd < Sk , and, as a Sc ×Sd -module, it is isomorphic to V 
V  . By Lemma 2.3.9 and Frobenius reciprocity, we get a surjective homomorphism indSk V 
V   → V / . But, using Proposition 2.3.5, we see that the dimensions of both modules   are equal to k dim V  dim V  . So V /  indSk V 
V  . Now the lemma c follows from the following standard general fact: if H is a subgroup of a finite group G, g ∈ G is not conjugate to an element of H, and V is a G-module induced from H, then the character of V on g is zero. Lemma 2.3.11 If / has two nodes on the same diagonal, and  = a 1k−a   be an arbitrary hook with
k-boxes, then  V is not a composition factor of / / 1 2     k = 0 V . In particular,  Proof The second statement follows from the first by Lemma 2.3.8. By assumption a 2 × 2 square is contained in /. It follows from Proposition 2.3.5 that V 22 is an S4 -submodule of V / (for S4 embedded not necessarily with respect to the first four letters, but such S4 is conjugate to the

22

Symmetric groups I

canonical one anyway). By Frobenius reciprocity and Lemma 2.3.9, there is a surjection indSk V 22 → V / , and the result now follows from the branching rule. Lemma 2.3.12 Let / be a skew hook, and  = k−b 1b . Then V  appears as a composition factor of V / if and only if b = L/, in which case its multiplicity is one. Proof It follows from Proposition 2.3.5 that translation of / does not change the corresponding Sk -module. So we may assume that  and  are minimal possible, as in the picture



Now, if b = L/, then  ⊆ , so, by the branching rule, V  does not appear as a composition factor of resSk V  , hence it does not appear in V / either. Let b = L/. Note that / has shape . So it follows from Proposition 2.3.5 that V / and V  are isomorphic as Sn−k -modules. So resSk ×Sn−k V   V 
V  = 1. Then resSn−k ×Sk V   V 
V  = 1. It remains to note that V /  V  = resSn−k ×Sk V   V 
V  . Theorem 2.3.6 follows from Lemmas 2.3.8 and 2.3.10–2.3.12. Remark 2.3.13 We sketch another interpretation of the graph  referring the reader to [KR, Lecture 4] for details. In fact, this interpretation is closer to what we are going to do later in the book. Let  =   be the Lie algebra of all  × -matrices over  with only finitely many non-zero entries. Thus, the matrix units Eij  i j ∈  form a basis of . The Lie algebra  acts on the Fock space  , which is the complex vector space, whose basis consists of the formal semi-infinite wedges vi0 ∧ vi1 ∧ vi2 ∧ · · · such that i0 > i1 >    and ik = −k for k 0. To write down the action we follow the usual rules for the action of Lie algebra on a wedge power of a module. For example, E2−1 · v0 ∧ v−1 ∧ v−2 ∧ · · · = v0 ∧ v2 ∧ v−2 ∧ · · · = −v2 ∧ v0 ∧ v−2 ∧ · · · 

2.3 Formulas of Young and Murnaghan–Nakayama

23

 In fact, more than just  acts on  . Let ak = j−i=k Eij be the kth diagonal. Even though ak is not an element of , we can still extend the action of  to it, at least if k = 0. For example, a−2 · v0 ∧ v−1 ∧ v−2 ∧ · · · = v2 ∧ v−1 ∧ v−2 ∧ · · · − v1 ∧ v0 ∧ v−2 ∧ · · ·  It is convenient to label semi-infinite wedges by partitions: to a partition  = 1 ≥ 2 ≥     we associate the vector v = vi0 ∧ vi1 ∧ vi2 ∧ · · · with ij = j − j. For example, v∅ = v0 ∧ v−1 ∧ v−2 ∧ · · · . Then v   ∈  is a basis of  , and we have in some sense recovered the vertices of  . For the edges, note that Eii+1 v = v where  is obtained from  by removing a removable node of residue i, if it exists, and otherwise v is interpreted as 0. Similarly, Ei+1i v = v≥ , where ≥ is obtained from  by adding an addable node of residue i, if it exists, and otherwise v≥ is interpreted as 0. Thus the action of the Chevalley generators of  on the basis vectors v

recovers the edges of  . Is it possible to explain this remarkable coincidence of two graphs, one coming from representation theory of Sn and the other from (completely different) representation theory of  ? If you want to know the answer, keep reading this book... We make one more observation along these lines. It is easy to see that for i > j we have Eij v =
v , where  is obtained from  by removing a skew hook of length j − i, starting at the node of residue i and ending at the node of residue j − 1; if no such hook exists, interpret v as 0. Moreover,
= −1L/ . It follows that for k > 0 we have  ak v = −1L/ v where the sum is over all  such that / is a skew hook with / = k. So the Murnaghan–Nakayama rule can be interpreted as follows: the value   c  of the irreducible character   on an element c with cycle-shape 1  2      l  is equal to the coefficient of v∅ in a1 a2    al v . Or better yet:   c  = v  a−1 a−2    a−l v∅ 

(2.19)

where · · is the contravariant form on  normalized so that v∅  v∅  = 1.

3 Degenerate affine Hecke algebra

In this chapter we define the degenerate affine Hecke algebra
n . As a vector space,
n is the tensor product FSn ⊗ F x1      xn of the group algebra FSn and the free commutative polynomial algebra F x1      xn . Moreover, FSn ⊗ 1 and 1 ⊗ F x1      xn are subalgebras of
n isomorphic to FSn and F x1      xn , respectively. Furthermore, there exists an algebra homomorphism
n → FSn , which is the “identity” on the subalgebra FSn , that is sends w ⊗1 to w, see Chapter 7. However, the relations between the elements of FSn and F x1      xn are not quite the ones coming from the natural action of Sn on F x1      xn – they are those modulo some “garbage”, which often can be kept under control, see for example Lemma 3.2.1. In particular, the center of
n is what we would like it to be: the ring of symmetric polynomials F x1      xn Sn . We introduce parabolic subalgebras of
n , the corresponding induction and restriction functors, and prove a Mackey-type theorem, which as usual is a result on induction followed by restriction. The only difference with Mackey Theorem for Sn is that here we have a filtration instead of a direct sum. Some important antiautomorphisms of
n are defined next. Informally, the automorphism  swaps the roles of the left- and the right-hand sides. For example, if we have proved something about the action of xn on
n -modules, we may then apply  and get a similar result about x1 . The antiautomorphism , however, will be used to define a dual of an
n -module. It will turn out later (see Chapter 5) that the irreducible
n -modules are self-dual. A non-trivial Theorem 3.7.5 establishes a nice relation between duality and induction. Finally, we introduce certain intertwining elements w for w ∈ Sn , whose main property is that they commute with the variables xi according to the natural action of the symmetric group element w (no “garbage” this time!)

24

3.1 The algebras

25

3.1 The algebras Let n = F x1      xn be the algebra of polynomials in x1      xn . If  = 1      n  ∈ n+ , set 

x = x1 1    xnn  We record the relations xi xj = xj xi 

(3.1)

for all 1 ≤ j < i ≤ n (the precise form of the relations is used in the proof of Theorem 3.2.2). The group algebra FSn of Sn over F will be denoted by n . Thus, n is generated by the basic transpositions s1      sn−1 subject to relations

si sj = sj si 

si2 = 1

(3.2)

si si+1 si = si+1 si si+1 

(3.3)

for all admissible i j with i − j > 2. The corresponding Bruhat ordering on Sn is denoted by ≤. We define a left action of Sn on n by algebra automorphisms so that w · xi = xwi 

(3.4)

for each w ∈ Sn , i = 1     n. Now we define the main object of study: the (degenerate) affine Hecke algebra
n . This was introduced by Drinfeld [D] and Lusztig [L]. The associative algebra
n is given by generators x1      xn and s1      sn−1 , subject to the same relations as n (3.1), and as n (3.2), (3.3), together with si xj = xj si 

(3.5)

si xi = xi+1 si − 1

(3.6)

for all admissible i j with j = i i + 1. We will call x1      xn polynomial generators and s1      sn−1 Coxeter generators. Note that
1  1  F x . Also, by agreement,
0  F . The relation (3.6) implies si xi+1 = xi si + 1

(3.7)

By induction, we deduce a convenient general formula si f = si · fsi +

f − si · f xi+1 − xi

(3.8)

26

Degenerate affine Hecke algebra

for f ∈ F x1      xn . In particular, for j ≥ 1 we have j si xij = xi+1 si −

j−1 

j−1−k xik xi+1 

(3.9)

k=0 j = xij si + si xi+1

j−1 

j−1−k xik xi+1 

(3.10)

k=0

3.2 Basis Theorem Our first goal is to construct a “PBW-type” basis of
n . There are obvious homomorphisms   n →
n and   n →
n under which the xi or sj map to the corresponding elements of
n . We also write x for the image under  of the basis element x ∈ n , and w for the image under  of the basis element w ∈ Sn ⊂ n . This notation will be justified shortly, when we show that  and  are both algebra monomorphisms. The following lemma is obvious from the defining relations: Lemma 3.2.1 Let f ∈ n , w ∈ Sn . Then in
n we have    wf = w · f w + fu u fw = ww−1 · f  + ufu u<w

u<w

for some fy  fy ∈ n of degrees less than the degree of f . It follows easily from this lemma that that
n is at least spanned by all x w,  ∈ n+  w ∈ Sn . We wish to prove that these elements are linearly independent too: Theorem 3.2.2 The x w   ∈ n+  w ∈ Sn form a basis for
n . Proof We give two proofs of this important result. First proof. Consider instead the algebra
˜ n given by generators x˜ i  s˜j , 1 ≤ i ≤ n 1 ≤ j < n, subject to the relations (3.1), (3.2), (3.5), (3.6), and (3.7). Thus we have all the relations of
n , except for the braid relations (3.3). Using these relations as the reduction system in Bergman’s diamond lemma ˜ for all  ∈ n+ and all [Be, 1.2] we see that
˜ n has a basis given by all x˜  w words w ˜ in the s˜j which do not involve subword of the form s˜j2 . Hence, the subalgebra ˜ n of
˜ n generated by the x˜ i is isomorphic to n . Also let ˜n denote the subalgebra of
˜ n generated by the s˜j , so that ˜n is isomorphic to the algebra on generators s˜1      s˜n−1 subject to relations s˜j2 = 1 for each j.

3.3 The center of
n

27

Now,
n is the quotient of
˜ n by the two-sided ideal  generated by the elements aij = s˜i s˜j − s˜j s˜i 

bi = s˜i s˜i+1 s˜i − s˜i+1 s˜i s˜i+1

for i j as in (3.3). Let  be the two-sided ideal of ˜n generated by the same elements aij  bi for all i j. Then ˜n /  n , and to prove the theorem it n . In turn, this follows if we can show suffices to show that  =  Pn  in
that r x˜ k ∈ ˜ n r for each k = 1     n and r ∈ aij  bi . This is an easy check. For example, ˜si+1 s˜i s˜i+1 − s˜i s˜i+1 s˜i ˜xi+2 = x˜ i ˜si+1 s˜i s˜i+1 − s˜i s˜i+1 s˜i  using only the relations in
˜ n . Second proof. By verifying the defining relations of
n , we can see that
n acts on the polynomials F y1      yn in n variables by the following formulas: f − sj · f xi  f = yi f sj  f = sj · f  + yj+1 − yj for all admissible i j and all f ∈ F y1      yn (secretly, we consider the action
of
n on the module indFSnn 1Sn induced from the trivial FSn -module). Now, to see that the elements x w are linearly independent, it suffices to show that they act by linearly independent linear transformations on F y1      yn . But this is clear if we consider the action on an element of the form y1N y22N    ynnN for N 0. By Theorem 3.2.2, we have a right from now on to identify n and n with the corresponding subalgebras of
n . Then
n is a free right n -module on a basis x   ∈ n . As another consequence, if m ≤ n, we can consider
m as the subalgebra of
n generated by x1      xm , s1      sm−1 . Finally, let us point out that there is the obvious variant of Theorem 3.2.2:
n also has wx w ∈ Sn   ∈ n+ as a basis. This follows using Lemma 3.2.1.

3.3 The center of n The following simple description of the center is very important. Theorem 3.3.1 The center of
n consists of all symmetric polynomials in x1      xn .

28

Degenerate affine Hecke algebra

Proof That the symmetric polynomials are indeed central is easily verified  using (3.9). Conversely, take a central element z = w∈Sn fw w ∈
n where each fw ∈ n . Let w be maximal with respect to the Bruhat order such that fw = 0. Assume w = 1. Then there exists i ∈ 1     n with wi = i. By Lemma 3.2.1, xi z − zxi looks like fw xi − xwi w plus a linear combination of terms of the form fu u for fu ∈ n and u ∈ Sn with u ≥ w in the Bruhat order. So, in view of Theorem 3.2.2, z is not central, giving a contradiction. Hence, we must have that z ∈ n . To see that z is actually a symmet ric polynomial, write z = ij≥0 aij x1i x2j , where the coefficients aij lie in F x3      xn . Applying Lemma 3.2.1 to s1 z = zs1 now gives that aij = aji for each i j, hence z is symmetric in x1 and x2 . Similar argument shows that z is symmetric in xi and xi+1 for all i = 1     n − 1.

3.4 Parabolic subalgebras Suppose that =  1      r  is a composition of n. Let S  S 1 × · · · × S r denote the corresponding Young subgroup of Sn . Then the subalgebra  of n generated by the sj for which sj ∈ S is isomorphic to FS   1 ⊗ · · · ⊗  r  We define the parabolic subalgebra
of the degenerate affine Hecke algebra
n in a similar way: it is the subalgebra generated by n and all sj for which sj ∈ S . It follows easily from Theorem 3.2.2 that the elements x w   ∈ n+  w ∈ S

form a basis for
. In particular,

1 ⊗ · · · ⊗
r  Note that the parabolic subalgebra
111 is precisely the subalgebra n . We will use the induction and restriction functors between
n and
. These will be denoted simply indn 
-mod →
n -mod

resn 
n -mod →
-mod

(3.11)

the former being the tensor functor
n ⊗
?, which is left adjoint to resn . More generally, we will consider induction and restriction between nested

3.5 Mackey Theorem

29

parabolic subalgebras, with obvious notation. We will also occasionally consider the restriction functor resnn−1 
n -mod →
n−1 -mod

(3.12)

3.5 Mackey Theorem Let   be compositions of n. Denote by D the set of minimal length left S -coset representatives in Sn , and by D −1 the set of minimal length right S -coset representatives. Then D  = D −1 ∩ D is the set of minimal length S  S -double coset representatives in Sn . We recall three known properties, see for example [DJ, Section 1], (1) For x ∈ D  , S ∩ xS x−1 and x−1 S x ∩ S are Young subgroups of Sn . So we can define compositions ∩ x and x−1 ∩  of n from S ∩ xS x−1 = S ∩x

x−1 S x ∩ S = Sx−1 ∩ 

and

(2) For x ∈ D   , the map w → x−1 wx restricts to a length preserving isomorphism S ∩x → Sx−1 ∩  (3) For x ∈ D  , every w ∈ S xS can be written as w = uxv for unique −1 elements u ∈ S and v ∈ S ∩ Dx−1 −1 ∩ . Moreover, S ∩ Dx−1 ∩ is the set of minimal length right coset representatives of Sx−1 ∩ in S . Now fix some total order ≺ refining the Bruhat order < on D  . For x ∈ D  , set  x =
y  (3.13) y∈D   yx

≺x =




y 

(3.14)

y∈D   y≺x

x = x / ≺x 

(3.15)

It follows from Lemma 3.2.1 that x and ≺x are invariant under right multiplication by n . Hence, since  =  n , we have defined a filtration of n as an    -bimodule. We want to describe the quotients x explicitly. To this end, note using the property (2) above that for each y ∈ D  , there exists an algebra isomorphism y−1 
∩y →
y−1 ∩

30

Degenerate affine Hecke algebra

with y−1 w = y−1 wy and y−1 xi  = xy−1 i for w ∈ S ∩y , 1 ≤ i ≤ n. If N is a left
y−1 ∩ -module, then by twisting the action with the isomorphism y−1 we get a left
∩y -module, which will be denoted y N . Lemma 3.5.1 Let us view
as an 

∩x -bimodule and
 as an 
x−1 ∩ 
 -bimodule in the natural ways. Then x
 is an 
∩x 
 bimodule, and x 
⊗
∩x x
 is an 

 -bimodule. Proof We define a bilinear map
×x
 → x = x / ≺x by h h  → hxh + ≺x  The map is checked to be
∩x -balanced, so it yields an 

 -bimodule map  
⊗
∩x x
 → x  To prove that  is bijective, note by the Property (3) above that x u ⊗ v   ∈ n+  u ∈ S  v ∈ S ∩ Dx−1 −1 ∩

is a basis of the induced module
⊗
∩x x
as a vector space, and the image of these elements under  is a basis of x . Now we can prove the Mackey Theorem. Theorem 3.5.2 (“Mackey Theorem”) Let M be an
 -module. Then resn indn M admits a filtration with subquotients isomorphic to ind ∩x x resx−1 ∩ M one for each x ∈ D  . Moreover, the subquotients can be taken in any order refining the Bruhat order on D  , in particular ind ∩ res ∩ M appears as a submodule. Proof This follows from Lemma 3.5.1 and the isomorphism






⊗
∩x x
  ⊗
 M  ind
∩x x res
−1 x

which is easy to check.

∩

M

3.7 Duality

31

3.6 Some (anti) automorphisms A check of relations shows that
n possesses an automorphism  and an antiautomorphism  defined on the generators as follows:   si → −sn−i    si → si 

xj → xn+1−j 

xj → xj 

(3.16) (3.17)

for all i = 1     n − 1 j = 1     n. If M is a finite dimensional
n -module, we can use  to make the dual space M ∗ into an
n -module denoted M  . Note  leaves invariant every parabolic subalgebra of
n , so also induces a duality on finite dimensional
-modules for each composition of n. Instead, given any
n -module M, we can twist the action with  to get a new module denoted M  . Moreover, for any composition  = 1      r  of n we denote by  ∗ the composition with the same non-zero parts but taken in the opposite order. For example 3 2 1∗ = 1 2 3. Then  induces an isomorphism of parabolic subalgebras
∗ →
 . So if M is an
 -module, we can inflate through  to get an
∗ -module denoted M  . If M = M1
· · ·
Mr is an outer tensor product module over
 =
1 ⊗ · · · ⊗
r then M   Mr
· · ·
M1 . These observations imply: Lemma 3.6.1 Let M ∈
m -mod and N ∈
n -mod. Then m+n m+n  indmn M
N  indnm N
M

3.7 Duality Throughout this section, is a composition of n and  = ∗ . Let d be the longest element of D  . Note that ∩ d = and d−1 ∩  = , so S dS = S d = dS . There is an isomorphism  = d−1 
→
 

(3.18)

see Section 3.5, and for an
 -module M, M denotes the
-module obtained by pulling back the action through . We begin by considering the situation for n . d

Lemma 3.7.1 Define a linear map   n → d  by

d−1 w if w ∈ dS ,  w = 0 otherwise,

32

Degenerate affine Hecke algebra

for each w ∈ Sn . Then:  (i) is a homomorphism of    -bimodules; (ii) ker  contains no non-zero left ideals of n ; (iii) the map

f   n → Hom n  d  

h → h



is an isomorphism of n   -bimodules. Proof (i) Follows easily using S d = dS . (ii) It is enough to show that  n t = 0 for any t ∈ n . By multiplying t with an appropriate group element on the left, we may assume that  t = y∈D yhy with each hy ∈  and hd = 0. Now  t = hd = 0, as required. (iii) We remind that h   n → d  denotes the map with h  t =  th. Given (i), it is straightforward to check that f  is a homomorphism of n   -bimodules. To see that it is an isomorphism, it suffices by dimension to show that it is injective. Suppose t lies in the kernel, then f  th =  ht = 0 for all h ∈ n . Hence t = 0 by (ii). Now we extend this result to
n . Lemma 3.7.2 Define a linear map 
n → d
 by

f d−1 w if w ∈ dS , fw = 0 otherwise, for each f ∈ n  w ∈ Sn . Then: (i) is a homomorphism of 

 -bimodules; (ii) the map f 
n → Hom

n  d
 

h → h

is an isomorphism of 
n 
 -bimodules. Proof (i) According to a special case of Lemma 3.5.1, the top factor d in the bimodule filtration of
n defined in (3.15) is isomorphic to d
 as an 

 -bimodule. The map is simply the composite of this isomorphism with the quotient map
n → d . (ii) Recall that w  w ∈ D −1 forms a basis for
n as a free left
-module, and d
 is isomorphic to
as a left
-module. It follows that the maps w  w ∈ D −1 form a basis for Hom

n  d
  as a free right
 -module, where w 
n → d
 is the unique left
-module homomorphism with w u = wu 1 for all u ∈ D −1 .

3.7 Duality

33

Given (i), it is straightforward to check that f is a homomorphism of 
n 
 -bimodules. So, to prove that f is an isomorphism, it suffices to find a basis of
n as a free right
 -module, which is mapped by f to the basis w  w ∈ D −1 of Hom

n  d
 . The analogous maps w ∈ Hom n  d   defined by w u = wu 1 for u ∈ D −1 form a basis for Hom n  d   as a free right  -module. So, in view of Lemma 3.7.1(iii), we can find a basis aw  w ∈ D −1 for n viewed as a right  -module such that f  aw  = w for each w ∈ D −1 , that is



uaw  =

1

if u = w,

0

otherwise

for every u ∈ D −1 . But
 =  n , so the elements aw  w ∈ D −1 also form a basis for
n as a right
 -module, and faw  = w since =  on n .

Corollary 3.7.3 There is a natural isomorphism Hom

n d M 
n ⊗
 M of
n -modules, for every left
 -module M. Proof Let f 
n → Hom

n  d
  be the bimodule isomorphism constructed in Lemma 3.7.2. Then there are natural isomorphisms f ⊗id


n ⊗
 M −→ Hom

n  d
  ⊗
 M  Hom

n  d
 ⊗
 M  Hom
n 
n  d M the second isomorphism depending on the fact that
n is a free left
-module, see for example [AF, 20.10]. Corollary 3.7.4 There is a natural isomorphism indn M    indn d M for every finite dimensional
 -module M. Proof The functor   indn   
-mod →
n -mod is right adjoint to resn . Hence it is isomorphic to Hom

n  ? by uniqueness of adjoint functors.

34

Degenerate affine Hecke algebra

Now combine this natural isomorphism with Corollary 3.7.3 (with and  swapped and d replaced by d−1 ). We finally record an important special case: Theorem 3.7.5 For M ∈
m -mod and N ∈
n -mod, we have m+n    indm+n mn M
N  ind nm N
M 

3.8 Intertwining elements We will need certain elements of
n which go back Cherednik. Given 1 ≤ i < n, define i = si xi − xi+1  + 1

(3.19)

A straightforward calculation gives: i2 = xi − xi+1 − 1xi+1 − xi − 1

1 ≤ i < n

(3.20)

i xi = xi+1 i  i xi+1 = xi i  i xj = xj i 

j = i i + 1

(3.21)

i j = j i  i i+1 i = i+1 i i+1 

i − j > 1

(3.22)

Property (3.22) means that for every w ∈ Sn we obtain a well-defined element w ∈
n , namely, w = i1    im where w = si1    sim is any reduced expression for w. According to (3.21), these elements have the property that w xi = xwi w 

(3.23)

for all w ∈ Sn and 1 ≤ i ≤ n. We note that only the properties (3.20) and (3.21) will be essential in what follows.

4 First results on
n -modules

The polynomial subalgebra n of
n is a maximal commutative subalgebra, and so we may try to “do Lie Theory” using n as an analogue of Cartan subalgebra. The main difference however is that n is not semisimple, so we have to consider generalized eigenspaces, rather than usual eigenspaces. We define the formal character of an
n -module M as the generating function for the dimensions of simultaneous generalized eigenspaces of the elements x1      xn on M. In Chapter 5 we will prove that the formal characters of irreducible
n -modules are linearly independent (as any reasonable formal characters should be). The “Shuffle Lemma”, which is a special case of the Mackey Theorem, gives a transparent description of what induction “does” to the formal characters. Our knowledge of the center of
n allows us to develop an easy theory of blocks. The central characters of
n (and so the blocks too) are labeled by the Sn -orbits on the n-tuples of scalars. Next we study the properties of what can be considered as one of the main technical tools of the theory: the so-called Kato module (cf. [Kt]). Miraculously, if we take the 1-dimensional module over the polynomial algebra n on which every xi acts with the same scalar, and then induce it to
n , we get an irreducible module. This is not hard to prove once you believe it is true. As a module over symmetric group, any Kato module is just the regular module, and thus we deal with a remarkable extension of the regular FSn -module to
n , which is irreducible. The simple definition of Kato modules as induced modules allows us to investigate them in great detail. We conclude the chapter with a study of some “finite dimensional approximations” of projective covers of Kato modules.

35

36

First results on
n -modules

4.1 Formal characters Let a ∈ F . Denote by La the 1-dimensional 1 -module with x1 v = av for v ∈ La. As n  1 ⊗ · · · ⊗ 1 , we obtain the irreducible n -modules by taking outer tensor products La1 
· · ·
Lan , for a1      an ∈ F . Now take any M ∈ n -mod. For any a = a1      an  ∈ F n , let Ma be the largest submodule of M, all of whose composition factors are isomorphic to La1 
· · ·
Lan . Alternatively, Ma is the simultaneous generalized eigenspace for the commuting operators x1      xn corresponding to the eigenvalues a1      an , respectively. Hence: Lemma 4.1.1 For any M ∈ n -mod, we have M = ⊕a∈F n Ma as a n -module. Note, for M ∈ n -mod, knowledge of the dimensions of the spaces Ma for all a is equivalent to knowing the coefficients ra when the class M of M in the Grothendieck group Kn -mod is expanded as M =



ra La1 
· · ·
Lan 

a∈F n

in terms of the basis La1 
· · ·
Lan   a ∈ F n . Now let M ∈
n -mod. Recall that
11 = n . We define the formal character of M by: ch M = resn11 M ∈ Kn -mod

(4.1)

Since the functor resn11 is exact, ch induces a homomorphism ch  K
n -mod → Kn -mod at the level of Grothendieck groups. We will later see that this map is actually injective (Theorem 5.3.1). We will also occasionally consider characters of modules over parabolic subalgebras
. The definitions are modified in this case in obvious ways. Lemma 4.1.2 Let a = a1      an  ∈ F n . Then ch indn11 La1 
· · ·
Lan  =



Law−1 1 
· · ·
Law−1 n 

w∈Sn

Proof This follows from the Mackey Theorem with =  = 1n .

4.2 Central characters

37

Lemma 4.1.3 (“Shuffle Lemma”) Let n = m + k, and let M ∈
m -mod, K ∈
k -mod. Assume  ch M = ra La1 
· · ·
Lam   a∈F m



ch K =

sb Lb1 
· · ·
Lbk  

b∈F k

Then ch indnmk M
K =

 



ra sb 

a∈F m b∈F k

Lc1 
· · ·
Lcn 

c

where the last sum is over all c = c1      cn  ∈ F n which are obtained by shuffling a and b, that is there exist 1 ≤ u1 < · · · < um ≤ n such that cu1      cum      cn  = b1      bk . cu1      cum  = a1      am , and c1      Proof Apply the Mackey Theorem with = 1n  and  = m k.

4.2 Central characters Recall by Theorem 3.3.1 that every element z of the center Z
n  of
n can be written as a symmetric polynomial fx1      xn . Given a ∈ F n , we associate the central character a  Z
n  → F

fx1      xn  → fa1      an 

(4.2)

Consider the action of Sn on F n by place permutation. We write a ∼ b if a and b lie in the same orbit with respect to this action. The following lemma is immediate. Lemma 4.2.1 For a b ∈ F n , a = b if and only if a ∼ b. Thus the central characters of
n are actually labeled by the set F n /∼ of Sn -orbits on F n . If  is such an orbit we set  = a for any a ∈ . Now let M be a finite dimensional
n -module and  ∈ F n /∼. We let M denote the generalized eigenspace of M over Z
n  that corresponds to the central character  , that is M = v ∈ M  z −  zk v = 0 for all z ∈ Z
n  and k 0 

38

First results on
n -modules

Observe this is an
n -submodule of M. Now, for any a ∈ F n with a ∈ , Z
n  acts on La1 
· · ·
Lan  via the central character  . So applying Lemma 4.2.1, we see that  Ma  M = a∈

recalling the decomposition of M as a n -module from Lemma 4.1.1. Therefore: Lemma 4.2.2 Any M ∈
n -mod decomposes as  M M= ∈F n /∼

as an
n -module. Thus the    ∈ F n /∼ exhaust the possible central characters that can arise in a finite dimensional
n -module, while Lemma 4.1.2 shows that every such central character does arise in some finite dimensional
n -module. If  ∈ F n / ∼, let us denote by
n -mod for the full subcategory of
n -mod, consisting of all modules M with M = M. Then Lemma 4.2.2 implies that there is an equivalence of categories 
n -mod  (4.3)
n -mod  ∈F n /∼

We say that
n -mod is the block of
n -mod corresponding to  (or to the central character  ). If M ∈
n -mod , we say that M belongs to the block corresponding to . If M = 0 is indecomposable then M ∈
n -mod for a unique  ∈ F n /∼.

4.3 Kato’s Theorem Let a ∈ F . Introduce the Kato module Lan  = indn11 La
· · ·
La

(4.4)

By Lemma 4.1.2, we know immediately that ch Lan  = n!La
· · ·
La  In particular, for each k = 1     n, the only eigenvalue of the element xk on Lan  is a.

4.3 Kato’s Theorem

39

Lemma 4.3.1 Let a ∈ F . Set L = La
· · ·
La, so Lan  =
n ⊗n L. The common a-eigenspace of the operators x1      xn−1 on Lan  is precisely 1 ⊗ L. Moreover, all Jordan blocks of x1 on Lan  are of size n.  Proof Note Lan  = w∈Sn w ⊗ L, since by Theorem 3.2.2 we know that
n is a free right n -module on basis w  w ∈ Sn . We first claim that the eigenspace of x1 is a sum of the subspaces of the  = s2      sn−1 . Well, any w can be written as form y ⊗ L, where y ∈ Sn−1  and 0 ≤ j < n. Note that xj+1 − av = 0 for any ys1 s2    sj for some y ∈ Sn−1 v ∈ L. Now the defining relations of
n imply x1 − ays1 s2    sj ⊗ v = −ys1    sj−1 ⊗ v + ∗ where ∗ stands for terms which belong to subspaces of the form y s1    sk ⊗ L  for y ∈ Sn−1 and 0 ≤ k < j − 1 Now assume that a linear combination   z = cyj ys1 s2    sj ⊗ v  0≤j
is an eigenvector for x1 . Choose the maximal j for which the coefficient cyj is non-zero. Then the calculation above shows that x1 − az = 0 unless j = 0. This proves our claim. Now apply the same argument to see that the common eigenspace of x1 and x2 is spanned by y ⊗ L for y ∈ s3      sn−1 , and so on, yielding the first claim of the lemma. Finally, define Vm = z ∈ Lan   x1 − am z = 0  It follows by induction from the calculation above that  Vm = span ys1 s2    sj ⊗ L  y ∈ Sn−1  j < m 

giving the second claim. Now we prove the main theorem on the structure of the Kato module Lan , compare [Kt]. Theorem 4.3.2 Let a ∈ F and =  1      r  be a composition of n: (i) Lan  is irreducible, and it is the only irreducible module in its block. (ii) All composition factors of resn Lan  are isomorphic to La 1 
· · ·
La r  and soc resn Lan  is irreducible. (iii) soc resnn−1 Lan   Lan−1 .

40

First results on
n -modules

Proof Denote La
· · ·
La by L. (i) Let M be a non-zero
n -submodule of Lan . Then resn11 M must contain a n -submodule N isomorphic to L. But the commuting operators x1      xn act on L as scalars, giving that N is contained in their common eigenspace on Lan . But by Lemma 4.3.1, this implies that N = 1 ⊗ L. This shows that M contains 1 ⊗ L, but 1 ⊗ L generates the whole of Lan  over
n . So M = Lan . To see that Lan  is the only irreducible in its block use Frobenius reciprocity and the fact just proved that Lan  is irreducible. (ii) The fact that all composition factors of resn Lan  are isomorphic to La 1 
· · ·
La r  follows by formal characters and (i). To see that soc resn Lan  is irreducible, note that the submodule
⊗ L of res
Lan  is isomorphic to La 1 
· · ·
La l . This module is irreducible, and so it is contained in the socle. Conversely, let M be an irreducible
-submodule of Lan . Then using Lemma 4.3.1 as in the proof of (i), we see that M must contain 1 ⊗ L, hence
⊗ L. (iii) By part (ii), Lan  has a unique
n−11 -submodule isomorphic to Lan−1 
La, namely
n−11 ⊗ L, which contributes a copy of Lan−1  to soc resnn−1 Lan . Conversely, take any irreducible
n−1 -submodule M of Lan . The common a-eigenspace of x1      xn−1 on M must lie in 1 ⊗ L by Lemma 4.3.1. Hence, M ⊆
n−11 ⊗ L, which completes the proof.

4.4 Covering modules Fix a ∈ F and n ≥ 1 throughout the section. We will construct for each m ≥ 1 an
n -module Lm an  with irreducible head isomorphic to Lan . Let  an  denote the annihilator in
n of Lan . Introduce the quotient algebra m an  =
n / an m

(4.5)

for each m ≥ 1. Obviously  an  contains xk − an! for each k = 1     n, whence each algebra m an  is finite dimensional. Moreover, by Theorem 4.3.2, Lan  is the unique irreducible m an -module up to isomorphism. Let Lm an  denote a projective cover of Lan  in the category m an -mod (for convenience, we also define L0 an  = 0 an  = 0). Then: Lemma 4.4.1 For each m ≥ 1,m an   Lm an ⊕n! There are obvious surjections 1 an   2 an      

(4.6)

La  = L1 a   L2 a      

(4.7)

n

n

n

where Rm an  and Lm an  are considered as
n -modules by inflation.

4.4 Covering modules

41

By an argument involving lifting idempotents (see for example [La]) we may assume that these surjections agree with the decompositions m an   Lm an ⊕n! for different ms. Lemma 4.4.2 Let M be an
n -module annihilated by  an k for some k. Then for all m ≥ k there is a natural isomorphism of
n -modules Hom
n m an  M  M Moreover, there is an isomorphism of functors lim Hom
n m an  ?  lim Hom
n Lm an  ?⊕n! − →m − →m from the category of
n -module annihilated by some power of  an  to the category of vector spaces. Proof The assumption implies that M is the inflation of an m an -module. So Hom
n m an  M  Homm an  m an  M  M all isomorphisms being the natural ones. The second statement follows from the remark preceding this lemma. Now let 1 and sgn be the trivial and the sign n -modules, respectively, and set


P = indnn 1




Q = indnn sgn

The following lemma gives another description of the modules Lm an . Lemma 4.4.3 We have Lm an   P/ an m P  Q/ an m Q Proof We prove the result for P, the argument for Q being similar. By definition of Lan , its restriction from
n to n is the regular module. So, using Frobenius reciprocity, we get Hom
n P/ an m P Lan   Hom
n P Lan 


 Homn 1 resnn Lan   F whence P/ an m P is an m an -module with irreducible head Lan . Therefore P/ an m P is a quotient of the principal indecomposable module Lm an . However, let JLm an  be the radical of Lm an , and let   Lm an  → Lm an /JLm an   Lan 

First results on
n -modules

42

be the natural projection. Recall again that on restriction to n , Lan  is the regular module, so it has a non-zero Sn -invariant vector v. Moreover, as the regular module splits out, there exists an Sn -invariant vector w ∈ Lm an  such that w = v. This vector w gives rise to the n -homomorphism 1 → Lm an , whose image is Fw. By Frobenius reciprocity, this homomorphism yields an
n -homomorphism P → Lm an  and hence an m an homomorphism P/ an m P → Lm an  which must be surjective by the choice of w. Let us finally consider two important special cases: n = 1 and m = 1. If n = 1, we easily check that Rm a = Lm a is just the Jordan block of size m with the eigenvalue a, that is the vector space on basis w1      wm with x1 wk = awk + wk+1 , interpreting wm+1 as 0. We can also describe the map Lm a  Lm+1 a from (4.7) explicitly: it is the identity on w1      wm but maps wm+1 to zero. If m = 1, we know by Lemma 4.4.1 that R1 an  has dimension n!2 . Let an  be the central character of Lan , cf. (4.2), and an  be the kernel of an  . By Theorem 3.3.1, an  consists of all symmetric polynomials in the xi − a without free term. By Theorem 1.0.2, m /m an  has dimension n!, and a basis which consists of the cosets of x1 − aa1    xn − aan  0 ≤ ai < i for all 1 ≤ i ≤ n  Now, in view of Theorem 3.2.2,
n /
n an  has dimension n!2 and a basis which consists of the cosets of wx1 − aa1    xn − aan  0 ≤ ai < i for all 1 ≤ i ≤ n w ∈ Sn  As
n an  annihilates Lan , we must have Jan  =
n an  by dimensions.

(4.8)

5 Crystal operators

In this key chapter we begin to study branching rules for affine Hecke algebras. These are results about restrictions of irreducible
n -modules to some large natural subalgebras, such as
n−1 or
n−1 ⊗
1 . The idea of “Robinson’s a-restriction”, which comes from symmetric groups, is that when you restrict an irreducible module from
n to
n−1 the blocks which can occur are of very limited form, and all possible blocks can be labeled by just one scalar parameter a ∈ F . So we can denote the corresponding block component of
the restriction res
nn−1 M by ea M (of course ea M could be 0). There is a better definition though which works for any module and shows that ea is actually a functor from
n -modules to
n−1 -modules: ea M is the generalized a-eigenspace of xn on M (considered as an
n−1 -module on restriction). An important result, which is a subtle generalization of the multiplicity freeness of the branching rule for symmetric groups in characteristic 0, states that the socle of the
n−1 -module ea M is irreducible when M is an irreducible
n -module. For affine Hecke algebras this was first proved by Grojnowski and Vazirani [GV], following earlier result [K2 ] for symmetric groups. Let us write e˜ a for the socle of ea M. Then e˜ a is a map from {iso-classes of irreducible
n -modules} to {iso-classes of irreducible
n−1 -modules}∪ 0 . We also define an important function
a on iso-classes of irreducible
n -modules. One of the equivalent definitions is:
a M = max m ≥ 0  e˜ am M = 0 . We prove that this function has many important representation theoretic interpretations:
a M is the multiplicity of e˜ a M as a composition factor of ea M, it also is the maximal size of a Jordan block of xn on M with eigenvalue a, and, finally, it is the dimension of the endomorphism algebra End
n−1 ea M. If we try to use induction instead of restriction to define the analogues fa  f˜a  a of ea  e˜ a 
a , we run into a problem. The trouble is that induction from
n to
n+1 does not preserve finite dimensionality, while inducing 43

44

Crystal operators

M ⊗ La from
n ⊗
1 to
n+1 , called !a below, is not “large enough”, and is quite “defective” in many respects. For example, it is not self-dual, while ea M is (for irreducible M). We will give “correct” definitions of fa and a in Chapter 8. Still, !a applied to an irreducible module does produce a module with irreducible head (always non-zero, which is another difference with ea ), and so we use it to define the map f˜a from iso-classes of irreducible
n -modules to iso-classes of irreducible
n+1 -modules . It has the following nice property: f˜a M = N if and only if e˜ a N = M. We use these operations f˜a to get a notation for irreducible
n -modules. This cannot be called labeling yet, because one irreducible can have several different notations.

5.1 Multiplicity-free socles Let M ∈
n -mod and a ∈ F . Define a M to be the generalized a-eigenspace of xn on M. Equivalently  a M = Ma  (5.1) a∈F n  an =a

recalling the decomposition from Lemma 4.1.1. Note since xn is central in the parabolic subalgebra
n−11 of
n , a M is invariant under this subalgebra. So, in fact, a can be viewed as an exact functor a 
n -mod →
n−11 -mod

(5.2)

being defined on morphisms simply as restriction. Slightly more generally, given m ≥ 0, define am 
n -mod →
n−mm -mod

(5.3)

so that am M is the simultaneous generalized a-eigenspace of the commuting operators xk for k = n − m + 1     n. In view of Theorem 4.3.2(i), am M can also be characterized as the largest submodule of resnn−mm M all of whose composition factors are of the form N
Lam  for irreducible N ∈
n−m -mod. The definition of am implies functorial isomorphisms Hom
n−mm N
Lam  am M  Hom
n indnn−mm N
Lam  M for N ∈
n−m -mod, M ∈
n -mod. Also from definitions we get: Lemma 5.1.1 Let M ∈
n -mod with  ra La1 
· · ·
Lan   ch M = a∈F n

(5.4)

5.1 Multiplicity-free socles Then we have ch am M =



45

rb Lb1 
· · ·
Lbn  

b

summing over all b ∈ F n with bn−m+1 = · · · = bn = a. Now for a ∈ F and M ∈
n -mod, define
a M = max m ≥ 0  am M = 0 

(5.5)

Lemma 5.1.1 shows that
a M can be worked out just from knowledge of the character ch M: it is the length of the “longest a-tail”. Lemma 5.1.2 Let M ∈
n -mod be irreducible, a ∈ F ,
=
a M. If N
Lam  is an irreducible submodule of am M for some 0 ≤ m ≤
, then
a N  =
− m. Proof The definitions imply immediately that
a N  ≤
− m. For the reverse inequality, the property (5.4) and the irreducibility of M imply that M is a quotient of indnn−mm N
Lam . So applying the exact functor a
, we see that a
M = 0 is a quotient of a
indnn−mm N
Lam  In particular, a
indnn−mm N
Lam  = 0. Now we get that
a N  ≥
− m applying the Shuffle Lemma and Lemma 5.1.1. Lemma 5.1.3 Let m ≥ 0, a ∈ F and N ∈
n -mod be irreducible with m
a N  = 0. Set M = indn+m nm N
La . Then: (i) am M  N
Lam ; (ii) hd M is irreducible with
a hd M = m; (iii) all other composition factors L of M have
a L < m. Proof (i) Clearly a copy of N
Lam  appears in am M. But by the Shuffle Lemma and Lemma 5.1.1, dimam M = dimN
Lam , hence am M  N
Lam  (ii) By (5.4), a copy of N
Lam  appears in am Q for any non-zero quotient Q of M, in particular for any constituent of hd M. But by (i), N
Lam  only appears once in am M, hence hd M must be irreducible. (iii) We have shown that am M = am hd M. Hence, am L = 0 for any other composition factor of M by exactness of am .

46

Crystal operators

Lemma 5.1.4 Let M ∈
n -mod be irreducible, a ∈ F , and
=
a M. Then a
M is isomorphic to N
La
 for some irreducible
n−
-module N with
a N  = 0 Proof Pick an irreducible submodule N
La
 of a
M. Then
a N  = 0 by Lemma 5.1.2. By (5.4) and the irreducibility of M, M is a quotient of indnn−

N
La
. Hence, a
M is a quotient of a
indnn−

N
La
 But this is isomorphic to N
La
 by Lemma 5.1.3(i). This shows that a
M  N
La
 Lemma 5.1.5 Let m ≥ 0, a ∈ F and N ∈
n -mod be irreducible. Set m M = indn+m nm N
La 

Then hd M is irreducible with
a hd M =
a N  + m, and all other composition factors L of M have
a L <
a N  + m. Proof Let
=
a N . By Lemma 5.1.4, we have that a
N = K
La
 for an irreducible K ∈
n−
-mod with
a K = 0. By (5.4) and the irreducibility of N , N is a quotient of indnn−

K
La
. So the transitivity of n+m m induction implies that indn+m nm N
La  is a quotient of ind n−

+m K

+m La . Now everything follows from Lemma 5.1.3. Theorem 5.1.6 Let M ∈
n -mod be irreducible and a ∈ F . Then, for any 0 ≤ m ≤
a M, soc am M is an irreducible
n−mm -module of the form L
Lam , with
a L =
a M − m. Proof Let
=
a M. Suppose that L
Lam  is a constituent of soc am M. By Lemma 5.1.2, we have
a L =
− m. So every such L contributes a non-trivial submodule to resn−

n−

−mm a
M. But a
M is irreducible of the form N
La
 by Lemma 5.1.4, and so by Theorem 4.3.2(ii), the socle n−

a
M is N
La
−m 
Lam . Hence soc am M must equal of resn−

−mm m L
La . We can also apply the theorem to study resnn−1 M, meaning the restriction of M to the subalgebra
n−1 ⊂
n , see (3.12). Define the functor ea = resn−11 n−1  a 
n -mod →
n−1 -mod 

(5.6)

5.2 Operators e˜ a and f˜a

47

Record the following obvious equalities:
a M = max m ≥ 0  eam M = 0   resnn−1 M = a∈F ea M Also, it is clear from (5.1) that  if ch M = ca La1 
· · ·
Lan 

(5.7) (5.8)

then

a∈F n

ch ea M =



ca1 an−1 a La1 
· · ·
Lan−1  

(5.9)

a∈F n−1

Corollary 5.1.7 For an irreducible M ∈
n -mod with
a M > 0, the socle of ea M is irreducible, and
a soc ea M =
a M − 1. Proof Let L be an irreducible submodule of ea M. The central element z = x1 + · · · + xn of
n acts as a scalar on the whole M by Schur’s Lemma, and similarly the central element z = x1 + · · · + xn−1 of
n−1 acts as a scalar on L. Hence xn = z − z acts on L as a scalar, too. Now it follows that the scalar must be a, and that L contributes a composition factor L
La to the socle of a M. It remains to apply Theorem 5.1.6. Corollary 5.1.8 For irreducible M ∈
n -mod, the socle of resnn−1 M is multiplicity-free.  Proof We have resnn−1 M = a∈F ea M with all but finitely many summands zero. Now, the socle of each non-zero ea M is irreducible by Corollary 5.1.7. Finally ea M and eb M are in different blocks for a = b, so their socles are definitely not isomorphic.

5.2 Operators e˜ a and f˜a Let M be an irreducible module in
n -mod. Define e˜ a M = soc ea M

f˜a M = hd indn+1 n1 M
La

(5.10)

Note f˜a M is irreducible by Lemma 5.1.5, and e˜ a M is irreducible or 0 by Corollary 5.1.7. Also from Corollary 5.1.7 we have
a M = max m ≥ 0  e˜ am M = 0 

(5.11)

48

Crystal operators

while a special case of Lemma 5.1.5 shows that
a f˜a M =
a M + 1

(5.12)

Lemma 5.2.1 Let M ∈
n -mod be irreducible, a ∈ F and m ≥ 0. (i) soc am M  ˜eam M
Lam  m ˜m (ii) hd indn+m nm M
La   fa M Proof (i) If m >
a M, then both parts in the equality above are zero. Let m ≤
a M. By Corollary 5.1.7 and Theorem 5.1.6, ˜ea M
La is a submodule of a M. By applying this m times we deduce that ˜eam M
La
m is a submodule of resn−mm eam M
Lam  is a submodule n−m11 am M, whence ˜ of am M by Frobenius reciprocity and Theorem 4.3.2(i). Now the result follows from Theorem 5.1.6. (ii) By exactness of induction and Theorem 4.3.2(i), f˜am M is a quotient of m indn+m nm M
La . Now the result follows from the simplicity of the head, see Lemma 5.1.5. Now we refine Corollary 5.1.7. Lemma 5.2.2 Let M ∈
n -mod be irreducible, a ∈ F and m ≥ 0. Then the socle of eam M is isomorphic to ˜eam M⊕m! . Proof Let m be the center of
m , and  = aa be the character of m , which comes from the action on the Kato module Lam . Recall that
m is a free (right) module over the polynomial subalgebra m of rank m!, and m is a free module over the ring of symmetric functions m of rank m! (see Theorem 1.0.2). So
m is a free module over m of rank m!2 . It follows that


U = ind mm  is a non-zero
m -module, all of whose composition factors are isomorphic to Lam , and the multiplicity of Lam  in U is m!. But by Frobenius reciprocity we have dim Hom
m U Lam  = m! so by Schur’s Lemma, U  Lam ⊕m! . Now, let L be an irreducible submodule of eam M. As in the proof of Corollary 5.1.7, Schur’s Lemma implies that any symmetric function in the variables x1      xn and any symmetric function in the variables x1      xn−m

5.3 Independence of irreducible characters

49

act on L with scalars. It follows that the symmetric functions in the variables xn−m+1      xn also act on L with scalars (one way to see this is to express the elementary symmetric functions in the variables xn−m+1      xn in terms of those in the variables x1      xn and the variables x1      xn−m ). This means that the center m of the subalgebra
m  1 ⊗
m ⊂
n−m ⊗
m acts on L with scalars, and it is clear that the corresponding central character is . Hence we have a non-zero Hn−m ⊗ Zm -homomorphism from L
 to M (whose image equals L). Frobenius reciprocity now yields a non-zero homomorphism


⊗


m L
 → M L
U  ind
n−m n−m ⊗ m

whose image contains L. As L
U is a direct sum of copies of the irreducible module L
Lam , it follows that the
n−m ⊗
m -submodule generated by L is isomorphic to L
Lam . Now the result follows from Lemma 5.2.1(i).

Lemma 5.2.3 Let M ∈
n -mod and N ∈
n+1 -mod be irreducible modules, and a ∈ F . Then f˜a M  N if and only if e˜ a N  M. Proof By Lemma 5.1.5, f˜a M  N is equivalent to Hom
n+1 indn+1 n1 M
La N  = 0 which in turn is equivalent to Hom
n1 M
La a N  = 0 thanks to (5.4). The last property means that M
La appears in the socle of a N , which is equivalent to M  e˜ i N in view of Lemma 5.2.1(i). From Lemma 5.2.3 we immediately deduce the following: Corollary 5.2.4 Let M N ∈
n -mod be irreducible. Then f˜a M  f˜a N if and only if M  N . Similarly, providing
a M
a N  > 0, e˜ a M  e˜ a N if and only if M  N .

5.3 Independence of irreducible characters We can now prove a very useful result (we follow the argument of [V1 , Section 5.5]): Theorem 5.3.1 The map ch  K
n -mod → Kn -mod is injective.

50

Crystal operators

Proof We need to show that the characters of the irreducible modules in
n -mod are linearly independent in KRepI n . Proceed by induction on n, the case n = 0 being trivial. Suppose n > 0 and there is a non-trivial -linear dependence  (5.13) cL ch L = 0 for some irreducible modules L ∈
n -mod. Choose any a ∈ F . We will show by downward induction on k = n     1 that cL = 0 for all L with
a L = k. Since every irreducible L has
a L > 0 for at least one a ∈ F , this is enough to complete the proof. Consider first the case that k = n. Then an L = 0 except if L  Lan , by Theorem 4.3.2(i). Applying an to the equation (5.13) and using Lemma 5.1.1, we deduce that the coefficient of ch Lan  is zero. Thus the induction starts. Now suppose 1 ≤ k < n and that we have shown cL = 0 for all L with
a L > k. Apply ak to the equation to deduce that  cL ch ak L = 0 L with
a L=k

Now each such ak L is isomorphic to ˜eak L
Lak , according to Lemmas 5.1.4 and 5.2.1(i). Moreover, for L  L , e˜ ak L  e˜ ak L by Corollary 5.2.4. So now the induction hypothesis on n gives that all such coefficients cL are zero, as required. Corollary 5.3.2 If L is an irreducible module in
n -mod, then L  L . Proof Since xi  = xi ,  leaves characters invariant. Hence it leaves irreducibles invariant, since they are determined up to isomorphism by their character according to the theorem. Now we can deduce the following criterion for irreducibility: Lemma 5.3.3 Let M ∈
m -mod and N ∈
n -mod be irreducible modules. Suppose: n+m (i) indm+n mn M
N  ind nm N
M; m+n (ii) M
N appears in resm+n mn ind mn M
N with multiplicity one.

Then indm+n mn M
N is irreducible. Proof Suppose for a contradiction that K = indm+n mn M
N is reducible. Then we can find a proper irreducible submodule S, and set Q = K/S. By Frobenius reciprocity, M
N appears in resm+n mn Q with non-zero multiplicity.

5.5 Alternative descriptions of
a

51

Hence, it cannot appear in resm+n mn S by assumption (ii). But assumption (i), Corollary 5.3.2, and Theorem 3.7.5 show that K  K  . Hence, K also has a quotient isomorphic to S   S, and the Frobenius reciprocity argument implies that M
N appears in resm+n mn S, giving a contradiction.

5.4 Labels for irreducibles We introduce some notation to label the isomorphism classes of irreducible representations. Write 1 for the (trivial) irreducible module of
0  F . If L is an irreducible module in
n -mod, we easily show using Lemma 5.2.3 repeatedly that L  f˜an    f˜a2 f˜a1 1 for at least one tuple a = a1  a2      an  ∈ F n . So if we define La = La1      an  = f˜an    f˜a2 f˜a1 1

(5.14)

we obtain a labeling of all irreducibles by tuples in I n . For example, La a     a (n times) is precisely the Kato module Lan  introduced in (4.4). Of course, the problem with this labeling is that a given irreducible L will in general be parametrized by several different tuples a ∈ F n . But basic properties of La are easy to read off from the notation: for instance the central character of La is a .

5.5 Alternative descriptions of a In this section we give three new interpretations of the functions
a . Theorem 5.5.1 Let a ∈ F and M be an irreducible module in
n -mod. Then:  (i) ea M =
a M˜ea M + cr Nr where the Nr are irreducible modules with
a Nr  <
a ˜ea M =
a M − 1; (ii)
a M is the maximal size of a Jordan block of xn on M with eigenvalue a; (iii) The algebra End
n−1 ea M is isomorphic to the algebra of truncated polynomials F x /x
a M . In particular, dim End
n−1 ea M =
a M

Crystal operators

52

Proof Let
=
a M and N = e˜ a
M. (i) By Lemma 5.1.4 and Frobenius reciprocity, there is a short exact sequence 0 −→ R −→ indnn−

N
La
 −→ M −→ 0 Moreover, by Lemma 5.1.3(iii),
a L <
for all composition factors L of R.Applying the exact functor a , we obtain the exact sequence 0 −→ a R −→ a indn−

N
La
 −→ a M −→ 0 As a N = 0, the Mackey Theorem yields
a indnn−

N
La
  indn−11 n−

−11 N
a La 

By considering characters, we see that a La
 =
La
−1 
La  Hence,
−1 a indnn−

N
La
 =
indn−11 
La  n−

−11 N
La

(5.15)

By Lemma 5.2.1(ii), the head of
−1 indn−11 
La n−

−11 N
La

is f˜a
−1 N 
La which is the same as ˜ea M
La, and all other composition factors of this module are of the form L
La with
a L <
− 1, thanks to Lemma 5.1.3. Moreover, all composition factors of a R are of the form L
La with
a L <
− 1. So we have now shown that  a M =
˜ea M
La + cr Nr
La for irreducibles Nr with
a Nr  <
a ˜ea M, which implies (i). (ii) We know that a
M  N
La
. So, applying the automorphism  to Lemma 4.3.1, we deduce that the maximal size of a Jordan block of xn on a
M is
. Hence the maximal size of a Jordan block of xn on a M is at least
. However, the argument given above in deriving (5.15) shows that the module a indnn−

N
La
 has a filtration with
factors, each of which is isomorphic to
−1 indn−11 
La n−

−11 N
La
−1 Since xn − a annihilates indn−11 
La, it follows that n−

−11 N
La n

xn − a annihilates a indn−

N
La . So certainly xn − a
annihilates its quotient a M. So the maximal size of a Jordan block of xn on a M is at most
.

5.5 Alternative descriptions of
a

53

(iii) The left multiplication by the element xn − a, which centralizes the subalgebra
n−1 of
n , induces an
n−1 -endomorphism  ea M → ea M By (ii),
−1 = 0 and
= 0. Hence, 1     
−1 give
linearly independent
n−1 -endomorphisms of ea M. They span a subalgebra of End
n−1 ea M isomorphic to the algebra of truncated polynomials F x /x
. However, ea M has irreducible head e˜ a M, and this appears in ea M with multiplicity
by (i). So the dimension of End
n−1 ea M is at most
. The following result is quite surprising. Corollary 5.5.2 Let M N ∈
n -mod be irreducible modules with M  N . Then, for every a ∈ F , we have Hom
n−1 ea M ea N  = 0 Proof Suppose there is a non-zero homomorphism  ea M → ea N . Then, since ea M has irreducible head e˜ a M, we see that ea N has e˜ a M as a composition factor. Hence, by Theorem 5.5.1(i),
a ˜ea N  ≥
a ˜ea M. However, ea N has irreducible socle e˜ a N , so ea M has e˜ a N as a composition factor, which gives the inequality the other way round. Thus
a ˜ea N  =
a ˜ea M. But then, e˜ a M is a composition factor of ea N with
a ˜ea M =
a ˜ea N , hence by Theorem 5.5.1(i) again, e˜ a M  e˜ a N . But this contradicts Corollary 5.2.4.

6 Character calculations

In the first section of this chapter we explain how to reduce the study of irreducible
n -modules to special blocks, namely the blocks corresponding to the orbits Sn · a such that ai s appearing in a cannot be split into two subsets B  C with the property ai − aj = ±1 for any ai ∈ B and any aj ∈ C (note ai = aj is allowed). The precise statement is Theorem 6.1.4. One implication of this result is that we may concentrate on the so-called integral representations of
n , see Chapter 7. Theorem 6.1.4 is also useful for concrete calculations in the small rank cases, that is the cases of representations of
n for n ≤ 4. The explicit character information for some small rank cases obtained in this chapter is exactly what we need in order to verify that the linear operators ea on the Grothendieck group, induced by the exact functors with the same name, satisfy Serre relations of an affine Kac–Moody algebra, see Lemma 9.2.4. We also use this small rank character information to define and investigate some generalizations f˜ar bas of the operators f˜a on the irreducible modules. These will be used only in a couple of technical points in Section 8.4.

6.1 Some irreducible induced modules Given a = a1      an  ∈ F n , let inda = inda1      an  = indn11 La1 
· · ·
Lan  By Lemma 4.1.2, every irreducible constituent of inda belongs to the block corresponding to the orbit Sn · a.

54

6.1 Some irreducible induced modules

55

Lemma 6.1.1 Let a ∈ F n . Then: (i) resn11 La has a submodule isomorphic to La1 
· · ·
Lan ; (ii) inda contains a copy of La in its head; (iii) every irreducible module in the block corresponding to the orbit Sn · a appears at least once as a constituent of inda. Proof (i) Proceed by induction on n, the case n = 1 being clear. Let n > 1. By Frobenius reciprocity, there is a non-zero, hence necessarily injective,
n−11 -module homomorphism from La1      an−1 
Lan  to resnn−11 La. Hence by inductive assumption we get a copy of La1 
· · ·
Lan−1 
Lan  in resn11 La. (ii) Use (i) and Frobenius reciprocity. (iii) By (ii), La appears in inda. But for any other b in the same orbit, indb has the same character as inda, hence they have the same set of composition factors thanks to Theorem 5.3.1. Hence, Lb appears in inda. Lemma 6.1.2 Let a ∈ F m  b ∈ F n be tuples such that ar − bs = 0 ±1 for all 1 ≤ r ≤ m and 1 ≤ s ≤ n. Then m+n indm+n mn La
Lb  ind nm Lb
La

is irreducible. Proof By the Shuffle Lemma, La
Lb appears in m+n resm+n mn ind mn La
Lb

with multiplicity 1. So, in view of Lemma 5.3.3, it suffices to show that m+n indm+n mn La
Lb  ind nm Lb
La

By the Mackey Theorem and central characters argument m+n resm+n mn ind nm Lb
La

contains La
Lb as a summand with multiplicity one, all other constituents lying in different blocks. Hence by Frobenius reciprocity, there exists a nonzero homomorphism m+n f  indm+n mn La
Lb → ind nm Lb
La

56

Character calculations

Every homomorphic image of indm+n mn La
Lb contains an
mn -submodule isomorphic to La
Lb. So, by Lemma 6.1.1(i), we see that the image of f contains a m+n -submodule V isomorphic to La1 
· · ·
Lam 
Lb1 
· · ·
Lbn  Next we claim that the image of f also contains a m+n -submodule isomorphic to Lb1 
· · ·
Lbn 
La1 
· · ·
Lam 

(6.1)

Indeed, let V = Fv, and consider m v. By (3.20) and the assumption that am − b1 = ±1, m2 acts on v by a non-zero scalar. So by (3.21), m V = 0 is a m+n -submodule isomorphic to La1 
· · ·
Lam−1 
Lb1 
Lam 
Lb2 
· · ·
Lbn  Next apply m−1      1 to move Lb1  to the first position, and continue in this way to complete the proof of the claim. Now, by the Shuffle Lemma, all composition factors of m+n resm+n 11 ind nm Lb
La

isomorphic to (6.1) necessarily lie in the irreducible
nm -submodule 1 ⊗ Lb
La of the induced module. Since this generates all of indm+n nm Lb
La as an
m+n -module, this shows that f is surjective. Hence f is an isomorphism by dimension, which completes the proof. Remark 6.1.3 Keep the assumptions of the lemma. Set  = Sm · a  = Sn · b  ∪  = Sm+n · a ∪ b   = Sm × Sn · a b Then the argument as above shows that the functor indm+n mn induces an equivalence of categories
m ⊗
n -mod  
m+n -mod ∪   Theorem 6.1.4 Let a ∈ F m  b ∈ F n be tuples such that ar − bs = ±1 for all 1 ≤ r ≤ m and 1 ≤ s ≤ n. Then m+n m+n La
Lb  indnm Lb
La indmn

6.2 Calculations for small rank

57

is irreducible. Moreover, every other irreducible
m+n -module lying in the m+n   same block as indm+n mn La
Lb is of the form ind mn La 
Lb  for  permutations a of a and b of b. Proof We prove the first statement by induction on m + n, the case m + n = 1 being trivial. For m + n > 1, we may assume by Lemma 6.1.2 that there exists c ∈ F that appears in both the tuples a and b. Note then that for every r = 1     m, either ar = c or ar − c = ±1, and similarly for every s = 1     n, either bs = c or bs − c = ±1. So by the induction hypothesis, we have that La  indnn−rr La 
Lcr 

and

 s Lb  indm m−ss Lb 
Lc 

for some r s ≥ 1, where a  b are tuples with no entries equal to c. By Lemma 6.1.2, m−s+r   r r indr+m−s rm−s Lc 
Lb   ind m−sr Lb 
Lc 

So using Theorem 4.3.2(i) and transitivity of induction, m+n   r+s indm+n  mn La
Lb  ind n−rm−sr+s La 
Lb 
Lc

(6.2)

and similarly m+n   r+s indm+n  nm Lb
La  ind n−rm−sr+s Lb 
La 
Lc

(6.3)

Now, the right-hand sides of (6.2) and (6.3) are irreducible and isomorphic to each other by the induction hypothesis and Lemma 6.1.2. This proves the first statement of the theorem. The second statement is an easy consequence of the first one and Lemma 6.1.1(iii).

6.2 Calculations for small rank We will need to know the characters of certain
n -modules for small n. Lemma 6.2.1 Let a b ∈ F with a − b = ±1. We have: (i) ch La b = La
Lb , and there is a non-split short exact sequence 0 −→ La b −→ indb a −→ Lb a −→ 0 (ii) If p = 2 then ch La a b = 2La
La
Lb  ch La b a = La
Lb
La  ch Lb a a = 2Lb
La
La 

Character calculations

58

and there are non-split short exact sequences 0 → La a b → ind321 La b
La → La b a → 0 0 → La b a → ind321 Lb a
La → Lb a a → 0 Proof (i) Set M = indb a. An easy calculation shows that N = x2 − aM is an
2 -submodule of M with character La
Lb . Then ch M/N = Lb
La . It follows that N  La b and M/N  Lb a. To prove that the extension is non-split apply Lemma 5.1.5. (ii) Set M = ind321 La b
La, and let La b = Fv and La = Fw. Then we have x1 v = av x2 v = bv s1 v = v, and x3 w = aw. Moreover, 1 ⊗ v ⊗ w s2 ⊗ v ⊗ w s1 s2 ⊗ v ⊗ w

is a basis of M. Using this and relations in
3 we easily check that N = x3 − aM is an irreducible 2-dimensional
3 -submodule of M with character 2La
La
Lb . Then M/N has character La
Lb
La so must be isomorphic to La b a. Next, we obtain the character of Lb a a, by twisting La a b with , see Section 3.6. To prove that the short exact sequences do not split apply Lemma 5.1.5. For a b ∈ F with a = b set ⎧ ⎪ ⎨ 0 if a − b = ±1, kab = 1 if a − b = ±1 and p > 2, ⎪ ⎩ 2 if a − b = ±1 and p = 2. Lemma 6.2.2 Let a b ∈ F with a − b = ±1, and set k = kab . Then Lak+1  b  Lak  b a and, for every r s ≥ 0 with r + s = k, r s Lar  b as+1   indk+2 k+11 La  b a 
La r s  indk+2 1k+1 La
La  b a 

with character r!s + 1!La
r
Lb
La
s+1 + r + 1!s!La
r+1
Lb
La
s 

(6.4)

6.2 Calculations for small rank

59

Proof The isomorphism k+2 r s r s indk+2 k+11 La  b a 
La  ind 1k+1 La
La  b a 

will follow from Corollary 5.3.2 and Theorem 3.7.5, once we prove that the first induced module is irreducible. Let k M = indk+2 k+11 La  b
La

In view of Lemma 6.2.1 and Theorem 4.3.2, we have k k+1 
Lb resk+2 k+11 M  La  b
La ⊕ La

(6.5)

So the
k+11 -submodule xk+2 − aM is isomorphic to Lak+1 
Lb. To prove that M is irreducible, it suffices to show that this submodule is not invariant under sk+1 , which is an explicit calculation. Hence, there is an irreducible
k+2 -module M with character k + 1!La
k+1
Lb + k!La
k
Lb
La  Moreover, e˜ a M  Lak  b and e˜ b M  Lak+1  by (6.5). So M  Lak  b a  Lak+1  b thanks to Lemma 5.2.3. Now consider the remaining irreducibles in the block. There are at most k remaining, namely Lar bas+1  for r ≥ 0 s ≥ 1 with r + s = k. Twisting with the automorphism  and using Lemma 3.6.1 gives a new irreducible module k indk+2 1k+1 La
Lb a  with character k + 1!Lb
La
k+1 + k!La
Lb
La
k  If p > 2, it must be Lb a a, and we are finished. Let p = 2. Consider M = ind431 La b a
La. By Lemma 6.2.1 and the Shuffle Lemma, ch M = 2La
La
Lb
La + 2La
Lb
La
La  Moreover the character of L must be -invariant, as otherwise we would produce two new modules in our block. So either ch L = ch M or ch L = 1/2ch M. We claim that the former happens. Indeed, La
La
Lb
La ⊆ res4 L implies by Frobenius reciprocity and Theorem 4.3.2 that La2 
Lb
La ⊆ res4211 L

Character calculations

60

Finally, we use Lemma 6.1.1(i) to identify the last module as La b a a and the previous one as Lb a a a.

6.3 Higher crystal operators In this section we will introduce certain generalizations of the crystal operators f˜j . To simplify notation, we will write simply ind in place of indn throughout the section. Lemma 6.3.1 Let a b ∈ F with a = b. For any r s ≥ 0 with r + s = kab , and m ≥ 0, ind Lar  b as 
Lam   ind Lam 
Lar  b as  is irreducible. Proof We may assume that a − b = ±1, as otherwise the result is immediate from Theorem 6.1.4. Now, we claim that ind Lar  b as 
Lam   ind Lam 
Lar  b as  Indeed, transitivity of induction and Lemma 6.2.2 give that ind Lar  b as 
Lam   ind Lar  b as 
La
Lam−1   ind La
Lar  b as 
Lam−1  and now repeating this argument m − 1 more times gives the claim. Hence, by Corollary 5.3.2 and Theorem 3.7.5, K = ind Lar  b as 
Lam  is self-dual. Now suppose for a contradiction that K is reducible. Then we can pick a proper irreducible submodule S of K, and set Q = K/S. Applying Lemmas 6.2.1 and 4.1.3, we see that ch K equals m    m r + t!s + m − t!La
r+t
Lb
La
s+m−t  t t=0 By Frobenius reciprocity, Q contains an
r+s+1m -submodule isomorphic to Lar  b as 
Lam  So by Lemma 6.1.1(i), the irreducible r+s+m+1 -module V = La
r
Lb
La
s+m

6.3 Higher crystal operators

61

appears in Q as a submodule, hence in fact by Theorem 4.3.2(i) it must appear with multiplicity r!s + m! (viewing Q as a module over
r1s+m ). It follows that V is not a composition factor of S. But this is a contradiction, since, as K is self-dual, S  S  is a quotient module of K, hence it must contain V by the argument above applied to the quotient Q = S. Lemma 6.3.2 Let a b ∈ F with a = b, and m be a non-negative integer. For any r ≥ 1 and s ≥ 0 with r +s = kab and any irreducible module M ∈
n -mod, hd ind M
Lam 
Lar  b as  is irreducible. Proof By the argument in the proof of Lemma 5.1.5, it suffices to prove this in the special case that
a M = 0. Let t = m + r + s + 1. Recall from the previous lemma that N = ind Lam 
Lar  b as  is an irreducible
t -module. Moreover by Lemma 6.2.1, ch Lar  b as  = r!s!La
r
Lb
La
s  So, since
a M = 0 and r > 0, the Mackey Theorem and a block argument give m r s resn+t nt ind M
La 
La  b a   M
N  ⊕ U

for some
nt -module U all of whose composition factors lie in different blocks to those of M
N . Now let H = hd ind M
Lam 
Lar  b as  ¯ ¯ It follows from above that resn+t nt H  M
N  ⊕ U , where U is some quotient module of U . Then: Hom
n+t H H  Hom
n+t ind M
Lam 
Lar  b as  H  Hom
nt M
N resn+t nt H  Hom
nt M
N M
N ⊕ U¯   Hom
nt M
N M
N   F Since H is completely reducible, this implies that H is irreducible, as required.

Character calculations

62

Now we can define the higher crystal operators. Let a b ∈ F with a = b, and r ≥ 1 s ≥ 0 satisfy r + s = kab . Then the special case m = 0 of Lemma 6.3.2 shows that f˜ar bas M = hd ind M
Lar  b as  is irreducible for every irreducible M ∈
n -mod. Lemma 6.3.3 Take a b ∈ F with a = b and set k = kab . Let M ∈
n -mod be irreducible. (i) There exists a unique integer r with 0 ≤ r ≤ k such that for every m ≥ 0 we have
a f˜am f˜b M = m +
a M − r (ii) Assume m ≥ k. Then a copy of f˜am f˜b M appears in the head of ind f˜am−k M
Lar  b ak−r  where r is as in (i). In particular, if r ≥ 1, then f˜am f˜b M  f˜ar bak−r f˜am−k M Proof Let
=
a M and write M = f˜a
N for an irreducible
n−
-module N with
a N  = 0. It suffices to prove (i) for any fixed choice of m, the result for all other m ≥ 0 then following immediately by (5.12). So take m ≥ k. Note that f˜am f˜b M = f˜am f˜b f˜a
N is a quotient of ind N
La

Lb
La
k
Lam−k  which by Lemma 6.2.1 has a filtration with factors isomorphic to Fr = ind N
La

Lar  b ak−r 
Lam−k 

0 ≤ r ≤ k

So f˜am f˜b M is a quotient of some such factor, and to prove (i) it remains to show that
a L =
+ m − r for any irreducible quotient L of Fr . The inequality
a L ≤
+ m − r is clear from the Shuffle Lemma. However, by transitivity of induction and Lemma 6.3.1 Fr  ind N
ind Lar  b ak−r 
La
+m−k  Moreover N
ind Lar  b ak−r 
La
+m−k  is irreducible by Lemma 6.3.1 again. So by Frobenius Reciprocity, it is a submodule of resn−
m+1+
L. Hence
a L ≥
+ m − r.

6.3 Higher crystal operators

63

For (ii), note by Lemma 6.3.1 and transitivity of induction that we also have Fr  ind N
Lam−k+

Lar  b ak−r  Now, by the Shuffle Lemma and Lemma 5.1.5, the only irreducible factors K of Fr with
a K =
+ m − r come from its quotient ind f˜am−k+
N
Lar  b ak−r   indf˜am−k M
Lar  b ak−r  Finally, in case r ≥ 1, the head of the last module is precisely f˜ar bak−r f˜am−k M see Lemma 6.3.2.

7 Integral representations and cyclotomic Hecke algebras

Starting from this chapter, we will consider only integral representations of
n . By this we mean those representations for which the eigenvalues of x1      xn are all “integral”, that is belong to I =  · 1 ⊂ F . In the beginning of the first section we explain why we can restrict ourselves to the category RepI
n of integral representations essentially without loss of generality. This category is very natural from another point of view. The affine algebra
n has a natural family of finite dimensional quotient algebras, which are degenerate analogues of cyclotomic Hecke algebras of [Ch, AK, BM]. So any module over such cyclotomic quotient can be inflated to an
n -module. Now it turns out that the category RepI
n consists precisely of all such inflations from all cyclotomic quotients. As we now work in RepI
n , the es,
s, f˜s, etc. will be labeled not by arbitrary a ∈ F but only by i ∈ I. So from now on we will only have ei s,
i s, f˜i s, etc. for i ∈ I. Note that the set I can be identified with the labeling set for simple roots of 1 the Kac–Moody algebra  of type Ap−1 if p > 0 and A if p = 0. Moreover the notation ei might suggest that we want to think about these functors as the Chevalley generators of the positive part of . At this stage, all the Lie theoretic notation we are going to bring in will seem completely artificial. For example, we will label the cyclotomic quotients by the dominant weights for , and in Chapter 8 we will introduce a “Lie theoretic” notation for blocks! However it will gradually become clear that relations Lie Theory are not superficial at all. Two main theorems of the chapter are Basis Theorem and Mackey Theorem for cyclotomic Hecke algebras. We also show that cyclotomic Hecke algebras are Frobenius. Finally, we draw the reader’s attention to the small Section 7.4, where the irreducible
n -modules factoring through to the cyclotomic quotient
n are characterized in terms of the functions
∗i , which are the “left-hand versions” of the
i . 64

7.1 Integral representations

65

7.1 Integral representations Recall the set I from (1.1). Let N be an irreducible
n -module, whose character we would like to understand. If N belongs to the block corresponding to the orbit Sn · a of Sn on F n , then, in view of Theorem 6.1.4, we may assume that ar − as ∈ I for all 1 ≤ r s ≤ n. Indeed, otherwise M can be decomposed as indnmk M
K for some irreducible
m -module M and some irreducible
k -module K, where n = m + k, and we are reduced to smaller ranks. Also, we may assume that a1 ∈ I, because this can be achieved by twisting the action of
n with the algebra automorphism
n →
n 

xi → xi + b sj → sj 

which just shifts the formal character by a constant b ∈ F . Thus, it is sufficient to understand the irreducible
n -modules, which are integral in the following sense. Definition 7.1.1 A n -module M is called integral if it is finite dimensional and all eigenvalues of x1      xn on M belong to I. An
n -module, or more generally an
-module for a composition of n, is called integral if it is integral on restriction to n . We write RepI
n (resp. RepI n , RepI
) for the full subcategory of
n -mod (resp. n -mod,
-mod) consisting of all integral modules. Lemma 7.1.2 Let M be a finite dimensional
n -module and fix j with 1 ≤ j ≤ n. Assume that all eigenvalues of xj on M belong to I. Then M is integral. Proof It suffices to show that the eigenvalues of xk belong to I if and only if the eigenvalues of xk+1 belong to I, for an arbitrary k with 1 ≤ k < n. Actually, by an argument involving conjugation with the automorphism , it suffices just to prove the “if ” part. So assume that all eigenvalues of xk+1 on M belong to I. Let a be an eigenvalue for the action of xk on M. Since xk and xk+1 commute, we can pick v lying in the a-eigenspace of xk so that v is also an eigenvector for xk+1 , of eigenvalue b say. By assumption we have b ∈ I. Now let k be the intertwining element (3.19). By (3.21), we have xk+1 k = k xk  So if k v = 0, we get that ak v = k xk v = xk+1 k v whence a is an eigenvalue of xk+1 , and so a ∈ I by assumption. Else, k v = 0 so k2 v = 0. So applying (3.20), we again get that a ∈ I.

Integral representations and cyclotomic Hecke algebras

66

Lemma 7.1.3 Let be a composition of n and M be an integral
-module. Then indn M is an integral
n -module. Proof By Theorem 3.2.2, indn M is spanned by elements w ⊗ m for w ∈ Sn and m ∈ M, in particular it is finite dimensional. Let Yj =



xj − i

1≤j≤n

i∈I

 (If I is infinite, take a product Ki=0 xj − i for sufficiently large K instead). By Lemma 7.1.2, it suffices to show that Y1N annihilates indn M for sufficiently large N . Consider Y1N w ⊗ m for w ∈ Sn  m ∈ M. We may write w = us1    sk for u ∈ S2n  Sn−1 and 0 ≤ k < n. Then Y1N commutes with u, so we just need to consider Y1N s1    sk ⊗ m. Now using the commutation relations, we check that Y1N s1    sk ⊗ m can be rewritten as an
n -linear combination of  elements of the form 1 ⊗ YjN m for 1 ≤ j ≤ n and N − k ≤ N  ≤ N . Since M is integral by assumption, we can choose N sufficiently large so that each such term is zero. It follows that the functors indn  resn restrict to well-defined functors indn  RepI
→ RepI
n 

resn  RepI
n → RepI


(7.1)

on integral representations. Similar remarks apply to more general induction and restriction between nested parabolic subalgebras of
n .

7.2 Some Lie theoretic notation We introduce some standard Lie theoretic notation, which at first will be used just as a book-keeping device, but later it will turn out to be deeply connected with the theory we are considering. Assume first that p > 0. In this case we set  = p − 1 and denote by  the 1 affine Kac–Moody algebra of type A over C, see [Kc, Ch. 4, Table Aff 1]. In particular, we label the Dynkin diagram by the index set I = 0 1     

as follows: 0

c H HH   c Hc c c 1

2

−1



if  ≥ 2, and

c<> c 0

1

if  = 1.

7.2 Some Lie theoretic notation

67

The weight lattice is denoted P, the simple roots are i  i ∈ I ⊂ P and the simple coroots are hi  i ∈ I ⊂ P ∗ . The Cartan matrix 
corresponding hi  j  0≤ij≤ is ⎞ ⎛ 2 −1 0 · · · 0 0 −1 ⎜−1 2 −1 · · · 0 0 0⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 −1 2 · · · 0 0 0 ⎟ ⎜ ⎟ ⎜   ⎟ if  ≥ 2, ⎜ ⎟ ⎜ ⎜0 0 0  2 −1 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎝0 0 0    −1 2 −1⎠ −1 0 0  0 −1 2 and



2 −2 −2 2

 if  = 1.

Let i i ∈ I ⊂ P denote fundamental dominant weights, so that hi  j  = ij , and let P+ ⊂ P denote the set of all dominant integral weights. Set c=

 

=

hi 

i=0

 

i 

(7.2)

i=0

Then the 0         form a -basis for P, and c i  = hi   = 0 for all i ∈ I. In the case p = 0, we make the following changes to these definitions. First, we let  = , and  denotes the Kac–Moody algebra of type A , see [Kc, Section 7.11]. So I = , corresponding to the nodes of the Dynkin diagram −1

c

0

c

1

c

Note certain notions, for example the element c from (7.2), only make sense if we pass to the completed algebra a , see [Kc, Section 7.12], though the intended meaning whenever we make use of them should be obvious regardless. Now, for either  <  or  = , we let U denote the -subalgebra of the universal enveloping algebra of  generated by the Chevalley generators ei  fi  hi i ∈ I. Recall these are subject only to the relations hi  hj = 0

ei  fj = ij hi 

(7.3)

hi  ej = hi  j ej 

hi  fj = −hi  j fj 

(7.4)

ad ei 1−hi k  ek = 0

ad fi 1−hi k  fk = 0

(7.5)

68

Integral representations and cyclotomic Hecke algebras

for all i j k ∈ I with i = k. We let U denote the -form of U generated by the divided powers n

ei = ein /n! and

n

fi

= fin /n!

Then U has the usual triangular decomposition U = U− U0 U+  We are particularly concerned here with the plus part U+ , generated by all n ei . It is a graded Hopf algebra over  via the principal grading where n degei  = n for all i ∈ I n ≥ 0.

7.3 Degenerate cyclotomic Hecke algebras For  ∈ P+ we set f =



x1 − ihi  ∈
n 

(7.6)

i∈I

(Note the product is finite even if  = ). Let  denote the two-sided ideal of
n generated by f , and define the (degenerate) cyclotomic Hecke algebra to be the quotient
n =
n /  Also, by agreement,
0  F . The algebra
n defined for  ∈ P+ should not be confused with the parabolic subalgebra
⊂
n defined earlier for a composition of n. The following result explains the relation between integral modules and cyclotomic Hecke algebras. It shows that an
n -module is integral if and only if it is an inflation of a finite dimensional
n -module for some “sufficiently large” . Lemma 7.3.1 Let M be a finite dimensional
n -module. Then M is integral if and only if  M = 0 for some  ∈ P+ . Proof If  M = 0, then the eigenvalues of x1 on M are all in I, by definition of  . Hence M is integral in view of Lemma 7.1.2. Conversely, suppose that M is integral. Then the minimal polynomial of x1 on M is of the form  i i∈I t − i for some i ≥ 0. So if we set  = 0 0 + 1 1 + · · · +   ∈ P+  we certainly have that  M = 0.

7.4 The ∗-operation

69

Lemma 7.3.1 allows us to introduce the functors pr   RepI
n →
n -mod

infl 
n -mod → RepI
n 

(7.7)

Here, infl is simply inflation along the canonical epimorphism
n →
n , while on a module M, pr  M = M/ M with the induced action of
n . The functor infl is right adjoint to pr  , that is there is a functorial isomorphism Hom
n pr  M N   Hom
n M infl N 

(7.8)

Note we will generally be sloppy and omit the functor infl in our notation. In other words, we generally identify
n -mod with the full subcategory of RepI
n consisting of all modules M with  M = 0.

7.4 The ∗-operation Suppose M is an irreducible module in RepI
n and 0 ≤ m ≤ n. Using Lemma 3.6.1 for the second equality in (7.10), define e˜ i∗ M = ˜ei M    f˜i∗ M = f˜i M   = hd indn+1 1n Li
M
∗i M =
i M   = max m ≥ 0  ˜ei∗ m M = 0 

(7.9) (7.10) (7.11)

We may think of the “starred” notions as left-hand versions of the original notions, which are right-hand versions. For example,
∗i M can be worked out just from knowledge of the character of M as the maximal k such that Li
k
   appears in ch M, while for
i M we would take · · ·
Li
k here. Recalling the definition of the ideal  generated by the element (7.6), Theorem 5.5.1(ii) has the following important corollary. Corollary 7.4.1 Let M be an irreducible module in RepI
n and  ∈ P+ . Then pr  M = M if and only if
∗i M ≤ hi   for all i ∈ I. Proof In view of Theorem 5.5.1(ii),
∗i M is the maximal size of a Jordan block of x1 on M with eigenvalue i. The result follows immediately.

70

Integral representations and cyclotomic Hecke algebras

7.5 Basis Theorem for cyclotomic Hecke algebras The goal in this section is to describe the Ariki–Koike explicit basis for
n , see [AK]. Our method does not use representation theory of Hn and so it is very different from that of [AK]. Let d = c . Then f is a monic polynomial of degree d. Write f = x1d + ad−1 x1d−1 + · · · + a1 x1 + a0  Set f1 = f and for i = 2     n, define inductively fi = si−1 fi−1 si−1  The first lemma follows easily by induction using (3.9). Lemma 7.5.1 For i = 1     n, we have fi = xid + terms lying in i−1 xie i for 0 ≤ e < d Given Z = z1 < · · · < zu ⊆ 1     n , let fZ = fz1 fz2    fzu ∈
n  Also, define "n =  Z  Z ⊆ 1     n   ∈ n+ with i < d whenever i

Z 

"+ n =  Z ∈ "n  Z = ∅  Lemma 7.5.2
n is a free right n -module on basis x fZ   Z ∈ "n  Proof Define a lexicographic ordering on n+ :  ≺  if and only if n = n      k+1 = k+1  k < k for some k = 1     n. Define a function   "n → n+ by  Z = 1      n , where

i i = i + d

if i

Z,

if i ∈ Z.

Using induction on n and Lemma 7.5.1, we prove for  Z ∈ "n x fZ = xZ + terms lying in x n for  ≺  Z

(7.12)

7.5 Basis Theorem for cyclotomic Hecke algebras

71

Since   "n → n+ is a bijection and we already know that the x   ∈ n+

form a basis for
n viewed as a right n -module by Theorem 3.2.2, (7.12) implies the lemma. Lemma 7.5.3 For n > 1 we have n−1 fn n = fn n . Proof It suffices to show that the left multiplication by the elements s1      sn−2 leaves the space fn n invariant. But this follows from the definition of fn and braid relations in Sn . Lemma 7.5.4 We have  =

n 

 n fi  n .

i=1

Proof We have  =
n f1
n =
n f1 n n =
n f1 n =  n n f 1 n =

n  

n si−1    s1 uf1 n

i=1 u∈S2n

=

n 

n si−1    s1 f1 n =

i=1

n 

n fi n 

i=1

as required. 

Lemma 7.5.5 For d > 0 we have  =

x  fZ  n 

 Z∈"+ n

Proof Proceed by induction on n, the case n = 1 being obvious. Let n > 1. Denote  =
n−1 f
n−1 , so    = x fZ n−1 (7.13)   Z ∈"+ n−1

by the induction hypothesis. Let =



x  fZ  n 

 Z∈"+ n

Obviously  ⊆  . So in view of Lemma 7.5.4, it suffices to show that x fi n ⊆  for each  ∈ n+ and each i = 1     n.

72

Integral representations and cyclotomic Hecke algebras

 Consider first x fn n . Write x = xnn x for  ∈ n−1 + . Expanding x in terms of the basis of
n−1 from Lemma 7.5.2, we see that   x  f n n ⊆ xnn x fZ n−1 fn n   Z ∈"n−1

which is contained in  thanks to Lemma 7.5.3. Finally, consider x fi n with i < n. Write x = xnn x for  ∈ n−1 + . By the induction hypothesis,   x fi n = xnn x fi n ⊆ xnn x fZ n   Z ∈"+ n−1 

Now we show by induction on n that xnn x fZ n ⊆  for each   Z  ∈ "+ n−1 . This is immediate if n < d, so take n ≥ d and consider the induction step. Expanding fn using Lemma 7.5.1, the set 

xnn −d x fZ fn n ⊆  

looks like the desired xnn x fZ n plus a sum of terms belonging to xnn −d+e  n with 0 ≤ e < d. It now suffices to show that each such xnn −d+e  n ⊆  . But by (7.13),   xnn −d+e x fZ n xnn −d+e  n ⊆  Z ∈"+ n−1

and each such term lies in  by induction, since 0 ≤ n − d + e < n . Theorem 7.5.6 The canonical images of the elements x w   ∈ n+ with 1      n < d w ∈ Sn

form a basis for
n . Proof By Lemmas 7.5.2 and 7.5.5, the elements x fZ   Z ∈ "+ n form a basis for  viewed as a right n -module. Hence Lemma 7.5.2 implies that the elements x   ∈ n+ with 1      n < d

form a basis for a complement to  in
n viewed as a right n -module. The theorem follows at once. Remark 7.5.7 It follows from Theorem 7.5.6 that
n0 is isomorphic to the group algebra FSn .

7.6 Cyclotomic Mackey Theorem

73

7.6 Cyclotomic Mackey Theorem We will need a special case of a Mackey Theorem for
n . Given any y ∈
n , we will denote its canonical image in
n by the same symbol. Thus, Theorem 7.5.6 says that Bn = x w   ∈ n+ with 1      n < d w ∈ Sn

(7.14)

 is a basis for
n . Also Theorem 7.5.6 implies that the subalgebra of
n+1  generated by x1      xn and w, for w ∈ Sn , is isomorphic to
n . We will





write ind
n+1 and res
n+1 for the induction and restriction functors between   n n  
n and
n+1 , to avoid confusion with the affine analogue from (3.11) and (3.12). So,


 ind
n+1  M =
n+1 ⊗
n M n

Lemma 7.6.1 (i)


n+1 is a free right
n -module on basis

xja sj    sn  0 ≤ a < d 1 ≤ j ≤ n + 1  (ii) As 
n 
n -bimodules




n+1 =
n sn
n ⊕

a xn+1
n 

0≤a
(iii) For 0 ≤ a < d, there are isomorphisms 
n sn
n 
n ⊗
n−1
n

and

a xn+1
n 
n

of 
n 
n -bimodules. Proof (i) By Theorem 7.5.6 and dimension considerations, we just need to  is generated as a right
n -module by the given elements. check that
n+1 This follows using (3.9). (ii) It suffices to notice, using (i) and (3.9), that xja sj    sn  0 ≤ a < d 1 ≤ j ≤ n

is a basis of
n sn
n as a free right
n -module. a
n 
n is clear from (i). Furthermore, (iii) The isomorphism xn+1 the map
n ×
n →
n sn
n  u v → usn v  is
n−1 -balanced, so it induces a homomorphism   
n ⊗
n−1
n →
n sn
n

74

Integral representations and cyclotomic Hecke algebras

 of 
n 
n -bimodules. By (i),
n ⊗
n−1
n is a free right
n -module on a basis

xja sj    sn−1 ⊗ 1  1 ≤ j ≤ n 0 ≤ a < d  But  maps these elements to a basis for
n sn
n as a free right
n -module, using a fact observed in the proof of (ii). This shows that  is an isomorphism.  We have now decomposed
n+1 as an 
n 
n -bimodule. So the same argument as for Theorem 3.5.2 gives:

Theorem 7.6.2 Let M be an
n -module. Then there is a natural isomorphism





n

n







⊕d n+1 res
n+1 ⊕ ind
n res
n M  ind
 M  M

of

n−1

n−1


n -modules.

7.7 Duality for cyclotomic algebras


We wish next to prove that the induction functor ind
n+1 commutes with the  n -duality. We need a little preliminary work. Lemma 7.7.1 For 1 ≤ i ≤ n and a ≥ 0, we have  sn    si xia si    sn

=

a xn+1 +

terms lying in


n sn
n +

a−2 

 k xn+1
n



k=0

Proof We apply induction on n = i i + 1    . In case n = i, the result follows from a calculation using (3.9). The induction step is similar, noting that sn  centralizes
n−1 .  Lemma 7.7.2 There exists an 
n 
n -bimodule homomorphism 
n+1 →  
n such that ker contains no non-zero left ideals of
n+1 .

Proof By Lemma 7.6.1(ii), we know that  d−1
n+1 = xn+1
n ⊕

d−2 

a xn+1
n ⊕
n sn
n

a=0


n 
n -bimodule.

 d−1 as an Let 
n+1 → xn+1
n be the projection on to the first summand of this bimodule decomposition, which by Lemma 7.6.1(iii)

7.7 Duality for cyclotomic algebras

75

is isomorphic to
n as an 
n 
n -bimodule. So we just need to show   has the property that hy = 0 for all h ∈
n+1 , then y = 0. that if y ∈
n+1 Using Lemma 7.6.1(i), we may write y=

d−1  a=0

a xn+1 ta +

d−1 n 

xja sj    sn uaj

a=0 j=1

for some elements ta  uaj ∈
n . As y = 0, we must have td−1 = 0. Now xn+1 y = 0 implies td−2 = 0. Similarly td−3 = td−4 = · · · = 0. Next, 2 sn y    , and using Lemma 7.7.1, we considering sn y xn+1 sn y xn+1 get ud−1n = ud−2n = · · · = 0. Now repeat the argument again, this time considering sn sn−1 y xn+1 sn sn−1 y    , to get that all uan−1 = 0. Continuing in this way we eventually arrive at the desired conclusion that y = 0. Now we are ready to prove the main result of the section: Theorem 7.7.3 There is a natural isomorphism   ⊗
n M  Hom
n 
n+1
n+1  M

for all
n -modules M. Proof We show that there exists an isomorphism   → Hom
n 
n+1 
n   
n+1  
n -bimodules. Then, applying the functor ? ⊗
n M, we obtain of 
n+1 natural isomorphisms ⊗id

   ⊗
n M −→ Hom
n 
n+1 
n  ⊗
n M  Hom
n 
n+1  M
n+1

as required. Note the existence of the second isomorphism here uses the fact  is a projective left
n -module and [AF, 20.10]. that
n+1 To construct , let be as in Lemma 7.7.2, and define h to be the map  , where h , for each h ∈
n+1  →
n  h → h h h 
n+1   → Hom
n 
n+1 
n  is then a wellWe can easily check that  
n+1   defined homomorphism of 
n+1 
n -bimodules. To see that it is an isomorphism, it suffices by dimensions to check it is injective. If h = 0 for some    h ∈
n+1 then for every x ∈
n+1 , xh = 0, that is the left ideal
n+1 h is contained in ker . So Lemma 7.7.2 implies h = 0.

Integral representations and cyclotomic Hecke algebras

76

Corollary 7.7.4
n is a Frobenius algebra, that is there is an isomorphism of left
n -modules
n  HomF 
n  F between the left regular module and the F -linear dual of the right regular module. Proof Proceed by induction on n. For the induction step,    
n 
n ⊗
n−1
n−1 
n ⊗
n−1 HomF 
n−1 F     
 Hom F 
  Hom
n−1
n  F  n n−1  F   Hom F 
n−1 ⊗
n−1

 HomF 
n  F  applying Theorem 7.7.3 and adjointness of ⊗ and Hom. For the next corollary, recall the duality induced by  317 on finite dimensional
n -modules. Since  leaves the two-sided ideal  invariant, it induces a duality also denoted  on finite dimensional
n -modules.


Corollary 7.7.5 The exact functor ind
n+1 is both left and right adjoint to  n




res
n+1  . Moreover, it commutes with duality in the sense that there is a natural n isomorphism





n

n

  n+1 ind
n+1  M   ind
 M

for all finite dimensional
n -modules M.





 Proof The fact that ind
n+1 =
n+1 ⊗
n ? is right adjoint to res
n+1 is immedi  n n  ate from Theorem 7.7.3, since Hom
n 
n+1  ? is right adjoint to restriction by adjointness of ⊗ and Hom. But on finite dimensional modules, a standard


  is also right adjoint to restriction. check shows that the functor   ind
n+1  n Now the remaining part of the corollary follows by uniqueness of adjoint functors. Let r ≥ 1. By Theorem 7.5.6, the subalgebra  r = sn−r+1  sn−r+2      sn−1  ⊆
n+r

is isomorphic to r . This subalgebra commutes with the subalgebra
n , and
n r is isomorphic to
n ⊗ r . From now on we will consider
n ⊗ r as a  in this way. Our goal is to generalize Theorem 7.7.3 and subalgebra of
n+r  Corollary 7.7.5 from subalgebra
n ⊂
n+1 to subalgebras
n r =
n ⊗

7.7 Duality for cyclotomic algebras

77

  r ⊂
n+r . First, we need to decompose
n+r as an 
n r 
n r -bimodule. To this end, note that the set of distinguished double coset representatives

Dnrnr = 1 = 0  1      m  where m = minn r, and i =

i−1 

n − j n + i − j

0 ≤ i ≤ m

j=0

Denote the corresponding double cosets by Ci = Sn × Sr i Sn × Sr 

0 ≤ i ≤ m

 from (7.14). Recall the basis Bn+r of
n+r

Theorem 7.7.6 Set d−1 d−1 X =
n r xn+1    xn+r
n r

and Y =



i+1 r    xn+r
n r xn+i+1 i
n r 

a

a

the sum running over all 0 ≤ i ≤ m and 0 ≤ ai+1 ≤ · · · ≤ ar < d such that a1 + · · · + ar < rd − 1 if i = 0. Then, as 
n r 
n r -bimodules, X 
n r , and  = X ⊕ Y
n+r

Moreover, X has basis b

b

n+r BX = x11    xn+r w ∈ Bn+r  w ∈ Sn × Sr 

bn+1 = · · · = bn+r = d − 1  and Y has basis BY = Bn+r \ BX  Proof Introduce a partial order on Bn+r by putting b

b

c

c

n+r n+r h1 = x11    xn+r w < h2 = x11    xn+r u

if and only if either w ∈ Ci and u ∈ Cj with i < j, or i = j and bn+1 + · · · + bn+r < cn+1 + · · · + cn+r .  = X + Y . It suffices to prove that every element We now prove that
n+r of Bn+r belongs to X + Y . Suppose this is false. Let b

b

n+r h = x11    xn+r w ∈ Bn+r

78

Integral representations and cyclotomic Hecke algebras

be a minimal element with respect to our partial order such that h ∈ X + Y . Assume that w ∈ Ci . Let V = spanh ∈ Bn+r  h < h ⊆ X + Y and write ≡ for equality modulo V . Let w = g1 i g2 for g1  g2 ∈ Sn × Sr . Using Lemma 3.2.1, we get b

b

b

n+1 n+r    xn+r g1 i g2 h = x11    xnbn xn+1

b

b

b

n+1 n+r ≡ x11    xnbn g1 xg−1    xg−1 g  n+1 n+r i 2 1

1

As sets, g1−1 n + 1     g1−1 n + r = n + 1     n + r . So, renaming bs with cs, we can write c

b

c

n+1 n+r h ≡ x11    xnbn g1 xn+1    xn+r  i g2 

Moreover, if g ∈ Sn ×Sr fixes all numbers, except possibly n+i+1     n+r, then g commutes with i , and so w = g1 i g2 = g1 gi g −1 g2 . So by changing g1 to g1 g and g2 to g −1 g2 , if necessary, we can achieve cn+i+1 ≤ · · · ≤ cn+r . Finally, using Lemma 3.2.1 again, we have c

b

c

n+1 n+r x11    xnbn g1 xn+1    xn+r  i g2

b

c

c

b

c

c

c

c

n+i+1 n+r n+1 n+i = x11    xnbn g1 xn+i+1    xn+r xn+1    xn+i i g2

c

n+i+1 n+r n+1 ≡ x11    xnbn g1 xn+i+1    xn+r i xn−i+1    xncn+i g2 

c

b

c

n+1 The last element is in X + Y , because x11    xnbn g1 and xn−i+1    xnn+i g2 belong    = X +Y. to
n r . Hence h ∈ X + Y , giving a contradiction. Thus
n+r

d−1 d−1 Next, note by Theorem 3.3.1 that the element xn+1    xn+r commutes with  
n r , so the statement about the basis of X follows immediately. It also follows that X 
n r . Now, it suffices to show that elements of Y are linear combinations of elements from BY . But Y is spanned by elements of the form a

b

a

c

i+1 r G = x11    xnbn g1 xn+i+1    xn+r i x11    xncn g2 

with g1  g2 ∈ Sn ×Sr , 0 ≤ ak  bk  ck < d, 0 ≤ ai+1 ≤ · · · ≤ ar < d, and a1 +· · ·+ ar < rd − 1 if i = 0. So it suffices to show that G is a linear combination of elements from BY . Write g1 = yz for y ∈ Sn and z ∈ Sr . Then b

c

c

a

a

c

i+1 n−i+1 n−i r G = x11    xnbn yx11    xn−i zxn+i+1    xn+r i xn−i+1    xncn g2 

7.7 Duality for cyclotomic algebras c

79

c

n−i Note yx11    xn−i ∈
n , so it can be written as a linear combination of b b the standard basis elements of the form x11    xnn y for y ∈ Sn . So we may assume that

a

b

c

a

i+1 n−i+1 r G = x11    xnbn yzxn+i+1    xn+r i xn−i+1    xncn g2 

Assume first that i = 0. Then b

a

a

b

a

a

1 r 1 r    xn+r g2 = x11    xnbn zxn+1    xn+r yg2  G = x11    xnbn yzxn+1

a

a

1 r In view of Lemma 3.2.1 applied to zxn+1    xn+r we conclude that G is a linear combination of elements in BY . cn−i+1    xncn we see that this Now, let i > 0. Applying Lemma 3.2.1 to i xn−i+1 can be written as a linear combination of elements of the form

d

d

d

n−i+1 n+1 n+i xn−i+1    xndn xn+1    xn+i 

where either  = i or the degree dn−i+1 + · · · + dn + dn+1 + · · · + dn+i < id − 1

(7.15)

So we may assume that a

b

a

d

d

d

i+1 n−i+1 n+1 n+i r G = x11    xnbn yzxn+i+1    xn+r xn−i+1    xndn xn+1    xn+i g2

and (7.15) holds. Using Lemma 3.2.1 to commute z past the element ai+1 dn+1 dn+i dn−i+1 ar    xn+r xn+1    xn+i and y past xn−i+1    xndn , we conclude again that G xn+i+1 is a linear combination of elements from BY , as the maximal possible degree rd − 1 on xn+1      xn+r cannot be achieved. Corollary 7.7.7 There exists an 
n r 
n r -bimodule homomorphism   →
n r such that ker contains no non-zero left ideals of
n+r . 
n+r Proof Using the notation of Theorem 7.7.6, let be the projection to X along  has the property that hy ∈ Y for Y . So we just need to show that if y ∈
n+r  all h ∈
n+r , then y = 0. Assume for a contradiction that there is a non-zero such y. By applying the antiautomorphism  to the bases BX and BY we see that X also has a basis d−1 d−1 BX = wx11    xnbn xn+1    xn+r  w ∈ Sn × Sr  0 ≤ b1      bn < d  b

and Y also has basis n+r  w ∈ Sn × Sr or bn+1 + · · · + bn+r < rd − 1  BY = wx11    xn+r

b

b

Write y as a linar combination of basis elements from BY :  bn+r b  y = cbw wx11    xn+r

Integral representations and cyclotomic Hecke algebras

80

Multiplying on the left with elements from Sn+r , we may assume that some terms with w = w0 , the longest element of Sn+r , appear. Order them lexicographically so that b

b

c

c

c

c

n+r n+r w0 x11    xn+r < w0 x11    cn+r n+r if bk < ck and cl = bl for any l > k. Let w0 x11    xn+r be the top term. Then, in view of Lemma 3.2.1, the product

y = xn+1

d−1−cn+1

d−1−cn+r

   xn+r

y

contains the term c

d−1 d−1 w0 x11    xncn xn+1    xn+r

(when decomposed in terms of the basis BX ∪ BY ). But then w0−1 y contains d−1 d−1 x11    xncn xn+1    xn+r ∈ BX  c

giving a contradiction. Corollary 7.7.8 Let r ≥ 1. Then: (i) There is a natural isomorphism  
n+r ⊗
n ⊗r M  Hom
n ⊗r 
n+r  M

for all
n ⊗ r -modules M.


(ii) The exact functor ind
n+r  ⊗ is both left and right adjoint to the functor


r

n

res
n+r . Moreover, it commutes with duality in the sense that there is  n ⊗r a natural isomorphism





  n+r ind
n+r  ⊗ M   ind
 ⊗ M n

for all finite dimensional

r


n ⊗ r -modules

n

r

M.

Proof The proof is the same as that of Theorem 7.7.3 and Corollary 7.7.5, except that it uses Corollary 7.7.7 instead of Lemma 7.7.2.

7.8 Presentation for degenerate cyclotomic Hecke algebras Cyclotomic Hecke algebras are usually given by generators and relations, and in this section we describe such a presentation for
n . Note this result will not be used anywhere in this book.

7.8 Presentation for degenerate cyclotomic Hecke algebras

81

Proposition 7.8.1 The algebra
n is generated by its elements x1 and s1      sn−1 , subject only to the following relations:  hi  = 0 (7.16) i∈I x1 − i x1 sl = sl x1 2 ≤ l < n

(7.17)

x1 s1 x1 s1 + s1  = s1 x1 s1 + s1 x1 

(7.18)

sk2 = 1 sk sm = sm sk  sk sk+1 sk = sk+1 sk sk+1 

(7.19)

for all admissible k and m, with k − m > 1. Proof Let  be the algebra given by generators y1 and t1      tn−1 and relations (7.16)–(7.19) with y1 instead of x1 and tk s instead of sk s. It suffices to show that there exist algebra homomorphisms    →
n 

y1 → x1  tk → sk

1 ≤ k < n

 
n → 

x1 → y1  sk → tk

1 ≤ k < n

The existence of  follows from the easily checked fact that elements sk and x1 in
n satisfy the same relations as the elements tk and y1 in . To construct  it suffices to construct an algebra homomorphism ˆ 
n → 

x1 → y1  sk → tk

1 ≤ k < n

(7.20)

ˆ   = 0, see (7.6). Well, let us define ˆ using (7.20) and the such that f recurrent formula ˆ k−1 tk−1 + tk−1 ˆ  xk → tk−1 x

2 ≤ k ≤ n

ˆ k  and All we have to check is that this makes sense, that is that s ˆ m  satisfy the defining relations of
n , of which only the relations x (3.1) and (3.6) are not immediate. Now we verify all the relations of the ˆ k  using induction on k = 1 2     n and form (3.1) and (3.6) involving x relations in .

8 Functors ei and fi

In this chapter we will finally define the “induction analogues” fi , f˜i , and i of ei , e˜ i , and
i , respectively. These will depend crucially on , that is on the cyclotomic algebra
n we are working with. In this respect they will differ from ei , e˜ i , and
i , which are the same as ei , e˜ i , and
i , if we consider a module over
n as a module over
n via inflation. This crucial role of the cyclotomic Hecke algebras in the definition of fi , f˜i , and i explains why we could not define these notions in Chapter 5. One natural definition of f˜i is easy to come up with if we recall that for an
n -module M, ei M is roughly speaking restriction from
n to
n−1 followed by a projection to a block. If M is integral, which we always assume, then it is also a module over
n for some sufficiently large , and we can think of  followed by a projection to a block. It ei as the restriction from
n to
n−1 is crucial here that if an
n -module factors through
n , then its restriction  . Now, we can define fi as induction from
n to
n−1 factors through
n−1  to
n+1 followed by a projection to a block. We can see immediately that this definition depends crucially on . Although the definition of fi in terms of induction of cyclotomic Hecke algebras is quite natural, we will need a different description of this functor, see Lemma 8.2.3. This “inverse limit” description is one of the key observations of Grojnowski. It allows us to describe the function i , defined originally in (8.17), as the “stabilization” point of the limit, which should be thought of as the analogue of the description of
i given in Theorem 5.5.1(ii). The tricky definition of fi is responsible for the fact that the analogue of Theorem 5.5.1 is quite difficult to get. This is achieved by the end of this chapter, see Theorem 8.5.9. Finally, we mention that important “divided power” generalizations of the functors ei , ei , and fi are studied in Section 8.3.

82

8.2 Definitions

83

8.1 New notation for blocks We will use a new notation for blocks in RepI
n . In view of Section 4.2 the blocks (or central characters) in the category RepI
n are labeled by the Sn -orbits on I n . If i ∈ I n , define its content conti ∈ P by conti =



i i 

i =  j = 1     n  ij = i 

where

(8.1)

i∈I

So conti is an element of the set #n of non-negative integral linear combi  nations  = i∈I i i of the simple roots such that i∈I i = n. Obviously, the Sn -orbit of i is uniquely determined by the content of i, so we obtain a labeling of the orbits of Sn on I n by the elements of #n . We will also use the notation  for the central character i , where i is any element of I n with conti = . We will write as usual RepI
n  for the corresponding block, M for the corresponding block component of the module M, etc. We can extend some of these notions to
n -modules, for  ∈ P+ . In particular, if M ∈
n -mod, we also write M for the summand M of M defined by first viewing M as an
n -module by inflation. Also write
n -mod for the full subcategory of
n -mod consisting of the modules M with M = M . Then we have a decomposition
n -mod 




n -mod 

(8.2)

∈#n

Note though that we should not yet refer to
n -mod as a block of Rep
n : the center of
n may be larger than the image of the center of
n , so we cannot yet assert that Z
n  acts on M by a single central character. Also we no longer know precisely which  ∈ #n have the property that
n -mod is non-trivial. These questions will be settled in Section 9.6.

8.2 Definitions Fix  ∈ P+ throughout the section. Recall from (5.6) the functors ei for i ∈ I.  -module (recall If M is an
n -module then ei M is automatically an
n−1  that we always identify the category of
m -modules with a full subcategory of
m -modules via inflation). So the restriction of the functor ei gives a functor  ei 
n -mod →
n−1 -mod 

(8.3)

Functors ei and fi

84

There is an alternative definition of this functor: if M is a module in 
n -mod for some fixed  = j∈I j j ∈ #n then


 res
n M − i if i > 0,  n−1 ei M = (8.4) 0 if i = 0. This description suggests how to define an analogous (additive) functor  fi 
n -mod →
n+1 -mod 

(8.5)

Using (8.2) and additivity, it suffices to define this on an object M belonging  to
n -mod for fixed  = j∈I j j ∈ #n . Then we set


fi M = ind
n+1  M + i 

(8.6)

n

To complete the definition of the functor fi , it is defined on a morphism


simply by restriction of the corresponding morphism ind
n+1 .  n

Remark 8.2.1 Note that the functor fi depends fundamentally on , unlike ei , which is just the restriction of its affine counterpart ei . Lemma 8.2.2 For  ∈ P+ and each i ∈ I: (i) ei and fi are both left and right adjoint to each other, hence they are exact and send projectives to projectives; (ii) ei and fi commute with duality, that is there is a natural isomorphism ei M    ei M 

fi M    fi M

for each finite dimensional
n -module M.





Proof We know that ei M and fi M are summands of res
n M and ind
n+1  M n n−1 respectively. Moreover, -duality leaves central characters invariant because xj  = xj for each j. Now everything follows easily, applying Corollary 7.7.5. From (5.8) and Lemma 8.2.2(i), we have    

 res
n M = ei M ind
n+1 fi M  M = n−1

i∈I

n

(8.7)

i∈I

We will use an alternative description of ei and fi . Recall the definition of the
1 -modules Lm i for i ∈ I m ≥ 0 from Section 4.4. The limits in the next lemma are taken with respect to the systems induced by the maps (4.7).

8.2 Definitions

85

Lemma 8.2.3 For every M ∈
n -mod and i ∈ I, there are natural isomorphisms ei M  lim pr  Hom
1 Lm i M − → fi M  lim pr  indn+1 n1 M
Lm i ← − where
1 denotes the subalgebra of
n−11 generated by xn and the
n−1 module structure on Hom
1 Lm i M is defined by hf v = hfv for f ∈ Hom
1 Lm i M, h ∈
n−1 , and v ∈ Lm i. Proof For ei , it suffices to consider the effect on M ∈
n -mod for   = j∈I j j ∈ #n with i > 0, both sides of what we are trying to prove clearly being zero if i = 0. Then, for all sufficiently large m, Lemma 4.4.2 (in the special case n = 1) implies that


Hom
1 Lm i M  res
n M − i  n−1

Hence


lim pr  Hom
1 Lm i M  res
n M − i = ei M − → n−1 To deduce the statement about fi , it now suffices by uniqueness of adjoint functors to show that lim pr  indn+1 n1 ?
Lm i is left adjoint to ← −  lim pr Hom
1 Lm i ? Let N ∈
n−1 -mod and M ∈
n -mod. As explained − → in the previous paragraph, the direct system pr  Hom
1 L1 i M $→ pr  Hom
1 L2 i M $→    stabilizes after finitely many terms. We claim that the inverse system n+1  pr  indn+1 n1 N
L1 i  pr ind n1 N
L2 i    

also stabilizes after finitely many terms. To see this, it suffices to show that the dimension of pr  indn+1 n1 N
Lm i is bounded above independently of m. For, let w be a vector which generates the “Jordan block” Lm i as an
1 module, and W = Fw be the 1-dimensional subspace of Lm i spanned by w. Then indn+1 n1 N
Lm i is generated as an
n+1 -module by the subspace  W = 1⊗N ⊗W, of dimension independent of m. Finally, we need to observe   that pr  indn+1 n1 N
Lm i is a quotient of the vector space
n+1 ⊗F W , whose dimension is independent of m.

86

Functors ei and fi

Now we can complete the proof of adjointness. Using the fact that the direct and inverse systems stabilize after finitely many terms, we have natural isomorphisms Hom
n lim pr  indnn−11 N
Lm i M ← −  lim Hom
n pr  indnn−11 N
Lm i M − →  lim Hom
n indnn−11 N
Lm i M − →  lim Hom
n−11 N
Lm i resnn−11 M − →  lim Hom
n−1 N Hom
1 Lm i M − →   N pr Hom
 Lm i M  lim Hom
n−1 1 − →   N lim pr Hom
 Lm i M  Hom
n−1 1 − → This completes the argument. Now we define the cyclotomic crystal operators on irreducible modules e˜ i = pr   e˜ i  infl 

(8.8)

f˜i = pr   f˜i  infl

(8.9)

for each i ∈ I and  ∈ P+ (cf. Section 5.2). Let B and B denote the set of isomorphism classes of irreducible modules in RepI
n and
n -mod for all n ≥ 0, respectively. Note that we may consider B as a subset of B via inflation. We have maps e˜ i  B → B ∪ 0  f˜i  B → B e˜ i  f˜i  B → B ∪ 0  Remark 8.2.4 As with ei and fi , there is an important difference between e˜ i and f˜i . The operator e˜ i is just the restriction of e˜ i from B to B, that is pr   e˜ i  infl M = e˜ i  infl M for all irreducible
n -modules M. This is certainly not the case for f˜i : even though f˜i also “tries” to be the restriction of f˜i , but the problem is that f˜i does not leave B invariant, so it will often be the case that f˜i M = 0, even though f˜i M is never zero.

8.3 Divided powers

87

Theorem 8.2.5 Let  ∈ P+ and i ∈ I. Then for any irreducible
n -module M we have: (i) ei M is non-zero if and only if e˜ i M = 0, in which case ei M is a self-dual indecomposable module with irreducible socle and head isomorphic to e˜ i M; (ii) fi M is non-zero if and only if f˜i M = 0, in which case fi M is a self-dual indecomposable module with irreducible socle and head isomorphic to f˜i M. Proof (i) By Corollary 5.1.7, ei M has irreducible socle e˜ i M whenever ei M is non-zero. The remaining facts follow since M is self-dual by Lemma 5.3.2, and ei commutes with duality by Lemma 8.2.2(ii).  -mod be irreducible modules. Then, by (ii) Let M ∈
n -mod, N ∈
n+1    f M N  = Hom
 M e N . By (i), the latter is Lemma 8.2.2(i), Hom
n+1 i i n zero unless M = e˜ i N , or equivalently N = f˜i M by Lemma 5.2.3, in which case the Hom-space is 1-dimensional. By Schur’s lemma we now have that hd fi M  f˜i M. Finally, note fi M is self-dual by Lemma 8.2.2(ii) so everything else follows.

8.3 Divided powers Continue with  ∈ P+ and fix i ∈ I. We can generalize the definitions of r ei  ei  fi to define functors denoted ei  ei r  fi r . It will be the case 1 that ei = ei  ei 1 = ei  fi 1 = fi . For the definitions, we make use of the covering modules Lm ir  from Section 4.4. r Let M be a module in RepI
n . If r > n, we set ei M = 0. Otherwise, let
r denote the subalgebra of
n generated by xn−r+1      xn  sn−r+1      sn−1  We have a direct system Hom
r L1 ir  M $→ Hom
r L2 ir  M $→    induced by the inverse system (4.7). Now define r

ei M = lim Hom
r Lm ir  M − → r

(8.10)

 As in the case r = 1, if M is an
n -module, then ei M is an
n−r -module, r so, on restriction, ei gives a functor  ei r 
n -mod →
n−r -mod 

Functors ei and fi

88

Lemma 4.4.3 yields another description of ei r . Let M be a module in
n -mod for some fixed  ∈ #n . Then we have a functorial isomorphism




ei r M  res
n M − ri Sr  n−r

(8.11)



where −Sr stands for the invariants of the symmetric group Sr = sn−r+1  sn−r+2      sn−1 

(8.12)

which is a subgroup of the multiplicative group of
n , commuting with
 
n−r . Equivalently, instead of considering res
n M − ri , we could take n−r the simultaneous generalized i-eigenspace of the last r polynomial generators  . We could also xn−r+1      xn on M, which is invariant with respect to
n−r take coinvariants in the definition (8.11), in view of the following functorial isomorphism:


ei r M  res
n M − ri Sr  n−r

(8.13)

This follows from the following lemma. Lemma 8.3.1 Let G be a finite group, A be an associative F -algebra, and  be the full subcategory of A ⊗ FG-mod which consists of the modules whose restrictions to FG are free. Then the functors −G and −G from  to A-mod are isomorphic.  Proof Let e = g∈G g ∈ FG and Aug be the augmentation ideal in FG. Take M ∈ . Then M G = eM and MG = M/ Aug ·M. Consider the map M  MG → M G  m + Aug ·M → em This is well-defined because e · Aug = 0. Since FG commutes with A, M is an A-homomorphism, and, by assumption, it is an isomorphism of A-modules. Moreover, this isomorphism is clearly functorial in M. Finally, in view of Lemma 4.4.3, we could also take the largest submodule on which Sr acts as the sign representation instead of taking Sr -invariants in (8.11) (or the largest quotient module on which Sr acts as the sign representation instead of taking Sr -coinvariants in (8.13)). To define fi r , which as usual only makes sense in the cyclotomic case, let M be an
n -module. We have an inverse system M
L1 ir   M
L2 ir      of
nr -modules induced by the maps from (4.7). Define r fi r M = lim pr  indn+r nr M
Lm i  ← −

(8.14)

8.3 Divided powers

89

As in the case r = 1, we verify that the limit stabilizes after finitely many  steps and so fi r is a functor
n -mod →
n+r -mod. As in the proof of Lemma 8.2.3 we show that fi r is right adjoint to ei r . Now we use the uniqueness of adjoints to get another description of fi r . Let M be a module in
n -mod for some fixed  ∈ #n , and let 1 and sgn stand for the trivial and sign r -modules, respectively. Then


fi r M  ind
n+r  ⊗ M
1 + ri n

 ind

r


n+r
n ⊗r

(8.15)

M
sgn + ri 

the isomorphisms being functorial. We collect the main properties of ei r and fi r in the following theorem. Theorem 8.3.2 Let  ∈ P+ , i ∈ I, r ≥ 1, and M be an irreducible
n -module. ei r and fi r are both left and right adjoint to each other; in particular, they are exact and send projectives to projectives. (ii) There exist isomorphisms of functors (i)

eir  ei ⊕r!  ei r  ei r ⊕r!  fi r  fi r ⊕r!  r

r

(iii) ei , ei r , and fi r commute with duality. (iv) ei r M is non-zero if and only if ˜ei r M = 0, in which case ei r M is a self-dual indecomposable module with irreducible socle and head isomorphic to ˜ei r M. (v) fi r M is non-zero if and only if f˜i r M = 0, in which case fi r M is a self-dual indecomposable module with irreducible socle and head isomorphic to f˜i r M. Proof (i) To see that the functors are both right and left adjoint to each other, use Lemma 8.3.1, Corollary 7.7.8(ii) and the alternative descriptions of the functors obtained in (8.11), (8.15). (ii) Lemmas 4.4.1 and 4.4.2 show that there is an isomorphism of functors ei r  ei r ⊕r! . So, by (i), both fi r ⊕r! and fir are adjoint to ei r . Therefore they are isomorphic by uniqueness of adjoint functors. (iii) is proved as Lemma 8.2.2(ii) but using Corollary 7.7.8(ii) instead of Corollary 7.7.5. (iv) In view of (ii), ei r M = 0 if and only if ei r M = 0, which, in view of (5.7) and (5.11), is equivalent to ˜ei r M = 0. The self-duality of ei r M

Functors ei and fi

90

follows from (iii) and self-duality of M. The result on the socle is contained in Lemma 5.2.2, and the result on the head follows from self-duality. (v) Arguement as in the proof of Theorem 8.2.5(ii). Since we have defined the exact functors above on module categories we get induced linear maps denoted by the same symbols at the level of Grothendieck groups, namely, r

ei  KRepI
n  → KRepI
n−r   ei r  K
n -mod → K
n−r -mod  fi r  K
n -mod → K
n+r -mod

Also, in view of Theorem 8.3.2(i), we get linear maps on Grothendieck groups of projective modules:  ei r  K
n -proj → K
n−r -proj  fi r  K
n -proj → K
n+r -proj

We record a corollary of Theorem 8.3.2(ii): Lemma 8.3.3 As operators on the corresponding Grothendieck groups, r

r

eir = r!ei  ei r = r!ei r  fi r = r!fi 

8.4 Functions i
Fix  ∈ P+ throughout this section. Let M be an irreducible
n -module. Define
i M = max m ≥ 0  ˜ei m M = 0  i M

= max m ≥

0  f˜i m M

= 0 

(8.16) (8.17)

As e˜ i is simply the restriction of e˜ i , we have
i M =
i M, see Remark 8.2.4 and equation (5.11). On the other hand, i depends crucially on . We will see shortly (Corollary 8.4.4) that i M <  always, so that the definition makes sense. Recall that we interpret
0 as F . Let 1
 F denote the trivial irreducible
0 -module. Lemma 8.4.1 For any i ∈ I we have
i 1
 = 0 and i 1
 = hi  .

8.4 Functions i

91

Proof The statement involving
i is obvious. For i , note that f˜im 1
= Lim  and
∗i Lim  = m


∗j Lim  = 0

for every j = i. Hence by Corollary 7.4.1, pr  Lim  = 0 if and only if m ≤ hi  . This implies the required result. Lemma 8.4.2 Let i j ∈ I with i = j and M be an irreducible module in RepI
n . Then
∗j f˜im M ≤
∗j M for every m ≥ 0. Proof Follows from the Shuffle Lemma. Lemma 8.4.3 Let i j ∈ I with i = j. Suppose that M is an irreducible
n module such that j M > 0. Then i f˜j M −
i f˜j M ≤ i M −
i M − hi  j  Proof Set
=
i M  = i M k = −hi  j  By Lemma 6.3.3, there exist unique r s ≥ 0 with r +s = k such that
i f˜j M =
− r We need to show that i f˜j M ≤  + s, which follows if we can show that pr  f˜im f˜j M = 0 for all m >  + s. It suffices to prove that
∗i f˜im+s f˜j M ≥
∗i f˜im M

(8.18)

for all m ≥ 0. Indeed, by the definition of , we have pr  f˜im M = 0 for any m > . In view of Corollary 7.4.1, this means that
∗j f˜im M > hj   for some j ∈ I. But by Lemma 8.4.2, such j can only equal i. Thus,
∗i f˜im M > hi   for all m > . So (8.18) implies that
∗i f˜im f˜j M > hi   for all m >  + s, hence by Corollary 7.4.1 once more, pr  f˜im f˜j M = 0 as required. To prove (8.18), note that r ≤
, so s +
≥ k. Hence, Lemma 6.3.3(ii) shows that there is a surjection n+m+s+1 indnm−rr+s+1 M
Lim−r 
Lir  j is   f˜im+s f˜j M r s r s By Lemma 6.2.1, resr+s+1 rs+1 Li  j i   Li 
Lj i  Hence by Frobenius reciprocity, there is a surjection r s r s indr+s+1 rs+1 Li 
Lj i   Li  j i 

Combining, we have proved existence of a surjection m s ˜m+s f˜j M indn+m+s+1 nms+1 M
Li 
Lj i   fi

Functors ei and fi

92

Hence by Frobenius reciprocity there is a non-zero map m s n+m+s+1 ˜m+s ˜ fj M indn+m nm M
Li 
Lj i  → resn+ms+1 fi

Since the left-hand module has irreducible head f˜im M
Lj is , we deduce that f˜im+s f˜j M has a constituent isomorphic to f˜im M on restriction to the subalgebra
n+m ⊆
n+m+s+1 . This implies the claim. Corollary 8.4.4 Let  ∈ P+ and M be an irreducible
n -module with central character  for some  ∈ #n . Then i M −
i M ≤ hi   −  Proof Proceed by induction on n, the case n = 0 being immediate by  Lemma 8.4.1. For n > 0, we may write M = f˜j N for some irreducible
n−1  module N with j N  > 0. By induction, i N  −
i N  ≤ hi   −  + j  The conclusion follows from Lemma 8.4.3.

8.5 Alternative descriptions of i
Now we wish to prove the analogue of Theorem 5.5.1 for the function i . This is considerably more difficult to do. Let M be an irreducible
n -module. Recall that fi M = lim pr  indn+1 n1 M
Lm i ← − and that the inverse limit stabilizes after finitely many terms. Define ˜ i M to be the stabilization point of the limit, that is the least m ≥ 0 such that fi M = ˜ i = i , see Corollary 8.5.7. pr  indn+1 n1 M
Lm i. Later it will turn out that  Lemma 8.5.1 Let M be an irreducible
n -module and i ∈ I. Then:   (i) fi M = ˜ i Mf˜i M + cr Nr where the Nr are irreducible
n+1   ˜ modules with
i Nr  <
i fi M.   f M is isomorphic to the algebra of truncated (ii) The algebra End
n+1 i polynomials kx /x˜ i M . 

Proof For any m ≥ 1 denote Vm = indn+1 n1 M
Lm i

8.5 Alternative descriptions of i

93

(i) Since pr  is right exact, the natural surjection Lm i  Lm−1 i and the natural embedding Lm i $→ Lm+1 i of the “Jordan blocks” (see Section 4.4) induce a commutative diagram m

m

m+1

m+1

pr  V1 −−−−→ pr  Vm −−−−→ pr  Vm−1 → 0 ⏐ ⏐ " pr  V1 −−−−→ pr  Vm+1 −−−−→ pr  Vm

→0

where the rows are exact. Note if m = 0 then m+1 = 0. It follows that if m is an isomorphism so is m for every m ≥ m. So by definition of ˜ i M, the maps 1  2      ˜ i M are not isomorphisms but all other m  m > ˜ i M, are isomorphisms. Now to prove (i), we show by induction on m = 0 1     ˜ i M that fi M = pr  Vm = mf˜i M + lower terms  -modules N with
i N <
i f˜i M. where the lower terms are irreducible
n+1 This is vacuous if m = 0. For m > 0, m is not an isomorphism, so m = 0. Hence, by Lemma 5.1.5, the image of m contains a copy of f˜i M plus lower terms. Now the induction step is immediate. (ii) Take m = ˜ i M. We easily show, using the explicit construction of Lm i in Section 4.4, that there is an endomorphism

 Lm i → Lm i of
1 -modules, such that the image of k is isomorphic to Lm−k i for each 0 ≤ k ≤ m. Functoriality yields algebra homomorphisms End
1 Lm i $→ End
n1 M
Lm i $→ End
n+1 Vm  So induces an
n+1 -endomorphism ˜ of Vm , such that the image of ˜ k is isomorphic to Vm−k for 0 ≤ k ≤ m. Now apply the right exact functor pr  to  -endomorphism get an
n+1 ˆ  f M → f M i i induced by ˜ . Note ˆ m = 0 and ˆ m−1 = 0 because its image coincides with the image of the non-zero map m in the proof of (i). Hence, 1 ˆ      ˆ m−1 are linearly independent, endomorphisms of fi M. Now the proof of (ii) is completed in the same way as in the proof of Theorem 5.5.1(iii). Corollary 8.5.2 Let M N be irreducible
n -modules with M  N . Then for    f M f N  = 0 every i ∈ I we have Hom
n+1 i i

Functors ei and fi

94

Proof Repeat the argument in the proof of Corollary 5.5.2, but using ˜ i and Lemma 8.5.1(i) in place of
i and Theorem 5.5.1(i). We now start proving that ˜ i = i . Note right away from the definitions that for any irreducible
n -module M we have ˜ i M = 0 if and only if i M = 0. Lemma 8.5.3 If M is an irreducible
n -module then  

˜ i M −
i M = c 

i=0

Proof Using Lemma 8.5.1(ii), decompositions (8.7), Frobenius reciprocity, Theorem 7.6.2, Schur’s Lemma, and Theorem 5.5.1(iii), we have that  

˜ i M =

i=0

 

  f M dim End
n+1 i

i=0


n+1  ind  M = dim End
n+1
n







n

n

n+1 = dim Hom
n M res
n+1  ind
 M

= dim End
n M

⊕c





+ dim Hom
n M ind
n res
n M n−1

=

n−1


n  res  M c  + dim End
n−1
n−1

= c  +

 


i M

i=0

and the conclusion follows. Lemma 8.5.4 Let M be an irreducible
n -module and i ∈ I. Then: (i) ei fi M  M =
i f˜i M˜ i M and fi ei M  M =
i M˜ i ˜ei M;   (ii) soc ei fi M  M ⊕˜ i M and soc fi ei M  M ⊕
i M . Proof (i) follows from Theorem 5.5.1(i) and Lemma 8.5.1(i). (ii) Let N be an irreducible
n -module. Then by Lemma 8.2.2(i),    f N f M Hom
n N ei fi M  Hom
n+1 i i

8.5 Alternative descriptions of i

95

and the first part of (ii) follows from Corollary 8.5.2 and Lemma 8.5.1(ii). The second part is similar. The proof of the next lemma is based on [V2 , Lemma 6.1]. Lemma 8.5.5 Let M be an irreducible
n -module and i ∈ I. There is an
n -module homomorphism   fi ei M → ei fi M such that the following composition is surjective: 

can

fi ei M −→ ei fi M −→ ei fi M/soc ei fi M Proof Let k = ˜ i M and n+1     indn+1 n1 M
Lk i  pr ind n1 M
Lk i = fi M

be the quotient map. Let
1 denote the subalgebra of
n generated by xn . Recall from Section 4.4 that viewed as an
1 -module, we have that Lk i 
1 /xn − ik . In particular, Lk i is a cyclic module generated by the image 1˜ of 1 ∈
1 . We first observe that for any m ≥
i M + k, the element xn − im annihilates the vector sn ⊗ u ⊗ v ∈ indn+1 n1 M
Lk i for any u ∈ M v ∈ Lk i. This is obtained using the relations (3.10) in
n+1 :  We ultimately appeal to the facts that xn − i
i M annihilates u (see Theorem 5.5.1(ii)) and xn+1 − ik annihilates v. Therefore, for any m ≥
i M + k, the following equality holds in fi M:
 (8.19) xn − im  sn ⊗ u ⊗ v = 0 Next, it is not difficult to check that there exists a unique
n−11 -homomorphism
 ˜  ei M

1 → resnn−11 ei fi M u ⊗ 1 →  sn ⊗ u ⊗ 1 for each u ∈ ei M ⊆ M. It follows from (8.19) that this homomorphism factors to induce a well-defined
n−11 -module homomorphism ei M
Lm i → resnn−11 ei fi M We then get from Frobenius reciprocity an induced map m  indnn−11 ei M
Lm i → ei fi M for each m ≥
i M + k. Each m factors through the quotient pr  indnn−11 ei M
Lm i so we get an induced map   fi ei M = lim pr  indnn−11 ei M
Lm i → ei fi M = ei fi M ← −

(8.20)

Functors ei and fi

96

It remains to show that the composite of  with the canonical epimorphism from ei fi M to ei fi M/soc ei fi M is surjective. By Mackey Theorem there exists an exact sequence
n+1  0 → M
Lk i → resn+1 n1 ind n1 M
Lk i
 sn resnn−11 M
Lk i → 0 → indn1 n−111 In other words, there is an
n1 -isomorphism from
 sn indn1 resnn−11 M
Lk i n−111 to


n+1 
 resn+1 n1 ind n1 M
Lk i / M
Lk i 

with h ⊗ u ⊗ v → hsn ⊗ u ⊗ v + M
Lk i for h ∈
n  u ∈ M v ∈ Lk i, where M
Lk i is embedded into n+1 resn+1 n1 ind n1 M
Lk i as 1 ⊗ M ⊗ Lk i. Since dim Lk i = k, Lemma 8.5.4 (ii) gives ⊕k resn1  soc ei fi M n M
Lk i  M

So, applying the exact functor ei to the isomorphism above, we get an isomorphism
 ∼   indnn−11 ei M
Lk i −→ ei indn+1 n1 M
Lk i /soc ei fi M h ⊗ u ⊗ v → hsn ⊗ u ⊗ v + soc ei fi M
   (Note that ‘soc ei fi M’ is not the socle of ei indn+1 n1 M
Lk i , so ‘soc ei fi M’   just means some submodule isomorphic to soc ei fi M.) It follows that there is a surjection  indnn−11 ei M
Lk i  ei fi M/soc ei fi M such that the diagram m

indnn−11 ei M
Lm i −−−−→ ⏐ ⏐ "

ei fi M ⏐ ⏐can "

indnn−11 ei M
Lk i −−−−→ ei fi M/soc ei fi M commutes for all m ≥
i M + k, where m is the map from (8.20) and the left-hand arrow is the natural surjection. Now surjectivity of implies surjectivity of can m and of can .

8.5 Alternative descriptions of i

97

Lemma 8.5.6 Let M be an irreducible
n -module with
i M > 0. Then ˜ i ˜ei M = ˜ i M + 1 Proof Let us first show that ˜ i ˜ei M ≥ ˜ i M + 1

(8.21)

Recall that i M = 0 if and only if ˜ i M = 0. Suppose first that i M = 0. Then i ˜ei M = 0 in view of Lemma 5.2.3. Now, ˜ i M = 0 and ˜ i ˜ei M = 0, so the conclusion certainly holds in this case. Next, assume that i M > 0, hence ˜ i M > 0. Note by Lemma 8.5.4, ei fi M/soc ei fi M  M =
i f˜i M˜ i M − ˜ i M =
i M˜ i M = 0 In particular, the map  in Lemma 8.5.5 is non-zero. Now Lemma 8.5.5 implies that the multiplicity of M as a composition factor of im  is strictly greater than
i M˜ i M, since at least one composition factor of soc im  ⊆ soc ei fi M must be sent to zero on composing with the second map can. Using another part of Lemma 8.5.4, this shows that
i M˜ i ˜ei M >
i M˜ i M and (8.21) follows. Now using (8.21) and Lemma 8.5.4, we see that in the Grothendieck group, ei fi M − fi ei M  M ≤ ˜ i M −
i M with equality if and only if equality holds in (8.21). By central character considerations, for i = j, we have ei fj M  M = fj ei M  M = 0. So using (8.7) we deduce that





n

n







n+1 n n res
n+1  ind
 M − ind
 res
 M  M ≤ n−1

n−1

  ˜ i M −
i M i=0

with equality if and only if equality holds in (8.21) for all i ∈ I. Now Lemma 8.5.3 shows that the right-hand side equals c , which does indeed equal the left-hand side thanks to Theorem 7.6.2. Corollary 8.5.7 For any irreducible
n -module M we have i M = ˜ i M Proof We proceed by induction on i M, the conclusion being known already in the case i M = 0. For the induction step, take an irreducible
n -module N with i N  > 0, so N = e˜ i M, where M = f˜i N is an irreducible

Functors ei and fi

98


n+1 -module with
i M > 0, i M < i N . Then by Lemma 8.5.6 and the induction hypothesis,

˜ i N  = ˜ i ˜ei M = ˜ i M + 1 = i M + 1 = i ˜ei M = i N  This completes the induction step. As a first consequence, we can improve Corollary 8.4.4: Lemma 8.5.8 Let M be an irreducible
n -module of central character  for  ∈ #n . Then i M −
i M = hi   −  Proof In view of Corollary 8.4.4, it suffices to show that  

i M −
i M = c 

i=0

But this is immediate from Lemma 8.5.3 and Corollary 8.5.7. We are finally ready to assemble the results of the section to obtain the full analogue of Theorem 5.5.1 for i : Theorem 8.5.9 Let i ∈ I and M be an irreducible
n -module. Then:  (i) fi M = i Mf˜i M + cr Nr where the Nr are irreducibles with i Nr  < i f˜i M = i M − 1;

(ii) i M is the least m ≥ 0 such that fi M = pr  indn+1 n1 M
Lm i;   f M is isomorphic to the algebra of truncated (iii) the algebra End
n+1 i 

polynomials kx /xi M . In particular,    f M =  M dim End
n+1 i i

Proof (i) Since i M = ˜ i M by Corollary 8.5.7, we know by Lemma 8.5.1(i) that  fi M = i Mf˜i M + cr Nr where the Nr are irreducibles with
i Nr  <
i f˜i M. Suppose that M ∈ Rep
n , for  ∈ #n . Then f˜i M and each Nr have central character + , i since they are all composition factors of fi M. So by Lemma 8.5.8, i Nr  = hi   −  − i  +
i Nr  i f˜i M = hi   −  − i  +
i f˜i M

8.6 More on endomorphism algebras

99

It follows that i Nr  < i f˜i M: (ii) This is just the definition of ˜ i M combined with Corollary 8.5.7. (iii) This follows from Lemma 8.5.1(ii) and Corollary 8.5.7. Now we can generalize some results from ei and fi to ei r and fi r , respectively. Proposition 8.5.10 Let i ∈ I, M N be irreducible
n -modules with M  N ,
=
i M, and  = i M. Then:
  (i) ei r M =
r ˜ei r M + cr Nr , where the Nr are irreducibles with
i Nr  <
i ˜ei r M =
− r;
   (i ) fi r M = r f˜i r M + cr Nr , where the Nr are irreducibles with i Nr  < i f˜i r M =  − r;  r M ei r N  = 0; (ii) Hom
n−r  ei    r  f  M fi r N  = 0. (ii ) Hom
n+r i Proof Parts (i) and (i ) follow by applying Theorems 5.5.1, 8.5.9 r times and using Lemma 8.3.3. Parts (ii) and (ii ) follow from (i) and (i ), see the proofs of Corollaries 5.5.2 and 8.5.2.

8.6 More on endomorphism algebras The results of this section will only be referred to in Section 11.2.  r M and We want to study the endomorphism algebras EndHn−r  ei   r   f  M for an irreducible
-module M and arbitrary r. For EndHn+r i n r = 1 the result is Theorems 5.5.1(iii) and 8.5.9(iii). First, we slightly refine Theorem 5.5.1(iii). Lemma 8.6.1 Let i ∈ I, M be an irreducible module in RepI
n , and
=
i M. Let be the endomorphism of ei M given by multiplying with xn − i. Then id = 0      
−1 is a basis of End
n−1 ei M,
= 0, and im
−1 = soc ei M  e˜ i M. Proof The only thing not proved in the proof of Theorem 5.5.1(iii) is the fact that the image of
−1 is the socle of ei M, which we know is isomorphic to e˜ i M. Assume the image is strictly bigger. As the head of ei M is also e˜ i M, we see that the module im
−1 is not irreducible and has irreducible head and socle both isomorphic to e˜ i M. Therefore there exists a non-trivial nilpotent endomorphism  of im
−1 with im  = soc im
−1 . It follows that  
−1

Functors ei and fi

100

is an element of End
n−1 ei M such that 0  1     
−1   
−1 are linearly independent. This contradiction completes the proof of the lemma. Proposition 8.6.2 Let i ∈ I, r ≥ 1, M be an irreducible
n -module, and
=
i M. Then  
 r dim EndHn−r M =   ei  r Proof We can work with the affine Hecke algebra
n instead of
n . If r >
, r with the equality obviously holds, so let r ≤
. As ei M is indecomposable
 r irreducible head e˜ ir M, and the multiplicity of e˜ ir M in ei M is
r , it follows that  
r dim EndHn−r ei M ≤  r For the converse inequality, recall the subgroup Sr from (8.12) and consider the set of elements: Y = yw an−r+1      an  = wxn−r+1 − ian−r+1    xn − ian  where w runs over all elements of Sr , and the am s run over all integers satisfying 0 ≤ an−k <
− k for 0 ≤ k < r. The elements of Y commute with
n−r , so multiplying eir M with y ∈ Y gives an endomorphisms fy of eir M. r Now, eir M = ei M⊕r! , so r

dim End
n−r eir M = r!2 dim EndHn−r ei M


 As Y  = r!
r , it suffices to prove that fy  y ∈ Y are linearly independent. Assume there is a non-trivial linear combination  c y fy = 0 (8.22) y∈Y

in Let X = y ∈ Y  cy = 0 . Let Xn be the set of the elements y = yw an−r+1     an  ∈ X, which have minimal possible value of an . Say, this value is bn . Now, (8.22) implies  cy fy xn − i
−1−bn M = 0 End
n−r eir M.

y∈X

By Lemma 8.6.1, if y ∈ X \ Xn then fy xn − i
−1−bn M = 0, and if y = yw an−r+1      an−1  bn  ∈ Xn then fy xn − i
−1−bn M = wxn−r+1 − ian−r+1    xn−1 − ian−1 xn − i
−1 M = wxn−r+1 − ian−r+1    xn−1 − ian−1 e˜ i M

8.6 More on endomorphism algebras

101

where e˜ i M = soc ei M. Thus, we have  cy wxn−r+1 − ian−r+1    xn−1 − ian−1 e˜ i M = 0 y∈Xn

where the summation is over all y = yw an−r+1      bn  ∈ Xn . Next, let Xn−1 be the set of the elements yw an−r+1      an−1  bn  ∈ Xn which have minimal possible value of an−1 . Say, this value is bn−1 . Then, arguing as above, we get  cy wxn−r+1 − ian−r+1    xn−2 − ian−2 e˜ i2 M = 0 y∈Xn−1

where the summation is over all y = yw an−r+1      bn−1  bn  ∈ Xn−1 , and e˜ i2 M is considered as a submodule of ei2 M by considering it as the socle of ei e˜ i M, where e˜ i M is the socle of ei M. Continuing in this manner r − 2 more times, we get a non-empty subset Xn−r+1 of Y such that cy = 0 for any y ∈ Xn−r+1 , and such that  cy w e˜ ir M = 0 (8.23) y∈Xn−r+1

where the summation is over all y = yw bn−r+1      bn  ∈ Xn−r+1 , and e˜ ir M is a submodule of eir M. r By Theorem 8.3.2(ii),(iv), soc eir M = soc ei M⊕r! = ˜eir m⊕r! . However, r r soc ir M  ˜ei M
Li , and the dimension of the Kato module Lir  is r!, so any irreducible submodule e˜ ir M of eir M is an
n−r -submodule of the
n−rr -module ˜eir M
Lir  = soc ir M. As Lir  over Sr is just the regular module, (8.23) implies that cy = 0 for all y ∈ Xn−r+1 , which is a contradiction.

Remark 8.6.3 Recently, Chuang and Rouquier [CR] have pushed this result r a little further and actually described End
n−r ei M as an algebra. Namely, End
n−r ei M  r / r ∩ 
+
 r

(8.24)

where r (and r ) are embedded into 
in the last r variables, and we use notation of Theorem 1.0.2. This result can be recovered by a more careful analysis of the proof above as follows. Note that the elements of the
subalgebra
r ⊂
n act naturally on the restriction res
nn−r M and hence on eir M as endomorphisms. In fact, all
n−r -endomorphisms of eir M come from
r , since the endomorphisms from the spanning set Y certainly do. Therefore End
n−r eir M 
r /Jr , where Jr is the annihilator in
r of eir M. We also know that End
n−r eir M is the r! × r! matrix algebra over
 r r End
n−r ei M, and dim End
n−r ei M =
r . So, if we can exhibit a central

102

Functors ei and fi


 subalgebra A in End
n−r eir M of dimension
r , it will follow that A 
 r End
n−r ei M. In fact, it is even sufficient to exhibit
r linearly independent central elements of End
n−r eir M – they will automatically span a central subalgebra of the right dimension. Such elements are easy to come up with. Consider the monomial symmetric polynomials in xn−r+1 − a     xn − a of degree at most
− r in each variable. These are central even in
r , and, in view of the linear independence of elements of Y , produce linearly independent endomorphisms. It remains to understand the subalgebra A of
r /J spanned by our special symmetric polynomials. It is clear from above that this subalgebra is exactly the image of the center of
r , which consists of all symmetric polynomials in xn−r+1 − a     xn − a. Now, note that
r /Jr embeds into

 /J
. Also, in view of Lemma 5.1.4 and (4.8), J
is the ideal generated by the symmetric polynomials in xn−
+1 − a     xn − a without the free term. + So A  r / r ∩

+
  r / r ∩ 

. Remark 8.6.4 Proposition 8.6.2, Lemma 8.5.8, and Remark 9.4.5 below imply for i ∈ I, r ≥ 1, and an irreducible
n -module M:    i M  r  f  dim EndHn+r M =  (8.25) i r  r  f  M as an algebra We do not know how to get a description of EndHn+r i in the spirit of (8.24).

9 Construction of U+ and irreducible modules

In this chapter we obtain some major results connecting representation theory of affine and cyclotomic Hecke algebras with affine Kac–Moody algebras. First, we form the Grothendieck group K of integral representations of all affine algebras
n  n ≥ 0, and show that the operations of parabolic restriction and induction define on K a structure of a commutative (but not cocommutative) graded Hopf algebra over . We will construct an explicit isomorphism of graded Hopf algebras between the Kostant–Tits -form U+ of the positive part of  and the graded dual K∗ , see Theorem 9.5.3. Now fix  and consider the Grothendieck group K of finite dimensional representations of all cyclotomic algebras
n  n ≥ 0. In this case the graded dual K∗ can be identified with the Grothendieck group of the projective modules over all
n  n ≥ 0, and there is a natural decomposition map   K∗ → K, sending a projective module to the linear combination of its composition factors with multiplicities. We will prove that  is injective (this does not follow from general nonsense!). This will allow us to identify K∗ as a sublattice of K. Further, K is embedded into K via inflation, and is in fact a right subcomodule of K, see (9.12). So it naturally becomes a left module over K∗  U+ . Using the explicit nature of the last isomorphism, we can check that the action of the generator eir /r! of U+ (on the Grothendieck group K) comes from the exact functor ei r (on the module category). Now we define the action of the whole U on K with the action of fir /r! coming, of course, from the exact functors fi r . The action of hi is given in terms of  and block data, see (9.19). It turns out that Chevalley relations are satisfied and so we do get an action of the -form U on K. Moreover, K∗ ⊂ K is invariant with respect to this action. The next step is to extend the scalars to  to get the action of U on K . The natural pairing between K∗ and K yields a non-degenerate 103

104

Construction of U+ and irreducible modules

symmetric bilinear form   on K , under which the two sublattices K K∗ ⊂ K are dual to each other. The important Theorem 9.5.1 identifies the U -module K , constructed purely in terms of representation theory of affine and cyclotomic Hecke algebras, as the irreducible integrable U -module V of highest weight . Moreover, under this identification: • the highest weight vector v+ corresponds to the class 1
of the trivial
0 -module in the Grothendieck group; • the form   is nothing but the Shapovalov–Jantzen contravariant form on V, satisfying v+  v+  = 1; • blocks of the algebras
n correspond to the weight spaces of V (this will follow only after Corollary 9.6.2 is proved); • the sublattice K∗ ⊂ K corresponds to the smallest sublattice U− v, of V containing v+ and invariant with respect to the Kostant–Tits -form U . In the end of this chapter we will prove Theorem 9.6.1, which claims that two modules over a cyclotomic Hecke algebra
n are in the same block if and only if their inflations are in the same block of the affine Hecke algebra
n . This, of course, would be immediate if we knew that under the natural projection
n 
n the center of
n mapped on to the center of
n . Even though this is probably true, we have no idea how to prove it. The following fact is proved instead: if, in the category RepI
n , we have an extension X of an irreducible module M by an irreducible module N , and both M and N factor through
n , then X also factors through
n . The proof of this fact relies on a nice property of the functors ei proved in Corollary 5.5.2. Note that Theorem 9.6.1 provides us with a convenient classification of blocks for cyclotomic Hecke algebras, as the blocks of affine Hecke algebras are easy to understand, see Section 4.2.

9.1 Grothendieck groups Let us write K =



KRepI
n 

(9.1)

n≥0

for the sum over all n of the Grothendieck groups of the categories RepI
n . Also, write K =  ⊗ K

9.1 Grothendieck groups

105

extending scalars. Thus K is a free -module with canonical basis given by B, the isomorphism classes of irreducible modules, and K is the -vector space on basis B. We will always view K as a lattice in K . We let K∗ denote the restricted dual of K, namely the set of functions f  K →  such that f vanishes on all but finitely many elements of B. Equivalently, K∗ is the graded dual of K. Thus K∗ is also a free -module, with canonical basis M  M ∈ B

dual to the basis B of K, that is M M  = 1, M N  = 0 for N ∈ B with N  M. Note for an arbitrary N ∈ RepI
n , M N  simply computes the multiplicity N  M of the irreducible module M as a composition factor of N . Finally, we write B∗ =  ⊗ B∗  which can be identified with the restricted (or graded) dual of B . Entirely similar definitions can be made for each  ∈ P+ . Set  K = K
n -mod (9.2) n≥0

Again, K is a free -module on the basis B of isomorphism classes of irreducible
n -modules. Moreover, infl induces canonical embeddings infl  K $→ K

infl  B $→ B

(9.3)

We will generally identify K and B with their images under these embeddings. We also define K∗ and K =  ⊗ K as above. Note that K, K , K∗ , K, etc. are all naturally graded. r Recall the functors ei and more generally the divided power functors ei for r ≥ 1, defined in Chapter 8. These induce linear maps r

ei  K → K

(9.4)

for each r ≥ 1. Similarly, ei r and fi r from Chapter 8 induce maps ei r  fi r  K → K

(9.5)

Note by Lemma 8.3.3 that r

eir = r!ei 

ei r = r!ei r  r

fi r = r!fi r 

(9.6)

Extending scalars, the maps ei  ei r  fi r induce linear maps on K and K , too.

106

Construction of U+ and irreducible modules

9.2 Hopf algebra structure Now we wish to give K the structure of a graded Hopf algebra over . Note that
induces the canonical isomorphism KRepI
m  ⊗ KRepI
n  → KRepI
mn 

(9.7)

for each m n ≥ 0. The exact functor indm+n mn induces a well-defined map indm+n mn  KRepI
mn  → KRepI
m+n  Composing with the isomorphism (9.7) and taking the direct sum over all m n ≥ 0, we obtain a homogeneous map %  K ⊗ K → K

(9.8)

By transitivity of induction, this makes K into an associative graded -algebra. By Corollary 5.3.2, the duality  induces the identity map at the level of Grothendieck groups, so Theorem 3.7.5 implies that the multiplication % is commutative. Moreover, there is a unit %   → K

(9.9)

mapping 1 to the class of the trivial module 1 ∈ KRepI
0  ⊂ K. Now we define the comultiplication. The exact functor resnmn−m induces a map resnmn−m  KRepI
n  → KRepI
mn−m  On composing with the isomorphism (9.7), we obtain maps nmn−m KRepI
n  → KRepI
m  ⊗ KRepI Hn−m   nn1 n2  KRepI
n  n = n1 +n2 =n

→



n1 +n2 =n

KRepI
n1  ⊗ KRepI
n2 

Now taking the direct sum over all n ≥ 0 gives a homogeneous map   K → K ⊗ K

(9.10)

Transitivity of restriction implies that  is coassociative, while the homogeneous projection on to KRepI
0    gives a counit
 K → 

(9.11)

9.2 Hopf algebra structure

107

Thus K is also a graded coalgebra over . Now finally: Theorem 9.2.1 K %  %
 is a commutative, graded Hopf algebra over . Proof It just remains to check that  is an algebra homomorphism, which follows using the Mackey Theorem (Theorem 3.5.2). Remark 9.2.2 For  ∈ P+ , there is in general no natural way to give K the structure of a Hopf algebra, unlike K. An exception arises in the special case  = 0 where
n0  n . In this case, the parabolic subalgebras  ⊆ n can be used to make  K0  = Kn -mod n≥0

into a graded Hopf algebra in exactly the same way as K above. We will not make use of this. The following lemma, explaining how to compute the action of ei on K explicitly in terms of , follows from the definitions: Lemma 9.2.3 Let M be a module in RepI
n . Write  nn−11 M = Mr ⊗ Nr r

for irreducible
n−1 -modules Mr and irreducible
1 -modules Nr . Then  ei M = Mr  r with Nr Li

Lemma 9.2.4 The operators ei  K → K satisfy the Serre relations, that is ei ej = ej ei ei2 ej + ej ei2 = 2ei ej ei ei3 ej + 3ei ej ei2 = 3ei2 ej ei + ej ei3

if i − j > 1 if i − j = 1  = p − 1 > 1 if i − j = 1  = p − 1 = 1

Proof In view of Lemma 9.2.3 and coassociativity of , this reduces to checking it on irreducible
n -modules for n = 2 3 4 respectively. For this, the character information in Lemmas 6.2.1 and 6.2.2 is sufficient.

108

Construction of U+ and irreducible modules

Now consider K for  ∈ P+ . This has a natural structure as Kcomodule: viewing K as a subset of K, the comodule structure map is the restriction   K → K ⊗ K

(9.12)

of . In other words, each K is a subcomodule of the right regular Kcomodule. This follows from the fact that the restriction from
n to
n−m  of an
n -module, which factors through
n , itself factors through
n−m . ∗ The dual maps to %  %
induce on K the structure of a cocommutative graded Hopf algebra. Moreover, each K is a left K∗ -module in ¯    . the natural way: f ∈ K∗ acts on the left on K as the map id ⊗f Similarly, K is itself a left K∗ -module, indeed in this case the action is even faithful. r

Lemma 9.2.5 The operator ei acts on K (resp. K for any  ∈ P+ ) in the same way as the basis element Lir  of K∗ . Proof Let M be an irreducible module in RepI
n or
n -mod. Write n−rr M =



Ms ⊗ Ns

s  -modules Ms and irreducible
r -modules Ns . By the for irreducible
n−r definition of the action of K∗ on K, it follows that

Lir  M =



Ms 

s with Ns Lir 

Hence, since resr11 Lir  = r!Li
r  we get that rLi acts in the same way as r!Lir  . So in view of (9.6), it just remains to check that Li acts in the same way as ei , which follows by Lemma 9.2.3. Lemma 9.2.6 There is a unique homomorphism   U+ → K∗ of graded r Hopf algebras such that ei  = Lir  for each i ∈ I and r ≥ 1. Proof Since K is a faithful K∗ -module, (9.6) and Lemmas 9.2.4 and 9.2.5 imply that the operators Lir  satisfy the same relations as the

9.3 Contravariant form

109

generators ei of U+ . This implies existence of a unique such algebra homomorphism. The fact that  is a coalgebra map follows because r

Li  = Li ⊗ 1 + 1 ⊗ Li by the definition of the comultiplication on K∗ .

9.3 Contravariant form Now we focus on a fixed  ∈ P+ . For an
n -module M, we let PM denote its projective cover in the category
n -mod. Since
n is a finite dimensional algebra, we can identify  K∗ = K
n -proj (9.13) n≥0

so that the basis element M corresponds to the isomorphism class PM for each irreducible
n -module M and each n ≥ 0. Moreover, under this identification, the canonical pairing    K∗ × K → 

(9.14)

PM  N  = dim Hom
n PM  N 

(9.15)

satisfies

for
n -modules M N with M irreducible. There are natural maps K
n -proj → K
n -mod P →



P  M M

n ≥ 0

where the summation is over all isomorphism classes of irreducible
n modules M. They induce a homogeneous map   K∗ → K

(9.16)

In Theorem 9.3.5 we will prove that  is injective. In view of Theorem 8.3.2(i), the actions of ei r and fi r are defined on K∗ . Lemma 9.3.1 The operators ei r  fi r on K∗ and K satisfy ei r x y = x fi r y

fi r x y = x ei r y

for each x ∈ K∗ and y ∈ K. Proof This follows from (9.15) and Theorem 8.3.2(i).

Construction of U+ and irreducible modules

110

Corollary 9.3.2 Suppose  ei r M = aMN N  N ∈B

for M ∈ B. Then ei r PN =





fi r M =

bMN N 

N ∈B

bMN PM 

fi r PN =

M ∈B



aMN PM

M ∈B

for N ∈ B. Proof In view of (9.15), we have PM  N  = M N for irreducible
n modules M N . Now the result follows from Lemma 9.3.1. Lemma 9.3.3 Let M be an irreducible
n -module, set
=
i M  = i M. Then for any m ≥ 0 we have  ei m PM = aN P˜ei m N N with
i N ≥m

for coefficients aN ∈ ≥0 . Moreover, in case m =
we have   
+ ei 
 PM = P˜ei 
M + aN P˜ei 
N 
N with
 N >
i

Proof By Corollary 9.3.2, we have  ei m PM = fi m K  M PK  K ∈B

So if the term PK appears in the right-hand side with non-zero multiplicity then i K ≥ m – otherwise fi m K = 0 by Theorem 8.3.2(v) and the definition (8.17) of i . For such K denote the (non-zero) modules f˜i m K by N . This gives  ei m PM = fi m ˜ei m N  M P˜ei m N  N ∈B with
i N ≥m

and the first part of the lemma follows.
 Now let m =
. If
i N  =
, then we have
i ˜ei 
N = 0, and so by Lemma 8.5.1(i), the only composition factor of fi m ˜ei m N whose
i =
is f˜i m ˜ei m N = N . It follows that N = M, and  
+  
 
fi  ˜ei  M  M = 
thanks to Theorem 8.5.9(i). We also need:

9.3 Contravariant form

111

Theorem 9.3.4 Given an irreducible
n -module M, the element PM ∈ K
n -proj can be written as an integral linear combination of terms of the form fi1 r1     fis rs  1
. Proof Proceed by induction on n, the conclusion being trivial for n = 0. So let n > 0 and the result be true for all smaller n. Suppose for a contradiction that we can find an irreducible
n -module M for which the result does not hold. Pick i with
=
i M > 0. Since there are only finitely many irreducible
n -modules, we may choose M so that the result holds for all irreducible
n -modules L with
i L >
. Write  aL PL fi 
 P˜ei 
M = L ∈Irr
n

for coefficients aL ∈ . By Corollary 9.3.2, aL = ei 
 L  ˜ei 
M  In particular, aL = 0 unless
i L ≥
. Moreover, if aL = 0 for L with
i L =
, then by Theorem 5.5.1(i), we have that ei 
 L  ˜ei 
L  ˜ei 
M whence L  M and aM = 1. This shows:



PM = fi 
 P˜ei 
M −

aL PL

(9.17)

L with
i L>


for some aL ∈ . But the inductive hypothesis and choice of M ensure that all terms on the right-hand side are integral linear combinations of terms fi1 r1     fis rs  1
, hence the same is true for PM , a contradiction. The next two theorems are very important. Theorem 9.3.5 The map   K∗ → K from (9.16) is injective. Proof We show by induction on n that the map   K
n -proj → K
n -mod is injective. This is clear if n = 0, so assume n > 0 and the result has been proved for all smaller n. Suppose we have a relation  aM PM  = 0 where M runs over the isomorphism classes of irreducible
n -modules, and not all coefficients aM are zero. We may choose i ∈ I and M such that aM = 0
=
i M > 0 and aN = 0 for all N with
i N  <
.

112

Construction of U+ and irreducible modules

Apply ei 
 to the sum. By Lemma 9.3.3, we get   
+ i N  aN P˜ei 
N  + X = 0
N with
 N =
i

where X is a sum of terms of the form P˜ei 
L  with
i L >
. Now the inductive hypothesis shows that X = 0 and that aN = 0 for each N with
i N  =
. In particular, aM = 0, a contradiction. In view of Theorem 9.3.5, we may identify K∗ with its image under , so from now on K∗ ⊆ K are two lattices in K . Extending scalars, the pairing (9.14) induces a bilinear form    K × K → 

(9.18)

with respect to which the operators ei and fi are adjoint. Theorem 9.3.6 The form    K × K →  is symmetric and nondegenerate. Proof It is non-degenerate, since bases PM and N are orthonormal to each other, see (9.15). So we just need to check that it is symmetric. Proceed by induction on n to show that    K
n -mod × K
n -mod →  is symmetric. The case n = 0 is clear. Let n > 0. In view of Theorem 9.3.4, any element of K
n -mod can be written as a linear combination of  -mod . So it suffices to show that elements of the form fi x for x ∈ K
n−1    fi x y = y fi x for any y ∈ K
n -mod . Well, we have fi x y = x ei y = ei y x = y fi x by the induction hypothesis and Lemma 9.3.1.

9.4 Chevalley relations Continue working with a fixed  ∈ P+ . We turn now to considering the relations satisfied by the operators ei  fi on K.

9.4 Chevalley relations

113

Lemma 9.4.1 The operators ei  fi  K → K satisfy the Serre relations (7.5). Proof We know the ei satisfy the Serre relations on all of K by Lemma 9.2.4, so they certainly satisfy the Serre relations on restriction to K. Moreover, ei and fi are adjoint operators for the bilinear form   according to Lemma 9.3.1, and this form is non-degenerate by Theorem 9.3.6. The lemma follows. Now we consider relations between the ei and fj . For i ∈ I and an irreducible
n -module M with central character  for  ∈ #n , define hi M = hi   − M 

(9.19)

Recall, according to Lemma 8.5.8 that we have equivalently that hi M = i M −
i MM 

(9.20)

More generally, define      hi i M −
i M  K → K M → M r r

 where mr denotes mm − 1    m − r + 1/r!. Extending linearly, each hri can be viewed as a diagonal linear operator K → K. The definition (9.19) implies immediately that: Lemma 9.4.2 As operators on K, hi  ej = hi  j ej and hi  fj = −hi  j fj for all i j ∈ I. Lemma 9.4.3 As operators on K, ei  fj = ij hi

(9.21)

for all i j ∈ I. Proof Let M be an irreducible
n -module. It follows immediately from Theorems 5.5.1(i) and 8.5.9(i) (together with central character considerations in case i = j) that M appears in ei fj M − fj ei M with multiplicity ij i M −
i M Therefore, it suffices simply to show that ei fj M − fj ei M is a multiple of M .

114

Construction of U+ and irreducible modules

By Lemma 8.2.3, for m 0 we have a surjection  indn+1 n1 M
Lm j  fj M

Apply pr   ei to get a surjection   pr  ei indn+1 n1 M
Lm j  ei fj M

(9.22)

By the Mackey Theorem, there is an exact sequence n 0 → M ⊕ij m → ei indn+1 n1 M
Lm j → ind n−11 ei M
Lm j → 0

For sufficiently large m we have pr  indnn−11 ei M
Lm j = fj ei M So on applying the right exact functor pr  and using the irreducibility of M, this implies that there is an exact sequence   0 −→ M ⊕m1 −→ pr  ei indn+1 n1 M
Lm j −→ fj ei M −→ 0

(9.23)

for some m1 . Now let N be any irreducible
n -module with N  M. Combining (9.22) and (9.23) shows that fj ei M − ei fj M  N ≥ 0

(9.24)

Summing over all i j and using (8.7) gives ind res M − res ind M  N ≥ 0 But Theorem 7.6.2 shows that equality holds here, hence it must hold in (9.24) for all i j ∈ I. To summarize, we have shown in (9.6), Lemmas 9.4.1, 9.4.2, and 9.4.3 that: Theorem 9.4.4 The action of the operators ei  fi  hi on K satisfy the Chevalley relations (7.3), (7.4), and (7.5). Moreover, the actions of ei r 
 fi r , and hri for all i ∈ I r ≥ 1 make K into a U -module so that K∗  K are U -submodules. Remark 9.4.5 A stronger result than Lemma 9.4.3 holds: the relation (9.21) holds in the module category, not just on the level of the Grothendieck groups. If i = j this means that the functors ei fj and fj ei are isomorphic (which is easy to see). If i = j this means the following. Let M be an
n -module with central character  for  ∈ #n . If hi   −  ≥ 0, then there is a functorial isomorphism ei fi M  fi ei M ⊕ M ⊕hi − 

9.5 Identification of K∗ , K∗ , and K

115

otherwise there is a functorial isomorphism ei fi M ⊕ M ⊕−hi −  fi ei M This is proved in [CR, 5.28] (see also [V2 ] for the case where M is irreducible).

9.5 Identification of K∗ , K
∗ , and K
 We recall some basic notions from representation theory of U from [Kc, Chapters 3, 9, 10]. A U -module V is called integrable if it decomposes as a direct sum of its weight spaces, and all elements ei  fi act on V locally nilpotently. Let int be the category of integrable U -modules V with the additional property that for every v ∈ V there exists N ≥ 0 such that ei1    eiN v = 0 for any i1      iN ∈ I. The category int is semisimple, and its irreducible modules are V   ∈ P+  where V is the U -module generated by a vector v with defining relations hi v = hi  

ei v = 0

1+hi 

fi

v = 0

(9.25)

We refer to the module V as the irreducible U -module of highest weight  and the vector v as its highest weight vector. Every V has a unique (up to scalar) non-degenerate symmetric contravariant bilinear form    V × V →  sometimes called the Shapovalov–Jantzen form. Thus we have uv w = v &uw

u ∈ U  v w ∈ V

where &  U → U 

ei → fi  fi → ei  hi → hi

i ∈ I

is the Chevalley anti-involution. The weight spaces of V are orthogonal to each other with respect to this form. Theorem 9.5.1 For any  ∈ P+ : (i) K is precisely the irreducible integrable highest weight U -module V of highest weight , with highest weight vector 1
; (ii) the bilinear form   from (9.18) on the highest weight module K coincides with the usual contravariant form satisfying 1
 1
 = 1;

116

Construction of U+ and irreducible modules

(iii) K∗ ⊂ K are integral forms of K containing 1
, with K∗ being the minimal lattice U− 1
and K being its dual under the contravariant form; (iv) The classes M of the irreducible
n -modules M ∈
n -mod form a basis of the  − -weight space V− . The same is true for the classes PM of projective indecomposable modules in
n -mod . Proof It makes sense to think of K as a U -module according to Theorem 9.4.4. The actions of ei and fi are locally nilpotent by Theorems 5.5.1(i) and 8.5.9(i). The action of hi is diagonal by definition. Thus K is an integrable module. Clearly 1
is a highest weight vector of highest weight . Moreover, K = U− 1
by Theorem 9.3.4. This completes the proof of (i), and (ii) follows immediately from Lemma 9.3.1. For (iii), we know already that K∗ ⊂ K are dual lattices of K , which are invariant under U . Moreover, Theorem 9.3.4 again shows K∗ = U− 1
. Finally, (iv) follows from (9.19). We need the following well-known result: Lemma 9.5.2 Let u ∈ U+ act as zero on every V,  ∈ P+ . Then u = 0. Proof We may assume that u is in the weight component U+  for some . Pick  with hi   0 for every i. By assumption, uv = 0 for every v ∈ V− . It follows that 0 = uv v  = v &uv  where & is the Chevalley anti-involution. As   is non-degenerate and v ∈ V− is arbitrary, we have &uv = 0. But, since we have chosen  to be very large, it follows from the definition of V in (9.25) that there is an isomorphism U− − → V− , given by multiplying v with elements of U− − . So &u = 0, whence u = 0. Theorem 9.5.3 The map   U+ → K∗ constructed in Lemma 9.2.6 is an isomorphism. Proof Note by Lemma 9.2.5 that the action of the ei ∈ U+ on K and on each K factors through the map . So if x ∈ ker , we have by Theorem 9.5.1 that x acts as zero on all integrable highest weight modules K,  ∈ P+ . Hence x = 0 in view of Lemma 9.5.2, and  is injective. r

9.6 Blocks

117

To prove surjectivity, take x ∈ K. As  is homogeneous and all graded components are finite dimensional, it suffices to show that u x = 0 for all u ∈ U+ implies that x = 0. Note u x = 1u x = 1 ux = 1 ux where the second equality follows because the right regular action of K∗ on itself is precisely the dual action to the left action of K∗ on K, and the third equality follows from Lemma 9.2.5. Hence, if u x = 0 for all u ∈ U+ , we have that 1 ux = 0 for all u ∈ U+ . Now, 1 is the identity in K∗ , so 1 ux is just the coefficient of 1 in ux, where 1 ∈ K is the class of the trivial
0 -module 1. Let us choose  ∈ P+ sufficiently large so that in fact x ∈ K ⊂ K. Then, using identification (9.13), we have 1
 ux = 0 for all u ∈ U+ , where   now is a canonical pairing between K∗ and K. Hence by Lemma 9.3.1, v1
 x = 0 for all v ∈ U− . But then Theorem 9.3.4 implies that x = 0.

9.6 Blocks Let  ∈ P+ . We now classify the blocks of the algebras
n , following [G3 ]. Theorem 9.6.1 Let M and N be irreducible
n -modules with M  N , and 0 −→ infl M −→ X −→ infl N −→ 0

(9.26)

be an exact sequence of
n -modules. Then pr  X = X. Proof Let us denote infl M by M  and infl N by N  . We may assume that M  and N  are in the same block for
n , as otherwise the sequence (9.26) splits, and the result is clear. Recall the ideal  =
n f
n from Section 7.3, and set  =  . Note that pr  ? = 
n / ⊗
n ? So applying this right exact functor to (9.26) yields the exact sequence




· · · −→ Tor 1 n 
n /  N   −→ M  −→ pr  X −→ N  −→ 0 We have to show that  = 0. This will follow if we can prove the claim that there is a surjection of left
n -modules


indnn−11 resnn−11 N   Tor 1 n 
n /  N  

118

Construction of U+ and irreducible modules

Indeed, if the claim is true and  = 0 then the space Hom
n−1 resnn−1 N   resnn−1 M   ⊇ Hom
n−11 resnn−11 N   resnn−11 M    Hom
n indnn−11 resnn−11 N   M   is non-zero, which is false in view of Corollary 5.5.2. Finally, for the claim, consider the resolution d1

d0

· · · −→
n ⊗
n−11
n −→
n −→
n / −→ 0 of the right
n -module
n / , where d0 is the natural projection, and d1 a ⊗ b = af b for a b ∈
n . Let  = ker d1 . Then we have two exact sequences of
n -bimodules 0 −→  −→
n −→
n / −→ 0 0 −→  −→
n ⊗
n−11
n −→  −→ 0 On tensoring with N  these yield the following exact sequences of left
n -modules


0 −→ Tor 1 n 
n /  N   −→  ⊗
n N  

−→
n ⊗
n N  −→
n / ⊗
n N  −→ 0 · · · −→
n ⊗
n−11
n ⊗
n N  −→  ⊗
n N  −→ 0

(9.27)

(9.28)

Now note that  is an isomorphism, as
n ⊗
n N   N  and
n / ⊗
n N   pr  N   N  


Hence from (9.27), Tor 1 n 
n /  N     ⊗
n N  . Now, the claim follows from (9.28), as
n ⊗
n−11
n ⊗
n N   indnn−11 resnn−11 N  

Recalling the definitions from Sections 4.2 and 8.1 and using Theorem 9.5.1, we immediately deduce the following corollary which determines the blocks of
n : Corollary 9.6.2 The blocks of cyclotomic algebras
n are precisely the subcategories
n -mod for  ∈ #n . Moreover,
n -mod is non-trivial if

9.6 Blocks

119

and only if the  − -weight space of the highest weight module K is non-zero. Remark 9.6.3 Recently, Chuang, and Rouquier [CR] have proved that the blocks of the algebras
n (for various n) corresponding to two weights 1 and 2 of V in the same W -orbit are derived equivalent. A key notion introduced in [CR] is that of an 2 -categorification. We refer the reader to the original paper for the precise definition, but, informally speaking, a categorification is a pair of endo-functors E and F on an appropriate category  adjoint to each other on both sides, together with natural transformations X ∈ EndE and T ∈ EndE 2 , and a scalar a ∈ F , satisfying the following additional properties: • linear operators e = E and f = F on the Grothendieck group K give a locally finite 2 -representation; • the classes of irreducible objects of  are weight vectors; • T  T = 1E2 on E 2 , T  1E X  T + T = X1E on E 2 , and 1E T  T 1E   1E T = T 1E   1E T  T 1E  on E 3 ; • X − a is locally nilpotent. The main result of [CR] is that an 2 -categorification leads to a complex of functors, which induces a self-equivalence of Db . This can be applied to the category  = ⊕n≥0
n -mod, functors E = ei , F = fi , and a = i. The natural transformations X and T come from the action of xn on ei M and the action of sn on ei2 M for an
n -module M. From what was proved in this chapter it follows easily that this gives us an 2 categorification. The main result of [CR] now gives a self-equivalence of the derived category, which on restriction to blocks gives the desired result.

10 Identification of the crystal

In this chapter we will push one step further the deep connection between degenerate affine and cyclotomic Hecke algebras on the one hand and Kac–Moody algebras on the other. Namely, we will explain how some natural representation theoretic data coming from affine algebras
n and cyclotomic algebras
n can be used to define crystals in the sense of Kashiwara. Moreover, we will identify these crystals with those corresponding to U− and V, respectively. These crystals are explicitly known, which provides us with rich new information on representation theory of affine and cyclotomic algebras. For the case  = 0 , when
n is the group algebra of the symmetric group, this will be exploited in the next chapter.

10.1 Final properties of B Recall the “starred” versions of e˜ i  f˜i 
i , from Section 7.4. Lemma 10.1.1 Let M ∈ RepI
m be irreducible. (i) For any i ∈ I, either
i f˜i∗ M =
i M or
i M + 1. (ii) For any i j ∈ I with i = j, we have
i f˜j∗ M =
i M. Proof We prove (i), the proof of (ii) being similar. By the Shuffle Lemma, we certainly have that
i f˜i∗ M ≤
i M + 1. Now let N = f˜i∗ M. Then obviously,
i ˜ei∗ N  ≤
i N . Hence,
i M ≤
i f˜i∗ M, as e˜ i∗ f˜i∗ M = M, see Lemma 5.2.3. Lemma 10.1.2 Let M ∈ RepI
m be irreducible and i j be elements of I (not necessarily distinct). Assume that
i f˜j∗ M =
i M. Then, writing
=
i M, we have e˜ i
f˜j∗ M  f˜j∗ e˜ i
M 120

10.1 Final properties of B

121

Proof Set n = m −
. Let N = e˜ i
M, so N is an irreducible
n -module with
i N  = 0 and M = f˜i
N . For 0 ≤ b ≤
, let Qb = ei
−b f˜j∗ M Theorem 5.5.1(i) implies that in the Grothendieck group, Qb is some number of copies of e˜ i
−b f˜j∗ M plus terms with strictly smaller
i . In particular,
i L ≤ b for all composition factors L of Qb , while Q0 consists only of copies of e˜ i
f˜j∗ M. We will show by decreasing induction on b =

− 1     0 that there is a non-zero
n+b+1 -module homomorphism b b  indn+b+1 1nb Lj
N
Li  → Qb 

In case b =
, Q
= f˜j∗ M is a quotient of indm+1 1m Lj
M (see (7.10)), and
N
Li , so the induction starts. Now we suppose M is a quotient of indm n
by induction that we have proved b = 0 exists for some b ≥ 1 and construct b−1 . n+b+1 b Consider resn+b+1 n+b1 ind 1nb Lj
N
Li . By the Mackey Theorem, this has a filtration 0 ⊂ F1 ⊂ F2 ⊂ F3 with successive quotients 1nb b F1  indn+b1 1nb−11 res1nb−11 Lj
N
Li  1nb w b F2 /F1  indn+b1 1n−1b1 res1n−11b Lj
N
Li  b F3 /F2  indn+b1 nb1 N
Li 
Lj

where w is the obvious permutation. As b = 0, Frobenius reciprocity implies that there is a copy of the
1nb -module Lj
N
Lib  in the image of b . Now b > 0, so the i-eigenspace of xn+b+1 acting on Lj
N
Lib  is non-trivial. We conclude that the map n+b+1 Lj
N
Lib  → ei Qb = Qb−1 ˜ b = ei b   ei ind1nb

is non-zero. If i = j, then it follows from the description of F3 /F2 and F2 /F1 above that ei F3 /F1  = 0 (for F2 /F1 we need to use the fact that
i N  = 0). So in this case, we necessarily have that ˜ b F1  = 0. Similarly if i = j, ei F2 /F1  = 0, so if ˜ b F1  = 0 we see that ˜ b factors to a non-zero homomorphism n+b1 b resn+b1 n+b ind nb1 N
Li 
Li → Qb−1 

But this implies that Qb−1 has a constituent L with
i L = b, which we know is not the case. Hence we have that ˜ b F1  = 0 in the case i = j too.

Identification of the crystal

122

Hence, the restriction of ˜ b to F1 gives us a non-zero homomorphism b+n1 1nb b resn+b1 n+b ind 1nb−11 res1nb−11 Lj
N
Li  → Qb−1 

Now finally as all composition factors of resbb−1 Lib  are isomorphic to Lib−1 , this implies the existence of a non-zero homomorphism b−1 b−1  indb+n  → Qb−1 1nb−1 Lj
N
Li

completing the induction. Now taking b = 0 we have a non-zero map 0  indn+1 1n Lj
N → Q0  But the left-hand side has irreducible head f˜j∗ N = f˜j∗ e˜ i
M, while all composition factors of the right-hand side are isomorphic to e˜ i
f˜j∗ M. This completes the proof. We will also need the “starred” versions of Lemmas 10.1.1 and 10.1.2, which are obtained by applying the lemmas to the module M  : Lemma 10.1.3 Let M ∈ RepI
m be irreducible. (i) For any i ∈ I, either
∗i f˜i M =
∗i M or
∗i M + 1. (ii) For any i j ∈ I with i = j, we have
∗i f˜j M =
∗i M. Lemma 10.1.4 Let M ∈ RepI
m be irreducible and i j be elements of I (not necessarily distinct). Assume that
∗i f˜j M =
∗i M. Then, writing a =
∗i M, we have ˜ei∗ a f˜j M  f˜j ˜ei∗ a M Corollary 10.1.5 Let M ∈ RepI 
m  be irreducible and i ∈ I. Let M1 = ∗
M e˜ i i M and M2 = ˜ei∗ 
i M M. Then
∗i M =
∗i M1  if and only if
i M =
i M2 . Proof Suppose
i M =
i M2 . Lemma 10.1.1 implies that


i f˜i∗ k M2 =
i f˜i∗ k+1 M2  k = 0 1    
∗i M − 1 Now, using Lemma 10.1.2, we obtain
M

M1  e˜ i i


M

M  e˜ i i

f˜i∗ 
i M M2  f˜i∗ 
i M e˜ i i ∗

∗


M


M

M2 

(10.1)

Note that
∗i M2  = 0, hence
∗i ˜ei i M2  = 0 Now (10.1) shows that
∗i M1  =
∗i M. We have shown that
i M =
i M2  implies
∗i M =
∗i M1 . The converse is obtained in exactly the same way, but using Lemmas 10.1.3 and 10.1.4 instead of Lemmas 10.1.1 and 10.1.2, respectively.

10.2 Crystals

123

Lemma 10.1.6 Let M ∈ RepI
m be irreducible and i ∈ I satisfy
i f˜i∗ M =
i M + 1. Then e˜ i f˜i∗ M = M. Proof Set
=
i M and N = e˜ i
M. By the Shuffle Lemma and Theorem 5.3.1, we have m+1

+1 indm+1  1m−

Li
N
Li  = ind m−

+1 N
Li

(in the Grothendieck group). Hence by Theorem 5.5.1(i),
indm+1 1m−

Li
N
Li 

equals f˜i
+1 N = f˜i M plus terms L for irreducible L with
i L ≤
.
˜∗ However, indm+1 1m−

Li
N
Li  surjects on to fi M. So the assumption ∗ ∗
i f˜i M =
+ 1 implies f˜i M  f˜i M, or, applying e˜ i , M = e˜ i f˜i∗ M. Again, we record the “starred” version of the above: Lemma 10.1.7 Let M ∈ RepI
m be irreducible and i ∈ I satisfy
∗i f˜i M =
∗i M + 1. Then e˜ i∗ f˜i M = M.

10.2 Crystals Let us now recall some definitions from [Ka]. A crystal (of type ) is a set B endowed with maps i 
i  B →  ∪ −

e˜ i  f˜i  B → B ∪ 0

i ∈ I

i ∈ I

wt  B → P such that: (C1) i b =
i b + hi  wtb for any i ∈ I; (C2) if b ∈ B satisfies e˜ i b = 0, then
i ˜ei b =
i b − 1, i ˜ei b = i b + 1, and wt˜ei b = wtb + i ; (C3) if b ∈ B satisfies f˜i b = 0, then
i f˜i b =
i b + 1, i f˜i b = i b − 1, and wt˜ei b = wtb − i ; (C4) for b1  b2 ∈ B, b2 = f˜i b1 if and only if b1 = e˜ i b2 ; (C5) if i b = −, then e˜ i b = f˜i b = 0.

124

Identification of the crystal

Informally, we can think of a crystal as a colored directed graph with the set of vertices B, and an arrow of color i ∈ I going from vertex b1 to vertex b2 if and only if b2 = f˜i b1 . Moreover, to every vertex b we associate its weight wtb and a bunch of numbers
i b i b satisfying certain axioms. For example, for each i ∈ I, we have the crystal Bi defined as a set to be bi n  n ∈  with



−n if j = i, n if j = i,
j bi n = j bi n = − if j = i; − if j = i;

e˜ j bi n = f˜j bi n =

bi n + 1

if j = i,

0 if j = i;

bi n − 1 if j = i, 0

if j = i;

and wtbi n = ni . We abbreviate bi 0 by bi . Also for  ∈ P, we have the crystal T equal as a set to t , with
i t  = i t  = −, e˜ i t = f˜i t = 0 and wtt  = . A morphism   B → B of crystals is a map   B ∪ 0 → B ∪ 0 such that: (H1) 0 = 0; (H2) if b = 0 for b ∈ B, then wtb = wtb,
i b =
i b and i b = b; (H3) for b ∈ B such that b = 0 and ˜ei b = 0, we have that ˜ei b = e˜ i b; (H4) for b ∈ B such that b = 0 and f˜i b = 0, we have that f˜i b = f˜i b. A morphism of crystals is called strict if  commutes with the e˜ i s and f˜i s, and an embedding if  is injective. Note that the trivial map sending “everything” to 0 is a strict morphism. As for embeddings, we can informally think of them as follows: there is an embedding from a crystal B to a crystal B if and only if we can identify B with a subset of B , so that the data wt
  are respected, and arrows in B are arrows in B . Then the embedding is strict if there are no arrows starting in B and going “outside”, no arrows starting “outside” and coming to B, and no arrows in B \ B conecting two elements in B.

10.2 Crystals

125

We also need the notion of a tensor product of two crystals B B . As a set, B ⊗ B is just B × B , but we write b ⊗ b instead of b b  for b ∈ B b ∈ B . This is made into a crystal by setting
i b ⊗ b  = max
i b
i b  − hi  wtb i b ⊗ b  = maxi b + hi  wtb  i b 

e˜ i b ⊗ b if i b ≥
i b   e˜ i b ⊗ b  =  b ⊗ e˜ i b if i b <
i b 

f˜i b ⊗ b if i b >
i b  f˜i b ⊗ b  =  b ⊗ f˜i b if i b ≤
i b  wtb ⊗ b  = wtb + wtb  Here, we understand b ⊗ 0 = 0 = 0 ⊗ b. The definition comes from representation theory of sl2 . Recall the sets of isomorphism classes of irreducible modules B and B from Section 9.1. We now explain how to make them into crystals in the above sense. For B, we use the operators e˜ i , f˜i from (5.10), (8.8) and functions
i , i from (5.5), (8.17) to define the maps e˜ i  f˜i 
i  i , respectively. For the corresponding functions on B use e˜ i , f˜i from (5.10), functions
i from (5.5), and i defined below. For the weight functions on B and B set wtM = −

(10.2)

for an irreducible M ∈ Rep
n  , and wt  N  =  − 

(10.3)

for an irreducible N ∈
n -mod , respectively. Finally, for M ∈ B, define i M =
i M + hi  wtM

(10.4)

Note we have defined all data purely in terms of representation theory of
n and
n . Lemma 10.2.1 The tuples B
i  i  e˜ i  f˜i  wt and B
i  i  e˜ i  f˜i  wt   for  ∈ P+ are crystals in the sense of Kashiwara.

Identification of the crystal

126

Proof Property (C1) is Lemma 8.5.8 for the case of B or the definition for the case of B. Property (C4) is Lemma 5.2.3. The remaining properties are immediate. Recall the embedding infl  B ∪ 0 → B ∪ 0 from (9.3). Lemma 10.2.2 The map B $→ B ⊗ T N → infl N ⊗ t is an embedding of crystals with image $ # % M ⊗ t ∈ B ⊗ T $
∗i M ≤ hi   for each i ∈ I  Proof Since e˜ i and f˜i are restrictions of e˜ i  f˜i from B to B, respectively, the first statement is immediate. The second is a restatement of Corollary 7.4.1.

10.3 Identification of B and B
 In this section we will identify the crystals B and B defined purely in terms of modular representation theory with the Kashiwara’s crystals of U− and V, respectively. Lemma 10.3.1 Let M ∈ RepI
m be irreducible and i j ∈ I with i = j. Set a =
∗i M. (i)
j M =
j ˜ei∗ a M. (ii) If
j M > 0, then
∗i ˜ej M =
∗i M and ˜ei∗ a e˜ j M  e˜ j ˜ei∗ a M Proof (i) Applying Lemma 10.1.1(ii) (with i and j swapped) repeatedly, we get
j ˜ei∗ a M =
j f˜i∗ ˜ei∗ a M = · · · =
j f˜i∗ a ˜ei∗ a M =
j M (ii) By Lemma 10.1.3(ii), we have
∗i f˜j N  =
∗i N  for any irreducible N . Applying this to N = e˜ j M gives the first part of the claim. Further, a =
∗i N , so by Lemma 10.1.4, we have ˜ei∗ a f˜j N = f˜j ˜ei∗ a N whence the second part of the claim.

10.3 Identification of B and B

127

Lemma 10.3.2 Let M ∈ RepI
m be irreducible and i ∈ I. Set a =
∗i M and L = ˜ei∗ a M.
 (i)
i M = max
i L a − hi  wtL  (ii) If
i M > 0,

a if
i L ≥ a − hi  wtL, ∗
i ˜ei M = a − 1 otherwise. (iii) If
i M > 0, ˜ei∗ b e˜ i M



e˜ i L if
i L ≥ a − hi  wtL, L

otherwise,

where b =
∗i ˜ei M. Proof Let
=
i M, n = m −
, and N = ˜ei 
M. (i) By twisting with , it suffices to prove that (for arbitrary M)

∗i M = max
∗i N 
− hi  wtN  

(10.5)

Define the weights 0 1 · · · ∈ P+ by taking hi  r =
∗i N  + r

and

hj  r 0

for j = i

Then r N = 0 for any r ≥ 0, see Corollary 7.4.1. Moreover, the same r corollary and Lemma 10.1.3 imply that for k = i N  we have
∗i f˜ik N  = hi  r and


∗i f˜ik+1 N  = hi  r + 1

r

As
i N  =
i N  = 0, Lemma 8.5.8 gives i N  = hi  r + wtN  =
∗i N  + r + hi  wtN  r

Moreover, by Lemma 10.1.3(i) we have
∗i f˜is N  ≥
∗i N  for any s. All of these applied consecutively to 0 1    imply that

if s ≤
∗i N  + hi  wtN ,
∗ N  ∗ ˜s
i fi N  = i s − hi  wtN  if s ≥
∗i N  + hi  wtN  for all s ≥ 0. For s =
this gives (10.5). (ii) As
i ˜ei M =
i M − 1, it follows from (10.5) and (10.5) applied to e˜ i M in place of M that

if
≤ hi  wtN  +
∗i N ,
∗ M ∗
i ˜ei M = i∗
i M − 1 otherwise.

Identification of the crystal

128

Note that wtN  = wtM +
i , so hi  wtN  = hi  wtM + 2
 Therefore


∗i ˜ei M

=

a

if hi  wtM +
+
∗i N  ≥ 0,

a−1

otherwise.

(10.6)

Next, wtL = wtM + ai , so hi  wtL = hi  wtM + 2a

(10.7)

Hence
i L ≥ a − hi  wtL is equivalent to hi  wtM +
i L + a ≥ 0 So (ii) follows if we show that hi  wtM +
∗i N  +
< 0 is equivalent to hi  wtM +
i L + a < 0 Now, by (i) and (10.7), we have hi  wtM +
∗i N  +
= maxhi  wtM +
∗i N  +
i L
∗i N  − a and, by (10.5) and (10.6), we have hi  wtM +
i L + a = maxhi  wtM +
i L +
∗i N 
i L −
 Moreover, obviously
∗i N  − a ≤ 0 and
i L −
≤ 0, and it remains to observe that
∗i N  − a = 0 if and only if
i L −
= 0, thanks to Corollary 10.1.5. (iii) If
i L ≥ a − hi  wtL then (ii) implies b = a, and so by Lemma 10.1.4 we have ˜ei∗ b e˜ i M = e˜ i ˜ei∗ b = e˜ i L However, if
i L < a − hi  wtL, then by (ii) implies b = a − 1, and so by Lemma 10.1.7 we have ˜ei∗ b e˜ i M = ˜ei∗ b e˜ i∗ M = ˜ei∗ a M = L

10.3 Identification of B and B

129

Now for each i ∈ I, define a map 'i  B → B ⊗ Bi

(10.8)

mapping each M ∈ B to ˜ei∗ a M ⊗ f˜ia bi , where a =
∗i M. Lemma 10.3.3 The following properties hold: for every M ∈ B, wtM is a negative sum of simple roots; 1 is the unique element of B with weight 0;
i 1 = 0 for every i ∈ I;
i M ∈  for every M ∈ B and every i ∈ I; for every i, the map 'i  B → B ⊗ Bi defined above is a strict embedding of crystals; (vi) 'i B ⊆ B × f˜in bi  n ≥ 0 ; (vii) for any M ∈ B other than 1 , there exists i ∈ I such that 'i M  = N ⊗ f˜in bi for some N ∈ B and n > 0.

(i) (ii) (iii) (iv) (v)

Proof Properties (i)–(iv) are immediate from our construction of B. The information required to verify (v) is exactly contained in Lemmas 10.3.1 and 10.3.2. Finally, (vi) is immediate from the definition of 'i , and (vii) holds because every such M has
∗i M > 0 for at least one i ∈ I. The properties in Lemma 10.3.3 exactly characterize the crystal B by [KS, Proposition 3.2.3]. Hence, we have proved: Theorem 10.3.4 The crystal B is isomorphic to Kashiwara’s crystal B associated to the crystal base of U− . In view of [Ka, Theorem 8.2], we can also identify our maps 'i with those of [Ka]. Taking into account [Ka, Proposition 8.1] we can then identify our functions
∗i on B with those in [Ka]. It follows from this, Lemma 10.2.2 and [Ka, Proposition 8.2] that: Theorem 10.3.5 For each  ∈ P+ , the crystal B is isomorphic to Kashiwara’s crystal B associated to the integrable highest weight U -module of highest weight . Remark 10.3.6 Recent result of Berenstein and Kazhdan [BeKa] might make much of the work of this section, as well as Section 10.1, unnecessary, providing [BeKa] is generalized from locally finite modules over finite dimensional

130

Identification of the crystal

semisimple Lie algebras to integrable modules over Kac–Moody algebras (which actually seems to be more or less automatic). Indeed, let V be an integrable -module, and B be a basis of V , which consists of weight vectors. For any i ∈ I define Vi
<
i b−1



and also for every b ∈ B and i ∈ I such that fi b = 0 there exists a unique element b ∈ B such that i b  = i b − 1 and fi b ∈ × ·b + V−i

<i b−1

˜ in the obvious way (edge of To a perfect basis B we can associate a crystal B color i connecting b and b as above, etc.) It follows from the main result of ˜ is the same [BeKa] that no matter which perfect basis we take, the crystal B and it is isomorphic to Kashiwara’s crystal associated to the crystal base of V . Now, by Theorems 5.5.1(i) and 8.5.9(i), the classes of the irreducible
n modules form a perfect basis of K. And the result above (providing it holds for Kac–Moody algebras), together with Theorem 9.5.1, imply Theorem 10.3.5, which in turn implies Theorem 10.3.4.

11 Symmetric groups II

In this chapter we specialize to the case  = 0 . In this case
n  FSn , so we will be getting results on symmetric groups. In particular, we obtain a classification of the irreducible FSn -modules, describe how the irreducibles split into blocks (“Nakayama’s Conjecture”), and prove some of the branching rules of [K1 , K2 , K5 , K4 , BK1 ]. As  = 0 is fixed throughout the chapter,   we will not use the superscripts and just write ei for ei 0 , f˜i for f˜i 0 , etc. These should not be confused with the corresponding notions for the affine Hecke algebra
n .

11.1 Description of the crystal graph In the case  = 0 , Misra and Miwa [MiMi] described the crystal B0  in terms of Young diagrams, which we now explain. We will use the terminology concerning partitions introduced in Chapter 1. In particular, recall the definition of the residue content cont of a partition  from (1.2). Fix a partition  of n. We define the important notions of normal and good nodes. Label all i-addable nodes of the diagram  by + and all i-removable nodes by −. Then the i-signature of  is the sequence of pluses and minuses obtained by going along the rim of the Young diagram from bottom left to top right and reading off all the signs. The reduced i-signature of  is obtained from the i-signature by successively erasing all neighboring pairs of the form −+. Note the reduced i-signature always looks like a sequence of +s followed by −s. Nodes corresponding to a “−” in the reduced i-signature are called i-normal, nodes corresponding to a “+” are called i-conormal. The leftmost i-normal node (corresponding to the leftmost − in the reduced i-signature) is 131

Symmetric groups II

132

called i-good, and, similarly, the rightmost i-conormal node is called i-cogood. A node is called normal (resp. conormal, good, cogood) if it is i-normal (resp. i-conormal, i-good, i-cogood) for some i. Note that the notions just introduced are pretty boring for the case p = 0: every removable node is normal and good and every addable node is conormal and cogood. Example 11.1.1 Let p = 5,  = 14 11 10 10 9 4 1. The residues are as follows:

0 4 3 2 1 0 4

1 0 4 3 2 1

2 1 0 4 3 2

3 2 1 0 4 3

4 3 2 1 0

0 4 3 2 1

1 0 4 3 2

2 1 0 4 3

3 2 1 0 4

4 0 1 2 3 3 4 2 1

The residue content of  is  = i i∈I where 0 = 11 1 = 12 2 = 12 3 = 12 4 = 12 The 4-addable and 4-removable nodes are labeled in the diagram:

+ −

+

−h

−

Hence, the 4-signature of  is − + − − +

11.1 Description of the crystal graph

133

and the reduced 4-signature is −. The corresponding node is circled in the above diagram. So, there is one 4-normal node, which is also 4-good; there are no 4-conormal or 4-cogood nodes. In general, we define
i  =  i-normal nodes in 

=  −’s in the reduced i-signature of  

(11.1)

i  =  i-conormal nodes in 

=  +’s in the reduced i-signature of  

(11.2)

Also set

e˜ i  =

f˜i  =

A

if
i  > 0 and A is the (unique) i-good node,

0 if
i  = 0, ⎧ B ⎪ ⎪ ⎨ if i  > 0 and B is the (unique) ⎪ ⎪ ⎩0

(11.3)

(11.4)

i-cogood node, if i  = 0.

The definitions imply that e˜ i  f˜i  are p-regular (or zero) in case  is itself p-regular. Finally define wt = 0 −



i i

(11.5)

i∈I

where cont = i i∈I . We have now defined a datum p 
i  i  e˜ i  f˜i wt which makes the set p of all p-regular partitions into a crystal in the sense of Section 10.2 (O.K., the axiom (C1) is perhaps not so obvious. If you cannot prove it directly, no problem – it will follow from Theorem 11.1.3 anyway). Example 11.1.2 The first nine levels of the crystal graph 2 are as follows:

Symmetric groups II

134

∅ 0 0 1 0 1

@ @

1

0

0 1 0

0 1 1

1

0

0 1 0 1

0 1 0 1

0 1 0 1 0 0 0 1 0 1 0 0 0

0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 0

0 1 0 1 0 1 0 1

A1 A

AA0

0 1 0 1 1 0

0 1 0 1 0 1

0 1 0 1 1 0 1 0

0 1 0 1 0

0 1 0 1 1

AA1

 1 



1

@0 @

1

0

1

0 0 1 0 1 0 1 0

0 0 1 0 1 0 1 0 0

0 1 0 1 0 1

0 1 0 1 0 1 1

@1 @ 0 1 0 1 0 1 1 0

@ 0 @ 0 1 0 1 0 1 0

@0 @ 0 1 0 1 0 1 0 1

@

1

@

0 1 0 1 0 1 0 1

  We now state the result of Misra and Miwa [MiMi]: Theorem 11.1.3 The set p equipped with
i  i  e˜ i  f˜i  wt as above is isomorphic (in the unique way) to the crystal B0  associated to the integrable highest weight U -module of fundamental highest weight 0 . We now introduce some more combinatorial notions relevant to modular representation theory of Sn , see [JK, 2.7]. Until the end of the section we assume that p > 0. The rim of a Young diagram  is its south-east border. A rim p-hook of  is a connected part of the rim with p nodes and such that its removal leaves a Young diagram of a partition. If  has no rim p-hooks, it is called a p-core. In general, the p-core ˜ of  is obtained by successively removing rim p-hooks, until it is reduced to a core (we check that ˜ is well defined). The p-weight of , denoted w, is then the total number of rim p-hooks that need to be removed from  before we reach the p-core.

11.1 Description of the crystal graph

135

It is easy to see using the abacus notation (see [JK, 2.7] for details) that, for   ∈ n, cont = cont if and only if ˜ =  ˜

(11.6)

We now give a Lie theoretic interpretation of the combinatorial notions introduced in the previous paragraph, following the ideas of [LLT, Section 5.3]. Let W be the (affine) Weyl group corresponding to . Recall from [Kc, Section 12] that the set of weights of V0  is P0  = w0 − k  w ∈ W k ∈ ≥0  Moreover, let Par m n denote the number of partitions of n as a sum of positive integers of m different colors. Then the multiplicity of the weight w0 − k in V0  equals Par p−1 k, see [Kc, Section 12.13]. As W leaves  invariant, the W -orbits on P0  are precisely X0  X1  X2     , where Xk = w0 − k  w ∈ W 

(11.7)

Elements of X0 are called extremal weights. Lemma 11.1.4 The weight  = 0 −  only if k = 0 + i∈I i i+1 − i2 .

 i∈I

i i ∈ P0  belongs to Xk if and

Proof We have  ∈ Xk if and only if  = w0 − k. Let ·  · be the normalized invariant form from [Kc, Section 6.2]. Note that k = −w0 − k  w0 − k/2 So it remains to notice that    −0 − i i  0 − i i /2 = 0 + i i+1 − i2  i∈I

i∈I

i∈I

Proposition 11.1.5 Let  be a p-regular partition. Then w = k if and only if wt ∈ Xk . In particular,  is a p-core if and only if wt is an extremal weight. Proof Let w = k and wt ∈ Xl . As the removal of a rim p-hook from a p-regular partition  leads to the subtraction of  from wt, it follows that l ≥ k. In particular, l = 0 implies that  is a p-core. So there is a p-core  with wt = wt − l. Attaching l rim p-hooks horizontally to the first row of  we get a partition  with ˜ =  and wt = wt, whence ˜ = , ˜ and so k = l.

136

Symmetric groups II

Now, Theorem 11.1.3 and Kac’s formula [Kc, (12.13.5)] for the character of the highest weight U -module of highest weight 0 imply a classical combinatorial result: for  ∈ p n,   ∈ p n  cont = cont = Par p−1 w

(11.8)

11.2 Main results on Sn By Theorem 10.3.5, the isomorphism classes of irreducible FSn -modules are parametrized by the nodes of the crystal graph B0 . By Theorems 11.1.3 and 10.3.5, we can identify B0  with p . In other words, we can use the set p n of p-regular partitions of n to label the irreducible modules over FSn for each n ≥ 0. Let us write D for the irreducible FSn -module corresponding to  ∈ p n. To be precise D = Li1      in  if  = f˜in    f˜i1 ∅, see (5.14). Here the operator f˜i is as defined in (11.4), corresponding under the identification p n = B0  to the crystal operator  denoted f˜i 0 in earlier sections, and ∅ denotes the empty partition, corresponding to 1
∈ B0 . Theorem 11.2.1 The modules D   ∈ p n form a complete set of pairwise non-isomorphic irreducible FSn -modules. Moreover, for   ∈ p n we have: (i) D is self-dual; (ii) modules D and D belong to the same block of FSn if and only if cont = cont; (iii) D is a projective module if and only if  is p-core. Proof We have already discussed the first statement of the theorem, being a consequence of our main results combined with Theorem 11.1.3. Now, (i) follows from Corollary 5.3.2, and (ii) is a special case of Corollary 9.6.2. For (iii), note that if D is projective then it is the only irreducible in its block, hence by (11.8), Par p−1 w = 1. So either w = 0, or p = 2 and w = 1. Now if w = 0 then  is p-core so the contravariant form on the (1-dimensional) wt-weight space of K0  is 1 (since wt is conjugate to 0 under the action of the affine Weyl group). Hence, D is projective by

11.2 Main results on Sn

137

Theorem 9.5.1(ii). To rule out the remaining possibility p = 2, w = 1, we check in that case that the contravariant form on the wt-weight space of K0  is 2. Remark 11.2.2 Our characterization of the irreducible FSn -module D corresponding to a p-regular partition  is explicit but rather unpractical. Indeed, in order to say what D is, we need to do the following. First, find a path of length n in the crystal graph B0  from  to the empty partition ∅ – in practice this amounts to “destroying”  by successively removing good nodes, say, of residues in      i1 , where in is the residue of the first removed node, in−1 is the residue of the second removed node, etc. This is cumbersome but explicit combinatorics. Moreover, some remarkable special sequences in      i1 can be read off the formal character of D – this will be explained in Remark 11.2.14. The second step however is more unpleasant: this sequence of residues is then used to define D as f˜in    f˜i1 D∅ where D∅ is the trivial FS0 -module 1 (recall that FS0 is interpreted as F ). This means that the operation f˜i should be performed n times, that is n times we should induce, project to a block and take the head of the resulting module. A much more direct approach to the construction of D is the classical approach of James [J]. For each partition  of n there is an explicit construction of a module S  , the corresponding Specht module. Its head D is proved to be irreducible when  is p-regular, and the D form a complete set of irreducible FSn -modules. Thus we only have to deal with taking the head once. The natural question arises if our D are the same as James’ D . The answer to this question is “yes”, but unfortunately the only proof we know is somewhat unsatisfactory. Namely, we compare the “socle branching rule” for James’ D , established in [K1 ] by completely different methods, with Theorem 11.2.7(i),(ii) below, and observe that the two branching rules are exactly the same. This is enough to identify the two labelings of irreducible modules. It would be interesting to find a more direct proof. Now, consider in more detail the natural surjection  
n 
n /0 =
n0  FSn (see Remark 7.5.7). Under this surjection the Coxeter generator sm of
n maps on to the Coxeter generators sm of FSn , for any 1 ≤ m < n. We now describe the images of the polynomial generators xk ∈
n . Recall the JM-elements Lk from (2.1).

Symmetric groups II

138

Lemma 11.2.3 For any 1 ≤ k ≤ n, we have xk  = Lk . Proof Induction on k. For k = 1 we know that x1  = 0 because 0 =
n x1
n . However, L1 is also zero. For the induction step, just observe that Lk = sk Lk−1 sk + sk (clear) and xk = sk xk−1 sk + sk (see (3.6)). We can now use JM-elements to “avoid” affine Hecke algebras in definitions concerning symmetric groups. We noted in Section 2.1 that the JM-elements commute (it also follows from the fact that they are images of commuting elements under ). Lemma 7.3.1 implies that JM-elements act on FSn -modules with “integral” eigenvalues, that is eigenvalues from I =  · 1 ⊂ F . If i = i1      in  ∈ I n and M ∈ FSn -mod we define its i-weight space Mi = v ∈ M  Lk − ik N v = 0 for N 0 and k = 1     n  Lemma 11.2.4 Any M ∈ FSn -mod decomposes as M = ⊕i∈I n Mi . Now define the formal character of an FSn -module M by  ch M = dim Mi ei 

(11.9)

i∈I n

an element of the free -module on the formal basis ei  i ∈ I n . From Theorem 5.3.1 we get: Lemma 11.2.5 Let D1      Dk be non-isomorphic simple FSn -modules. then ch D1      ch Dk are -linearly independent. It follows from Theorem 3.3.1 that the symmetric polynomials in the JMelements form a central subalgebra in FSn (Murphy [Mu2 , 1.9] proves that this central subalgebra actually equals the center of FSn , but we will never need this fact). The characters of this central subalgebra are captured by the following  data. Let  = i∈I i i ∈ #n (see Section 8.1). For an FSn -module M set  M = Mi (11.10) i∈I n with conti=

(see (8.1)). Theorem 11.2.6 For  =

 i∈I

i i ∈ #n the following are equivalent:

(i) there exists a finite dimensional FSn -module M with M = 0; (ii) 0 −  is a weight of V0 ; (iii) there exists a p-regular partition  of n whose residue content is i i∈I .

11.2 Main results on Sn

139

Moreover, two FSn -modules M and N are in the same block if and only if M = M and N = N for some  ∈ #n . Proof Follows from Theorem 11.2.1(ii) and Corollary 9.6.2. Now, the functor ei  FSn -mod → FSn−1 -mod is defined on a module M as the generalized i-eigenspace of the last JM-element Ln acting on M. This is considered as an FSn−1 -module via restriction to FSn−1 , see (5.1) and (5.6). The more general divided power functor r

ei  FSn -mod → FSn−r -mod on a module M is defined as follows (see Section 8.3). Let Sr be the subgroup r of Sn from (8.12). Now ei M is the space of Sr -invariants in the simultaneous generalized i-eigenspace of the last r JM-elements Ln−r+1      Ln . Again, this is considered as an FSn−r -module via restriction. As explained in Section 8.3, taking Sr -coinvariants or Sr -anti(co)invariants instead of invariants yields isomorphic functors. As in Section 8.3, in the definitions above, instead of taking generalized eigenspaces, we could just project to appropriate blocks of FSn−r . More precisely, assume first that M belongs to a fixed block of FSn , i.e. M = M for  ∈ #n . Then the simultaneous generalized i-eigenspace of the last S r JM-elements on M equals resSnn−r M − ri (restriction followed by the projection to the block of Sn−r corresponding to  − ri ). Finally, we extend r ei to an arbitrary M by additivity. This approach is used to define the dual functors r

fi  FSn -mod → FSn+r -mod  On a module M ∈ FSn -mod , we have 
S r M
1Sr  + ri  fi M = indSn+r n ×Sr see (8.15). Then the functor is extended to an arbitrary module by additivity. Taking the sign module sgnSr instead of the trivial module 1Sr leads to an 1 isomorphic functor. Set fi = fi . The next two theorems summarize earlier results concerning restriction and induction in the special case of the symmetric groups. Theorem 11.2.7 Let  ∈ p n. We have resFSn−1 D  e0 D ⊕ e1 D ⊕ · · · ⊕ ep−1 D 

Symmetric groups II

140

and, for each i ∈ I, ei D = 0 if and only if  has an i-good node A, in which case ei D is a self-dual indecomposable module with irreducible socle and head isomorphic to DA . Moreover, if i ∈ I and  has an i-good node A, then: the multiplicity of DA in ei D is
i ,
i A  =
i  − 1, and
i  <
i  − 1 for all other composition factors D of ei D ; (ii) The algebra EndFSn−1 ei D  is isomorphic to the algebra of truncated polynomials F x /x
i  . In particular,

(i)

dim EndFSn−1 ei D  =
i  (iii) HomFSn−1 ei D  ei D  = 0 for all  ∈ p n with  = ; (iv) ei D is irreducible if and only if
i  = 1. In particular, the restriction resFSn−1 D is completely reducible if and only if
i  ≤ 1 for every  i ∈ I, and resFSn−1 D is irreducible if and only if i∈I
i  = 1. Proof The first statement follows from (8.7) and Theorem 8.2.5(i), combined as usual with Theorem 11.1.3. For the remaining properties, (i),(ii), and (iii) follow from Theorem 5.5.1 and Corollary 5.5.2. Finally, (iv) follows from (i) as ei D is a module with irreducible socle and head both isomorphic to DA . Theorem 11.2.8 Let  ∈ p n. We have indFSn+1 D  f0 D ⊕ f1 D ⊕ · · · ⊕ fp−1 D  and, for each i ∈ I, fi D = 0 if and only if  has an i-cogood node B, in which case fi D is a self-dual indecomposable module with irreducible socle and B head isomorphic to D . Moreover, if i ∈ I and  has an i-cogood node B, then: B

the multiplicity of D in fi D is i , i B  = i  − 1, and i  < i  − 1 for all other composition factors D of fi D ; (ii) the algebra EndFSn+1 fi D  is isomorphic to the algebra of truncated polynomials F x /xi  . In particular,

(i)

dim EndFSn+1 fi D  = i  (iii) HomFSn+1 fi D  fi D  = 0 for all  ∈ p n with  = ; (iv) fi D is irreducible if and only if i  = 1. In particular, indFSn+1 D is completely reducible if and only if i  ≤ 1 for every i ∈ I.

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141

Proof The argument is the same as in Theorem 11.2.7, but using (8.7), Theorem 8.2.5(ii), Corollary 8.5.2, and Theorem 8.5.9. Remark 11.2.9 We state without proof some further branching results mentioned in the introduction, which do not seem to follow from the methods developed here. Let  ∈ p n. The following is proved in [K4 ] (resp. [BK1 ]). Suppose that A is an i-removable (resp. i-addable) node of  such that A A (resp. A ) is p-regular. Then, ei D  DA (resp. fi D  D is the number of i-normal (resp. i-conormal) nodes to the right (resp. left) of A, counting A itself, or 0 if A is not i-normal (resp. i-conormal). r

r

We also have somewhat less strong results on ei and fi . Theorem 11.2.10 Let  ∈ p n, i ∈ I and r ≥ 1. We have eir D  ei D ⊕r!  r

r

and ei D = 0 if and only if  has at least r i-normal nodes, in which case r ei D is a self-dual indecomposable module with irreducible socle and head isomorphic to D , where  is obtained from  by removing r bottom i-normal nodes. Moreover:
 r ,
i  =
i  − r, and (i) the multiplicity of D in ei D is
i  r r
i  <
i  − r for all other composition factors D of ei D ; r  (ii) The endomorphism algebra EndFSn−r ei D  is isomorphic to the algebra r / r ∩ 
i  +
i   of (8.24). In particular,  
 r dim EndFSn−r ei D  = i  r r

r

(iii) HomFSn−r ei D  ei D  = 0 for all  ∈ p n with  = ; r (iv) ei D is irreducible if and only if r =
i . Proof The first statement follows from Theorem 8.3.2. For the remaining properties, (i) and (iii) follow from Proposition 8.5.10(i) and (ii) respectively; (ii) comes from Remark 8.6.3 and Proposition 8.6.2, and (iv) follows from (i).

Theorem 11.2.11 Let  ∈ p n, i ∈ I and r ≥ 1. We have fir D  fi D ⊕r!  r

r

and fi D = 0 if and only if  has at least r i-conormal nodes, in which case r fi D is a self-dual indecomposable module with irreducible socle and head

Symmetric groups II

142

isomorphic to D , where  is obtained from  by adding r top i-conormal nodes. Moreover: 
r , i  = i  − r, and (i) the multiplicity of D in fi D is i  r r i  < i  − r for all other composition factors D of fi D ;
  r  (ii) dim EndFSn+r fi D  = i r  r

r

(iii) HomFSn+r fi D  fi D  = 0 for all  ∈ p n with  = ; r

(iv) fi D is irreducible if and only if r = i . Proof Similar to the proof of Theorem 11.2.10 using Theorem 8.3.2, Proposition 8.5.10(i ),(ii ), and (8.25). Let M be an FSn -module. Define
i M = max r ≥ 0  eir M = 0  i M = max r ≥ 0  fir M = 0 

(11.11)

From Theorems 11.2.10 and 11.2.11 we have: Lemma 11.2.12 Let  ∈ p n. Then: (i)
i D  = max r ≥ 0  e˜ ir M = 0 =
i . (ii) i D  = max r ≥ 0  f˜ir M = 0 = i . Note that
i M can be computed just from knowledge of the character of r M: it is the maximal r such that ei  appears with non-zero coefficient in ch M. Less obviously, i M can also be read off from the character of M. By additivity of fi , we may assume that M = M for  ∈ #n . Then, from (9.20) we get: i M =
i M + i0 − 2i + i−1 + i+1 

(11.12)

r

We record the effect of ei on formal characters: Lemma 11.2.13 Let M ∈ FSn -mod and ch M =  i i∈I n−r ai1 in−r ir  e .



r

i∈I n

ai ei . Then ch ei M =

Proof For r = 1 this follows from the definition. For r > 1, use the fact that r eir = ei ⊕r! . Remark 11.2.14 We describe an inductive algorithm to determine the label of an irreducible FSn -module D purely from knowledge of its character 
 ch D. Pick i ∈ I such that
=
i D is non-zero. Let E = ei D. In view of

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143

Theorem 11.2.10(iv), E is an irreducible FSn−
-module, isomorphic to e˜ i
D. 
 By Lemma 5.2.3, D  f˜i E. Moreover, the formal character of E is explicitly known by Lemma 11.2.13. By induction, the label  of E can be computed purely from knowledge of its character. Then, D  f˜i
D  D , where  is obtained from  by adding the rightmost
of the i-conormal nodes. We would of course like to be able to reverse this process: given a p-regular partition  of n, we would like to be able to compute the character of the irreducible FSn -module D . We can compute a quite effective lower bound for this character inductively using Remark 11.2.9. But only over  is this lower bound always correct: indeed if p = 0 then from combinatorics we always have
i ≤ 1 and so  i i  e 1 n (11.13) ch D = i1 in  i1

i2

in

summing over all paths ∅ −→ 1 −→ · · · −→  in the characteristic zero crystal graph (that is Young’s partition lattice) from ∅ to . Of course, we already know this result from Chapter 2. Remark 11.2.15 There is an interesting class of modules in characteristic p where the lower bound just mentioned gives a correct answer for the formal character. These modules are called completely splittable and can be characterized as FSn -modules whose restriction to any standard subgroup in Sn is completely reducible. For more information on this see [K3 , M, R]. Remark 11.2.16 Reducing the entries ik in (11.13) modulo p gives the formal characters of the Specht module in characteristic p – this follows from the branching rule for Specht modules [J, 9.3]. Next we explain a useful inductive method for finding some composition factors of FSn -modules using their formal characters and induction. It is based on the following: 


Lemma 11.2.17 Let M be an FSn -module and set
=
i M. If ei M  D = m > 0 then f˜i
D = 0 and M  f˜i
D = m. Proof Follows from Theorem 11.2.10(iv) and Lemmas 11.2.5, 5.2.3. Using the known characters of Specht modules (see Remark 11.2.16) and Lemma 11.2.17 provides new non-trivial information on decomposition numbers, which is difficult to obtain by other methods.

144

Symmetric groups II

Example 11.2.18 Let p = 3. By [J, Tables], the composition factors of the 2 Specht module S 6421 are D121 , D94 , D92  , D742 , D652 , D643 , and 2 2 D6421 , all appearing with multiplicity 1. As
1 S 642   = 1 and e1 S 642  = S 6421 by Remark 11.2.16, application of Lemma 11.2.17 implies that the 2 2 following composition factors appear in S 642  with multiplicity 1: D121  , 2 2 2 D941 , D932 , D842 , D6 2 , D64  , and D642  . Given i = i1      in  ∈ I n we can gather consecutive equal terms to write it in the form m

i = j1 1    jrmr 

(11.14)

where js = js+1 for all 1 ≤ s < r. For example 2 2 2 1 1 = 23 12 . Now, for an FSn -module M, the weight (11.14) is called extremal if m

m

s+1 ms =
js ejs+1    ejr r M

for all s = r r − 1     1 (do not confuse with extremal weights for  as in Section 11.1). Informally speaking this means that among all weights i of M we first choose those with the longest jr -string in the end, then among these we choose the ones with the longest jr−1 -string preceding the jr -string in the end, etc. By definition Mi = 0 if i is extremal for M. Example 11.2.19 The formal character of the Specht module S 52 in characteristic 3 is e0210201 + 2e0120201 + 2e02120 +e0212010 + 2e012

2 010

2 1

+ 4e012

2 02 1

+ e0120210 + e0120120 

The extremal weights are 0122 02 1, 0122 010, 0120210, 0120120 Our main result about extremal weights is: m

Theorem 11.2.20 Let i = i1      in  = j1 1    jrmr  be an extremal weight for an irreducible FSn -module D written in the form (11.14). Then D = f˜in    f˜i1 D∅ , and dim Di = m1 !    mr !. In particular, the weight i is not extremal for any irreducible D  D . Proof We apply induction on r. If r = 1, then by considering possible weights appearing in the Specht module S  , of which D is a quotient, we

11.2 Main results on Sn

145

conclude that n = 1 and D = D1 . So for r = 1 the result is obvious. Let r > 1. By definition of an extremal weight, mr =
jr D . So, in view of Theorem 11.2.10, we have ejr r D = ˜ejr r D ⊕mr !  m

m

m

m

r−1  is clearly an extremal weight for the irreducible Moreover, j1 1    jr−1 mr  module e˜ jr D . So the inductive step follows:

m

Corollary 11.2.21 If M is an FSn -module and i = i1      in  = j1 1    jrmr  is an extremal weight for M written in the form (11.14), then the multiplicity of D = f˜in    f˜i1 D∅ as a composition factor of M is dim Mi /m1 !    mr !. Example 11.2.22 In view of Corollary 11.2.21, the extremal weight 0122 02 1 in Example 11.2.19 yields the composition factor D52 of S 52 , while the extremal weight 0120120 yields the composition factor D7 . It turns out that these are exactly the composition factors of S 52 , see For example [J, Tables]. For more non-trivial examples let us consider a couple of Specht modules 2 for n = 11 in characteristic 3. For S 631  , Corollary 11.2.21 yields compo2 6312  731 821 ,D , and D but ‘misses’ D11 , and for S 432  sition factors D 2 we get hold of D432  , D5321 , D821 , and D83 , but ‘miss’ 2D11 and 2 D541  , cf. [J, Tables]. We record here some other useful general facts about formal characters: Lemma 11.2.23 For any weight i represented in the form (11.14) and any FSn -module M we have that dim Mi is divisible by m1 !    mr !. Proof We can lift M to an
n -module. By Theorem 4.3.2, each composition m factors of resnm1 mr M isomorphic to Lj1 1 
· · ·
Ljrmr  contributes the multiplicity of m1 !    mr ! to the i-weight space of M, and no other composition factors contribute to this weight. Lemma 11.2.24 Let M be an FSn -module. Assume i j i1      in−2 ∈ I and i = j: (i) Assume that i − j > 1. Then for any 1 ≤ r ≤ n − 2 we have dim Mi1 ir ijir+1 in−2  = dim Mi1 ir jiir+1 in−2  

146

Symmetric groups II

(ii) Assume that i − j = 1 and p > 2. Then for any 1 ≤ r ≤ n − 3 we have 2 dim Mi1 ir ijiir+1 in−3  = dim Mi1 ir iijir+1 in−3  + dim Mi1 ir jiiir+1 in−3   (iii) Assume that i − j = 1 and p = 2. Then for any 1 ≤ r ≤ n − 4 we have dim Mi1 ir iiijir+1 in−4  +3 dim Mi1 ir ijiiir+1 in−4  = dim Mi1 ir jiiiir+1 in−4  +3 dim Mi1 ir iijiir+1 in−4   Proof Follows from the Serre relations satisfied by the operators ei , see Lemma 9.2.4. Combining Lemma 11.2.24 with Corollary 11.2.21 and Theorem 11.2.9 we obtain further non-trivial results on branching, which are difficult to get by other methods. To illustrate: 2

Example 11.2.25 We explain how to see that D732  appears as a composition factor of e2 D6432 . We have
2 D6432  = 2. Note that e˜ 22 D6432 = 2 D53 2 . So by Theorem 11.2.10(iv), the weights ending on 2 2 appearing in the character of D6432 are all obtained just by adding 2 2 to the end 2 2 of the weights appearing in the character of D53 2 . Next,
0 D53 2  = 1. 532 2 which ends at So there is an extremal weight in the character of D 0, say i1      i12  0. Then the weight i1      i12  0 2 2 appears in the character of D6432 . By Lemma 11.2.24(ii), the weight i1      i12  2 0 2 also appears in the character. This weight contributes i1      i12  2 0 to the character of e2 D6432 . Note that the character of e2 D6432 could not involve weights ending on 0 0 because then the character of D6432 would involve a weight ending on 0 0 2 and then, by Lemma 11.2.24(ii), the weight ending on 0 2 0, which contradicts the fact that
0 D6432  = 0. Moreover, the character of e2 D6432 could not involve weights ending on 2 2 0 because then the character of D6432 would involve a weight ending on 2 2 0 2 and then, by Lemma 11.2.24(ii), the weight ending on 2 0 2 2, which contradicts the fact that
2 ˜e0 e˜ 22 D6432  = 0. The two facts just observed and the choice of i1      i12 imply that i1      i12  2 0 is an

11.2 Main results on Sn

147

extremal weight for the module e2 D6432 . So, by Corollary 11.2.21, this 2 module has D732  = f˜0 f˜2 f˜i12    f˜i1 D∅ as a composition factor. Finally we discuss some properties of blocks, assuming now that p = 0. The affine Weyl group W acts on the -module V0  = ⊕n≥0 KFSn -mod  the generator si of W acting by the familiar formula si = exp−ei  expfi  exp−ei 

i ∈ I

(11.15)

The resulting action preserves the Shapovalov form, and leaves the lattices ⊕n≥0 KFSn -mod and ⊕n≥0 KFSn -proj invariant. Moreover, W permutes the weight spaces of V0  in the same way as its defining action on the weight lattice P, the orbits being Xk , see (11.7) and the discussion at the end of Section 11.1. So using Proposition 11.1.5 we see: Theorem 11.2.26 Let B and B be blocks of symmetric groups with the same p-weight. Then B and B are isometric in the sense that there is an isomorphism between their Grothendieck groups that is an isometry with respect to the Cartan form. Remark 11.2.27 The existence of such isometries was first noticed by Enguehard [E]. Implicit in Enguehard’s paper is the following conjecture, made formally by Rickard: blocks B and B of symmetric groups with the same p-weight should be derived equivalent. Moreover, it is known by work of Marcus [Ma] and Chuang–Kessar [CK] that the Abelian Defect Group Conjecture of Broué for symmetric groups follows from the Rickard’s conjecture above. The conjecture has been proved by Rickard for blocks of p-weight ≤ 5 and in full generality recently by Chuang and Rouquier [CR], cf. Remark 9.6.3. The complex of functors leading to the derived equivalence can be guessed by looking at the formula (11.15). There is one situation when there is actually a Morita equivalence between blocks of the same p-weight. This is a theorem of Scopes [Sc], though we are stating the result in a more Lie theoretic way following [LM, Section 8]: Theorem 11.2.28 Let   + i       + ri be an i -string of weights of V0  (so  − i and  + r + 1i are not weights of V0 ). Then the r r functors fi and ei define mutually inverse Morita equivalences between the blocks parametrized by  and by  + ri .

Symmetric groups II

148 r

r

Proof Since ei and fi are both left and right adjoint to one another, it r r suffices to check that ei and fi induce mutually inverse bijections between the isomorphism classes of irreducible modules belonging to the respective blocks. This follows from Theorems 11.2.10(iv) and 11.2.11(iv). Remark 11.2.29 We can use Lie Theory to explicitly compute the determinant of the Cartan matrix of a block. The details of the proof appear in [BK6 ] (see also [BO2 ] for a different approach). Note that, in view of Theorem 11.2.26, the determinant of the Cartan matrix only depends on the p-weight of the block. Moreover, by Theorem 9.5.1, we can work instead in terms of the Shapovalov form on V0 . Using the explicit construction of the latter module over  given in [CKK], we show: if B is a block of p-weight w of FSn , then the determinant of the Cartan matrix of B is pN , where     r1 + r2 + · · · p − 2 + r1 p − 2 + r2  N= p−1 r1 r2 =1r1 2r2  'w

PART II Projective representations

Throughout Part II of the book F will stand for an algebraically closed field of characteristic p = 2. The main goal of Part II is to develop a theory of projective or spin representations of symmetric and alternating groups which is parallel to the “classical” theory developed in Part I. Although in some places we do not go quite as far as in Part I, the similarity of the two theories is compelling. Informally speaking we just have to replace the Kac–Moody algebra of type 1 2 Ap−1 with the twisted Kac–Moody algebra of type Ap−1 . Having this in mind, we will always indicate which chapter in Part I a given chapter in Part II is parallel to (starting with Chapter 14). The reader is then advised to first browse the corresponding chapter of Part I and especially read a motivation and informal explanations in the beginning of that chapter, since those will not be duplicated in Part II. Out of two synonymous terms projective representation and spin representation we will stick with spin representation (or spin module), while the term projective representation or projective supermodule will be reserved for the usual homological algebra notion (direct summand of a free module).

12 Generalities on superalgebra

The language of superalgebra will be used throughout Part II of this book – this is convenient when dealing with spin representations of symmetric and alternating groups. In this chapter we review some known results concerning superalgebras and their modules. We recommend [Le, Chapter I], [Man, Chapter 3, Sections 1 and 2], and especially [Jos] as basic references. Alternatively, for reader’s convenience we sketch the proofs of the results which are going to be used later. When dealing with superalgebras, it is natural to assume that the characteristic of the ground field is different from 2. This restriction will not be a problem later when we study spin representations of Sn and An , because in characteristic 2 such representations are linear and hence have been treated in the first part of this book.

12.1 Superalgebras and supermodules By a (vector) superspace we mean a 2 -graded vector space V = V0¯ ⊕ V1¯ over F . If dim V0¯ = m and dim V1¯ = n we write sdim V = m n and dim V = m+n. Elements of V0¯ are called even and elements of V1¯ are called odd. A vector is called homogeneous if it is either even or odd. Given a homogeneous vector 0 = v ∈ V , we denote its degree by v¯ ∈ 2 . A subspace W of V is called a subsuperspace if it is homogeneous, that is W = W ∩ V0¯  + W ∩ V1¯ . Define the linear map V  V → V

v → −1v¯ v

for homogeneous vectors v. Note it is typical in superalgebra to write expressions which only make sense for homogeneous elements, and the expected meaning for arbitrary elements is obtained by extending linearly from the 151

152

Generalities on superalgebra

homogeneous case. It is easy to see that a subspace of V is a subsuperspace if and only if it is stable under V . Given superspaces V and W , we view the direct sum V ⊕ W and the tensor product V ⊗ W as superspaces with V ⊕ W i = Vi ⊕ Wi , and V ⊗ W 0¯ = V0¯ ⊗ W0¯ ⊕ V1¯ ⊗ W1¯ , V ⊗ W 1¯ = V0¯ ⊗ W1¯ ⊕ V1¯ ⊗ W0¯ . Also, we make the vector space HomF V W  of all linear maps from V to W into a superspace by declaring that HomF V W i consists of the homogeneous maps of degree i for each i ∈ 2 , that is, the linear maps  V → W with Vj  ⊆ Wi+j for j ∈ 2 . Elements of HomF V W 0¯ will be referred to as even linear maps. The dual superspace V ∗ is HomF V F , where we view F as a superspace ¯ concentrated in degree 0. A superalgebra is a vector superspace  with the additional structure of an associative unital F -algebra such that i j ⊆ i+j for i j ∈ 2 . By forgetting the grading we may consider any superalgebra  as a usual algebra–this algebra will be denoted . By a superideal of  we mean a homogeneous ideal. Left and right superideals are defined similarly. The superalgebra  is called simple if if it has no non-trivial superideals. A superalgebra homomorphism   → is an even linear map that is an algebra homomorphism in the usual sense; its kernel is a superideal. An antiautomorphism    →  of a superalgebra  is an even linear map which satisfies ab = ba (note there is no sign). Given two superalgebras  and , we view the tensor product of superspaces  ⊗ as a superalgebra with multiplication defined by ¯



a ⊗ ba ⊗ b  = −1ba¯ aa  ⊗ bb 

a a ∈  b b ∈  (12.1)

We note that  ⊗  ⊗ , an isomorphism being given by the supertwist map T   ⊗ → ⊗ 

¯

a ⊗ b → −1a¯ b b ⊗ a

a ∈  b ∈ 

It is important that the tensor product of two superalgebras  and is not the same as the tensor product of  and as usual algebras (with natural 2 -grading) – the product rule is different! Example 12.1.1 If V is a superspace with sdim V = m n then V  = EndF V  is a superalgebra with sdim V  = m2 + n2  2mn. Moreover, if W is another finite dimensional superspace, we have an isomorphism of superalgebras V  ⊗ W   V ⊗ W 

(12.2)

12.1 Superalgebras and supermodules

153

Under this isomorphism f ⊗ g corresponds to the endomorphism of V ⊗ W , mapping v ⊗ w to −1g¯ v¯ fv ⊗ gw. In what follows we identify the algebras V  ⊗ W  and V ⊗ W  using this isomorphism. In particular, it makes sense to speak of the element f ⊗ g ∈ EndF V ⊗ W . Moreover, the algebra V  is defined uniquely up to an isomorphism by the superdimension m n of V . So we can speak of the superalgebra mn , which can also be identified with the obvious superalgebra of matrices. Now, (12.2) becomes mn ⊗ kl  mk+nlml+nk 

(12.3)

Finally, we note that mn is a simple superalgebra, as mn  = m+n , the algebra of m + n × m + n matrices, which is simple. Example 12.1.2 Let V be a finite dimensional superspace and J be a degree 1¯ involution in EndF V . Such one exists if and only if dim V0¯ = dim V1¯ . Consider the superalgebra ¯

V J  = f ∈ EndF V   fJ = −1f Jf  Note that all degree 1¯ involutions in EndF V  are conjugate to each other by an (invertible) element in EndF V 0¯ . Hence another choice of J will yield an isomorphic superalgebra. So we can speak of the superalgebra V , defined up to an isomorphism. Let sdim V = n n. Pick a basis v1      vn of V0¯ , and set vi = Jvi  for 1 ≤ i ≤ n. Then v1      vn is a basis of V1¯ . With respect to the basis v1      vn  v1      vn , the elements of V J  have matrices of the form   A B  (12.4) −B A where A and B are arbitrary n × n matrices, with B = 0 for even endomorphisms and A = 0 for odd ones. In particular, sdim V  = n2  n2 . The superalgebra V J  can be identified with the superalgebra n of all matrices of the form (12.4). Moreover, it is easy to see that the isomorphism (12.2) restricts to an isomorphism: V  ⊗ W J   V ⊗ W idV ⊗J 

(12.5)

or, in terms of matrix superalgebras, mn ⊗ k  m+nk 

(12.6)

154

Generalities on superalgebra

Let
be a primitive 4th root of 1 in F . It is easy to check explicitly that the map       a b  a b aa −
bb a b −
ab ⊗  → −b a −b a −a b −
ab aa +
bb from 1 ⊗ 1 to 11 is a superalgebra isomorphism. So (12.3) and (12.6) imply by induction that m ⊗ n  mnmn 

(12.7)

Finally, n is easily checked to be a simple superalgebra, even though it is not simple as a usual algebra. In fact, n  = n × n . Example 12.1.3 Define the Clifford superalgebra n to be the superalgebra given by odd generators c1      cn , subject to the relations ci2 = 1 ci cj = −cj ci

1 ≤ i ≤ n

(12.8)

1 ≤ i = j ≤ n

(12.9)

It is easy to see that n+m  n ⊗ m  The isomorphism is defined as follows: the generators c1      cn are mapped to c1 ⊗ 1     cn ⊗ 1, and cn+1      cn+m are mapped to 1 ⊗ c1      1 ⊗ cm , respectively. It follows that n  1⊗n  In particular, sdim n = 2n−1  2n−1 . Note that 1  1 . So from (12.3), (12.6), and (12.7) we have 2k  2k−1 2k−1 and 2k−1  2k−1 for k = 1 2    . In particular, n is a simple superalgebra. Example 12.1.4 Define the Grassman superalgebra n to be the superalgebra given by odd generators d1      dn , subject to the relations di2 = 0 di dj = −dj di

1 ≤ i ≤ n

(12.10)

1 ≤ i = j ≤ n

(12.11)

As for Clifford algebra, it is easy to see that there is a natural isomorphism n  ⊗n 1  More generally, let j = j1      jn  ∈ F n be fixed, and j be the superalgebra given by odd generators a1      an , subject to the relations a2i = ji ai aj = −aj ai

1 ≤ i ≤ n

(12.12)

1 ≤ i = j ≤ n

(12.13)

12.1 Superalgebras and supermodules

155

We call j the Clifford–Grassman superalgebra. Using the fact that F is algebraically closed, it is easy to see that ⊗0

j  1

⊗n−0

⊗ 1



(12.14)

where 0 =  i  1 ≤ i ≤ n ji = 0 . Let  be a superalgebra. A (left) -supermodule is a vector superspace V which is a left -module in the usual sense, such that i Vj ⊆ Vi+j for i j ∈ 2 . Right supermodules are defined similarly. A homomorphism f  V → W of (left) -supermodules V and W means a (not necessarily homogeneous) linear map such that ¯

fav = −1f a¯ afv

a ∈  v ∈ V 

The category of finite dimensional -supermodules is denoted -smod. A homomorphism f  V → W of right -supermodules V and W means a (not necessarily homogeneous) linear map such that fva = fva

a ∈  v ∈ V 

Note there is no sign here as a “does not go past f ”. If is a subsuperalgebra of , and W is a -supermodule we write ind W as a functor for the induced supermodule  ⊗ W . We may consider ind from the category of -supermodules to the category of -supermodules. This functor is left adjoint to the restriction functor res going in the other direction. If  is free as a right -supermodule the induction functor is exact. Any -supermodule V can be considered as a usual -module denoted V . Thus, for V ∈ -smod we have V  ∈ -mod. The following result establishes an isomorphism of vector spaces between Hom V W  and Hom V  W . Lemma 12.1.5 Let V W ∈ -smod, and f  V → W be a linear map. Define a linear map f −  V → W

¯

v → −1f v¯ fv

Then f ∈ Hom V W  if and only if f − ∈ Hom V  W . Let  be an antiautomorphism of the superalgebra . If V is a finite dimensional -supermodule, then we can use  to make the dual space V ∗ into an -supermodule by defining af v = fav

a ∈  f ∈ V ∗  v ∈ V 

(12.15)

156

Generalities on superalgebra

We will denote the resulting module by V  . There is a natural even isomorphism Hom V W  → Hom W   V  

(12.16)

for any V W ∈ -smod. The isomorphism sends ∈Hom V W  to the dual map ∗ ∈ Hom W   V   defined by  ∗ f v = f v for all f ∈ W  . We have the (left) parity change functor "  -smod → -smod 

(12.17)

For an object V , "V is the same underlying vector space but with the opposite 2 -grading. The new action of a ∈  on v ∈ "V is defined in terms of the old action by a · v = −1a¯ av. On a morphism f , "f is the same underlying linear map as f . Note that the identity map on V defines an odd isomorphism from V to "V . It is more subtle however whether V and "V are evenly isomorphic, see for example Lemma 12.2.8 below. We will write V W if the -supermodules V and W are isomorphic, and V W if V and W are evenly isomorphic. Given two superalgebras  , an  -bisupermodule is a left -supermodule V , which is also a right -supermodule (with respect to the same grading of V ) such that avb = avb for all a ∈  b ∈  v ∈ V . A homomorphism f  V → W of  -bisupermodules is a map which is both a homomorphism of left -supermodules and a homomorphism of right supermodules. If V is an  -bisupermodule, "V denotes the -supermodule, defined as in the previous paragraph, with the right -action on "V being the same as the original action on V . For a superalgebra , the subcategory -smodev of -smod, consisting of the same objects but only even morphisms, is an abelian category in the usual sense. This allows us to make use of all the basic notions of homological algebra by restricting our attention to even morphisms. For example, by a short exact sequence in -smod, we mean a sequence 0 −→ V1 −→ V2 −→ V3 −→ 0

(12.18)

with all the maps being even. All functors between categories of superobjects that we will ever consider send even morphisms to even morphisms. So they will give rise to the corresponding functors between the underlying even subcategories.

12.2 Schur’s Lemma and Wedderburn’s Theorem

157

We define the Grothendieck group K-smod to be the quotient of the free -module with generators given by all finite dimensional -supermodules by the -submodule generated by: (1) V1 − V2 + V3 for every short exact sequence of the form (12.18); (2) V − "V for every -supermodule V . We will write V for the image of the -module V in KA-smod. Similarly, we define the Grothendieck group K-proj, where -proj denotes the full subcategory of -smod consisting of projective -modules. As usual, the Grothendieck groups K-smod and K-proj are free -modules with canonical bases corresponding to the isomorphism classes of irreducible and projective indecomposable supermodules, respectively (see below). The embedding -proj ⊂ -smod induces the natural Cartan map   K-proj → K-smod

12.2 Schur’s Lemma and Wedderburn’s Theorem By a subsupermodule of an -supermodule we mean a subsuperspace which is -stable. An -supermodule is irreducible (or simple) if it is non-zero and has no non-zero proper -subsupermodules. An -supermodule M is called completely reducible if any subsupermodule of M is a direct summand of M. It might happen that a supermodule V is irreducible, but the module V  is reducible – in this case V  will have a non-trivial proper -invariant subspace which is not homogeneous. We need to understand this situation better. First of all, we say that an irreducible -supermodule V is of type M if the -module V  is irreducible, and otherwise we say that V is of type Q. Lemma 12.2.1 Let V be a finite dimensional irreducible -supermodule of type Q. Then there exist bases v1      vn of V0¯ and v1      vn of V1¯ such that V  = span v1 + v1      vn + vn ⊕ span v1 − v1      vn − vn  a direct sum of two non-isomorphic irreducible -submodules. Moreover, the linear map JV  V → V defined by vi → −vi  vi → vi is an endomorphism of V as an -supermodule. Proof Denote  = V . Let W be an irreducible -submodule of V . Since V is an irreducible supermodule, W is not -stable. Moreover, W 

158

Generalities on superalgebra

is also an irreducible -submodule of V . We have W ∩ W  = 0 and W + W  is -stable. Hence V = W ⊕ W . Let w1      wn be a basis for W . Then w1      wn  is a basis for W . Take vi = wi + wi  and vi = wi − wi  for 1 ≤ i ≤ n. To verify that JV is an endomorphism of V , note that JV− = − idW  ⊕ idW  is an endomorphism of V  and use Lemma 12.1.5. Finally, assume that W  W  as -modules. Then in view of Lemma 12.1.5, dim End V  = dim End V  = 4. It follows that dim End V 0¯ = dim End V 1¯ = 2. Now we can construct a non-zero homogeneous endomorphism of V with non-trivial kernel as a linear combination of two linearly independent elements of End V 0¯ . The existence of such endomorphism contradicts the irreducibility of V . Remark 12.2.2 Let
be a primitive 4th root of 1 in F , V be a type Q irreducible -supermodule, and JV be the map constructed in Lemma 12.2.1. Then
JV is a degree 1¯ involution in End V . Now we have the following analogue of Schur’s lemma: Lemma 12.2.3 (Schur’s lemma) Let V be a finite dimensional irreducible -supermodule. Then

if V is of type M , span idV

End V  = span idV  JV if V is of type Q , where JV is as in Lemma 12.2.1. Moreover, if W is another irreducible -supermodule with V  W , then Hom V W  = 0. Proof Apply Lemmas 12.1.5, 12.2.1, and usual Schur’s Lemma. Example 12.2.4 Let V be a superspace. Then V is naturally an irreducible type M supermodule over V . Moreover, if sdim V = n n, then V is naturally an irreducible type Q supermodule over V . This explains our terminology of types. So, in view of Example 12.1.3, n has the irreducible supermodule Un of dimension 2n/2 and type M if n is even, and of dimension 2n+1/2 and type Q if n is odd. This supermodule is called the Clifford supermodule. As n is a simple superalgebra, Un is the unique irreducible n -supermodule up to isomorphism.

12.2 Schur’s Lemma and Wedderburn’s Theorem

159

Lemma 12.2.5 Let V ∈ -smod. Then V is completely reducible if and only if V  ∈ -mod is completely reducible. Proof If V is completely reducible then V  is completely reducible by Lemma 12.2.1. Conversely, let V  be completely reducible, and W ⊆ V be a subsupermodule. We have to show that there exists a subsupermodule X ⊆ V with V = W ⊕ X. By the complete reducibility of V , we have an -submodule Y ⊆ V  with V = W ⊕ Y . However, Y might not be homogeneous. Let  be the projection to W along Y , and consider the linear map   =  + V V /2  V → V Set X = ker   . Note that W = idW , and im   = im  = W . So V = W ⊕ X. Moreover, it is easy to see that X is V -invariant, so homogeneous, and that X is -invariant. A finite dimensional superalgebra  is called semisimple if the left regular -supermodule   is semisimple. Corollary 12.2.6 Let  be a finite dimensional superalgebra. Then  is semisimple if and only if  is semisimple. Lemma 12.2.7 Let  be a finite dimensional superalgebra. Then the Jacobson radical J can be characterized as the unique smallest superideal  of  such that / is a semisimple superalgebra. Proof Set  = J. Observe that  is a superideal since  is invariant under the algebra automorphism  of . Let  be any superideal of  that is minimal with respect to the property that / is a semisimple superalgebra. By Corollary 12.2.6, /  is semisimple, so  ⊆  . However, / is a superalgebra that is semisimple as an algebra. So, by Corollary 12.2.6, it is a semisimple superalgebra, so  =  by minimality of  . The superideal  = J from Lemma 12.2.7 is called the Jacobson radical of the superalgebra  and denoted J. Lemma 12.2.8 Let V ∈ -smod be an irreducible supermodule. (i) V is evenly isomorphic to "V if and only if V is of type Q. (ii) Assume that V is of type M, and let W = V ⊕m ⊕ "V ⊕n . Then End W   mn . (iii) Assume that V is of type Q, and let W = V ⊕n . Then End W   n .

160

Generalities on superalgebra

Proof (i) V is evenly isomorphic to "V if and only if V is oddly isomorphic to itself, which by virtue of Lemma 12.2.3, is equivalent to V being of type Q. (ii) follows from the analogous result for the usual modules, as V   "V  is irreducible -module, and Schur’s Lemma 12.1.5. (iii) It is clear from Schur’s Lemma that End V   1 . So, if we write the superspace W in the form W  V ⊗ E, where E is a superspace of dimension n 0, then we get an embedding of the algebra 1 ⊗ n0 into End W . Now the result follows by dimensions and (12.6). Theorem 12.2.9 (Wedderburn’s Theorem) Any finite dimensional simple superalgebra is isomorphic to some mn or n . Moreover, the following conditions on a finite dimensional superalgebra  are equivalent: (i) (ii) (iii) (iv)

 is semisimple; every -supermodule is completely reducible;  is a direct product of finitely many simple F -superalgebras; J = 0.

Proof If  is a finite dimensional simple superalgebra, take an irreducible -supermodule V with sdim V = m n. The action of  on V must be faithful by simplicity of . So, by Schur’s Lemma 12.2.3, we get an embedding of  into V , if V is of type M, and into V  if V is of type Q. Now, the usual Wedderburn Theorem applied to the algebra  and module V  shows that the dimension of  equals dim V  in the first case and dim V  in the second case. For the second part of the theorem, the usual proof for algebras more or less works. Indeed, (i)⇔(ii) is clear, for every -supermodule is a quotient of a free -supermodule. (ii)⇒(iii) Decompose   = n1 L1 ⊕ · · · ⊕ nr Lr where L1      Lr are pairwise non-isomorphic irreducible supermodules. For every a ∈ , we define ¯ fa ∈ End   by fa b = −1a¯ b ba for b ∈ A. Then a → fa is an isomorphism of superalgebras between  and End  . Now (iii) follows from Lemma 12.2.3 and Lemma 12.2.8. (iii)⇒(iv) is clear for example by Lemma 12.2.7. (iv)⇒(i) follows from Lemma 12.2.7 and Corollary 12.2.6. As  =    →  is an algebra automorphism, we can twist any -module V with  to get a new -module V  , which is the same vector space V with the new action a · v = av.

12.2 Schur’s Lemma and Wedderburn’s Theorem

161

Corollary 12.2.10 Let  be a finite dimensional superalgebra, and V1      Vn be a complete set of pairwise non-isomorphic irreducible -supermodules such that V1      Vm are of type M and Vm+1      Vn are of type Q. For i = m + 1     n, write Vi  = Vi+ ⊕ Vi− as a direct sum of irreducible -modules. Then ±      Vn±

V1      Vm  Vm+1

is a complete set of pairwise non-isomorphic irreducible -modules. Moreover, Vi   Vi  and Vi±   Vi∓ . Proof In view of Lemma 12.2.7 and Corollary 12.2.6 the result reduces to the semisimple case, and then to the simple case, when it follows from Wedderburn’s Theorem. If s ∈  is a homogeneous invertible element, conjugation by s defines an automorphism s of the algebra 0¯ . Given an 0¯ -module V we write V s for the twisted 0¯ -module V s . The following result is similar to Clifford Theory for index 2 subgroups. Proposition 12.2.11 Let  be a finite dimensional superalgebra, and V1      Vn be a complete set of pairwise non-isomorphic irreducible -supermodules such that V1      Vm are of type M and Vm+1      Vn are ±      Vn± be a complete set of pairwise of type Q. Let V1      Vm  Vm+1 non-isomorphic irreducible -modules constructed as in Corollary 12.2.10. Assume that 1¯ contains an invertible element s. Then, on restriction to 0¯ , the modules Vi , 1 ≤ i ≤ m, split as a direct sum Wi+ ⊕ Wi− of two nonisomorphic irreducible modules such that Wi−  Wi+ s , and the modules Vi± , m < i ≤ n, are irreducible with Vi+  Vi− = Wi . Moreover, W1±      Wm±  Wm+1      Wn

is a complete set of pairwise non-isomorphic irreducible 0¯ -modules. Proof From Wedderburn’s Theorem, the restriction of an irreducible -supermodule to 0¯ always has at most two composition factors. So if we can decompose an irreducible -supermodule as a direct sum of two non-trivial 0¯ -modules, those modules must be irreducible. Let 1 ≤ i ≤ m. By assumption, sVi 0¯ = Vi 1¯ . Moreover, Vi 0¯ and Vi 1¯ are 0¯ -invariant, and sVi 0¯  Vi 0¯ s . So it remains to notice that Vi 0¯ and Vi 1¯ are non-isomorphic by Wedderburn and irreducible by the previous paragraph.

Generalities on superalgebra

162

Let m < i ≤ n. As Vi+  Vi−  and  is trivial on 0¯ , the modules Vi+ and are isomorphic on restriction to 0¯ . Now use the first paragraph to see that they are irreducible 0¯ -modules. Vi−

±

That Wi form a complete set of irreducible 0¯ -modules up to isomorphism again follows from Wedderburn. Let  be a finite dimensional superalgebra. Indecomposable summands of the left regular module   are called principal indecomposable -supermodules. Theory of principal indecomposable supermodules is analogous to the usual one. In particular, we can show (we leave this as an exercise) that principal indecomposable supermodules are projective in the category -smod, have irreducible heads, and are determined up to an isomorphism by their heads. We refer to the principal indecomposable supermodule with head L as the projective cover of L, and denote it by PL . Proposition 12.2.12 Let  be a finite dimensional superalgebra, and V1      Vn be a complete set of pairwise non-isomorphic irreducible -supermodules such that V1      Vm are of type M and Vm+1      Vn are of type Q. Set sdim Vi = di  di . Then 



m n   di PVi ⊕ di "PVi  ⊕ di PVi  i=1

i=m+1

Proof Reduce to the semisimple case and use Wedderburn. Given left supermodules V and W over superalgebras  and respectively, the (outer) tensor product V
W is the superspace V ⊗ W considered as an  ⊗ -supermodule via ¯

a ⊗ bv ⊗ w = −1b¯v av ⊗ bw

a ∈  b ∈  v ∈ V w ∈ W  (12.19)

If f  V → V  (resp. g  W → W  ) is a homomorphism of - (resp. -) supermodules, then f ⊗ g  V ⊗ W → V  ⊗ W  is a homomorphism of  ⊗ supermodules. If  is an antiautomorphism of  and  is an antiautomorphism of , ¯ then    ⊗  a ⊗ b → −1a¯ b a ⊗ b is an antiautomorphism of  ⊗ . Under these circumstances there is a natural isomorphism isomorphism ¯

˜ V A ⊗ W B  f ⊗ g → −1f g¯ f ⊗ g V
W  →

(12.20)

12.2 Schur’s Lemma and Wedderburn’s Theorem

163

Lemma 12.2.13 Let V be an irreducible -supermodule and N be an irreducible -supermodule. (i)

If both V and W are of type M, then V
W is an irreducible  ⊗ -supermodule of type M.

(ii) If one of V or W is of type M and the other is of type Q, then V
W is an irreducible  ⊗ -supermodule of type Q. (iii) If both V and W are of type Q, then V
W  X ⊕ "X for a type M irreducible  ⊗ -supermodule X. Moreover, all irreducible  ⊗ -supermodules arise as constituents of V
W for some choice of irreducibles V W . Proof Wedderburn’s Theorem reduces the lemma to the situation where  and are simple, in which case the result follows from (12.3), (12.6), and (12.7). If V ∈ -smod and W ∈ -smod are irreducible, denote by V  W an irreducible component of V
W . Thus,

V
W 

V  W ⊕ "V  W  if V and W are both of type Q,

V  W

otherwise.

We stress that V  W is in general only well-defined up to isomorphism. Example 12.2.14 Recall the Clifford–Grassman superalgebra j from Example 12.1.4. We already know that the Clifford algebra 1 has only one irreducible supermodule up to isomorphism, namely U1 , and it is of type Q. It is also easy to see that the only irreducible supermodule over ¯ it does not matter up 1 is the trivial supermodule F (in degree 0¯ or 1, isomorphism) on which the generator d1 acts as 0. It follows from the isomorphism (12.14) and Lemma 12.2.13 that j has only one irreducible supermodule n−0

Uj  F 0  U1

In particular, Un  U1n . Note that dim Uj = 2) M if and only if n − 0 is even.

 n−0 +1 * 2

. Also Uj is of type

164

Generalities on superalgebra

The following simple observation explains why the operation  is convenient: Lemma 12.2.15 Let  and be finite dimensional superalgebras. Then there is an isomorphism K-smod ⊗ K -smod → K ⊗ -smod L ⊗ L → L  L

(12.21)

Proof Follows using Lemma 12.2.13. Lemma 12.2.16 PL admits an odd involution if and only if L does, that is if and only if L is of type Q. Proof If PL admits an odd involution J , then J factors through the radical of PL to give an odd involution of L. Conversely, assume that L is of type Q, that is L admits an odd involution J . Then we can define the action of the Clifford algebra 1 on V by requiring that the generator c1 acts as J . This makes V into an irreducible  ⊗ 1 -supermodule of type M. In fact, it is clear that this supermodule is isomorphic to V  U1 . ⊗ Now, we claim that res 1 PV U1  PV . This completes the proof since then the odd involution on PV comes from the action of 1 ⊗ c1 ∈  ⊗ 1 on PV U1 . ⊗ For the claim, note first that res 1 PV U1 is projective as an -supermodule, so it remains to prove that its head is isomorphic to V . This follows from the following calculation for any irreducible -supermodule W : ⊗1

Hom res

PV U1  W 

Hom⊗1 PV U1  Hom  ⊗ 1  W  Hom⊗1 PV U1  Hom  W 
HomF 1  F  Hom⊗1 PV U1  W
U1  Hom⊗1 V  U1  W
U1  Hom V W 

13 Sergeev superalgebras

By general theory, studying spin representations of the symmetric group Sn is equivalent to studying representations of its twisted group algebra n . It is convenient to consider n as a superalgebra with respect to the natural grading, where even (resp. odd) elements come from even (resp. odd) permutations. Even if we are only interested in the usual n -modules, Corollary 12.2.10 shows that, at least as far as irreducibles are concerned, we do not loose anything by working in the category of supermodules, providing we keep track of types of irreducible supermodules. Moreover, we even gain an additional insight into the usual irreducible modules, in view of Proposition 12.2.11. This additional information is exactly what we need in order to deal with spin representations of the alternating groups. It is interesting that the superalgebra approach is not useful for the linear representations of Sn , while spin representation theory of Sn has intrinsic features of a “supertheory”. An important idea due to Sergeev is that instead of the superalgebra n it is more convenient to consider n = n ⊗ n , where n is the Clifford superalgebra, and ⊗ is the tensor product of superalgebras. On the one hand, nothing much is going to happen to representation theory when we tensor our superalgebra with a simple superalgebra (classically we get a Morita equivalence and in the “superworld” we get either a Morita equivalence or something almost as good as a Morita equivalence). On the other hand, it is well known that the Clifford algebra plays a special role in the theory of spin representations of symmetric groups, so why not bring it in voluntarily? Finally, the new superalgebra n turns out to have at least two important advantages over n : it has a nice q-analogue (which we will not pursue here) and it has a natural “affinization”, so we have a chance to work out a theory parallel to the one developed in the first part of this book for symmetric groups.

165

Sergeev superalgebras

166

13.1 Twisted group algebras It is well known that H 2 Sn  F ×   /2Z for n ≥ 5 , see e.g. [Su, Chapter 3 (2.21)] (recall that throughout Part II we assume p > 2). So there are two twisted group algebras of the symmetric group Sn up to isomorphism. One of them is the usual group algebra FSn . Studying FSn -modules is of course equivalent to studying representations of Sn over F . The non-trivial twisted group algebra n of Sn is the associative F -algebra with basis tg  g ∈ Sn

and multiplication tg th = g htgh for any g h ∈ Sn , where  is a non-trivial 2-cocycle in Z2 Sn  F × . Studying n -modules is equivalent to studying spin representations of Sn . It is shown in [Su, p. 303] that the twisted group algebra n is generated by the elements t1      tn−1 subject only to the relations ti2 = 1 ti ti+1 ti = ti+1 ti ti+1 ti tj = −tj ti

1 ≤ i < n

(13.1)

1 ≤ i ≤ n − 2

(13.2)

1 ≤ i j < n i − j > 1

(13.3)

(Actually, Suzuki has the relations Ti2 = −1, Ti T√i+1 Ti = −Ti+1 T√ i Ti+1 , Ti Tj = T . To get the relations above, take t −T = −1T  t = − −1T2  t3 = 1 1 2 √ j i √ −1T3  t4 = − −1T4 , etc.) Inside the algebra n we have a subalgebra n

= span tg  g ∈ An 

If n > 7, this is the only non-trivial twisted group algebra of the alternating group An , see [Su, p. 304], and the exceptional cases (for n = 6 7) are of course easy to understand, see for example [At, MAt]. We consider n as a superalgebra with respect to the following grading  n 0¯ =

n

 n 1¯ = span tg  g ∈ Sn \ An 

It follows from the defining relations that there exists a superalgebra antiautomorphism   n → n  ti → −ti

1 ≤ i < n

(13.4)

In fact, tg  = −1g¯ tg−1 . The anti-involution  can be used to define duality on n -modules and supermodules as in (12.15). For 1 ≤ i < j ≤ n, define “transpositions” i j = −j i = −1j−i−1 tj−1    ti+1 ti ti+1    tj−1 

(13.5)

13.1 Twisted group algebras

167

The defining relations of n imply i j 2 = 1 i j k l = −k l i j

if i j ∩ k l = ∅

i j j k i j = j k i j j k = k i

for distinct i j k.

Finally, for distinct 1 ≤ i1      ir ≤ n, define ‘r-cycles’ i1  i2      ir = −1r−1 i2      ir  i1 = ir−1  ir ir−2  ir    i1  ir  For 1 ≤ k ≤ n, the analogue of the Jucys–Murphy element (2.1) is Mk =

k−1 

i k 

(13.6)

i=1

in particular, M1 = 0. Note that Mk2 = k − 1 +

k−1 

i j k 

(13.7)

ij=1 i =j

and

⎧ ⎪ ⎨ −Mk ti ti Mk = −Mk−1 ti + 1 ⎪ ⎩ −Mk+1 ti + 1

if i = k − 1 k, if i = k − 1, if i = k.

(13.8)

It follows that Mk Ml = −Ml Mk if k = l. Now using these facts, it is easy to show: Lemma 13.1.1 (i) for 1 ≤ k l ≤ n, Mk2 and Ml2 commute; (ii) ti commutes with Mk2 for k = i i + 1; 2 2 and Mi2 Mi+1 . (iii) ti commutes with Mi2 + Mi+1 This implies: Lemma 13.1.2 The symmetric polynomials in M12  M22      Mn2 belong to the center of n . Remark 13.1.3 It can be shown (see [BK5 ]) that the space of all even central elements of n equals the set of symmetric polynomials in the M12      Mn2 . But n could also have odd central elements. We will not need these facts.

Sergeev superalgebras

168

13.2 Sergeev superalgebras We define the Sergeev superalgebra n (cf. [S1 , N]) to be the tensor product of superalgebras

n = n ⊗ n  where n is the twisted group superalgebra defined in Section 13.1 and n is the Clifford superalgebra defined in Section 12.1. Let us write ti for ti ⊗ 1 ∈ n and cj for 1 ⊗ cj ∈ n . The following result is clear: Lemma 13.2.1 The superalgebra n is generated by (odd generators) t1      tn−1 and c1      cn , subject only to the relations (13.1)–(13.3), (12.8), (12.9), together with ti cj = −cj ti for all admissible i and j. The antiautomorphism  of n defined in (13.4) can be extended to an antiautomorphism of n , which we also denote by :   n → n 

ti → −ti  cj → cj

1 ≤ i < n 1 ≤ j ≤ n

(13.9)

As usual,  defines a duality on n -supermodules. Recall the Clifford supermodule Un from Example 12.2.4, which is the unique irreducible n -supermodule up to isomorphism, is of type M if n is even, type Q if n is odd, and dim Un = 2)n+1/2* . Consider the exact functors !n  n -smod → n -smod

!n = ?
Un 

"n  n -smod → n -smod

"n = Homn Un  ?

So, given a n -supermodule W , !n W  is just the outer tensor product W
Un of (12.19), and, for a n -supermodule V , "n V  is the superspace Homn Un  V  considered as a n -supermodule with respect to the action t u = t u for t ∈ n , u ∈ Un , ∈ Homn Un  V . Also let

ind nn−1  n−1 -smod → n -smod



ind nn−1  n−1 -smod → n -smod

res nn−1  n -smod → n−1 -smod res nn−1  n -smod → n−1 -smod



denote the induction and restriction functors, where n−1 ⊂ n and n−1 ⊂ n are the natural subalgebras generated by all but the last generators. These functors are exact, which is clear for restriction, and for induction we need to use the fact that the right supermodules  n  n−1 and  n  n−1 are free. The following proposition shows that !n and "n establish Morita equivalence between n and n when n is even and “almost” Morita equivalence

13.2 Sergeev superalgebras

169

when n is odd. It also relates restriction/induction functors for s and s. Recall that " denotes the parity change functor (12.17). Proposition 13.2.2 The functors !n and "n are exact, commute with duality, and are left and right adjoint to one another. Moreover: (i) Suppose that n is even. Then !n and "n are inverse equivalences of categories, so induce a type-preserving bijection between the isomorphism classes of irreducible n -supermodules and irreducible

n -supermodules. Also,



!n−1  res nn−1  res nn−1  !n 



(13.10)



"n−1  res nn−1  res nn−1  "n ⊕ "  res nn−1  "n 

(13.11)



!n+1  ind n+1  ind n+1  !n  n n

"n+1  ind n+1 n



ind n+1 n

 "n ⊕ "  ind n+1 n

(13.12)

 "n 

(13.13)

(ii) Suppose that n is odd. Then !n  "n  Id ⊕ " and "n  !n  Id ⊕ " Furthermore, the functor !n induces a bijection between isomorphism classes of irreducible n -modules of type M and irreducible n -modules of type Q, while the functor "n induces a bijection between isomorphism classes of irreducible n -modules of type M and irreducible n -modules of type Q. Finally,





res nn−1  !n  !n−1  res nn−1 ⊕ "  !n−1  res nn−1  res nn−1

 "n 

(13.14)

"n−1  res nn−1 



(13.15)



ind n+1  !n  !n+1  ind n+1 ⊕ "  !n+1  ind n+1  n n n

(13.16)



ind n+1  "n  "n+1  ind n+1  n n

(13.17)

Proof It is clear that !n is exact, and for "n this follows from the fact that n is simple. Let n be even. For any n -supermodule V define the map V  !n "n V  = Homn Un  V 
Un → V

⊗ u → u

and for any n -supermodule W define the map W  W → "n  !n W  = Homn Un  W
Un 

w →

w

170

Sergeev superalgebras

where w u = w ⊗ u for u ∈ Un . It is easy to see that V and W are natural isomorphisms, so they define isomorphisms of functors !n  "n  Id and Id  "n  !n . Then !n and "n are clearly adjoint to each other. Now, assume n is odd. Let J be an odd involution in Endn Un , see Remark 12.2.2. For any n -supermodule V define the map V  !n "n V  = Homn Un  V 
Un → V ⊕ "V  ¯

⊗ u →  u −1 Ju and for any n -supermodule W define the map W  W ⊕ "W  → "n !n W  = Homn Un  W
Un  w w  →

ww 

where ww u



= w ⊗ u + −1w¯ w ⊗ Ju

for each u ∈ Un . As n is simple, any n -supermodule is just a direct sum of Un ’s. Using this, we can easily see that V is surjective, W is injective, and sdim Homn Un  V 
Un = sdim V ⊕ "V  sdim W ⊕ "W  = sdim Homn Un  W
Un  Finally, observe that V and W are supermodule homomorphisms, natural with respect to V and W , respectively. Thus, !n  "n  Id ⊕ "

and

"n  !n  Id ⊕ "

(13.18)

Let W be an irreducible n -supermodule. If W is type M, then !n W is irreducible of type Q, thanks to Lemma 12.2.13(ii). Moreover, by Lemma 12.2.13 (iii), if W is type Q, then !n W  V ⊕ "V for a type M irreducible

n -supermodule V . It now follows from (13.18) that !n induces a bijection between isoclasses of irreducible n -supermodules of type M and irreducible

n -supermodules of type Q, while "n induces a bijection between isoclasses of irreducible n -supermodules of type M and irreducible n -supermodules of type Q. Now, let %  Id −→ Id ⊕ "

  Id ⊕ " −→ Id

be the obvious natural transformations. Then it is easy to see   % and    give the unit and the counit of the adjointness needed to prove that !n is left adjoint to√"n (cf. [ML, IV.1, √ Theorem 2(v)]). A more tedious check shows that  = 2−1  % and  = 2  −1 give the unit and the counit of the

13.2 Sergeev superalgebras

171

adjointness needed to prove that !n is right adjoint to "n . Indeed, let us write " = "n ,  = n , etc., and prove, for example, that the composition of natural transformations "

"

" −→ "!" −→ " is the identical natural transformation. For any -supermodule V , let  ∈ "V  = Hom U V . Then the first arrow takes  to the function √   u → 2−1 V u 0 ∈ "!"V  = Hom  U Hom  U V 
U  It suffices to prove that 1 1  = √ "V   0 − −1¯ √ "V  0   J 2 2

(13.19)

since the application of  to the right-hand side gives , as desired. To prove (13.19), evaluate the right-hand side at u to get 1 1 1 √ 0J u = √  ⊗ u + √   J ⊗ Ju 2 2 2 √ −1 which is indeed equal to 2V  0, since 1 √ 2

0 u − −1

¯

√
1 1 V √  ⊗ u + √   J ⊗ Ju = 2u 0 2 2 Next, for general n we prove that !n commutes with duality. It is enough to show that !n commutes with duality, that is   !n    !n , since then, using (12.16) and the fact that "n is left adjoint to !n , the composite functor   "n   is right adjoint to !n , but we already know that "n is right adjoint to !n , so uniqueness of adjoints gives that   "n    "n , that is "n commutes with duality too. To prove that !n commutes with duality, note that the antiautomorphism  of n induces the antiautomorphism  of the subalgebra n with ci  = ci for each i = 1     n. As Un is the only irreducible n -supermodule up to isomorphism, there exists a homogeneous isomorphism   Un → Un . Then, using (12.20) and idW  ⊗, we get natural isomorphisms ˜ W 
Un → ˜ W 
Un  W
Un  → It just remains to check the isomorphisms (13.10)–(13.17). Well, (13.10)  follows from definitions noting that resnn−1 Un  Un−1 if n is even. Then (13.11) follows from (13.10) on composing on the left with "n−1 and on the right with "n . Next, (13.15) follows from the definition and an application of  Frobenius reciprocity, using the observation that Un  indnn−1 Un−1 if n is odd.

Sergeev superalgebras

172



To prove (13.15), note that Un  indnn−1 Un−1 when n is odd. So for a

n -supermodule V , using Frobenius reciprocity, we get



"n−1  res nn−1 V   Homn−1 Un−1  res nn−1 V  



 Homn indnn−1 Un−1  res nn−1 ⊗n V 



 res nn−1 Homn Un  V   res nn−1  "n V  Now (13.14) follows from (13.15) by composing with !n−1 and !n . Finally, (13.12), (13.13), (13.16), and (13.17) follow from (13.15), (13.14), (13.11), and (13.10), respectively, by uniqueness of adjoints. Now we describe another reincarnation of n (cf. [S2 ]). The symmetric group Sn acts on the generators c1      cn of the Clifford algebra n by place permutations: ci · w = cw−1 i . This action can be extended to the action of Sn on n by superalgebra automorphisms. Recall that n denotes the group algebra FSn , which will be considered as a superalgebra concentrated in ¯ Now denote by n  n the superalgebra, which as a superspace degree 0. is just n ⊗ n , where the Clifford algebra n is graded as usual, and the multiplication is given by w ⊗ cw ⊗ c  = ww ⊗ c · w c 

w w ∈ Sn  c c ∈ n 

Lemma 13.2.3 There is an isomorphism of superalgebras ∼

  n −→ n  n 

1 1 ⊗ cj → 1 ⊗ cj  ti ⊗ 1 → √ si ⊗ ci − ci+1  −2

for j = 1     n, i = 1     n − 1. Proof Obviously, the elements 1 ⊗ cj ∈ n  n satisfy the defining relations of n . A simple calculation shows that the ti ⊗ 1 satisfy the relations (13.1)–(13.3). Finally, we have 1 ⊗ cj ti ⊗ 1 = −ti ⊗ 11 ⊗ cj  for all admissible i and j. By Lemma 13.2.1,  as above exists. Now,  is surjective, as ci − ci+1 is invertible in n , and it remains to compare dimensions.

13.2 Sergeev superalgebras

173

We will from now on identify n with n  n according to the lemma. Let us write si for si ⊗ 1 ∈ n  n = n and cj for 1 ⊗ cj ∈ n  n = n for 1 ≤ i < n 1 ≤ i ≤ n. Then the following is easy to check: Lemma 13.2.4 The superalgebra n is generated by (even generators) s1      sn−1 and (odd generators) c1      cn , subject only to the relations (3.2), (3.3), (12.8), (12.9), together with si ci = ci+1 si  si cj = cj si

(13.20)

for all admissible i and j with j = i i + 1. The antiautomorphism  of n defined in (13.9) can now be defined via:   n → n 

si → si  cj → cj

1 ≤ i < n 1 ≤ j ≤ n

(13.21)

The JM-elements Mk ∈ n from (13.6) can be considered as elements of

n because n is a subalgebra of n . We will need to know the image of Mk under . Define the Jucys–Murphy elements of n as follows:  Lk = 1 + cj ck j k ∈ n 1 ≤ k ≤ n (13.22) 1≤j
Lemma 13.2.5 (i) For 2 ≤ k ≤ n we have Mk  = 2√1−2 1 − ck−1 ck Lk ck−1 − ck ; (ii) For 1 ≤ k ≤ n we have Mk2  = 21 L2k . Proof (i) is proved by induction on k using relations Mk+1 = −tk Mk tk + tk  which follow from (13.8), and Lk+1 = sk Lk sk + 1 + ck ck+1 sk 

(13.23)

which are easy to check directly. (ii) is checked using (i) and relations ck Lk = −Lk ck , ck Lj = Lj ck for k = j.

14 Affine Sergeev superalgebras

This chapter is parallel to Chapter 3. The role of the degenerate affine Hecke algebra is played by the affine Sergeev superalgebra n , introduced and studied by Nazarov [N]. The superalgebra n is in the same relation to the Sergeev superalgebra n as
n to the group algebra n of the symmetric group. We will sometimes suppress the word “super”, as “everything is super” anyway. So we may speak of a subalgebra n of n rather than a subsuperalgebra, etc.

14.1 The superalgebras Let n denote the superalgebra with even generators x1      xn and odd generators c1      cn , where the xi are subject to the polynomial relations (3.1), the ci are subject to the Clifford superalgebra relations (12.8), (12.9), and there are the mixed relations c i xj = x j c i 

ci xi = −xi ci

(14.1)

for all 1 ≤ i j ≤ n with i = j. For  = 1      n ∈ n and  = 1      n  ∈   n2 , we write x and c for the monomials x1 1    xnn and c1 1    cnn , respectively. Then it is easy to see that the elements x c   ∈ n   ∈ n2

form a basis of n . In particular, n  1 ⊗ · · · ⊗ 1 (n times). Note that the polynomial algebra n can be identified with the subalgebra of n generated by the xi , and the Clifford algebra n can be identified with the subalgebra of

174

14.2 Basis Theorem for n

175

n generated by the ci . Thus, n is the twisted tensor product n ⊗ n . We define a left action of Sn on n by algebra automorphisms so that w · xi = xwi 

w · ci = cwi

(14.2)

for each w ∈ Sn , i = 1     n. The affine Sergeev superalgebra n has even generators s1      sn−1 , x1      xn and odd generators c1      cn , subject to the relations (3.1), (12.8), (12.9), (14.1), (3.2), (3.3), together with the new relations: si cj = cj si 

(14.3)

si ci = ci+1 si  si ci+1 = ci si 

(14.4)

si xj = xj si 

(14.5)

si xi = xi+1 si − 1 − ci ci+1

(14.6)

for all admissible i j with j = i i + 1. We call x1      xn polynomial generators, c1      cn Clifford generators, and s1      sn−1 Coxeter generators. Note that 1  1 . Also, by agreement, 0  F . The relation (14.6) implies si xi+1 = xi si + 1 − ci ci+1 

(14.7)

By induction, we deduce more general formulas for j ≥ 1: j si xij = xi+1 si −

j−1 

j−1−k j−1−k xik xi+1 + −xi k xi+1 ci ci+1 

(14.8)

j−1−k xik xi+1 − xik −xi+1 j−1−k ci ci+1 

(14.9)

k=0 j si xi+1 = xij si +

j−1  k=0

14.2 Basis Theorem for n There are obvious homomorphisms   n → n ,   n → n , and (  n → n under which the elements xi , ck , and sj map to the corresponding elements of n . We write n for n , n for n , n for n , n for ( n , x for x , c for c , and w for w (for w ∈ Sn ⊂ n ). This notation will be justified shortly, when we show that , , and ( are monomorphisms. The following is obvious from the defining relations: Lemma 14.2.1 Let f ∈ n , w ∈ Sn . Then in n we have    fu u fw = ww−1 · f + ufu wf = w · fw + u<w

for some

fy  fy

u<w

∈ n of x-degrees less than the x-degree of f .

176

Affine Sergeev superalgebras

It follows easily from this lemma that that n is at least spanned by all x c w,  ∈ n+   ∈ n2  w ∈ Sn . Theorem 14.2.2 The x c w   ∈ n+   ∈ n2  w ∈ Sn form a basis for n . Proof The proof using Bergman’s Diamond Lemma is entirely similar to the first proof of Theorem 3.2.2. (The second proof seems to be harder to mimic). By Theorem 14.2.2, we have a right from now on to identify n , n , n , and n with the corresponding subalgebras of n . Then n is a free right n -module on basis x c   ∈ n   ∈ n2 . As another consequence, if m ≤ n, we can consider m as the subalgebra of n generated by x1      xm , c1      cm , and s1      sm−1 . Sometimes, we will use another embedding of m into n . In order to distinguish between the two, denote m = si  xj  cj  n + 1 − m ≤ j ≤ n n + 2 − m ≤ i < n ⊂ n 

(14.10)

Finally, let us point out that there are obvious variants of Theorem 14.2.2. For example, n also has wx c  w ∈ Sn   ∈ n+   ∈ n2 as a basis. This follows using Lemma 14.2.1.

14.3 The center of n Theorem 14.3.1 The center of n consists of all symmetric polynomials in x12      xn2 . Proof That the symmetric polynomials in x12      xn2 are indeed central is easily verified. Indeed, it is clear that they commute with the Clifford generators. Moreover, as si commutes with all xj for j = i i+1, we just need to check 2 2 b + xi2 b xi+1 a , that si commutes with a polynomial of the form xi2 a xi+1 which is done using (14.8) and (14.9).  Conversely, take a central element z = w∈Sn fw w ∈ n where each fw ∈ n . Let w be maximal with respect to the Bruhat order such that fw = 0. Assume w = 1. Then there exists i ∈ 1     n with wi = i. By Lemma 14.2.1, 2 w plus a linear combination of terms of the xi2 z − zxi2 looks like fw xi2 − xwi   form fu u for fu ∈ n and u ∈ Sn with u ≥ w in the Bruhat order. So in view of Theorem 14.2.2, z is not central, giving a contradiction. Hence, we must have that z ∈ n . Commuting with polynomial generators, we see that z is actually in n , and commuting with Clifford generators, we

14.5 Mackey Theorem for n

177

then deduce that z ∈ F x12      xn2 . To see that z is a symmetric polynomial,  write z = ij≥0 aij x12 i x22 j where the coefficients aij lie in F x3      xn . Applying Lemma 14.2.1 to s1 z = zs1 now gives that aij = aji for each i j, hence z is symmetric in x1 and x2 . Similar argument shows that z is symmetric in xi and xi+1 for all i = 1     n − 1.

14.4 Parabolic subalgebras of n Let =  1      r  be a composition of n. We define the parabolic subalgebra  of n as the subalgebra generated by n and all sj for which sj ∈ S . It follows from Theorem 14.2.2 that the elements x c w   ∈ n+   ∈ n2  w ∈ S

form a basis for  . In particular,    1 ⊗ · · · ⊗  r (tensor product of superalgebras). Note that the parabolic subalgebra 111 is precisely the subalgebra n . We will use the induction and restriction functors between n and  . These will be denoted simply indn   -smod → n -smod

resn  n -smod →  -smod

(14.11)

the former being the tensor functor n ⊗ ? which is left adjoint to resn . More generally, we will consider induction and restriction between nested parabolic subalgebras, with obvious notation. We will also occasionally consider the restriction functor resnn−1  n -smod → n−1 -smod 

(14.12)

14.5 Mackey Theorem for n Recall the notation of Section 3.5. Fix some total order ≺ refining the Bruhat order < on D  . For x ∈ D  , set   y  (14.13) x = y∈D   yx

≺x =



 y 

(14.14)

y∈D   y≺x

x = x / ≺x 

(14.15)

178

Affine Sergeev superalgebras

It follows from Lemma 14.2.1 that x and ≺x are invariant under right multiplication by n . Hence, since  =  n , we have a filtration of n as an    -bisupermodule. Note using the property (2) from Section 3.5 that for each y ∈ D  , there exists an algebra isomorphism y−1   ∩y → y−1 ∩ with y−1 w = y−1 wy, y−1 ci  = cy−1 i , and y−1 xi  = xy−1 i for w ∈ S ∩y , 1 ≤ i ≤ n. If N is a left y−1 ∩ -supermodule, then by twisting the action with the isomorphism y−1 we get a left  ∩y -supermodule, which will be denoted y N . Lemma 14.5.1 Let us view  as an    ∩x -bisupermodule and  as an x−1 ∩   -bisupermodule in the natural ways. Then x  is an  ∩x   -bisupermodule, and x   ⊗ ∩x x  as    -bisupermodules. Proof Similar to the proof of Lemma 3.5.1. Theorem 14.5.2 (“Mackey Theorem”) Let M be an  -supermodule. Then resn indn M admits a filtration with subquotients  to ind ∩x x resx−1 ∩ M one for each x ∈ D  . Moreover, the subquotients can be taken in any order refining the Bruhat order on D  , in particular ind ∩ res ∩ M appears as a subsupermodule. Proof Similar to the proof of Theorem 3.5.2, but using Lemma 14.5.1 instead of Lemma 3.5.1.

14.6 Some (anti) automorphisms of n A check of relations shows that n possesses an automorphism  and an antiautomorphism  defined on the generators as follows:   si → −sn−i    si → si 

cj → cn+1−j 

cj → cj 

for all i = 1     n − 1 j = 1     n.

xj → xn+1−j 

xj → xj 

(14.16) (14.17)

14.8 Intertwining elements for n

179

If M ∈ n -smod, we can use  to make the dual space M ∗ into an n -module denoted M  , see Section 12.1. Note that  leaves invariant every parabolic subalgebra of n , so it also induces a duality on finite dimensional  supermodules for each composition of n. Given M ∈ n -smod, we can twist the action of n with  to get a new n -supermodule M  . The following result is proved similarly to Lemma 3.6.1. Lemma 14.6.1 Let M ∈ m -smod and N ∈ n -smod. Then m+n   M
N  indm+n indmn nm N
M 

Moreover, if M and N are irreducible, the same holds for  in place of
.

14.7 Duality for n -supermodules This section should be entirely parallel to Section 3.7. In particular, essentially the same arguments lead to: Theorem 14.7.1 For M ∈ m -smod and N ∈ n -smod, we have m+n    indm+n mn M
N  ind nm N
M 

Moreover, if M and N are irreducible, the same holds for  in place of
.

14.8 Intertwining elements for n For 1 ≤ i < n set 2  + xi + xi+1  + ci ci+1 xi − xi+1  i = si xi2 − xi+1

(14.18)

A tedious calculation using (14.8) and (14.9) gives: 2 2 − xi2 − xi+1 2  i2 = 2xi2 + 2xi+1

(14.19)

i xi = xi+1 i  i xi+1 = xi i  i xj = xj i 

(14.20)

i ci = ci+1 i  i ci+1 = ci i  i cj = cj i 

(14.21)

i k = k i  i i+1 i = i+1 i i+1

(14.22)

for all admissible i j k with j = i i+1 and i−k > 1. Property (14.22) means that for every w ∈ Sn we have a well-defined element w ∈ n , namely, w = i1    im

180

Affine Sergeev superalgebras

where w = si1    sim is any reduced expression for w. According to (14.20), these elements have the property that w xi = xwi w 

(14.23)

for all w ∈ Sn and 1 ≤ i ≤ n. We note that only the properties (14.19) and (14.20) will be essential in what follows: Lemma 14.8.1 Let V be an n -module, 1 ≤ j < n, and v ∈ V be a vector 2 v = bv for some a b ∈ F . Then j2 v = 0 if and only if with xj2 v = av, xj+1 2 2a + 2b = a − b , which in turn is equivalent to a = c2 − c and b = c2 + c for some c ∈ F . Proof By (14.19), j2 acts on v with the scalar 2a + 2b − a − b2 . The second part is an elementary exercise.

15 Integral representations and cyclotomic Sergeev algebras

For notational reasons, when studying representation theory of n , it is more convenient to start with the definition of integral representations, although in principle we could postpone doing this until after the chapter on character calculations, as was done in the classical case. So Chapter 15 is parallel to Chapter 7. Note that the definition of an integral representation is now less obvious. If you want to ‘discover’ this notion, you need to play with calculations made in Chapter 18 first.

15.1 Integral representations of n Recall that p stands for the characteristic of the ground field F , which is 0 or odd prime. Denote

 if p = 0, (15.1)  = p − 1/2 if p > 0. We define the set I ⊂ F as follows:

≥0 I = 0 1     

if p = 0, if p > 0.

(15.2)

For i ∈ I we set qi = ii + 1

(15.3)

Note that qi = qj if and only if i = j (for i j ∈ I). Definition 15.1.1 An n -supermodule M is called integral if it is finite dimensional and all eigenvalues of x12      xn2 on M are of the form qi 181

182

Integral representations and cyclotomic Sergeev algebras

for i ∈ I. An n -supermodule, or more generally an  -supermodule for a composition of n, is called integral if it is integral on restriction to n . We write RepI n (resp. RepI n , RepI  ) for the full subcategory of n -smod (resp. n -smod,  -smod) consisting of all integral supermodules. Lemma 15.1.2 Let M be a finite dimensional n -supermodule and fix j with 1 ≤ j ≤ n. Assume that all eigenvalues of xj on M belong to I. Then M is integral. Proof It suffices to show that the eigenvalues of xk2 are of the form qi 2 are of the same form, for an for i ∈ I if and only if the eigenvalues of xk+1 arbitrary k with 1 ≤ k < n. Actually, by an argument involving conjugation with the automorphism , it suffices just to prove the ‘if ’ part. So assume 2 on M are of the form qi for i ∈ I. Let a be that all eigenvalues of xk+1 an eigenvalue of xk on M. We have to prove that a2 is of the form qi. As since 0 = q0, we may assume that a = 0. Since xk and xk+1 commute, we can pick v lying in the a-eigenspace of xk so that v is also an eigenvector for xk+1 , of eigenvalue b say. By assumption we have b2 = qi for some i ∈ I. Now let k be the intertwining element (14.18). By (14.20), we have xk+1 k = k xk  So if k v = 0, we get that ak v = k xk v = xk+1 k v whence a is an eigenvalue of xk+1 , and so a = qi  for some i ∈ I by assumption. Else, k v = 0 so k2 v = 0. So applying Lemma 14.8.1, we again get that a = qi  for some i ∈ I. Lemma 15.1.3 Let be a composition of n and M be an integral  -supermodule. Then indn M is an integral n -supermodule. Proof The proof is similar to that of Lemma 7.1.3 but uses the element  Yj = xj2 − qi 1 ≤ j ≤ n i∈I

instead of the Yj there and Lemma 15.1.2 instead of Lemma 7.1.2. It follows that the functors indn  resn restrict to well-defined functors indn  RepI  → RepI n 

resn  RepI n → RepI 

(15.4)

on integral representations. Similar remarks apply to more general induction and restriction between nested parabolic subalgebras of n .

15.2 Some Lie theoretic notation

183

15.2 Some Lie theoretic notation 2

If p > 0, let  denote the twisted affine Kac–Moody algebra of type A2 (over C), see [Kc, Ch. 4, Table Aff 2]. In particular we label the Dynkin diagram by the index set I = 0 1      as follows: 0

1

c< c

−2

2

c

c

−1



c< c

0

1

c< c

if  ≥ 2, and

if  = 1.

The weight lattice is denoted P, the simple roots are i  i ∈ I ⊂ P and simple coroots are hi  i ∈ I ⊂ P ∗ . The Cartan matrix
the corresponding  hi  j  0≤ij≤ is ⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ ⎟ ⎟    2 −1 0 ⎟ ⎟ ⎟    −1 2 −2 ⎠    0 −1 2

⎛

2 −2 0 ⎜−1 2 −1 ⎜ ⎜ ⎜ 0 −1 2 ⎜ ⎜ ⎜ ⎜ ⎜ 0 0 0 ⎜ ⎜ ⎝ 0 0 0 0 0 0

··· ··· ···  



0 0 0

2 −4 −1 2

0 0 0

if  ≥ 2, and

 if  = 1.

Let i  i ∈ I ⊂ P denote fundamental dominant weights, so that hi  j  = ij , and let P+ ⊂ P denote the set of all dominant integral weights. Set c = h0 +

 

=

2hi 

i=1

−1 

2i +  

(15.5)

i=0

Then the 0         form a -basis for P, and c i  = hi   = 0 for all i ∈ I. In the case p = 0, we make the following changes to these definitions. First,  denotes the Kac–Moody algebra of type B , see [Kc, Section 7.11]. So I = 0 1 2    , corresponding to the nodes of the Dynkin diagram 0

1

c< c

2

c

184

Integral representations and cyclotomic Sergeev algebras


 The Cartan matrix hi  j  ij≥0 is ⎛

2 ⎜ −1 ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎝

−2 2 −1

0 −1 2

0

−1

⎞ 0 −1 2  

⎟ ⎟ ⎟ ⎟ ⎟  ⎟ ⎟ ⎠  

Note certain notions, for example the element c from (15.5), only make sense if we pass to the completed algebra b , see [Kc, Section 7.12], though the intended meaning whenever we make use of them should be obvious regardless. n The meaning of U , ei  fi  hi , U , U− , U0 , ei , etc. is the same as in Section 7.2.

15.3 Cyclotomic Sergeev superalgebras For  ∈ P+ we set  

h0 

f = x1

x12 − qihi  ∈ n 

(15.6)

i∈I\ 0

Let  denote the two-sided ideal of n generated by f , and define the cyclotomic Sergeev superalgebra to be the quotient n = n /  Also, by agreement, 0  F . Lemma 15.3.1 Let M be a finite dimensional n -supermodule. Then M is integral if and only if  M = 0 for some  ∈ P+ . Proof If  M = 0, then the eigenvalues of x12 on M are all of the form qi for i ∈ I, by definition of  . Hence M is integral in view of Lemma 15.1.2. Conversely, suppose that M is integral. Then the minimal polynomial of  x12 on M is of the form i∈I t − qii for some i ≥ 0. So if we set  = 20 0 + 1 1 + · · · +   ∈ P+ , we certainly have that  M = 0. Lemma 15.3.1 allows us to introduce the functors pr   RepI n → n -smod

infl  n -smod → RepI n 

(15.7)

15.4 Basis Theorem for cyclotomic Sergeev superalgebras

185

Here, infl is simply inflation along the canonical epimorphism n → n , while on a supermodule M, pr  M = M/ M with the induced action of n . The functor infl is right adjoint to pr  , that is there is a functorial isomorphism Homn pr  M N   Homn M infl N 

(15.8)

Note we will generally be sloppy and omit the functor infl in our notation. In other words, we generally identify n -smod with the full subcategory of RepI n consisting of all supermodules M with  M = 0.

15.4 Basis Theorem for cyclotomic Sergeev superalgebras Let d = c . Then f is a monic polynomial of degree d. Write f = x1d + ad−1 x1d−1 + · · · + a1 x1 + a0  Set f1 = f and for i = 2     n, define inductively fi = si−1 fi−1 si−1  Note using (15.6): cj fi = ±fi cj

1 ≤ i j ≤ n

(15.9)

The first lemma follows easily by induction using relations in n , especially (14.8). Lemma 15.4.1 For i = 1     n, we have fi = xid + terms lying in i−1 xie i for 0 ≤ e < d Given Z = z1 < · · · < zu ⊆ 1     n , let fZ = fz1 fz2    fzu ∈ n  Also, define "n =  Z  Z ⊆ 1     n   ∈ n+ with i < d whenever i "+ n

=  Z ∈ "n  Z = ∅ 

Lemma 15.4.2 n is a free right n -module on basis x fZ   Z ∈ "n 

Z 

186

Integral representations and cyclotomic Sergeev algebras

Proof The proof is similar to that of Lemma 7.5.2 with i in place of i , and uses Lemma 15.4.1 instead of Lemma 7.5.1. Lemma 15.4.3 For n > 1 we have n−1 fn n = fn n . Proof It suffices to show that the left multiplication by the elements s1      sn−2 and c1      cn−1 leaves the space fn n invariant. But this follows from the definition of fn and relations in n . Lemma 15.4.4 We have  =

n 

 n fi n .

i=1

Proof We have  = n f1 n = n f1 n n = n f1 n =  n n f1 n =

n  

n si−1    s1 uf1 n

i=1 u∈S2n

=

n 

n si−1    s1 f1 n =

i=1

=

n 

n   n f i n i=1

n n fi n =

i=1

n 

n fi n 

i=1

using (15.9) for the last equality. Lemma 15.4.5 For d > 0 we have  =



x fZ n 

Z∈"+ n

Proof The proof is similar to that of Lemma 7.5.5 using Lemmas 15.4.4, 15.4.2, 15.4.3, 15.4.1 instead of Lemmas 7.5.4, 7.5.2, 7.5.3, 7.5.1, respectively.

Theorem 15.4.6 The canonical images of the elements x c w   ∈ n+ with 1      n < d  ∈ n2  w ∈ Sn

form a basis for n . Proof By Lemmas 15.4.2 and 15.4.5, the elements x fZ   Z ∈ "+ n form a basis for  viewed as a right n -module. Hence Lemma 15.4.2 implies that the elements x   ∈ n+ with 1      n < d

15.5 Cyclotomic Mackey Theorem

187

form a basis for a complement to  in n viewed as a right n -module. The theorem follows at once. Remark 15.4.7 It follows from Theorem 15.4.6 that n0 is isomorphic to the Sergeev superalgebra n . If we identify the two using the map c w +  → c w, then the projection n  n / = n can be described as follows: it is identity on n , i.e. c w ∈ n ⊂ n is mapped to c w ∈ n , and xk is mapped to the kth Jucys–Murphy element Lk ∈ n , defined in (13.22). This is easy to see because L1 = 0 by definition, and x1 must go to 0. Moreover, the exression (13.23) of Lk+1 in terms of Lk is the same as that of xk+1 in terms of xk .

15.5 Cyclotomic Mackey Theorem Given any y ∈ n , we will denote its canonical image in n by the same symbol. Thus, Theorem 15.4.6 says that Bn = x c w   ∈ n+ with 1      n < d  ∈ n2  w ∈ Sn

(15.10)

 is a basis for n . Also Theorem 15.4.6 implies that the subalgebra of n+1 generated by x1      xn  c1      cn and w, for w ∈ Sn , is isomorphic to n . 



We will write indn+1 and resn+1 for the induction and restriction functors   n n   between n and n+1 . So, 

 indn+1  M = n+1 ⊗n M n

 Lemma 15.5.1 (i) n+1 is a free right n -supermodule on basis

xja cjb sj    sn  0 ≤ a < d b ∈ 2  1 ≤ j ≤ n + 1  (ii) As n  n -bisupermodules,  n+1 = n sn n ⊕



a b xn+1 cn+1 n 

0≤a
(iii) Let 0 ≤ a < d. Then there are isomorphisms of n  n -bisupermodules a a  n  xn+1 n  n  xn+1 cn+1 n  "n  n sn n  n ⊗n−1

188

Integral representations and cyclotomic Sergeev algebras

Proof (i) By Theorem 15.4.6 and dimension considerations, we just need to  check that n+1 is generated as a right n -module by the given elements. This follows from relations, especially (14.8). (ii) It suffices to notice, using (i) and (14.8), that xja cjb sj    sn  0 ≤ a < d b ∈ 2  1 ≤ j ≤ n

is a basis of n sn n as a free right n -module. a a (iii) The isomorphisms xn+1 n  n and xn+1 cn+1 n  "n are clear from (i). Furthermore, the map n × n → n sn n  u v → usn v  is n−1 -balanced, so it induces an (even) homomorphism    n ⊗n−1 n → n sn n  of n  n -bisupermodules. By (i), n ⊗n−1 n is a free right n -supermodule on basis

xja cjb sj    sn−1 ⊗ 1  1 ≤ j ≤ n b ∈ 2  0 ≤ a < d  But  maps these elements to a basis for n sn n as a free right n -supermodule, using a fact observed in the proof of (ii). This shows that  is an isomorphism.  We have now decomposed n+1 as an n  n -bisupermodule. So the same argument as for Theorem 3.5.2 gives:

Theorem 15.5.2 Let M be an n -supermodule. Then there is a natural isomorphism 



n

n





⊕d n+1 resn+1 ⊕ indn resn M  ind   M  M ⊕ "M

of

n−1

n−1

n -supermodules.

15.6 Duality for cyclotomic superalgebras Lemma 15.6.1 For 1 ≤ i ≤ n and a ≥ 0, we have sn    si xia si    sn

a = xn+1 + ∗

a sn    si xia ci si    sn = xn+1 cn+1 + ∗∗  k  k  where ∗ and ∗∗ lie in n sn n + a−2 k=0 xn+1 n + xn+1 cn+1 n 

15.6 Duality for cyclotomic superalgebras

189

Proof We apply induction on n = i i + 1    . In case n = i, the result follows from a calculation using (14.8). The induction step is similar, noting that sn  centralizes n−1 . Lemma 15.6.2 There is an n  n -bisupermodule homomorphism   → n such that ker contains no non-zero left ideals of n+1 . n+1



Proof By Lemma 15.5.1(ii), we know that  d−1  n+1 = xn+1 n ⊕

d−2 

a xn+1 n ⊕

a=0

d−1 

a xn+1 cn+1 n ⊕ n sn n

a=0

as an n  n -bisupermodule. Let  d−1  → xn+1 n  n+1

be the projection on to the first summand of this decomposition, which by Lemma 15.5.1(iii) is isomorphic to n as an n  n -bisupermodule. So  has the property that hy = 0 for all we just need to show that if y ∈ n+1  h ∈ n+1 , then y = 0. Using Lemma 15.5.1(i), we may write y as d−1 

a a xn+1 a + xn+1 cn+1 a  +

a=0

d−1 n 

xja sj    sn aj + xja cj sj    sn aj 

a=0 j=1

for some elements a  a  aj  aj ∈ n . As y = 0, we must have d−1 = 0. 2 y = 0 implies d−3 = 0, Now xn+1 y = 0 implies d−2 = 0, then xn+1 etc., proving that all a are zero. Now cn+1 y = 0 implies that all a 2 are zero. Next, considering sn y, xn+1 sn y, xn+1 sn y    , and using Lemma 15.6.1, we get d−1n = d−2n = · · · = 0 and d−1n = d−2n = · · · = 0. Next, repeat the argument again, this time considering sn sn−1  xn+1 sn sn−1     , to get that all an−1 = 0 and all an−1 = 0. Continuing in this way we eventually arrive to the desired conclusion that y = 0. Theorem 15.6.3 There is a natural isomorphism   ⊗n M  Homn n+1 n+1  M

for all n -supermodules M. Proof It suffices to show that there exists an even isomorphism     n+1 → Homn n+1  n   of n+1  n -bisupermodules. This is done as in the proof of Theorem 7.7.3 using Lemmas 15.6.2 instead of Lemmas 7.7.2.

190

Integral representations and cyclotomic Sergeev algebras

Corollary 15.6.4 n is a Frobenius superalgebra, that is there is an even isomorphism of left n -supermodules n  HomF n  F . Proof Similar to the proof of Corollary 7.7.4. For the next corollary, recall the duality induced by  1417 on finite dimensional n -supermodules. Since  leaves the two-sided ideal  invariant, it induces a duality also denoted  on finite dimensional n -supermodules. 

Corollary 15.6.5 The exact functor indn+1 is both left and right adjoint to  n



resn+1  . Moreover, it commutes with duality in the sense that there is a natural n isomorphism 



n

n

  n+1 indn+1  M   ind   M

for all finite dimensional

n -supermodules

M.

Proof Similar to the proof of Corollary 7.7.5.

16 First results on n -modules

This chapter is parallel to Chapter 4. The role of the subalgebra n is played by n . However, it will be more convenient to consider only integral modules everywhere, as defined in Chapter 15.

16.1 Formal characters of n -modules Let i ∈ I, see (15.2). Denote by Li the vector superspace on basis v1  v−1 , ¯ v¯ −1 = 1, ¯ made into an 1 -supermodule via where v¯ 1 = 0, c1 v1 = v−1  c1 v−1 = v1  x1 v1 = qiv x1 v−1 = − qiv−1  only To be precise, we need to first make a choice of qi, as it is defined up to ±. But it is not a big deal, since changing from qi to − qi leads to an isomorphic supermodule (the isomorphism swaps v1 and v−1 ). Note that Lqi is irreducible of type M if i = 0, and irreducible of type Q if i = 0. It is also easy to see that the modules Li  i ∈ I form a complete set of pairwise non-isomorphic irreducible 1 -supermodules in the category RepI 1 . Now, n  1 ⊗ · · · ⊗ 1 , and so the next result follows from Lemmas 12.2.13 and 12.2.15: Lemma 16.1.1 The n -supermodules Li1   · · ·  Lin   i1      in  ∈ I n

form a complete set of pairwise non-isomorphic irreducible supermodules in the category RepI n . Moreover, let 0 denote the number of j = 1     n such that ij = 0. Then Li1   · · ·  Lin  is of type M if 0 is even and type Q if 0 is odd. Finally, dim Li1   · · ·  Lin  = 2n−)0 /2*  191

192

First results on n -modules

Now take any M ∈ RepI n . For any i = i1      in  ∈ I n , let Mi be the largest submodule of M all of whose composition factors are isomorphic to Li1   · · ·  Lin . Alternatively, Mi is the simultaneous generalized eigenspace for the commuting operators x12      xn2 corresponding to the eigenvalues qi1      qin , respectively. Hence: Lemma 16.1.2 For any M ∈ RepI n we have M = ⊕i∈I n Mi as an n -supermodule. Since xj2 acts as qij  on Li1   · · ·  Lin , and the dimension of each Li1 · · ·Lin  is known from Lemma 16.1.1, knowledge of the dimensions of the spaces Mi for all i is equivalent to knowing the coefficients ri when the class M of M in the Grothendieck group KRepI n  is expanded as  ri Li1   · · ·  Lin  M = i∈I n

in terms of the basis Li1   · · ·  Lin   i ∈ I n . Now let M ∈ RepI n . Recall that 11 = n . We define the formal character of M by: ch M = resn11 M ∈ KRepI n 

(16.1)

Since the functor resn11 is exact, ch induces a homomorphism ch  KRepI n  → KRepI n  at the level of Grothendieck groups. As in Section 4.1, we get as special cases of Mackey Theorem: Lemma 16.1.3 Let i = i1      in  ∈ I n . Then  ch indn11 Li1   · · ·  Lin  = Liw−1 1   · · ·  Liw−1 n  w∈Sn

Lemma 16.1.4 (“Shuffle Lemma”) Let n = m + k, and let M ∈ RepI m , K ∈ RepI k . Assume  ch M = ri Li1   · · ·  Lim   i∈I m



ch K =

sj Lj1   · · ·  Ljk  

j∈I k

Then ch indnmk M  K =

 i∈I m j∈I k



ri sj 

c

Lh1   · · ·  Lhn 

16.2 Central characters and blocks

193

where the last sum is over all h = h1      hn  ∈ I n , which are obtained by shuffling i and j.

16.2 Central characters and blocks Recall by Theorem 14.3.1 that every element z of the center Zn  can be written as a symmetric polynomial fx12      xn2 . Given i ∈ I n , we associate the central character i  Zn  → F

fx12      xn2  → fqi1      qin 

If i ∈ I n , define its content conti ∈ P by  conti = i i where i =  j = 1     n  ij = i 

(16.2)

i∈I

So conti is an element of the set #n of non-negative integral linear combi  nations  = i∈I i i of the simple roots such that i∈I i = n. Obviously, the Sn -orbit of i is uniquely determined by the content of i, so we obtain a labeling of the orbits of Sn on I n by the elements of #n . We will also use the notation  for the central character i where i is any element of I n with conti = . It is clear that  is well defined. Now let M ∈ RepI n , and  ∈ #n . Denote M = v ∈ M  z −  zk v = 0 for all z ∈ Zn  and k 0  Observe this is an n -subsupermodule of M. Now, for any i ∈ I n with conti = , Zn  acts on Li1   · · ·  Lin  via the central character  . So applying Lemma 4.2.1, we see that  M = Mi  (16.3) conti=

recalling the decomposition of M from Lemma 16.1.2. Therefore: Lemma 16.2.1 Any M ∈ RepI n decomposes as  M= M ∈#n

as an n -supermodule. Thus the    ∈ #n exhaust the possible central characters that can arise in an integral n -supermodule, while Lemma 16.1.3 shows that every such central character does arise in some integral n -supermodule.

194

First results on n -modules

If  ∈ #n , let us denote by RepI n  the full subcategory of RepI n consisting of all supermodules M with M = M. Then Lemma 16.2.1 implies that there is an equivalence of categories  RepI n   (16.4) RepI n  ∈#n

We say that RepI n  is the block of RepI n corresponding to  (or to the central character  ). If M ∈ RepI n  , we say that M belongs to the block corresponding to . If M = 0 is indecomposable then M ∈ RepI n  for a unique  ∈ #n . We can extend some of these notions to n -supermodules, for  ∈ P+ . In particular, if M ∈ n -smod, we also write M for the summand M of M defined by first viewing M as an n -supermodule by inflation. Also write n -smod for the full subcategory of n -smod consisting of the modules M with M = M . Then we have a decomposition   n -smod  (16.5) n -smod  ∈#n

Note though that we should not yet refer to n -smod as a block of n -smod: the center of n may be larger than the image of the center of n , so we cannot yet assert that Zn  acts on M by a single central character. Also we no longer know precisely which  ∈ #n have the property that n -smod is non-trivial. These questions will be settled in Section 20.6.

16.3 Kato’s Theorem for n Let i ∈ I. Introduce the Kato (super)module Lin  = indn11 Li  · · ·  Li

(16.6)

By Lemma 16.1.3, ch Lin  = n!Li  · · ·  Li  In particular, for each k = 1     n, the only eigenvalue of the element xk2 on Lin  is qi. We have to work a little harder than in Section 4.3 in order to prove “Kato’s Theorem”.
n Lemma 16.3.1 Let n ≥ 2, 1 ≤ k <  n, i ∈ I \ 0 , and v ∈ Li \ 0 . Then
xk 1 − ck ck+1  + 1 − ck ck+1 xk+1 v = 0.

16.3 Kato’s Theorem for n

195

Proof The elements of n which are involved in the inequality act only on the positions k and k + 1 in the tensor product. So we may assume that n = 2 and k = 1. Let w = a1 v1 ⊗ v1 + a2 v1 ⊗ v−1 + a3 v−1 ⊗ v1 + a4 v−1 ⊗ v−1 for a1  a2  a3  a4 ∈ F , and denote b = qi. Then the result is clear from: x1 1 − c1 c2 w = ba1 + ba4 v1 ⊗ v1 + ba2 + ba3 v1 ⊗ v−1 +−ba3 + ba2 v−1 ⊗ v1 + −ba4 + ba1 v−1 ⊗ v−1  1 − c1 c2 x2 w = ba1 − ba4 v1 ⊗ v1 + −ba2 + ba3 v1 ⊗ v−1 +ba3 + ba2 v−1 ⊗ v1 + −ba4 − ba1 v−1 ⊗ v−1 

Lemma 16.3.2 Let i ∈ I. Set L = Lin , so Lin  = n ⊗n L. 2 on Lin  is (i) If i = 0, the common qi-eigenspace of x12      xn−1 precisely 1 ⊗ L, which is contained in the qi-eigenspace of xn2 too. Moreover, all Jordan blocks of x12 on Lin  are of size n. (ii) If i = 0, the common 0-eigenspace of x1      xn−1 on L0n  is precisely 1 ⊗ L, which is contained in the 0-eigenspace of xn too. Moreover, all Jordan blocks of x1 on L0n  are of size n.

 Proof We prove (i), (ii) being similar. Note Lin  = w∈Sn w ⊗ L, since by Theorem 14.2.2 we know that n is a free right n -module on basis w  w ∈ Sn . We first claim that the qi-eigenspace of x12 is a sum of the subspaces of  = s2      sn−1 . Well, any w can be written the form y ⊗ L, where y ∈ Sn−1  2 and 0 ≤ j < n. Note that xj+1 − qiv = 0 as ys1 s2    sj for some y ∈ Sn−1 for any v ∈ L, by definition of L. So the defining relations of n imply x12 − qiys1 s2    sj ⊗ v = − ys1    sj−1 ⊗ xj 1 − cj cj+1  − 1 − cj cj+1 xj+1 v + ∗  where ∗ stands for terms in subspaces of the form y s1    sk ⊗ L for y ∈ Sn−1 and 0 ≤ k < j − 1. Now assume that a linear combination    cyjv ys1 s2    sj ⊗ v z =  0≤j
is an eigenvector for x12 . Choose the maximal j for which the coefficient cyjv is non-zero, and for this j choose the maximal (with respect to the

196

First results on n -modules

Bruhat order) y such that cyjv is non-zero. Then the calculation above and Lemma 16.3.1 show that x12 − qiz = 0 unless j = 0. This proves our claim. Now apply the same argument to see that the common eigenspace of x12 and x22 is spanned by y ⊗ L for y ∈ s3      sn , and so on, yielding the first claim in (i). Finally, define Vm = w ∈ Lin   x12 − qim w = 0  It follows by induction from the calculation above and Lemma 16.3.1 that  Vm = span ys1 s2    sj ⊗ v  y ∈ Sn−1  j < m v ∈ L 

giving the second claim. The main theorem on the structure of the Kato module Lin  is Theorem 16.3.3 Let i ∈ I and =  1      r  be a composition of n. Lin  is irreducible, and it is the only irreducible module in its block. Moreover, Lin  is of type Q if i = 0 and n is odd, and of type M otherwise. (ii) All composition factors of resn Lin  are isomorphic to (i)

Li 1   · · ·  Li r  and soc resn Lin  is irreducible.
n−1  Li   Li . (iii) soc resnn−1 Lin   resn−11 n−1 Proof Denote Li  · · ·  Li by L. (i) Let M be a non-zero n -submodule of Lin . Then resn11 M must contain a n -submodule N isomorphic to L. But the commuting operators x12      xn2 (or x1      xn if i = 0) act on L as scalars, giving that N is contained in their common eigenspace on Lin . But by Lemma 16.3.2, this implies that N = 1 ⊗ L. This shows that M contains 1 ⊗ L, but 1 ⊗ L generates the whole of Lin  over n . So M = Lin . To see that Lin  is the only irreducible in its block use Frobenius reciprocity and the fact just proved that Lin  is irreducible. Finally, in view of Lemma 16.1.1, it remains to see that the type of the n -supermodule Lin  is the same as the type of the n -supermodule L. The functor indn11 determines a map Endn L → Endn Lin 

16.4 Covering modules for n

197

We just need to see that this is an isomorphism, which we do by constructing the inverse map. Let f ∈ Endn Lin . Then f leaves 1 ⊗ L invariant by Lemma 16.3.2, so f restricts to an n -endomorphism of L. (ii) The fact that all composition factors of resn Lin  are isomorphic to Li 1   · · ·  Li r  follows by formal characters and (i). To see that soc resn Lin  is irreducible, note that the submodule  ⊗ L of res Lin  is isomorphic to Li 1   · · ·  Li l . This module is irreducible, and so it is contained in the socle. Conversely, let M be an irreducible  -submodule of Lin . Then using Lemma 16.3.2 as in the proof of (i), we see that M must contain 1 ⊗ L, hence  ⊗ L. (iii) By part (ii), Lin  has a unique n−11 -submodule isomorphic to n−11 ⊗ L, which contributes a copy of Lin−1 
 Li, namely,  Lin−1   Li to soc resnn−1 Lin , since resn−11 n−1
n−1  resn−11 Li   Li n−1 is completely reducible. Conversely, take an irreducible n−1 -submodule M of Lin . The common 2 (or x1      xn−1 if i = 0) on M must lie in qi-eigenspace of x12      xn−1 1 ⊗ L by Lemma 16.3.2. Hence, M ⊆ n−11 ⊗ L which completes the proof.

16.4 Covering modules for n Fix i ∈ I and n ≥ 1 throughout the section. We will construct for each m ≥ 1 an n -supermodule Lm in  with irreducible head isomorphic to Lin . Let  in  denote the annihilator in n of Lin . Introduce the quotient superalgebra m in  = n / in m

(16.7)

for each m ≥ 1. Obviously  in  contains xk2 − qin! for each k = 1     n, whence each superalgebra m in  is finite dimensional. Moreover, by Theorem 16.3.3, Lin  is the unique irreducible m in -module up to isomorphism. Let Lm in  denote a projective cover of Lin  in the category m in -smod (for convenience, we also define L0 in  = 0 in  = 0). Lemma 16.4.1 For each m ≥ 1, ⎧ n n ⊕n!2n−1 ⎪ if i = 0, ⎪ ⎨Lm i  ⊕ "Lm i  n−2/2 n n n ⊕n!2 m i   Lm i  ⊕ "Lm i  if i = 0 and n is even, ⎪ ⎪ ⎩L in ⊕n!2n−1/2 if i = 0 and n is odd, m

198

First results on n -modules

as left n -modules. Moreover, Lm in  admits an odd involution if and only if i = 0 and n is odd. Proof The superdimension of Lin  is known from Lemma 16.1.1 and the definition. Now use Lemma 12.2.16 and Proposition 12.2.12. The obvious surjections 1 in   2 in     

(16.8)

of superalgebras and properties of projective covers lead to even surjections Lin  = L1 in   L2 in      

(16.9)

where Lm in  are considered as n -modules by inflation. Moreover, in case i = 0 and n is odd, we can choose the odd involutions m

 Lm in  → Lm in 

(16.10)

given by Lemma 16.4.1 in such a way that they are compatible with the maps in (16.9) (see the argument in Lemma 12.2.16). Lemma 16.4.2 Let M be an n -supermodule annihilated by  in k for some k. Then for all m ≥ k there is a natural isomorphism of n -supermodules Homn m in  M  M Proof The assumption implies that M is the inflation of an m in supermodule. So Homn m in  M  Homm in  m in  M  M all isomorphisms being natural. We can also obtain an analogue of Lemma 4.4.3 for n , but we will not need this result. In the important special case n = 1 we easily check that the ideal  im is generated by x12 − qim if i = 0 or x1m if i = 0. It follows that

4m if i = 0, (16.11) dim m i = 2m if i = 0. Moreover, Lemma 16.4.1 shows in this case that

Lm i ⊕ "Lm i if i = 0, m i  Lm i if i = 0.

(16.12)

16.4 Covering modules for n

199

Hence, dim Lm i = 2m in either case. Using this, it follows easily that Lm i can be described alternatively as the vector superspace on basis w1      wm , w1      wm  where each wk is even and each wk is odd, with 1 -module structure uniquely determined by c1 wk = wk x1 wk = qiwk + wk+1  for each k = 1     m, interpreting wm+1 as 0. Using this explicit description, we check that Lm i is uniserial with m composition factors all  Li. We can also describe the map Lm i  Lm+1 i from (16.9) explicitly: it  to zero. is the identity on w1      wm  w1      wm but maps wm+1 and wm+1 Also, the map m from (16.10) can be chosen so that √ √ wk → −1wk  wk → − −1wk for each k = 1     m.

17 Crystal operators for n

This chapter is parallel to Chapter 5.

17.1 Multiplicity-free socles For M ∈ RepI n and i ∈ I, define i M to be the generalized qi-eigenspace of xn2 on M. Equivalently,  Mi  (17.1) i M = i∈I n  in =i

recalling the decomposition from Lemma 16.1.2. Note since xn2 is central in the parabolic subalgebra n−11 of n , i M is invariant under this subalgebra. So, in fact, i can be viewed as an exact functor i  RepI n → RepI n−11 

(17.2)

Slightly more generally, given m ≥ 0, define im  RepI n → RepI n−mm

(17.3)

so that im M is the simultaneous generalized qi-eigenspace of the commuting operators xk2 for k = n − m + 1     n. In view of Theorem 16.3.3(i), im M can also be characterized as the largest submodule of resnn−mm M all of whose composition factors are of the form N  Lim  for irreducible N ∈ RepI n−m . The definition of im implies a functorial isomorphism Homn−mm N
Lim  im M  Homn indnn−mm N
Lim  M for N ∈ RepI n−m , M ∈ RepI n . For irreducible N this implies Homn−mm N  Lim  im M  Homn indnn−mm N  Lim  M 200

(17.4)

17.1 Multiplicity-free socles

201

Also from definitions we get: Lemma 17.1.1 Let M ∈ RepI n with  ch M = ri Li1 
· · ·
Lin   i∈I n

Then we have ch im M =



rj Lj1 
· · ·
Ljn  

j

summing over all j ∈ I n with jn−m+1 = · · · = jn = i. Now for i ∈ I and M ∈ RepI n , define
i M = max m ≥ 0  im M = 0 

(17.5)

The following results and their proofs are completely analogous to the corresponding results of Section 5.1. The only change we usually need to make is to go from “
” to “” and apply the n -analogues of the lemmas used in Section 5.1. Lemma 17.1.2 Let M ∈ RepI n be irreducible, i ∈ I,
=
i M. If N Lim  is an irreducible submodule of im M for some 0 ≤ m ≤
, then
i N  =
−m. Proof Analogous to the proof of Lemma 5.1.2. Lemma 17.1.3 Let m ≥ 0, i ∈ I and N ∈ RepI n be irreducible with m
i N  = 0. Set M = indn+m nm N  Li . Then: (i) im M  N  Lim ; (ii) hd M is irreducible with
i hd M = m; (iii) all other composition factors L of M have
i L < m. Proof Analogous to the proof of Lemma 5.1.3. Lemma 17.1.4 Let M ∈ RepI n be irreducible, i ∈ I, and
=
i M. Then i
M is isomorphic to N  Li
 for some irreducible n−
-supermodule N with
i N  = 0 Proof Analogous to the proof of Lemma 5.1.4. Lemma 17.1.5 Let m ≥ 0, i ∈ I and N ∈ RepI n be irreducible. Set m M = indn+m nm N  Li 

Crystal operators for n

202

Then hd M is irreducible with
i hd M =
i N  + m, and all other composition factors L of M have
i L <
i N  + m. Proof Analogous to the proof of Lemma 5.1.5. Theorem 17.1.6 Let M ∈ RepI n be irreducible and i ∈ I. Then, for any 0 ≤ m ≤
i M, soc im M is an irreducible n−mm -supermodule of the same type as M, and is isomorphic to L  Lim , for some irreducible n−m -supermodule L with
i L =
i M − m. Proof An argument analogous to the one given in the proof of Theorem 5.1.6 shows that soc im M  L  Lim  for some irreducible n−m -supermodule L with
i L =
i M − m. We just need to show that L  Lim  has the same type as M. Note by Lemma 17.1.5, indnn−mm L  Lim  has irreducible head, necessarily isomorphic to M by Frobenius reciprocity. So applying (17.4) we have that Endn−mm L  Lim   Homn−mm L  Lim  im M  Homn indnn−mm L  Lim  M  Endn M which implies the statement concerning types. Corollary 17.1.7 For an irreducible supermodule M ∈ RepI n , the socle of resnn−11 M is multiplicity free. Define the functor resi = resn−11 n−1  i  RepI n → RepI n−1 

(17.6)

Record the following obvious equalities for M ∈ RepI n :
i M = max m ≥ 0  resm i M = 0   n resn−1 M = resi M i∈I

(17.7) (17.8)

17.2 Operators e˜i and f˜i

203

Corollary 17.1.8 For an irreducible M ∈ RepI n with
i M > 0,

L if M is of type M and i = 0, soc resi M  L ⊕ "L otherwise for some irreducible n−1 -supermodule L of the same type as M if i = 0 and of the opposite type to M if i = 0. Proof Set  = 1 if M is of type M and i = 0,  = 2 otherwise. By Theorem 17.1.6, the socle of i M is isomorphic to L  Li for some irreducible n−1 -supermodule L, and ⊕ resn−11 n−1 L  Li  L 

indeed it is exactly as in the statement of the corollary, including the statement about the type. Now take any irreducible subsupermodule K of resi M. Consider the n−11 -subsupermodule 1 K ⊆ resi M, where 1 is the subalgebra generated by cn  xn . All composition factors of 1 K are isomorphic to K  Li. As the socle of 1 K must be isomorphic to L  Li, this implies K  L. We deduce  that soc resi M  L⊕ for some  ≥ . By Lemma 17.1.2,
i L =
− 1 where
=
i M, and i
−1 L is irreducible by Lemma 17.1.4. So at least  copies of i
−1 L appear in soc resn−

n−

−1 i
M But i
M  N  Li
 for some irreducible N , thus applying Theorem 16.3.3 (iii) and the facts about type in Theorem 17.1.6, n−


soc resn−

n−

−1 i
M  soc resn−

−1 N  Li 
−1  resn−

−11   Li n−

−1 N  Li

 N  Li
−1 ⊕  Hence  ≤  also.

17.2 Operators e˜i and f˜i Let M be an irreducible n -supermodule. Define f˜i M = hd indn+1 n1 M  Li

(17.9)

Note f˜i M is irreducible by Lemma 17.1.5. To define e˜ i M, note by Theorem 17.1.6 that either i M = 0 or soc i M has form N  Li for an

204

Crystal operators for n

irreducible n−1 -supermodule N . In the former case we define e˜ i M = 0, and in the latter case we define e˜ i M = N . Thus, e˜ i M is defined from soc i M  ˜ei M  Li

(17.10)

Note right away from Lemma 17.1.2 that
i M = max m ≥ 0  e˜ im M = 0 

(17.11)

while a special case of Lemma 17.1.5 shows that
i f˜i M =
i M + 1

(17.12)

Lemma 17.2.1 Let M ∈ RepI n be irreducible, i ∈ I and m ≥ 0. (i) soc im M  ˜eim M  Lim  m ˜m (ii) hd indn+m nm M  Li   fi M Proof Analogous to the proof of Lemma 5.2.1. Lemma 17.2.2 Let M ∈ RepI n and N ∈ RepI n+1 be irreducible supermodules, and i ∈ I. Then f˜i M  N if and only if e˜ i N  M. Proof Analogous to the proof of Lemma 5.2.3. From Lemma 17.2.2 we immediately deduce the following: Corollary 17.2.3 Let M N ∈ RepI n be irreducible. Then f˜i M  f˜i N if and only if M  N . Similarly, providing
i M
i N  > 0, e˜ i M  e˜ i N if and only if M  N .

17.3 Independence of irreducible characters Theorem 17.3.1 The map ch  KRepI n  → KRepI n  is injective. Proof Analogous to the proof of Theorem 5.3.1. Corollary 17.3.2 If L is an irreducible n -supermodule, then L  L . Proof Analogous to the proof of Corollary 5.3.2

17.4 Labels for irreducibles

205

Lemma 17.3.3 Let M ∈ RepI m and N ∈ RepI n be irreducible supermodules. Suppose: n+m (i) indm+n mn M  N  ind nm N  M; m+n (ii) M  N appears in resm+n mn ind mn M  N with multiplicity one.

Then indm+n mn M  N is irreducible. Proof Analogous to the proof of Lemma 5.3.3. We can also show at this point that the type of an irreducible supermodule is determined by the type of its central character: Lemma 17.3.4 Suppose L ∈ RepI n is irreducible with central character   , where  = i∈I i i ∈ #n . Then, L is of type Q if 0 is odd, type M if 0 is even. Proof Proceed by induction on n, the case n = 0 being trivial. If n > 1, choose i ∈ I so that e˜ i L = 0. By definition, e˜ i L has central character  − i . So by the induction hypothesis, e˜ i L is of type Q if 0 − i0 is odd, type M otherwise. But by Lemmas 12.2.13 and 16.1.1, ˜ei L  Li is of the opposite type to e˜ i L if i = 0, of the same type if i = 0. Hence, ˜ei L  Li is of type Q if 0 is odd, type M otherwise. Finally, the proof is completed by Theorem 17.1.6, since this shows that L has the same type as soc i L = ˜ei L  Li.

17.4 Labels for irreducibles Write 1 for the trivial irreducible supermodule over 0  F . If L is an irreducible n -supermodule, it follows from Lemma 17.2.2 that L  f˜in    f˜i2 f˜i1 1 for at least one tuple i = i1  i2      in  ∈ I n . So if we define Li = Li1      in  = f˜in    f˜i2 f˜i1 1

(17.13)

we obtain a labeling of all irreducibles by tuples in I n . For example, Li i     i (n times) is precisely the Kato module Lin  introduced in (16.6). A given irreducible L will in general be parametrized by several different tuples i ∈ I n . Some basic properties of Li are easy to read off from the notation: for instance the central character of Li is i .

18 Character calculations for n

This chapter is parallel to Chapter 6. The explicit character calculations in Section 18.2 determine the type of the Kac–Moody algebra acting on the Grothendieck group of n -supermodules.

18.1 Some irreducible induced supermodules Given i = i1      in  ∈ I n , let indi = indi1      in  = indn11 Li1   · · ·  Lin  By Lemma 16.1.3 and (16.3), every irreducible constituent of indi belongs to the block corresponding to the orbit Sn · i. Lemma 18.1.1 Let i ∈ I n . Then: (i) resn11 Li has a submodule isomorphic to Li1   · · ·  Lin ; (ii) indi contains a copy of Li in its head; (iii) every irreducible module in the block corresponding to the orbit Sn · i appears at least once as a constituent of indi. Proof Analogous to the proof of Lemma 6.1.1. Lemma 18.1.2 Let i ∈ I m  j ∈ I n be tuples such that ir − js ∈ 0 ±1 for all 1 ≤ r ≤ m and 1 ≤ s ≤ n. Then m+n indm+n mn Li  Lj  ind nm Lj  Li

is irreducible. 206

18.1 Some irreducible induced supermodules

207

Proof By the Shuffle Lemma, Li  Lj appears in m+n resm+n mn ind mn Li  Lj

with multiplicity 1. So, in view of Lemma 17.3.3, it suffices to show that m+n indm+n mn Li  Lj  ind nm Lj  Li

By the Mackey Theorem and central characters argument, m+n resm+n mn ind nm Lj  Li

contains LiLj as a summand with multiplicity one, all other constituents lying in different blocks. Hence by Frobenius reciprocity, there exists a nonzero homomorphism m+n f  indm+n mn Li  Lj → ind nm Lj  Li

Every homomorphic image of indm+n mn Li  Lj contains an mn -submodule isomorphic to Li  Lj. So, by Lemma 18.1.1(i), we see that the image of f contains a m+n -submodule V isomorphic to Li1   · · ·  Lim   Lj1   · · ·  Ljn  Next we claim that the image of f also contains an m+n -submodule isomorphic to Lj1   · · ·  Ljn   Li1   · · ·  Lim 

(18.1)

Indeed, by Lemma 14.8.1 and the assumption that im − j1 ∈ 0 ±1 , m2 acts on v by a non-zero scalar. So by (14.20), (14.21), m V = 0 is a m+n -submodule isomorphic to Li1   · · ·  Lim−1   Lj1   Lim   Lj2   · · ·  Ljn  Next apply m−1      1 to move Lj1  to the first position, and continue in this way to complete the proof of the claim. Now, by the Shuffle Lemma, all composition factors of m+n resm+n 11 ind nm Lj  Li

isomorphic to (18.1) necessarily lie in the irreducible nm -submodule 1 ⊗ Lj  Li of the induced module. Since this generates all of indm+n nm Lj  Li as an m+n -module, this shows that f is surjective. Hence f is an isomorphism by dimension.

Character calculations for n

208

Remark 18.1.3 Keep assumptions of the lemma. Set  = Sm · i  = Sn · j  ∪  = Sm+n · i ∪ j   = Sm × Sn · i j Then the argument as above shows that the functor indm+n mn induces an equivalence of categories RepI m ⊗ n    RepI m+n  ∪   Theorem 18.1.4 Let i ∈ I m  j ∈ I n be tuples such that ir − js ∈ ±1 for all 1 ≤ r ≤ m and 1 ≤ s ≤ n. Then m+n indm+n mn Li  Lj  ind nm Lj  Li

is irreducible. Moreover, every other irreducible m+n -supermodule lying in m+n   the same block as indm+n mn Li  Lj is of the form ind mn Li   Lj  for   permutations i of i and j of j. Proof Analogous to the proof of Theorem 6.1.4.

18.2 Calculations for small rank Recall the notation hi  j  from Section 15.2. Theorem 18.2.1 Let i j ∈ I, i − j ∈ ±1 , and k = −hi  j . (i) For all r s ≥ 0 with r + s ≤ k we have ch Lir jis  = r!s!Lir  Lj  Lis  Moreover, if additionally we have r > 0, then there is a non-split short exact sequence r−1 s 0 −→ Lir jis  −→ indr+s+1 ji   Li r+s1 Li

−→ Lir−1 jis+1  −→ 0 (ii) We have Lik+1 j  Lik ji. Moreover, for all r ≥ 0 and s > 0 with r + s = k + 1 we have ch Lir jis  =r!s!Lir  Lj  Lis + r + 1!s − 1!Lir+1  Lj  Lis−1  and r s−1 r s−1 Lir jis   indk+1   Li  indk+1  k1 Li ji 1 k Li  Li ji

18.2 Calculations for small rank

209

Proof The general scheme here is similar to the classical case, see Lemmas 6.2.1 and 6.2.2. (i) We first proceed by induction on n = 0 1     k to show that ch Lin j = n!Lin  Lj  the induction base being clear. For n > 0, let n−1 Mn = indn+1 j  Lj n1 Li

(18.2)

We know by the inductive hypothesis and the Shuffle Lemma that ch Mn = n!Lin  Lj + n − 1!Lin−1  Lj  Li  Now consider the n1 -submodule 2 − qiM  Lin   Lj Nn = xn+1

(18.3)

of Mn . We claim that Nn is stable under the action of sn , hence all of n+1 . This is a brutal calculation outlined in Lemmas 18.2.2–18.2.5. So there exists an irreducible n+1 -supermodule Nn with character n!Lin  Lj . This must be Lin j, by Lemma 18.1.1(i), completing the proof of the induction step. Now we explain how to deduce the characters of the remaining irreducibles in the block. The quotient module Mn /Nn has character ch Mn /Nn = n − 1!Lin−1  Lj  Li  so Mn /Nn  Lin−1 ji. Twisting with the automorphism  proves that there exist irreducibles with characters n!Lj  Lin and

n − 1!Li  Lj  Lin−1 

which must be Ljin  and Lijin−1  respectively by Lemma 18.1.1(i) once more. This covers everything unless n = 4, when we necessarily have that i = 0 j = 1 and p = 3. In this case, we have shown already that there exist four irreducibles with characters ch L04 1 = 24L04  L1  ch L03 10 = 6L03  L1  L0  ch L104  = 24L1  L04  ch L0103  = 6L0  L1  L03  So by Lemma 18.1.1, there must be exactly one more irreducible module in the block, namely L00100, since none of the above involves the character L02  L1  L02 . Considering the character of ind541 L0010  L0 shows that ch L00100 is either 4L02  L1  L02 or 4L02 

210

Character calculations for n

L1  L02 + 6L03  L1  L0 , but the latter is not -invariant so cannot occur as there would then be too many irreducibles. Now that the characters are known, it is a routine matter using the Shuffle Lemma and Lemma 17.1.5 to prove the existence of the required non-split sequence. n−1 j  Li. We first claim that Mn (ii) Let n = k + 1, and Mn = indn+1 n1 Li is irreducible. To prove this, arguing as in (i), it suffices to show that the n1 -submodule 2 xn+1 − qiMn  Lin   Lj

of Mn is not invariant under sn . Again, this is a brutal calculation, see Lemmas 18.2.2–18.2.5. Hence, in view of (i), there is an irreducible n+1 module Mn with character n!Lin  Lj + n − 1!Lin−1  Lj  Li  Therefore e˜ i Mn  Lin−1 j and e˜ j Mn  Lin . So we deduce that Mn  Lin−1 ji  Lin j thanks to Lemma 17.2.2. Now consider the remaining irreducibles in the block. There are at most k remaining, namely Lir jis  for r ≥ 0 s ≥ 2 with r + s = k + 1. Considering r s−1   Li and arguing in a similar way the known characters of indn+1 n1 Li ji to (i), the remainder of the lemma follows without further calculation. We now sketch the calculations skipped in the proof of Theorem 18.2.1. From now on till the end of this section i j are as in the assumptions of Theorem 18.2.1, and Mn , Nn are as in (18.2), (18.3), respectively. Until the end of Section 18.2 we write a for qi and b for qj. Lemma 18.2.2 N1 is s1 -invariant. Proof By assumption i j are not both zero, so Lj  Li = Lj
Li, see Lemma 16.1.1. Recall the basis v1  v−1 of Li from Section 16.1. We use the same notation for a similar basis of Lj. Then M1 has basis 1 ⊗ v ⊗ v  s1 ⊗ v ⊗ v    ∈ ±1

 Note that x22 − a1 ⊗ v ⊗ v  = 0 and also, using (14.8), we have √ √ x22 − as1 ⊗ v ⊗ v  = b − as1 ⊗ v ⊗ v  +  b +  a1 ⊗ v ⊗ v √ √ +  b −  a1 ⊗ v− ⊗ v−  (18.4)

18.2 Calculations for small rank

211

Denote v = x22 − as1 ⊗ v ⊗ v 

  ∈ ±1 

Then v    ∈ ±1

is a basis of N1 . Note for future reference: √ √ x1 v =  av  x2 v =  bv  c1 v = −v− 

c2 v = −v− 

Using (18.4) and the assumption that i − j ∈ ±1 , we check that √ √ √ √ b −  a  b+ a v + v−−  s1 v = b−a b−a In particular, N1 is s1 -invariant.

(18.5)

(18.6)

Lemma 18.2.3 N2 is s2 -invariant if and only if i j ∈ 0 1  − 1   Proof We have found in the proof of Lemma 18.2.2 that Li j has basis v    ∈ ±1

with the action given by (18.5) and (18.6). So N2 = x32 − aind321 Li j
Li is spanned by the vectors z = x32 − as2 ⊗ v ⊗ v  y = x32 − as1 s2 ⊗ v ⊗ v for    ∈ ±1 . If i = 0, we have N2 = N2 . If i = 0, then Li j
Li  Li j  Li⊕2 , so N2  N2⊕2 . In any case, it is enough to check the lemma for N2 instead of N2 . Using relations we get: √ √ z = b − as2 ⊗ v ⊗ v +  b +  a1 ⊗ v ⊗ v  (18.7) √ √ +  a −  b1 ⊗ v− ⊗ v−  y = b − a s1 s2 ⊗ v ⊗ v √ √ √ √  b +  a  b +  a  1 ⊗ v ⊗ v  + b−a √ √ √ √  b +  a  b −  a  + 1 ⊗ v−− ⊗ v  b−a √ √ √ √  a −  b − b +  a  + 1 ⊗ v− ⊗ v−  b−a √ √ √ √  a −  b  b +  a  + 1 ⊗ v− ⊗ v−  b−a

(18.8)

212

Character calculations for n

Now, a calculation entirely similar to the n = 1 case shows that √ √ √ √  b+ a  a −  b s2 z = z + z−−  b−a b−a Next, we check that √ √ s2 y =  b +  a s1 s2 ⊗ v ⊗ v √ √ +  b −  a s1 s2 ⊗ v−− ⊗ v √ √ √ √  b +  a  b +  a  s2 ⊗ v ⊗ v + b−a √ √ √ √  b +  a  b −  a  s2 ⊗ v−− ⊗ v + b−a √ √ √ √  a −  b − b +  a  s2 ⊗ v− ⊗ v− + b−a √ √ √ √  a −  b  b +  a  s2 ⊗ v− ⊗ v−  + b−a So, N2 is s2 -invariant if and only if √ √ √ √ b −  a  b+ a y + y−− s2 y = b−a b−a √ √ √ √  b +  a  b +  a  + z b − a 2 √ √ √ √  b +  a  b −  a  + z−− b − a 2 √ √ √ √  a −  b − b +  a  + z−− b − a 2 √ √ √ √  a −  b  b +  a  + z−−  b − a 2

(18.9)

(18.10)

(18.11)

Now, a calculation, using (18.10), (18.7), and (18.8), shows that (18.11) is equivalent to i j ∈ 0 1  − 1  . Lemma 18.2.4 Let i j = 0 1 or  − 1 . Then N3 is s3 -invariant if and only if p = 3. Proof As in the proof of Lemma 18.2.3, we can work with the module N3 = x42 − aind431 N2
Li ⊂ M3 = ind431 N2
Li

18.2 Calculations for small rank

213

Recall the basis z  y of N2 found in the proof of Lemma 18.2.3. The action of the generators of 3 on the basis elements is given by (18.11) together with

s2 z

s1 z = y  s1 y = z  √ √ √ √  b+ a  a −  b z + z−−  = b−a b−a c1 z = −z−  c1 y = y−  c2 z = z− 

x1 z x2 z

c2 y = −y− 

c3 z = −z−  c3 y = −y−  √ =  az  x1 y =  ay − z + z−−  √ √ =  az  x2 y =  ay + z + z−−  √ √ x3 z =  bz  x3 y =  by  √

r for the elements x42 − asr sr+1    s3 ⊗ z ⊗ v Let us write zr and y 2 and x4 − asr sr+1    s3 ⊗ y ⊗ v of N3 , respectively. We can easily see that N3 is spanned by the vectors 1 2 3  y  y  z1  z2  z3  y

Let us also denote by y and z the elements 1 ⊗ y ⊗ v and 1 ⊗ z ⊗ v of M3 , respectively. A calculation using the action formulas on N2 above shows that √ √ z3 = b − as3 ⊗ z ⊗ v +  a +  bz √ √ +  a −  bz−−  √ √ 3 = b − as3 ⊗ y ⊗ v +  a +  by y √ √ +  a −  by−−  z2 = b − as2 s3 ⊗ z ⊗ v √ √ √ 1  √  a +  b b +  az + b−a √ √ √ √ +  a +  b a −  bz−− √ √ √ √ +  a −  b− b +  az−−  √ √ √ √ +  a −  b− a −  bz−− 

214

Character calculations for n

2 y = b − as2 s3 ⊗ z ⊗ v √ √ √ 1  √  a +  b b +  ay + b−a √ √ √ √ +  a +  b b −  ay−− √ √ √ √ +  a −  b− b +  ay−−  √ √ √ √ +  a −  b b +  ay−−  √ √ √ √ √ √ 1  a +  b b +  a b +  az + 2 b − a √ √ √ √ √ √ +  a +  b b +  a b −  az−− √ √ √ √ √ √ +  a +  b a −  b− b +  az−− √ √ √ √ √ √ +  a +  b a −  b b +  az−− √ √ √ √ √ √ +  a −  b− b +  a− b +  az−− √ √ √ √ √ √ +  a −  b− b +  a b +  az−− √ √ √ √ √ √ +  a −  b− a −  b b +  az−−  √ √ √ √ √ √ +  a −  b− a −  b b −  az−−−−  2 2 Next we calculate s3 y , and note that for s3 y to belong to N3 it must equal  √ √ 2 √ 2 1  √  b +  ay +  b −  ay−− b−a  √ √ √ √ 1  b +  a b +  az2 + 2 b − a √ √ √ √ +  b +  a b −  az2−− √ √ √ √ +  a −  b− b +  az2−− √ √ √ √ +  a −  b b +  az2−− √ √ √ √ 3 +  a +  b b +  ay √ √ √ √ 3 +  a +  b b −  ay−− √ √ √ √ 3 +  a −  b− b +  ay−−  √ √ √ √ 3 +  a −  b b +  ay−−

18.2 Calculations for small rank

215

 √ √ √ √ √ √ 1  a +  b b +  a b +  az3 3 b − a √ √ √ √ √ √ +  a +  b b +  a b −  az3−− √ √ √ √ √ √ +  a +  b a −  b− b +  az3−− √ √ √ √ √ √ +  a +  b a −  b b +  az3−− √ √ √ √ √ √ +  a −  b− b +  a− b +  az3−− √ √ √ √ √ √ +  a −  b− b +  a b +  az3−− √ √ √ √ √ √ +  a −  b− a −  b b +  az3−−  √ √ √ √ √ √ +  a −  b− a −  b b −  az3−−−−  +

r Using the formulas for zr and y given above and considering the coefficient of z in the last expression we conclude that p = 3. Now it is easy to check that in the case p = 3 the equality above does hold, and, r are s3 -invariant. In fact, for p = 3 moreover, all other vectors zr and y we get the following formulas: 1 2 2 1 = y  s1 z1 = z2  s1 y = y  s1 z2 = z1  s1 y 3 3 1 1 s1 y = z3  s1 z3 = y  s2 y = z1  s2 z1 = y  2 3 3 2 s2 y = y  s2 z2 = z3  s2 y = y  s2 z3 = z2 

√ 1 √ 1 1 s3 y = −  2y − 2y−− − z1 − z1−− − z1−− + z1−− − z3 − z3−− √ 3 √ 3 − z3−− + z3−− +  2y + 2y−− √ 3 √ 3 √ 3 +  2y−− −  2y−− −  2y−− √ 3 √ 3 √ 3 +  2y−− +  2y−− +  2y−−−−  √ √ 3 3 s3 z1 = −  2z1 +  2z1−− − y + y−− 3 3 − y−− − y−−  √ √ 2 2 2 s3 y = −  2y − 2y−− − z2 − z2−− 3 3 − z2−− + z2−− − y − y−− √ √ 3 3 − y−− + y−− +  2z3 + 2z3−−

Character calculations for n

216

√ √ √ +  2z3−− −  2z3−− −  2z3−− √ √ √ +  2z3−− +  2z3−− +  2z3−−−−  √ √ s3 z2 = −  2z2 +  2z2−− − z3 + z3−− − z3−− − z3−−  √ 3 √ 3 3 s3 y = −  2y +  2y−−  √ √ s3 z3 = −  2z3 +  2z3−− 

Lemma 18.2.5 Let p = 3 and i j = 0 1. Then N4 is s4 -invariant and N5 is not s5 -invariant. Proof By now you have either got the point or decided to skip these calculations altogether. Anyway, we proceed as above using formulas at the end of the proof of Lemma 18.2.4.

18.3 Higher crystal operators Lemma 18.3.1 Let i j ∈ I with i = j. For any r s ≥ 0 with r + s = −hi  j , and m ≥ 0, ind Lir  j is   Lim   ind Lim   Lir  j is  is irreducible. Proof We may assume that i − j = ±1, as otherwise the result is immediate from Theorem 18.1.4. Now, we claim that ind Lir  j is   Lim   ind Lim   Lir  j is  Indeed, transitivity of induction and Theorem 18.2.1(ii) give that ind Lir  j is   Lim   ind Lir  j is   Li  Lim−1   ind Li  Lir  j is   Lim−1  and now repeating this argument m − 1 more times gives the claim. Hence, by Corollary 17.3.2 and Theorem 14.7.1, K = ind Lir  j is   Lim 

18.3 Higher crystal operators

217

is self-dual. Now suppose for a contradiction that K is reducible. Then we can pick a proper irreducible submodule S of K, and set Q = K/S. By Theorem 18.2.1(i) and the Shuffle Lemma, ch K equals m    m r + t!s + m − t!Lir+t  Lj  Lis+m−t  t t=0 By Frobenius reciprocity, Q contains an r+s+1m -submodule isomorphic to Lir  j is   Lim  Hence by Lemma 18.1.1(i), the irreducible r+s+m+1 supermodule V = Lir  Lj  Lis+m appears in Q as a submodule. It now follows from Theorem 16.3.3(i) and the above formula for ch K that V must appear with multiplicity r!s + m! (viewing Q as a module over r1s+m ). It follows that V is not a composition factor of S. But this is a contradiction, since, as K is self-dual, S  S  is a quotient module of K and hence must contain V by the argument above applied to the quotient Q = S. Lemma 18.3.2 Let i j ∈ I with i = j, and m be a non-negative integer. For any r ≥ 1, s ≥ 0 with r + s = −hi  j  and any irreducible supermodule M ∈ RepI n , hd ind M  Lim   Lir  j is  is irreducible. Proof By the argument as in the proof of Lemma 5.1.5, it suffices to prove this in the special case that
i M = 0. Let t = m + r + s + 1. Recall from the previous lemma that N = ind Lim   Lir  j is  is an irreducible t -supermodule. Moreover by Theorem 18.2.1(i), ch Lir  j is  = r!s!Lir  Lj  Lis  So, since
i M = 0 and r > 0, the Mackey Theorem and a block argument imply that m r s resn+t nt ind M  Li   Li  j i   M  N  ⊕ U

for some nt -module U all of whose composition factors lie in different blocks to those of M  N . Now let H = hd ind M  Lim   Lir  j is 

218

Character calculations for n

It follows from above that ¯ resn+t nt H  M  N  ⊕ U where U¯ is some quotient module of U . Then: Homn+t H H  Homn+t ind M  Lim   Lir  j is  H  Homnt M  N resn+t nt H  Homnt M  N M  N ⊕ U¯   Homnt M  N M  N  Since H is completely reducible and M  N is irreducible, this implies that H is irreducible, as required. Let i j ∈ I with i = j, and r ≥ 1 s ≥ 0 satisfy r + s = −hi  j . Then the special case m = 0 of Lemma 18.3.2 shows that f˜ir jis M = hd ind M  Lir  j is  is irreducible for every irreducible M ∈ RepI n . Lemma 18.3.3 Take i j ∈ I with i = j and set k = −hi  j . Let M ∈ RepI n be irreducible. (i) There exists a unique integer r with 0 ≤ r ≤ k such that for every m ≥ 0 we have
i f˜im f˜j M = m +
i M − r (ii) Assume m ≥ k. Then a copy of f˜im f˜j M appears in the head of ind f˜im−k M  Lir  j ik−r  where r is as in (i). In particular, if r ≥ 1, then f˜im f˜j M  f˜ir jik−r f˜im−k M Proof The proof is similar to that of Lemma 6.3.3, using (17.12), Theorem 18.2.1(i), Lemma 18.3.1, Lemma 17.1.5, and Lemma 18.3.2 instead of (5.12), Lemma 6.2.1, Lemma 6.3.1, Lemma 5.1.5, and Lemma 6.3.2, respectively.

19 Operators ei and fi

This chapter is more or less parallel to Chapter 8. Some additional complications arise because the dimension of the 1 -supermodule Li is 2 rather than 1, and so sometimes we need to “halve” resi and indi . Whether we need to halve them or not depends on the properties of the operation . Note also that our “halving” procedure is not functorial and so ei and fi end up being not functors, but rather functions from irreducible n -supermodules to   and n+1 , respectively. isomorphism classes of supermodules over n−1

19.1 i-induction and i-restriction Fix  ∈ P+ . Recall the functors i from (17.1) and resi from (17.6) for i ∈ I.  Note if M is an n -supermodule, then resi M is automatically an n−1 supermodule. So the restriction of the functor resi gives a functor  -smod  resi  n -smod → n−1

There is an alternative definition of resi :  for some fixed  = j∈I j j ∈ #n

 resn M − i  n−1 resi M = 0

n -smod

(19.1)

if M is a supermodule in then if i > 0, if i = 0.

(19.2)

This description makes it clear how to define an analogous (additive) functor  indi  n -smod → n+1 -smod 

(19.3)

Using (16.5) and additivity, it suffices to define this on an object M belonging  to n -smod for fixed  = j∈I j j ∈ #n . Then we set 

indi M = indn+1  M + i  n

219

(19.4)

Operators ei and fi

220

To complete the definition of the functor indi , it is defined on a morphism f 

simply by restriction of the corresponding morphism indn+1  f . We stress that n

the functor indi depends fundamentally on the fixed choice of , unlike resi , which is just the restriction of its affine counterpart resi . Lemma 19.1.1 For  ∈ P+ and each i ∈ I: (i) indi and resi are both left and right adjoint to each other, hence they are exact and send projectives to projectives; (ii) indi and resi commute with duality, that is there are natural isomorphisms indi M    indi M 

resi M    resi M

for each finite dimensional n -supermodule M. Proof Similar to the proof of Lemma 8.2.2 using Corollary 15.6.5 instead of Corollary 7.7.5. By (17.8) and Lemma 19.1.1, we have that 

resn M = n−1

 i∈I

resi M



indn+1  M = n



indi M

(19.5)

i∈I

In order to give an alternative description of resi and indi recall the 1 -supermodules m i from (16.7). Let 1  1 denote the subalgebra of n generated by cn  xn , see (14.10). The limits in the next lemma are taken with respect to the systems induced by the maps (16.8). Lemma 19.1.2 For every finite dimensional n -supermodule M and i ∈ I, there are natural isomorphisms indi M  lim pr  indn+1 n1 M
m i ← − resi M  lim pr  Hom1 m i M − → (in the second case, the n−1 -module structure is defined by hf r = hfr for f ∈ Hom1 m i M and r ∈ m i). Proof For resi , it suffices to consider the effect on M ∈ n -smod for   = j∈I j j ∈ #n with i > 0, both sides of what we are trying to prove clearly being zero if i = 0. Then, for all sufficiently large m, Lemma 16.4.2 (in the special case where n = 1) implies that 

Hom1 m i M  resn M − i  n−1

19.2 Operators ei and fi

221

Hence, 

lim pr  Hom1 m i M  resn M − i = resi M − → n−1 To deduce the statement about induction, it now suffices by uniqueness of adjoint functors to show that lim pr  indn+1 n1 ?
m i is left adjoint ← −  -smod and M ∈ n -smod. First to lim pr  Hom1 m i ? Let N ∈ n−1 − → observe as explained in the previous paragraph that the direct system pr  Hom1 1 i M $→ pr  Hom1 2 i M $→    stabilizes after finitely many terms. We claim that the inverse system n+1  pr  indn+1 n1 N
1 i  pr ind n1 N
2 i    

also stabilizes after finitely many terms. To see this, it suffices to show that n+1 the dimension of pr  indn1 N
m i is bounded above independently of m. Well, each m i is generated as an 1 -module by a subspace W isomorphic (as a vector space) to the head 1 i of m i. Then indn+1 n1 N
m i is generated as an n+1 -supermodule by the subspace W  = 1 ⊗ N ⊗ W, also of dimension independent of m. Finally, pr  indn+1 n1 N
m i is a quotient of  ⊗F W  , whose dimension is independent of m. the vector space n+1 Now we can complete the proof of adjointness. Using the fact from the previous paragraph that the direct and inverse systems stabilize after finitely many terms, we have natural isomorphisms Homn lim pr  indnn−11 N
m i M ← −  lim Homn pr  indnn−11 N
m i M − →  lim Homn indnn−11 N
m i M − →  lim Homn−11 N
m i resnn−11 M − →  lim Homn−1 N Hom1 m i M − →   N pr Hom   m i M  lim Homn−1 1 − →   N lim pr Hom   m i M Homn−1 1 − → This completes the argument.

19.2 Operators ei
and fi
We wish to refine the functors resi (resp. indi ) to give operators, denoted ei (resp. fi ) from irreducible n -supermodules to isomorphism classes of   n−1 - (resp. n+1 -) supermodules.

Operators ei and fi

222

Actually, ei is the restriction of an operator denoted ei on the irreducible supermodules in RepI n , which we define first. Recall the definition of Lm i from Section 16.4. Let M ∈ RepI n be irreducible. For each m ≥ 1, we define an n−1 -supermodule Hom1 Lm i M

(19.6)

as follows. If M is of type M or i = 0, this is simply Hom1 Lm i M viewed as an n−1 -module in the same way as in Lemma 19.1.2. But if M is of type Q and i = 0, we can pick an odd involution M  M → M and also have the odd involutions m  Lm i → Lm i from (16.10). Let M

⊗

m

 Hom1 Lm i M → Hom1 Lm i M ¯

denote the map defined by  M ⊗ m f v = −1f M f m v It is easy to check that  M ⊗ m 2 = 1, hence the ±1-eigenspaces of M ⊗ m split Hom1 Lm i M into a direct sum of two isomorphic n−1 -supermodules (there is an obvious odd automorphism swapping the two eigenspaces). Now in this case, we define Hom1 Lm i M to be the 1-eigenspace (say). In either case, we have a direct system Hom1 L1 i M $→ Hom1 L2 i M $→    induced by the inverse system (16.9). Now define ei M = lim Hom1 Lm i M − →

(19.7)

 Note if M is an n -supermodule then each Hom1 Lm i M is an n−1 supermodule, so

ei M = lim pr  Hom1 Lm i M − →

(19.8)

We take (19.8) as our definition of the operator ei in the cyclotomic case. Comparing (19.7) and (19.8) with Lemma 19.1.2 and using (16.12), we get: Lemma 19.2.1 Let i ∈ I and M ∈ RepI n or M ∈ n -smod be an irreducible supermodule. Then

 if i = 0 and M is of type M , e M  resi M  i  ei M ⊕ "ei M otherwise The action of ei and ei on the level of characters is similar to the classical case (cf. (5.9)):

19.2 Operators ei and fi

223

Corollary 19.2.2 Let M ∈ RepI n or M ∈ n -smod be an irreducible supermodule, and  ch M = ci Li1   · · ·  Lin   i∈I n

Then 

ch ei M =



ci1 in−1 i Li1   · · ·  Lin−1  

i∈I n−1

Now we turn to the definition of fi M which, just like indi M, only makes sense in the cyclotomic case. Let M be an irreducible n -supermodule. We need to extend the definition of the operation  to give meaning to the notation M  Lm i, for each m ≥ 1. If either M is of type M or i = 0, then M  Lm i = M
Lm i. But if M is√of type Q and i = 0, pick an odd involution M  M → M. Then the ± −1-eigenspaces of M ⊗ m acting on the left of M
Lm i split it into a direct √ sum of two isomorphic n1 supermodules. Let M  Lm i denote the −1-eigenspace (say) for each m. We then have an inverse system M  L1 i  M  L2 i     of n1 -supermodules induced by the maps from (16.9). Now we can define fi M = lim pr  indn+1 (19.9) n1 M  Lm i ← − Comparing the definition with the proof of Lemma 19.1.2, we see that the inverse limit stabilizes after finitely many steps, hence that fi M is a well -supermodule. In fact, Lemma 19.1.2 and defined finite dimensional n+1 (16.12) imply: Lemma 19.2.3 Let i ∈ I and M be an irreducible n -supermodule. Then

if i = 0 and M is of type M , f M  indi M  i fi M ⊕ "fi M otherwise Lemma 19.2.4 Let i ∈ I and M ∈ RepI n be irreducible. Then ei M is nonzero if and only if e˜ i M = 0, in which case it is a self-dual indecomposable supermodule with irreducible socle and head isomorphic to e˜ i M. Proof To see that ei M has irreducible socle e˜ i M whenever it is non-zero, combine Lemma 19.2.1 with Corollary 17.1.8. The remaining facts follow since M is self-dual by Lemma 17.3.2, and resi commutes with duality by Lemma 19.1.1(ii).

224

Operators ei and fi

Now we define the cyclotomic crystal operators on irreducible supermodules e˜ i = pr   e˜ i  infl 

(19.10)

f˜i = pr   f˜i  infl

(19.11)

for each i ∈ I and  ∈ P+ . Let B and B denote the set of isomorphism classes of irreducible supermodules in RepI n and n -smod for all n ≥ 0, respectively. Note that we may consider B as a subset of B via inflation. We have maps e˜ i  B → B ∪ 0  f˜i  B → B e˜ i  f˜i  B → B ∪ 0  The reader is advised to consult with Remark 8.2.4 at this point. Theorem 19.2.5 Let  ∈ P+ and i ∈ I. Then for any irreducible n -supermodule M: (i) ei M is non-zero if and only if e˜ i M = 0, in which case it is a selfdual indecomposable supermodule with irreducible socle and head isomorphic to e˜ i M; (ii) fi M is non-zero if and only if f˜i M = 0, in which case it is a selfdual indecomposable supermodule with irreducible socle and head isomorphic to f˜i M. Proof (i) This is immediate from Lemma 19.2.4.  -supermodule. Let M equal 1 if i = 0 and M (ii) Let N be an irreducible n+1 is of type M, 2 otherwise, and define N similarly. Then, by Lemmas 19.1.1(i), 19.2.1, and 19.2.3,   f M N  = dim Homn+1 i

1   ind M N  dim Homn+1 i M

=

1 dim Homn M resi N  M

=

N dim Homn M ei N  M

By (i), the latter is zero unless M = e˜ i N , or equivalently N = f˜i M by Lemma 17.2.2. Taking into account Lemmas 12.2.3 and 17.3.4, we deduce

19.3 Divided powers

225

that hd fi M  f˜i M. Finally, note fi M is self-dual by Lemma 19.1.1(ii) so everything else follows. Remark 19.2.6 Let us also point out, as follows easily from the definitions, that ei M and fi M admit odd involutions if either i = 0 and M is of type Q, or i = 0 and M is of type M.

19.3 Divided powers We recall the covering modules Lm ir  from Section 16.4 and the embedding (14.10). Let M be an irreducible supermodule in RepI n . If r > n, we set r ei M = 0. Otherwise, we have a direct system Homr L1 ir  M $→ Homr L2 ir  M $→    induced by the inverse system (16.9), where Hom is defined in exactly the same way as in Section 19.2 using the generalized maps m from (16.10) in case i = 0. Now define r

ei M = lim Homr Lm ir  M − →

(19.12) r

As in Section 19.2, if M is an irreducible n -supermodule then ei M is an  -supermodule, so that n−r r

ei M = lim pr  Hom1 Lm ir  M (19.13) − → We take (19.13) as our definition of ei r in the cyclotomic case. To define fi r , which as usual only makes sense in the cyclotomic case, let M be an irreducible n -supermodule. We have an inverse system M  L1 ir   M  L2 ir      of nr -supermodules induced by the maps from (16.9), again interpreting  as in Section 19.2. Now define r fi r M = lim pr  indn+r nr M  Lm i  ← −

(19.14)

Lemma 19.3.1 Let i ∈ I r ≥ 1 and M be an irreducible supermodule in n -smod. Then resi r M is evenly isomorphic to ⎧ r−1 ⎪ ei r M ⊕ "ei r M⊕2 r! if i = 0, ⎪ ⎪ ⎪ ⎨e r M ⊕2r−1/2 r! if i = 0, r is odd, and i ⎪ M is of type M , ⎪ ⎪ ⎪ ⎩  r  r ⊕2)r−1/2* r! ei  M ⊕ "ei  M otherwise;

Operators ei and fi

226

and indi r M is evenly isomorphic to ⎧  r  r ⊕2r−1 r! ⎪ if i = 0, ⎪fi  M ⊕ "fi  M ⎪ ⎪ ⎨f  r M ⊕2r−1/2 r! if i = 0, r is odd, and i ⎪ M is of type M , ⎪ ⎪ ⎪ ⎩  r  r ⊕2)r−1/2* r! fi  M ⊕ "fi  M otherwise. Proof Using Lemma 16.4.1 and the definitions, it suffices to show that resi r M  lim pr  Homr m ir  M − → r indi r M  lim pr  indn+r nr M
m i  ← −

For resi r , this follows from Lemma 16.4.2 in exactly the same way as in the proof of Lemma 19.1.2. Now indi r is left adjoint to resi r , so the statement for induction follows from uniqueness of adjoint functors on checking that r  r the functor lim pr  indn+r nr ?
m i  is left adjoint to lim pr Hom r m i  ? − → ← − The latter follows as in the proof of Lemma 19.1.2. r



Since we have defined the operators ei and ei r on irreducible superr  modules we get induced operators also denoted ei and ei r at the level of Grothendieck groups, namely, r

ei  KRepI n  → KRepI n−r  and  ei r  Kn -smod → Kn−r -smod

in the affine and cyclotomic cases respectively. Similarly fi r induces an operator  -smod fi r  Kn -smod → Kn+r

on Grothendieck groups. Lemma 19.3.2 As operators on the Grothendieck groups, we have that eir = r r!ei , ei r = r!ei r and fi r = r!fi r . Proof If i = 0, by Lemmas 19.2.1 and 19.2.3, we have that resi r = 2r ei r  and by Lemma 19.3.1, we have that resi r = 2r r!ei r  as operators on the Grothendieck group. The result for e follows, the proof for f being similar. Now, let i = 0. The idea is the same here as for i = 0, but we have to check several cases. We explain one of them. Let r be odd and M ∈ n -smod be

19.3 Divided powers

227

irreducible of type M. By Lemmas 19.2.1, 19.2.3, 19.3.1, and 17.3.4 we get resi r M = 2r−1/2 ei r M = 2r−1/2 r!ei r M  and the result for e follows in this case. Let us finally note that we have only defined the operators ei r and fi r on irreducible supermodules. However, the definitions could be made more generally on pairs M M , where M is an n -supermodule (or an integral n -supermodule in the case of ei ) and M  M → M is either the identity map or else an odd involution of M. In case M = idM , the definitions of ei r M and fi r M are exactly the same as in the case where M is irreducible of type M above. In the case where M is an odd involution, the definitions of ei r M and fi r M are exactly the same as in the case where M is irreducible of type Q above, substituting the given map M for the canonical odd involution of M in the situation above. This remark applies especially to give us ei r PM  fi r PM , where PM is the projective cover of an irreducible n -supermodule: in this case, if M is of type Q, the odd involution M of M lifts to a unique odd involution also denoted M of the projective cover, see Lemma 12.2.16. On doing this, we have that ei r PM = ei r PM 

fi r PM = fi r PM

(19.15)

  where the equalities are written in Kn−r -smod and Kn+r -smod respectively. To prove this, we need to observe that all composition factors of PM are of the same type as M by Lemma 17.3.4. Note Lemma 19.3.1 is also true if M is replaced by its projective cover PM , the proof being the same as above. In particular, this shows that ei r PM is a summand of resi r PM , and similarly for fi . So Lemma 19.1.1(i) gives that ei r PM and fi r PM are also projective modules. Hence ei r and fi r induce operators with the same names on the Grothendieck groups of projective modules too:  -proj ei r  Kn -proj → Kn−r  fi r  Kn -proj → Kn+r -proj

Moreover, by the same argument as in the proof of Lemma 19.3.2, we have: Lemma 19.3.3 As operators on the Grothendieck group Kn -proj, we have that ei r PM = r!ei r PM  for all irreducible n -modules M.

fi r PM = r!fi r PM 

228

Operators ei and fi

19.4 Alternative descriptions of i Theorem 19.4.1 Let i ∈ I and M be an irreducible supermodule in RepI n , Then:  (i) ei M =
i M˜ei M + ca Na , where the Na are irreducibles with
i Na  <
i ˜ei M; (ii)
i M is the maximal size of a Jordan block of xn2 (resp. xn if i = 0) on M with eigenvalue qi; (iii) Endn−1 ei M  Endn−1 ˜ei M⊕
i M as vector superspaces. Proof Let
=
i M. (i) The proof is similar to that of Theorem 5.5.1(i), except that we use  and Lemmas 17.1.4, and 17.1.3 instead of
and Lemmas 5.1.4, and 5.1.3, respectively, to deduce  i M =
˜ei M  Li + ca Na  Li for irreducibles Na with
i Na  <
i ˜ei M. The conclusion follows on applying Lemma 19.2.1. (ii) Similar to the proof of Theorem 5.5.1(ii) but using Lemma 16.3.2, instead of Lemma 4.3.1. (iii) Similar to the proof of Theorem 5.5.1(iii), but more delicate, so we provide details. Let z = xn2 − qi if i = 0 and z = xn if i = 0. Consider the effect of left multiplication by z on the n−1 -supermodule R = resi M Note R is equal to either ei M or ei M ⊕ "ei M, by Lemma 19.2.1. In the latter case, xn2 (resp. xn ) acts as a scalar on soc R  e˜ i M ⊕ "˜ei M, hence it leaves the two indecomposable summands invariant. This shows that in any case left multiplication by z (which centralizes the subalgebra n−1 of n ) induces an n−1 -endomorphism  ei M → ei M But by (ii),
−1 = 0 and
= 0. Hence, 1     
−1 give
linearly independent even n−1 -endomorphisms of ei M. In view of Remark 19.2.6, we automatically get from these
linearly independent odd endomorphisms in the case where e˜ i M is of type Q, so we have now shown that dim Endn−1 ei M ≥
dim Endn−1 ˜ei M However, ei M has irreducible head e˜ i M, and this appears in ei M with multiplicity
by (i), so the reverse inequality also holds.

19.6 Functions i

229

Corollary 19.4.2 Let M N be irreducible n -supermodules with M  N . Then, for every i ∈ I, Homn−1 ei M ei N  = 0 Proof Similar to the proof of Corollary 5.5.2.

19.5 The *-operation Suppose M is an irreducible supermodule in RepI n and 0 ≤ m ≤ n. Using Lemma 14.6.1 for the second equality in (19.17), define e˜ i∗ M = ˜ei M   

(19.16)

f˜i∗ M = f˜i M   = hd indn+1 1n Li  M

(19.17)


∗i M =
i M   = max m ≥ 0  ˜ei∗ m M = 0 

(19.18)

We may think of the “starred” notions as left-hand versions of the original notions which are right hand. For example,
∗i M can be worked out just from knowledge of the character of M as the maximal k such that Lik     appears in ch M, while for
i M we would take · · ·  Lik here. Recalling the definition (15.7) of pr  , Theorem 19.4.1(ii) has the following important corollary: Corollary 19.5.1 Let M be an irreducible supermodule in RepI n and  ∈ P+ . Then pr  M = M if and only if
∗i M ≤ hi   for all i ∈ I. Proof In view of Theorem 19.4.1(ii),
∗i M is the maximal size of a Jordan block of x12 (resp. x1 ) on M with eigenvalue qi (resp. 0) if i = 0 (resp. i = 0). The result follows immediately.

19.6 Functions i
Let M be an irreducible n -supermodule. Define
i M = max m ≥ 0  ˜ei m M = 0 

(19.19)

i M = max m ≥ 0  f˜i m M = 0 

(19.20)

As e˜ i is simply the restriction of e˜ i , we have
i M =
i M, see (17.11). However, the integer i M depends on the fixed choice of . ¯ and let 1
 F Recall that 0 is interpreted as F (concentrated in degree 0), denote the irreducible 0 -supermodule.

Operators ei and fi

230

Lemma 19.6.1
i 1
 = 0 and i 1
 = hi  . Proof Similar to the proof of Lemma 8.4.1 but uses Corollary 19.5.1, instead of Corollary 7.4.1. Lemma 19.6.2 Let i j ∈ I with i = j and M be an irreducible supermodule in RepI n . Then
∗j f˜im M ≤
∗j M for every m ≥ 0. Proof Follows from the Shuffle Lemma. Lemma 19.6.3 Let i j ∈ I with i = j. Let M ∈ n -smod be irreducible with j M > 0. Then i f˜j M −
i f˜j M ≤ i M −
i M − hi  j  Proof Similar to the proof of Lemma 8.4.3 but uses Lemma 18.3.3, Corollary 19.5.1, Lemma 19.6.2, Lemma 18.3.3(ii), and Theorem 18.2.1(i) instead of Lemma 6.3.3, Corollary 7.4.1, Lemma 8.4.2, Lemma 6.3.3(ii), and Lemma 6.2.1, respectively. Corollary 19.6.4 Let  ∈ P+ and M be an irreducible n -supermodule with central character  for some  ∈ #n . Then i M −
i M ≤ hi   −  Proof Similar to the proof of Corollary 8.4.4.

19.7 Alternative descriptions of i
Let M be an irreducible n -supermodule. Recall that fi M = lim pr  indn+1 n1 M  Lm i ← −

indi M = lim pr  indn+1 n1 M
m i ← −

and that the inverse limits stabilize after finitely many terms. Define ˜ i M to be the stabilization point of the limit, i.e. the least m ≥ 0 such that fi M = pr  indn+1 n1 M  Lm i or equivalently indi M = pr  indn+1 n1 M
m i

19.7 Alternative descriptions of i

231

Lemma 19.7.1 Let M be an irreducible n -supermodule and i ∈ I. Then:   (i) fi M = ˜ i Mf˜i M + ca Na where the Na are irreducible n+1   ˜ supermodules with
i Na  <
i fi M;   ⊕˜ i M  f M  End   f˜ M (ii) Endn+1 as vector superspaces. i i n+1 Proof (i) Similar to the proof of Lemma 8.5.1(i). (ii) Take m = ˜ i M. We easily show using the explicit construction of Lm i in Section 16.4 that there is an even endomorphism  Lm i → Lm i of 1 -modules, such that the image of k is  Lm−k i for each 0 ≤ k ≤ m. Frobenius reciprocity induces superalgebra homomorphisms End1 Lm i $→ Endn1 M  Lm i $→ Endn+1 indn+1 n1 M  Lm i So induces an even n+1 -endomorphism ˜ of indn+1 n1 M  Lm i, such that the image of ˜ k is  indn+1 M  L i for 0 ≤ k ≤ m. Now apply the right m−k n1  -endomorphism exact functor pr  to get an even n+1 ˆ  pr  indn+1 M  Lm i → pr  indn+1 M  Lm i n1 n1 induced by ˜ . Note ˆ m = 0 and ˆ m−1 = 0 because its image coincides with the image of the non-zero map m in the proof of (i). Hence, 1 ˆ      ˆ m−1 are linearly independent, even endomorphisms of fi M. Now the proof of (ii) is completed in the same way as in the proof of Theorem 19.4.1(iii). Corollary 19.7.2 Let M N be irreducible n -supermodules with M  N .    f M f N  = 0 Then, for every i ∈ I, Homn+1 i i Proof Similar to the proof of Corollary 8.5.2. As in the classical case we want to prove that ˜ i M = i M. Note right away from the definitions that for any irreducible n -supermodule M we have ˜ i M = 0 if and only if i M = 0. Lemma 19.7.3 If M is an irreducible n -supermodule then ˜ 0 M −
0 M + 2

  i=1

˜ i M −
i M = c 

Operators ei and fi

232

Proof By Frobenius reciprocity and Theorem 15.5.2, we have that 



n

n



n+1 n+1 n+1  ind  M Hom   M res  ind  M Endn+1    n n

Homn M M ⊕ "M

⊕c





⊕ Homn M indn resn M n−1

n−1

Homn M M ⊕ "M

⊕c



n  res  M ⊕ Endn−1  n−1

Hence, by Schur’s Lemma, 



n+1 n  ind  M − dim End   res  M dim Endn+1 n n−1 n−1

2c  if M is of type M, = 4c  if M is of type Q.

Now if M is of type M, then 

 indn+1  M  f0 M ⊕ n



resn M  e0 M ⊕ n−1

 

fi M ⊕ "fi M

i=1   ei M ⊕ "ei M i=1

by Lemmas 19.2.1 and 19.2.3 and (19.5). Hence by Lemma 19.7.1 (ii) and Theorem 19.4.1(iii), 

n+1  ind  M = 2 dim Endn+1 ˜ 0 M + 4  n



 n  res  M = 2
M + 4 dim Endn−1 0  n−1

 

˜ i M

i=1  


i M

i=1

and the conclusion follows in this case. The argument for M of type Q is similar. Lemma 19.7.4 Let M be an irreducible n -supermodule and i ∈ I. Let ci = 1 if i = 0, ci = 2 otherwise. Then resi indi M  M = 2ci
i f˜i M˜ i M indi resi M  M = 2ci
i M˜ i ˜ei M soc resi indi M  M ⊕ "M⊕ci ˜ i M  

soc indi resi M  M ⊕ "M⊕ci
i M  

19.7 Alternative descriptions of i

233

Proof The statement about composition multiplicities follows from Theorem 19.4.1(i) and Lemma 19.7.1(i), taking into account how resi and indi are related to ei and fi as explained in Lemmas 19.2.1 and 19.2.3. Now consider the statement about socles. We consider only resi indi M, the other case being entirely similar but using results from Section 19.4 instead. By adjointness, it suffices to be able to compute    ind N ind M Homn+1 i i

for any irreducible n -supermodule N . But in view of Lemma 19.2.3, this    f N f M, which is known can be computed from knowledge of Homn+1 i i by Corollary 19.7.2 (if M  N ) and Lemma 19.7.1(ii) (if M  N ). The details are similar to those in the proof of Lemma 19.7.3, so we omit them. Lemma 19.7.5 Let M be an irreducible n -supermodule and i ∈ I. There are maps 

can

indi resi M −→ resi indi M −→ resi indi M/soc resi indi M whose composite is surjective. Proof Let k = ˜ i M and n+1     indn+1 n1 M
k i  pr ind n1 M
k i = ind i M

be the quotient map. Set z = xn2 − qi (resp. z = xn if i = 0). Recall from Section 16.4 that viewed as an 1 -module, we have that k i  1 /zk . In particular, k i is a cyclic module generated by the image 1˜ of 1 ∈ 1 . We first observe that for any m ≥
i M + k, zm annihilates the vector sn ⊗ u ⊗ v ∈ indn+1 n1 M
k i for any u ∈ M v ∈ k i. This follows from the relations in n+1 , for example in the case where i = 0 we ultimately appeal to the facts that xn2 − qi
i M 2 annihilates u (see Theorem 19.4.1(ii)) and xn+1 − qik annihilates v.  Therefore, for any m ≥
i M + k, the following equality holds in fi M:
 zm  sn ⊗ u ⊗ v = 0 (19.21) Next, it is not difficult to check that there exists a unique n−11 -homomorphism ˜ resi M
1 → resnn−11 resi indi M u ⊗ 1 → sn ⊗ u ⊗ 1

Operators ei and fi

234

for each u ∈ resi M ⊆ M. It follows from (19.21) that this homomorphism factors to induce a well-defined n−11 -module homomorphism resi M
m i → resnn−11 resi indi M We then get from Frobenius reciprocity an induced map m  indnn−11 resi M
m i → resi indi M

(19.22)

for each m ≥
i M + k. Each m factors through the quotient pr  indnn−11 resi M
m i so we get an induced map   indi resi M = lim pr  indnn−11 resi M
m i ← − → resi indi M = resi indi M It remains to show that the composite of  with the canonical epimorphism from resi indi M to resi indi M/soc resi indi M is surjective. By Mackey Theorem there exists an exact sequence n+1 indn+1 0 → M
k i → resn1 n1 M
k i n sn → indn1 n−111 resn−11 M
k i → 0

In other words, there is an even n1 -isomorphism from n sn indn1 n−111 resn−11 M
k i

to n+1 resn+1 n1 ind n1 M
k i/M
k i

which maps h ⊗ u ⊗ v → hsn ⊗ u ⊗ v + M
k i for h ∈ n  u ∈ M v ∈ k i, where M
k i is embedded into resn+1 n1 indn+1 n1 M
k i as 1 ⊗ M ⊗ k i. Recall from (16.11) that dim k i = 2kci , where ci is as in Lemma 19.7.4. Hence, applying Lemma 19.7.4, we get kci  soc resi indi M resn1 n M
k i  M ⊕ "M

So applying the exact functor resi to the isomorphism above we get an isomorphism ∼

  indnn−11 resi M
k i → resi indn+1 n1 M
k i/soc resi ind i M

h ⊗ u ⊗ v → hTn ⊗ u ⊗ v + soc resi indi M

19.7 Alternative descriptions of i

235

It follows that there is a surjection  indnn−11 resi M
k i  resi indi M/soc resi indi M such that the diagram m

indnn−11 resi M
m i −−−−→ ⏐ ⏐ "

resi indi M ⏐ ⏐can "

indnn−11 resi M
k i −−−−→ resi indi M/soc resi indi M commutes for all m ≥
i M + k, where m is the map from (19.22) and the left-hand arrow is the natural surjection. Now surjectivity of immediately implies surjectivity of can m and of can . Lemma 19.7.6 Let M be an irreducible n -supermodule with
i M > 0. Then ˜ i ˜ei M = ˜ i M + 1 Proof Let us first show that ˜ i ˜ei M ≥ ˜ i M + 1

(19.23)

Recall i M = 0 if and only if ˜ i M = 0. Suppose first that i M = 0. Then i ˜ei M = 0 in view of Lemma 17.2.2. Now, ˜ i M = 0 and ˜ i ˜ei M = 0, so the conclusion certainly holds in this case. Next, assume that i M > 0, hence ˜ i M > 0. Note by Lemma 19.7.4, resi indi M/soc resi indi M  M = 2ci
i f˜i M˜ i M − 2ci ˜ i M = 2ci
i M˜ i M = 0 In particular, the map  in Lemma 19.7.5 is non-zero. So Lemma 19.7.5 implies that im   M > 2ci
i M˜ i M since at least one composition factor of soc im  ⊆ soc resi indi M must be sent to zero on composing with the second map can. Using another part of Lemma 19.7.4, this shows that 2ci
i M˜ i ˜ei M > 2ci
i M˜ i M and (19.23) follows.

Operators ei and fi

236

Now using (19.23) and Lemma 19.7.4, we see that in the Grothendieck group, resi indi M − indi resi M  M ≤ 2ci ˜ i M −
i M with equality if and only if equality holds in (19.23). By central character considerations, for i = j, resi indj M  M = indj resi M  M = 0 So using (19.5) we deduce that 



n

n





n+1 n n resn+1  ind   M − ind   res  M  M n−1

n−1

≤ 2˜ 0 M −
0 M + 4

  ˜ i M −
i M i=1

with equality if and only if equality holds in (19.23) for all i ∈ I. Now Lemma 19.7.3 shows that the right-hand side equals 2c , which does indeed equal the left-hand side, thanks to Theorem 15.5.2. Corollary 19.7.7 For any irreducible M ∈ n -smod, i M = ˜ i M. Proof Similar to the proof of Corollary 8.5.7. Lemma 19.7.8 Let M be an irreducible n -supermodule with central character  for  ∈ #n . Then i M −
i M = hi   −  Proof In view of Corollary 19.6.4, it suffices to show that 0 M −
0 M + 2

 

i M −
i M = c 

i=1

But this is immediate from Lemma 19.7.3 and Corollary 19.7.7. Theorem 19.7.9 Let i ∈ I and M be an irreducible n -supermodule. Then:  (i) fi M = i Mf˜i M + ca Na where the Na are irreducibles with i Na  < i f˜i M = i M − 1; (ii) i M is the least m ≥ 0 such that fi M = pr  indn+1 n1 M  Lm i;   ⊕i M ˜ as vector superspaces. (iii) End fi M  End fi M n+1

n+1

Proof Similar to the proof of Theorem 8.5.9.

19.7 Alternative descriptions of i

237

Lemma 19.7.10 Let  ∈ P+ , i ∈ I, r ≥ 1, and M be an irreducible n supermodule: (i) ei r M is non-zero if and only if ˜ei r M = 0. (ii) fi r M is non-zero if and only if f˜i r M = 0. Proof Apply Lemma 19.3.2 and Theorems 19.4.1(i), 19.7.9(i).

20 Construction of U+ and irreducible modules

This chapter is parallel to Chapter 9.

20.1 Grothendieck groups revisited Let us write K =



KRepI n 

K =  ⊗ K

(20.1)

n≥0

Thus K is a free -module with canonical basis given by B, the isomorphism classes of irreducible supermodules (see Section 19.2 for this notation), and K is the -vector space on basis B. We let K∗ denote the restricted dual of K, namely, the set of functions f  K →  such that f vanishes on all but finitely many elements of B. Thus K∗ is also a free -module, with canonical basis M  M ∈ B

dual to the basis B of K, that is M M  = 1, M N  = 0 for N ∈ B with N  M. Finally, we write B∗ =  ⊗ B∗ . Entirely similar definitions can be made for each  ∈ P+ :  K = Kn -smod (20.2) n≥0

denotes the Grothendieck groups of the categories n -smod for all n. Again, K is a free -module on the basis B of isomorphism classes of irreducible supermodules. Moreover, infl induces a canonical embedding infl  K $→ K and infl  B $→ B We will generally identify K with its image under this embedding. We also define K∗ and K =  ⊗ K as above. 238

20.2 Hopf algebra structure

239

Recall the operators ei and more generally the divided power operators r ei for r ≥ 1, defined on irreducible supermodules in RepI n in (19.7) and (19.12) respectively. These induce linear maps r

ei  K → K

(20.3)

for each r ≥ 1. Similarly, the operators ei r and fi r from (19.13) and (19.14) respectively induce maps ei r  fi r  K → K

(20.4)

Recall by Lemma 19.3.2 that r

eir = r!ei 

ei r = r!ei r  fi r = r!fi r 

(20.5)

r

Extending scalars, the maps ei  ei r  fi r induce linear maps on K and K .

20.2 Hopf algebra structure Now we wish to give K the structure of a graded Hopf algebra over . To do this, recall the canonical isomorphism KRepI m  ⊗ KRepI n  → KRepI mn 

(20.6)

from (12.21), for each m n ≥ 0. The exact functor indm+n mn induces a welldefined map indm+n mn  KRepI mn  → KRepI m+n  Composing with the isomorphism (20.6) and taking the direct sum over all m n ≥ 0, we obtain a homogeneous map %  K ⊗ K → K

(20.7)

By transitivity of induction, this makes K into an associative graded -algebra. By Corollary 17.3.2,  induces the identity map on K, so Theorem 14.7.1 implies that the multiplication % is commutative (in the usual unsigned sense). Moreover, there is a unit %   → K 1 → 1 ∈ KRepI 0  ⊂ K The exact functor resnn1 n2 induces a map resnn1 n2  KRepI n  → KRepI n1 n2 

(20.8)

240

Construction of U+ and irreducible modules

On composing with the isomorphism (20.6), we obtain maps nn1 n2  KRepI n  → KRepI n1  ⊗ KRepI n2  

n =

n1 +n2 =n

nn1 n2  KRepI n 



→

n1 +n2 =n

KRepI n1  ⊗ KRepI n2 

Now taking the direct sum over all n ≥ 0 gives a homogeneous map   K → K ⊗ K

(20.9)

Transitivity of restriction implies that  is coassociative, while the homogeneous projection on to KRepI 0    gives a counit
 K → 

(20.10)

Thus K is also a graded coalgebra over . Theorem 20.2.1 K %  %
 is a commutative, graded Hopf algebra over . Proof It just remains to check that  is an algebra homomorphism, which follows using the Mackey Theorem. Note in checking the details, we need to use Lemma 17.3.4 to take the definition of  into account correctly. We record the following lemma explaining how to compute the action of ei on K explicitly in terms of : Lemma 20.2.2 Let M be a supermodule in RepI n . Write  nn−11 M = Ma ⊗ Na a

for irreducible n−1 -supermodules Ma and irreducible 1 -supermodules Na . Then  Ma  ei M = a with Na Li

Proof This is immediate from Lemma 19.2.1. Lemma 20.2.3 The operators ei  K → K satisfy the Serre relations, that is for i j ∈ I: ei ej = ej ei  if i − j > 1 ei2 ej

+ ej ei2 = 2ei ej ei  if i − j = 1 i = 0 j = 

20.3 Shapovalov form

241

e03 e1 + 3e0 e1 e02 = 3e02 e1 e0 + e1 e03  if  = 1 3 2 2 3 e−1 e + 3e−1 e e−1 = 3e−1 e e−1 + e e−1  if  = 1

e05 e1 + 5e0 e1 e04 + 10e03 e1 e02 = 10e02 e1 e03 + 5e04 e1 e0 + e1 e05  if  = 1 Proof In view of Lemma 20.2.2 and coassociativity of , this reduces to checking it on irreducible n -supermodules for n = 2 3 4 4 6 respectively. For this, the character information in Theorem 18.2.1 is sufficient. Next, K has a natural structure as K-comodule. The comodule structure map is the restriction of  to K ⊂ K:   K → K ⊗ K The dual maps to %  %
induce on K∗ the structure of a cocommutative graded Hopf algebra. Each K is a left K∗ -module in the natural way: ¯    . Similarly, f ∈ K∗ acts on the left on K as the map id ⊗f K is itself a left K∗ -module, indeed in this case the action is even faithful. r

Lemma 20.2.4 The operator ei acts on K (resp. ei r on K for any  ∈ P+ ) in the same way as the basis element Lir  of K∗ . Proof The proof is similar to the proof of Lemma 9.2.5, but uses (20.5) and Lemma 20.2.2 instead of (9.6) and Lemma 9.2.3, respectively. Lemma 20.2.5 There is a unique homomorphism   U+ → K∗ of graded r Hopf algebras such that ei → Lir  for each i ∈ I and r ≥ 1. Proof Similar to the proof of Lemma 9.2.6.

20.3 Shapovalov form Fix  ∈ P+ . For a finite dimensional n -supermodule M, we let PM denote its projective cover in the category n -smod. Since n is a finite dimensional superalgebra, we can identify  Kn -proj (20.11) K∗ = n≥0

Construction of U+ and irreducible modules

242

so that the basis element M corresponds to the isomorphism class PM for each irreducible n -supermodule M and each n ≥ 0, see Section 12.2. Moreover, under this identification, the canonical pairing    K∗ × K →  satisfies PM  N  =

dim Homn PM  N  1 2

dim Homn PM  N 

if M is of type M, if M is of type Q,

(20.12)

(20.13)

for n -supermodules M N with M irreducible (since the right-hand side clearly computes the composition multiplicity N  M ). There is a homogeneous map   K∗ → K

(20.14)

induced by the natural maps Kn -proj → Kn -smod for each n. As explained at the end of Section 19.3, we can define an action of ei r and fi r on the projective indecomposable supermodules, hence on K∗ . We know by Lemma 19.3.3 that (20.5) holds for the operations on K∗ as well as on K. Also, the actions of ei r and fi r commute with  by (19.15). Lemma 20.3.1 The operators ei  fi on K∗ and K satisfy ei x y = x fi y

fi x y = x ei y

for each x ∈ K∗ and y ∈ K. Proof Let M be an irreducible n -supermodule and N be an irreducible  -supermodule. We check that fi PM  N  = PM  ei N  in the spen+1 cial case that i = 0, M is of type Q and N is of type M. In this case, by Lemmas 19.2.3, 19.1.1(i), and 19.2.1, we have 1 1   ind PM  N  fi PM  N  = indi PM  N  = dim Homn+1 i 2 2 1 = dim Homn PM  resi N  = PM  resi N  2 = PM  ei N  All the other situations that need to be considered follow similarly.

20.3 Shapovalov form Corollary 20.3.2 Suppose  ei r M = aMN N 

243



fi r M =

N ∈B

bMN N 

N ∈B

for M ∈ B. Then ei r PN =



bMN PM 

fi r PN =

M ∈B



aMN PM

M ∈B

for N ∈ B. Lemma 20.3.3 Let M be an irreducible supermodule in n -smod. Set
=
i M,  = i M. Then, for any m ≥ 0,  ei m PM = aN P˜ei m N N with
i N ≥m

for coefficients aN ∈ ≥0 . Moreover, in case m =
,   
+ P˜ei 
M + aN P˜ei 
N  ei 
 PM =
N with
 N >
i

Proof Similar to the proof of Lemma 9.3.3, but using Corollary 20.3.2, Lemma 19.7.10(ii), (19.20), Lemma 19.7.1(i), and Theorem 19.7.9(i) instead of Corollary 9.3.2, Theorem 8.3.2(v), (8.17), Lemma 8.5.1(i), and Theorem 8.5.9(i), respectively. We also need: Theorem 20.3.4 Given an irreducible n -supermodule M, the element PM of Kn -proj can be written as an integral linear combination of terms of the form fi1 r1     fia ra  1
. Proof Similar to the proof of Theorem 9.3.4, but using Corollary 20.3.2 and Theorem 19.4.1(i) instead of Corollary 9.3.2 and Theorem 5.5.1(i), respectively. Theorem 20.3.5 The map   K∗ → K from (20.14) is injective. Proof Similar to the proof of Theorem 9.3.5, but using Lemma 20.3.3 instead of Lemma 9.3.3.

Construction of U+ and irreducible modules

244

In view of Theorem 20.3.5, we may identify K∗ with its image under , so K∗ ⊆ K are two different lattices in K . Extending scalars, the pairing (20.12) induces a bilinear form    K × K → 

(20.15)

with respect to which the operators ei and fi are adjoint. Theorem 20.3.6 The form    K × K →  is symmetric and non-degenerate. Proof Similar to the proof of Theorem 9.3.6 but using Theorem 20.3.4 instead of Theorem 9.3.4.

20.4 Chevalley relations Lemma 20.4.1 The operators ei  fi  K → K satisfy the Serre relations (7.5) for  as in Section 15.2. Proof Similar to the proof of Lemma 9.4.1. Now, for i ∈ I and an irreducible n -supermodule M with central character  for  ∈ #n , define hi M = hi   − M 

(20.16)

By Lemma 19.7.8 we have equivalently hi M = i M −
i MM  More generally, define      hi i M −
i M  K → K M → M  (20.17) r r
 Extending linearly, each hri can be viewed as a diagonal linear operator K → K. The definition (20.16) implies immediately that: Lemma 20.4.2 As operators on K, hi  ej = hi  j ej and hi  fj = −hi  j fj for all i j ∈ I.

20.4 Chevalley relations

245

Lemma 20.4.3 As operators on K, the relation ei  fj = ij hi holds for each i j ∈ I. Proof Let M be an irreducible n -supermodule. It follows immediately from Theorems 19.4.1(i) and 19.7.9(i) (together with central character considerations in case i = j) that M appears in ei fj M − fj ei M with multiplicity ij i M −
i M Therefore, it suffices simply to show that ei fj M − fj ei M is a multiple of M . Let us show equivalently that resi indj M − indj resi M is a multiple of M . For m 0, we have a surjection  indn+1 n1 M
m j  ind j M

Apply pr   resi to get a surjection   pr  resi indn+1 n1 M
m j  resi ind j M

(20.18)

By the Mackey Theorem and (16.12), there is an exact sequence 0 → M ⊕ "M⊕ij mcj → resi indn+1 n1 M
m j → indnn−11 resi M
m j → 0 where cj is as in Lemma 19.7.4. For sufficiently large m, we have pr  indnn−11 resi M
m j = indj resi M So on applying the right exact functor pr  and using the simplicity of M, this implies that there is an exact sequence 0 → M ⊕m1 ⊕ "M ⊕m2 → pr  resi indn+1 n1 M
m j → indj resi M → 0

(20.19)

for some m1  m2 . Now let N be any irreducible n -supermodule with N  M. Combining (20.18) and (20.19) shows that indj resi M − resi indj M  N ≥ 0

(20.20)

Now summing over all i j and using (19.5) gives that ind res M − res ind M  N ≥ 0 But Theorem 15.5.2 shows that equality holds here, hence it must hold in (20.20) for all i j ∈ I. This completes the proof.

246

Construction of U+ and irreducible modules

To summarize, we have shown in (20.5), Lemmas 20.4.1, 20.4.2, and 20.4.3 that: Theorem 20.4.4 The action of the operators ei  fi  hi on K satisfy the Chevalley relations (7.3), (7.4), and
 (7.5) for  as in Section 15.2. Hence, the actions of ei r  fi r , and hri for all i ∈ I r ≥ 1 make K into a U -module so that K∗  K are U -submodules.

20.5 Identification of K∗ , K
∗  and K
 Theorem 20.5.1 For any  ∈ P+ : K is precisely the integrable highest weight U -module of highest weight , with highest weight vector 1
; (ii) the bilinear form   from (20.15) on the highest weight module K coincides with the usual Shapovalov form satisfying 1
 1
 = 1; (iii) K∗ ⊂ K are integral forms of K containing 1
, with K∗ being the minimal lattice U− 1
and K being its dual under the Shapovalov form; (iv) the classes M of the irreducible supermodules M ∈ n -smod form a basis of the  − -weight space V− . The same is true for the classes PM of projective indecomposable supermodules in n -smod .

(i)

Proof It makes sense to think of K as a U -module according to Theorem 20.4.4. The actions of ei and fi are locally nilpotent by Theorems 19.4.1(i) and 19.7.9(i). The action of hi is diagonal by definition. Hence, K is an integrable module. Clearly 1
is a highest weight vector of highest weight . Moreover, K = U− 1
by Theorem 20.3.4. This completes the proof of (i), and (ii) follows immediately from Lemma 20.3.1. For (iii), we know already that K∗ ⊂ K are dual lattices of K  which are invariant under U . Moreover, Theorem 20.3.4 again shows that K∗ = U− 1
. Finally, (iv) follows from (20.16). Theorem 20.5.2 The map   U+ → K∗ , which was constructed in Lemma 20.2.5 is an isomorphism. Proof Similar to the proof of Theorem 9.5.3.

20.6 Blocks of cyclotomic Sergeev superalgebras

247

20.6 Blocks of cyclotomic Sergeev superalgebras Fix  ∈ P+ . Theorem 20.6.1 Let M and N be irreducible n -supermodules with M  N , and 0 −→ infl M −→ X −→ infl N −→ 0

(20.21)

be an exact sequence of n -supermodules. Then pr  X = X. Proof Similar to the proof of Theorem 9.6.1 but using Corollary 19.4.2 instead of Corollary 5.5.2. Recalling the definitions from Section 16.2 and using Theorem 20.5.1, we immediately deduce the following corollary which determines the blocks of n : Corollary 20.6.2 The blocks of cyclotomic Sergeev superalgebras n are precisely the subcategories n -mod for  ∈ #n . Moreover, the subcategory n -mod is non-trivial if and only if the  − -weight space of the highest weight module K is non-zero.

21 Identification of the crystal

This chapter is very short since essentially no changes need to be made to the statements and the proofs in Chapter 10. We only state the main results. Recall the sets of isomorphism classes of irreducible supermodules B and B from Section19.2. We can make them into crystals in the sense of Section10.2. For B, we use the operators e˜ i , f˜i from (19.10), (19.11) and functions
i , i from (19.19), (19.20) to define the maps e˜ i  f˜i 
i  i , respectively. For the corresponding functions on B use e˜ i , f˜i from (17.10), (17.9), functions
i from (17.5), and i defined below. For the weight functions on B and B set wtM = −

(21.1)

for an irreducible M ∈ RepI n  , and wt  N  =  − 

(21.2)

for an irreducible N ∈ n -smod , respectively. Finally, for M ∈ B, define i M =
i M + hi  wtM Lemma 21.0.3 The tuples B
i  i  e˜ i  f˜i  wt and B
i  i  e˜ i  f˜i  wt   for  ∈ P+ are crystals in the sense of Kashiwara. 248

(21.3)

Identification of the crystal

249

Theorem 21.0.4 The crystal B is isomorphic to Kashiwara’s crystal B associated to the crystal base of U− . Theorem 21.0.5 For each  ∈ P+ , the crystal B is isomorphic to Kashiwara’s crystal B associated to the integrable highest weight U module of highest weight .

22 Double covers

In this chapter we specialize to the case  = 0 . In this case n  n , see Remark 15.4.7. So, in view of Chapter 13, we will be getting results on spin representations of symmetric groups. The fact that we deal with the category of supermodules will usually provide important additional insights; for example, this will allow us to treat spin representation theory of alternating groups without any extra work. We will obtain a classification of the irreducible spin representations of Sn and An over an algebraically closed field F of characteristic p ≥ 0, describe how the irreducibles split into blocks (spin version of “Nakayama’s Conjecture”), and prove some of the branching rules. As  = 0 is fixed throughout the chapter, we will not use the superscripts   and just write ei for ei 0 , f˜i for f˜i 0 , etc. These should not be confused with the corresponding notions for the affine Sergeev algebra n .

22.1 Description of the crystal graph Kang [Kg] has given a convenient combinatorial description of the crystal B0  in terms of Young diagrams, which we now explain. For any n ≥ 0, let  = 1  2      be a partition of n. Recall  ∈ >0 ∪ 

from (15.1). We call  a p-strict partition if p divides r whenever r = r+1 for r ≥ 1. If p = 0, we interpret this as a strict partition, that is a partition whose non-zero parts are all distinct. We say that a p-strict partition  is p-restricted if in addition

r − r+1 < p if pr  r − r+1 ≤ p if p  r for each r ≥ 1. Let  p n denote the set of all p-restricted p-strict partitions & of n, and  p = n≥0  p n. 250

22.1 Description of the crystal graph

251

Let  be a p-strict partition. As in Chapter 1, we identify  with its Young diagram, but the way we are going to label the nodes of  with residues is different from the one used in Chapter 1. Residues are now the elements of the set I = 0 1      , see (15.2). The labeling depends only on the column and follows the repeating pattern 0 1      − 1   − 1     1 0 starting from the first column and going to the right, see Example 22.1.1 below. The residue of the node A is denoted res A. Define the residue content of  to be the tuple cont = i i∈I

(22.1)

where for each i ∈ I, i is the number of nodes of residue i contained in the diagram . Let  be a p-strict partition and i ∈ I be some fixed residue. A node A = r s ∈  is called i-removable (for ) if one of the following holds: (R1) res A = i and A =  − A is again a p-strict partition; (R2) the node B = r s + 1 immediately to the right of A belongs to , res A = res B = i, and both B =  − B and AB =  − A B are p-strict partitions. Similarly, a node B = r s following holds:

 is called i-addable (for ) if one of the

(A1) res B = i and B =  ∪ B is again a p-strict partition; (A2) the node A = r s − 1 immediately to the left of B does not belong to , res A = res B = i, and both A =  ∪ A and AB =  ∪ A B

are p-strict partitions. We note that (R2) and (A2) above are only possible in case i = 0. Now label all i-addable nodes of the diagram  by + and all i-removable nodes by −. Then, the i-signature of  is the sequence of pluses and minuses obtained by going along the rim of the Young diagram from bottom left to top right and reading off all the signs. The reduced i-signature of  is obtained from the i-signature by successively erasing all neighboring pairs of the form +−. Note the reduced i-signature always looks like a sequence of −s followed by +s. Nodes corresponding to a − in the reduced i-signature are called i-normal, nodes corresponding to a + are called i-conormal. The rightmost i-normal node (corresponding to the rightmost − in the reduced i-signature) is

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252

called i-good, and the leftmost i-conormal node (corresponding to the leftmost + in the reduced i-signature) is called i-cogood. A node is called removable (resp. addable, normal , conormal, good, cogood ) if it is i-removable (resp. i-addable, i-normal, i-conormal, i-good, i-cogood) for some i. Example 22.1.1 Let p = 5, so  = 2. The partition  = 16 11 10 10 9 5 1 belongs to 5 , and its residues are as follows:

0 0 0 0 0 0 0

1 1 1 1 1 1

2 2 2 2 2 2

1 1 1 1 1 1

0 0 0 0 0 0

0 0 0 0 0

1 1 1 1 1

2 2 2 2 2

1 1 1 1 1

0 0 1 2 1 0 0 0 0 0 0

The 0-addable and 0-removable nodes are as labeled in the diagram:

− −h −

−h

−h +

+

Hence, the 0-signature of  is − − + + − − − and the reduced 0-signature is − − −

22.1 Description of the crystal graph

253

Note the nodes corresponding to the −s in the reduced 0-signature have been circled in the above diagram. So, there are three 0-normal nodes, the rightmost of which is 0-good; there are no 0-conormal or 0-cogood nodes. We define
i  =  i-normal nodes in 

=  −’s in the reduced i-signature of   i  =  i-conormal nodes in 

=  +’s in the reduced i-signature of  

(22.2)

(22.3)

Also set

e˜ i  =

f˜i  =

A if
i  > 0 and A is the i-good node, 0 if
i  = 0,

(22.4)

B if i  > 0 and B is the i-cogood node, 0 if i  = 0.

(22.5)

Finally define wt = 0 −



i i

(22.6)

i∈I

where cont = i i∈I . The definitions imply that e˜ i  f˜i  are p-restricted (or zero) in case  is itself p-restricted. So we have now defined a datum p 
i  i  e˜ i  f˜i  wt which makes the set p of all p-restricted p-strict partitions into a crystal in the sense of Section 10.2 (a combinatorial exercise, but follows from the theorem below anyway). We can now state the main result of Kang [Kg, 7.1] 2 for type A2 (if p = 0 this is an easier result for type B ): Theorem 22.1.2 The set p equipped with
i  i  e˜ i  f˜i  wt as above is isomorphic (in the unique way) to the crystal B0  associated to the integrable highest weight U -module of fundamental highest weight 0 .

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254

Example 22.1.3 The crystal graph of 3 = B0 , up to degree 10, is as follows: ∅ 0 0

PP 1 PP 0 1

0  

 H H0 H  0  

0 1 0

0 1 0 0

PP 1 P



0 1 0 0 0

0 1 0 0 1

1

0 0 1 0 0 1 0

0 1 0 0 0 1

 H  HH1 0  H 0 1 0 0 0 1 0

0 1 0 0 1 0 1

   0

1

0 1 0 0 1 0 1 0

  0   0 1 0 0 0 1 0 0

0

0 1 0 0 1 0 1 0 0 0

0

@@0

0 1 0 0 1 0 1 0 0 1

 

1

0 1 0 0 1 0 0 1 0

0 1 0 0 0 1 0 0 1 1

0 1 0 0 1 0 0

0 1 0 0 1 0 0 1

1

0 1 0 0 1 0 1 0 0 0

Z Z0 Z

0 @ @ 0 1 0 0 0 1 0 0 1 0

0 1 0 0 1 0 0 1 0 0

Finally, we discuss here the extension of Morris’ notion of p-bar core [Mo] to an arbitrary p-strict partition . By a p-bar of , we mean one of the following: (B1) the rightmost p nodes of row i of  if i ≥ p and either pi or  has no row of length i − p; (B2) the set of nodes in rows i and j of  if i + j = p. If  has no p-bars, it is called a p-bar core. In general, the p-bar core ˜ of  is obtained by successively removing p-bars, reordering the rows each time

22.2 Representations of Sergeev superalgebras

255

so that the result still lies in p , until it is reduced to a core. The p-bar weight of , denoted w, is then the total number of p-bars that get removed. There is a notion of a p-bar abacus due to Morris and Yassin [MoY1 ], which implies easily that for p-strict partitions   of n we have ˜ cont  = cont if and only if ˜ = 

(22.7)

This was first observed in [LT2 , Section 4]. The Lie theoretic interpretation of these combinatorial notions is exactly the same as in the classical situation, see the end of Section 11.1. In particular, ⎧ ⎪   + · · · + −2 −1 + 2−1  ⎪ ⎨ 01 1 if  > 1, 2 2 2 w = − 2 0 0 − 1 − 1 − · · · − −1 − 2 ⎪ ⎪ ⎩ if  = 1, 20 1 − 21 0 0 − 1 − 212 if cont = 0  1       . Also, bearing in mind Theorem 22.1.2, we can state Kac’ formula [Kc, (12.13.5)] for the character of the highest weight U -module of highest weight 0 as follows: for  ∈ p n,  ∈ p n  cont  = cont = Par  w

(22.8)

where Par  N  denotes the number of partitions of N as a sum of positive integers of  different colors.

22.2 Representations of Sergeev superalgebras Now that we have an explicit description of the crystal B0 , we formulate a more combinatorial description of our main results for the representation theory of the Sergeev superalgebras n . Recall from Remark 15.4.7 that this is precisely the cyclotomic Sergeev superalgebra n0 . By Theorems 22.1.2 and 21.0.5, we can identify B0  with p . In other words, we can use the set p n of p-restricted p-strict partitions of n to parametrize the irreducible n -supermodules for each n ≥ 0. Let us write M for the irreducible n -supermodule corresponding to  ∈ p n. To be precise, M = Li1      in  if  = f˜in    f˜i1 ∅. Here the operator f˜i is as defined in (22.5), corresponding  under the identification p n = B0  to the crystal operator denoted f˜i 0 in (19.11), and ∅ denotes the empty partition, corresponding to 1
∈ B0 . For  ∈ p n, we also define b =  r ≥ 1  p  r > 0 

(22.9)

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256

the number of (non-zero) parts of  that are not divisible by p. The definition of residues immediately gives that b ≡ 0

mod 2

(22.10)

where 0 denotes the number of 0s in the residue content of . Theorem 22.2.1 The supermodules M   ∈ p n form a complete set of pairwise non-isomorphic irreducible n -supermodules. Moreover, for  ∈ p n: (i) M  M ; (ii) M is of type M if b is even, type Q if b is odd; (iii) M  and M belong to the same block if and only if cont  = cont; (iv) M is projective if and only if  is a p-bar core. Proof We have already discussed the first statement of the theorem, being a consequence of our main results combined with Theorem 22.1.2. For the rest, (i) follows from Corollary 17.3.2, (ii) is a special case of Lemma 17.3.4 combined with (22.10), and (iii) is a special case of Corollary 20.6.2. For (iv), note that if M is projective then it is the only irreducible in its block, hence by (22.8), Par  w = 1. So either w = 0, or  = 1 and w = 1. Now if w = 0 then  is a p-bar core so the Shapovalov form on the (1-dimensional) wt-weight space of K0 Z is 1 (since wt is conjugate to 0 under the action of the affine Weyl group). Hence, M is projective by Theorem 20.5.1(ii). To rule out the remaining possibilty  = 1 and w = 1, we check in that case that the Shapovalov form on the wt-weight space of K0  is 3. The next two theorems summarize earlier results concerning restriction and induction. Theorem 22.2.2 Let  ∈  p n. There exist n−1 -supermodules ei M for each i ∈ I, unique up to isomorphism, such that:

(i) res nn−1 M is isomorphic to

2e0 M ⊕ 2e1 M ⊕ · · · ⊕ 2e M if b is odd, e0 M ⊕ 2e1 M ⊕ · · · ⊕ 2e M

if b is even;

(ii) for each i ∈ I, ei M = 0 if and only if  has an i-good node A, in which case ei M is a self-dual indecomposable supermodule with irreducible socle and head isomorphic to MA .

22.2 Representations of Sergeev superalgebras

257

Moreover, if i ∈ I and  has an i-good node A, then: (iii) the multiplicity of MA  in ei M is
i ,
i A  =
i  − 1, and
i   <
i  − 1 for all other composition factors M  of ei M; (iv) End n−1 ei M  End n−1 MA ⊕
i  as a vector superspace; (v) Hom n−1 ei M ei M  = 0 for all ∈ p n with = ; (vi) ei M is irreducible if and only if
i  = 1. Hence, the restriction

res nn−1 M is completely reducible if and only if
i  ≤ 1 for every i ∈ I. Proof The existence of such supermodules ei M follows from (19.5), Lemma 19.2.1 and Theorem 19.2.5(i), combined as usual with Theorem 22.1.2. Uniqueness follows from Krull–Schmidt and the block classification from Theorem 22.2.1(iii). For the remaining properties, (iii), (iv), and (v) follow from Theorem 19.4.1 and Corollary 19.4.2. Finally, (vi) follows from (iii) as ei M is a module with irreducible socle and head, both isomorphic to MA . Theorem 22.2.3 Let  ∈  p n. There exist n+1 -supermodules fi M for each i ∈ I, unique up to isomorphism, such that:

M is isomorphic to (i) ind n+1 n

2f0 M ⊕ 2f1 M ⊕ · · · ⊕ 2f M if b is odd, f0 M ⊕ 2f1 M ⊕ · · · ⊕ 2f M if b is even; (ii) for each i ∈ I, fi M = 0 if and only if  has an i-cogood node B, in which case fi M is a self-dual indecomposable supermodule with irreducible socle and head isomorphic to MB . Moreover, if i ∈ I and  has an i-cogood node B, then: (iii) the multiplicity of MB  in fi M is i , i B  = i  − 1, and i   < i  − 1 for all other composition factors M  of fi M; (iv) End n+1 fi M  End n+1 MB ⊕i  as a vector superspace; (v) Hom n+1 fi M fi M  = 0 for all ∈ p n with = ; (vi) fi M is irreducible if and only if i  = 1. Hence, the induction

M is completely reducible if and only if i  ≤ 1 for ind n+1 n every i ∈ I. Proof The argument is the same as Theorem 22.2.2, but using (19.5), Lemma 19.2.3, Theorem 19.2.5(ii), Corollary 19.7.2, and Theorem 19.7.9.

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258

There is one n -supermodule that deserves special mention, the so-called basic spin supermodule. Recall from Section 13.2 that the subalgebra of

n generated by s1      sn−1 is isomorphic to the group algebra n of the symmetric group Sn . It has the trivial 1-dimensional module denoted 1, on which each si acts as multiplication by 1. For n ≥ 1, we define

In = indnn 1

(22.11)

giving a n -supermodule of dimension 2n . Also define the p-restricted p-strict partition

pa  b if b = 0, n = (22.12) pa−1  p − 1 1 if b = 0, where n = ap + b with 0 ≤ b < p. Lemma 22.2.4 If p  n then In  Mn ; if pn then In is an indecomposable module with two composition factors both isomorphic to Mn . In particular,

2n if p  n, dim Mn  = 2n−1 if p  n. Proof This is obvious if n = 1 2 and easy to check directly if n = 3. Now for n > 3 we proceed by induction using Theorem 22.2.2 together with the observation that

res nn−1 In  In − 1 ⊕ "In − 1 We consider the four cases n ≡ 0 1 or 2 mod p and n ≡ 0 1 2 mod p separately. Suppose first that n ≡ 0 1 2 mod p. Considering the crystal graph shows that f˜i n−1 = 0 only for i = 0 and for one other i ∈ I, for which f˜i n−1 =

n . By the induction hypothesis, res nn−1 In  2Mn−1 . Hence by Theorem 22.2.2, In can only contain Mn  and Mf˜0 n−1  as composition factors. But the latter case cannot hold since by Theorem 22.2.2 again,

res nn−1 Mf˜0 n−1  is not isotypic. Hence all composition factors of In are  Mn , and we easily get that in fact In  Mn  by a dimension argument. Next suppose that n ≡ 0 mod p. This time, f˜0 n−1 = n and all other ˜ fi n−1 are zero. Hence, by the induction hypothesis and the branching rules,

In only involves Mn  as a constituent. But we have that res nn−1 Mn  = e0 Mn   Mn−1  so in fact that In must have Mn  as a constituent

22.3 Spin representations of Sn

259

with multiplicity two. Further consideration of the endomorphism ring of In shows moreover that it is an indecomposable module. The argument in the remaining two cases n ≡ 1 mod p and n ≡ 2 mod p is entirely similar.

22.3 Spin representations of Sn Here we consider representation theory of the twisted group algebra n , see Section 13.1. By Theorem 22.2.1, we have a parametrization M   ∈ p n of the irreducible n -supermodules. Proposition 13.2.2 shows that the functors !n and "n set up a natural correspondence between classes of irreducible n and n -supermodules, type-preserving if n is even and typereversing if n is odd. Hence we have a parametrization D   ∈ p n

of the irreducible n -supermodules, letting D be an irreducible n -supermodule corresponding to M under the correspondence. Also, recalling the definition (22.9), define a = n − b

(22.13)

for  ∈ p n. We observe by (22.10) that a ≡ 1 + · · · + 

mod 2

(22.14)

where 1 + · · · +  counts the number of nodes in the Young diagram  of residue different from 0. Finally, it is an easy combinatorial exercise to see that a ≡ n − hp  mod 2

(22.15)

where the p -height hp  of  is the number of parts in  not divisible by p. Theorem 22.3.1 The supermodules D   ∈ p n form a complete set of pairwise non-isomorphic irreducible n -supermodules. Moreover, for  ∈ p n, (i) D  D ; (ii) D is of type M if a is even, type Q if a is odd; (iii) D  and D belong to the same block if and only if cont  = cont; (iv) D is projective if and only if  is a p-bar core.

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260

Proof Observe that (i)–(iv) follow directly from Theorem 22.2.1 using Proposition 13.2.2. Remark 22.3.2 The p-blocks of the ordinary irreducible spin representations of Sn were described by Humphreys [H], in terms of the notion of p-bar core. However, unlike the case of Sn , Humphreys’ result does not imply Theorem 22.3.1(iii) because of the lack of information on decomposition numbers. See, however, Remark 22.3.20. Recall the definition of n ∈ p n in (22.12). We call the irreducible n -supermodule Dn  the basic spin supermodule. The following result is closely related to [W]. Lemma 22.3.3 Dn  is of dimension 2)n/2* , unless pn when its dimension is 2)n−1/2* . Moreover, Dn  is equal to the reduction modulo p of the basic spin module DnC of  n C over C, except if pn and n is even when the reduction modulo p of DnC has two composition factors, both isomorphic to Dn . Proof The statement about dimension is immediate from Lemma 22.2.4 and Proposition 13.2.2. The final statement is easily proved by working in terms of n and using the explicit construction given in (22.11). To motivate the next two theorems, note that the map D → M for each  ∈ p n extends linearly to an isomorphism ∼

K n -smod −→ K n -smod of Grothendieck groups. Using this identification, we can lift the operators  ei and fi on K0  = n≥0 K n -smod defined earlier to define similar  operators on n≥0 K n -smod. Then all our earlier results about K0 , for instance Theorems 20.4.4 and 20.5.1, could be restated purely in terms of the representations of n instead of n . In fact, we can do slightly better and define the operators ei and fi on irreducible n -supermodules, not just on the Grothendieck group. Theorem 22.3.4 Let  ∈  p n. There exist n−1 -supermodules ei D for each i ∈ I, unique up to isomorphism, such that:

(i) res nn−1 D is isomorphic to

e0 D ⊕ 2e1 D ⊕ · · · ⊕ 2e D if a is odd, e0 D ⊕ e1 D ⊕ · · · ⊕ e D if a is even;

22.3 Spin representations of Sn

261

(ii) for each i ∈ I, ei D = 0 if and only if  has an i-good node A, in which case ei D is a self-dual indecomposable supermodule with irreducible socle and head isomorphic to DA . Moreover, if i ∈ I and  has an i-good node A, then: (iii) the multiplicity of DA  in ei D is
i ,
i A  =
i  − 1, and
i   <
i  − 1 for all other composition factors D  of ei D; (iv) End n−1 ei D  End n−1 DA ⊕
i  as a vector superspace; (v) Hom n−1 ei D ei D  = 0 for all ∈ p n with = ;

(vi) ei D is irreducible if and only if
i  = 1. Hence, res nn−1 D is completely reducible if and only if
i  ≤ 1 for every i ∈ I. Proof If n is odd, we simply define ei D = "n−1 ei M for each i ∈ I  ∈ p n. If n is even, take

ei D =

⎧ if a is even and i = 0, ⎪ ⎪ ⎪ " e M ⎪ ⎨ n−1 i or a is odd and i = 0, ⎪ ⎪ if a is even and i = 0, ⎪ ⎪ ⎩ "n−1 ei M or a is odd and i = 0.

We need to explain the notation "n−1 used in the last two cases: here, ei M admits an odd involution in view of Remark 19.2.6 and Theorem 22.2.1(ii), and also the Clifford supermodule Un−1 has an odd involution since n is even. So in exactly the same way as in the definition of (19.6), we can introduce the space "n−1 ei M = Homn−1 Un−1  ei M It is then the case that "n−1 ei M  "n−1 ei M ⊕ ""n−1 ei M Equivalently, by Proposition 13.2.2(ii) ei D can be characterized by ei M  !n−1 ei D if a is even and i = 0, or a is odd and i = 0. With these definitions, it is now a straightforward matter to prove (i)–(vi) using Theorem 22.2.2 and Proposition 13.2.2. Finally, the uniqueness statement is immediate from Krull–Schmidt and the description of blocks from Theorem 22.3.1(iii).

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262

Theorem 22.3.5 Let  ∈  p n. There exist n+1 -supermodules fi D for each i ∈ I, unique up to isomorphism, such that:

(i) ind n+1 D is isomorphic to n

f0 D ⊕ 2f1 D ⊕ · · · ⊕ 2f D if a is odd, f0 D ⊕ f1 D ⊕ · · · ⊕ f D if a is even; (ii) for each i ∈ I, fi D = 0 if and only if  has an i-cogood node B, in which case fi D is a self-dual indecomposable supermodule with irreducible socle and head isomorphic to DB . Moreover, if i ∈ I and  has an i-cogood node B, then: (iii) the multiplicity of DB  in fi D is i , i B  = i  − 1, and i   < i  − 1 for all other composition factors D  of fi D; (iv) End n+1 fi D  End n+1 DB ⊕i  as a vector superspace; (v) Hom n+1 fi D fi D  = 0 for all ∈ p n with = ; (vi) fi D is irreducible if and only if i  = 1. Hence the induction D is completely reducible if and only if i  ≤ 1 for every ind n+1 n i ∈ I. Proof This is deduced from Theorem 22.2.3 by similar argument to the proof of Theorem 22.3.4. Remark 22.3.6 Over C, the branching rules in the preceeding two theorems are the same as Morris’ branching rules, see [Mo]. Using this observation, we can show that our labeling of irreducibles over C agrees with the standard labeling. It is possible to define the formal characters of n -supermodules, as well as operators ei and fi in terms of the Jucys–Murphy elements Mk ∈ n (13.6) without referring to the Sergeev superalgebra n . (Of course, the proofs do depend on the representation theory of n developed so far). Recall the notation  = p − 1/2 (resp.  =  if p = 0) and I = 0 1      . Given a tuple i = i1      in  ∈ I n and an n -supermodule V , we define the i-weight space of V : $
# % i i + 1 N Vi = v ∈ V $ Mk2 − k k v = 0 for N 0 and k = 1     n  2  Lemma 22.3.7 Any V ∈ n -smod decomposes as V = i∈I n Vi 

22.3 Spin representations of Sn

263

Proof A similar statement for n with L2k in place of Mk2 and qi = ii + 1 follows from Lemmas 16.1.2, 15.3.1, and Remark 15.4.7. in place of ii+1 2 Now the desired fact follows from Lemma 13.2.5(ii).  Now fix i ∈ I n , and let  = i∈I i i be the content of i, see (16.2). Consider the Clifford–Grassman superalgebra i from Example 12.1.4. By Example 12.2.14, it has a unique irreducible supermodule Ui of dimension n−0 +1 2) 2 * . Now suppose that M is a n -supermodule. The weight space Mi is obviously invariant under the action of the subalgebra # of n generated by the JM-elements Mk . Put jk =

ik ik + 1 2

1 ≤ k ≤ n

Then the Mk satisfy the relations Mk2 = jk

1 ≤ k ≤ n

Mk Ml = −Ml Mk

1 ≤ k = l ≤ n

on every #-irreducible constituent of Mi . Note that, since ik ∈ I, we have jk = 0 if and only if ik = 0, so j  i and dim Ui = dim Uj = 2)

n−0 +1 * 2



This shows that dim Mi is divisible by dim Ui. Now define the formal character of M by ch M =

 dim Mi i∈I n

dim Ui

ei 

(22.16)

an element of the free -module on basis ei i ∈ I n . The next lemma explains why we want to divide by dim Ui: we set this definition up so that the formal characters of the n -module M and the corresponding n -module D were essentially the same. Lemma 22.3.8 Let  ∈ p n, and  ch M = ai Li1   · · ·  Lin   i∈I n

ch D =



i∈I n

Then ai = bi for all i ∈ I n .

bi ei 

264

Double covers

Proof Let i ∈ I n . We calculate the dimension of "n Mi . We know n+1 that dim Un = 2) 2 * and Un is of type M if and only if n is even. Now, by Lemma 13.2.5(ii), Mk2  = L2k /2, and so the Mk2  commute with the Clifford generators cl in n , whence dim "n Mi =

ai dim Li1   · · ·  Lin  dim Un 0

= ai 2n−) 2 *−)

n+1 2 *+n

2 n



where n = n mod 2. Now, D = "n M, unless both n and 0 are odd, in which case "n M  D ⊕ D. So the formula above implies that dim Mi = 2)

n−0 +1 * 2

ai = dim Uiai 

which implies the desired result. Corollary 22.3.9 The characters of the pairwise inequivalent irreducible n -supermodules are linearly independent. Proof Follows from Lemma 22.3.8 and Theorem 17.3.1. Recall the notation #n from Section 16.2. Given  = n -supermodule M, we set  M = Mi 



i i ∈ #n and a

i∈I n with conti=

 Corollary 22.3.10 The decomposition M = ∈#n M is precisely the decomposition of M into blocks as an n -supermodule. Proof By (16.3), (21.2), and (22.6), we have cont = conti for any i ∈ I n with Li1   · · ·  Lin  appearing in ch M. So the result follows from Lemma 22.3.8 and Theorems 22.2.1, 22.3.1(iii). For  ∈ #n and a n -supermodule M, we say that M belongs to the block  if M = M . It is also now clear that the type of an irreducible n -supermodule D can be read off from its formal character: Corollary 22.3.11 Let D be an irreducible n -supermodule belonging to the block . Then D is of type M if n − 0  is even, type Q if n − 0  is odd.

22.3 Spin representations of Sn

265

Next, we proceed to define i-induction and i-restriction purely in terms of n . Let M be a n -supermodule belonging to the block  ∈ #n . Given i ∈ I, define

resi M = res nn−1 M − i 

(22.17)



indi M = ind n+1 M + i  n

(22.18)

where resi M is interpreted as zero in case i = 0. These definitions extend in an obvious way to give exact functors resi and indi , which are adjoint to each other. We note in particular that resi M is the generalized eigenspace of eigenvalue ii + 1/2 for the action of Mn2 . Hence Lemma 22.3.7 and adjointness imply: Lemma 22.3.12 For an Sn-supermodule M,   resi M ind n+1 M  indi M res nn−1 M  n i∈I

i∈I

The next lemma shows how to define of the operators ei and fi without referring to the Sergeev superalgebra. Lemma 22.3.13 Let D be an irreducible n -supermodule, and i ∈ I. (i) There is a n−1 -supermodule ei D, unique up to isomorphism, such that

ei D ⊕ ei D if i = 0 and D is of type Q, resi D  ei D if i = 0 or D is of type M. (ii) There is an n+1 -supermodule fi D, unique up to isomorphism, such that

indi D 

f i D ⊕ fi D fi D

if i = 0 and D is of type Q, if i = 0 or D is of type M.

Moreover operators ei and fi defined in (i) and (ii) agree with the operators ei and fi defined in Theorems 22.3.4 and 22.3.5, respectively. Proof Consider for example (i) for D = D with a odd. It follows from the description of blocks of n and (22.7) that e0 D and 2ei D for i = 0, as defined in Theorem 22.3.4 are precisely res0 D and resi D, respectively, as defined in (22.17). Now the result follows from Theorem 22.3.4 and Krull– Schmidt. Note again that the ei  fi i ∈ I are operators on irreducible n -supermodules, but note they are not functors defined on arbitrary supermodules. However,

266

Double covers

extending linearly, they induce operators also denoted ei  fi at the level of characters or equivalently on the level of Grothendieck groups, see Corollary 22.3.9. The effect of ei on characters is exactly the same as in the classical case (cf. (5.9)): Lemma 22.3.14 If ch M =



ci e i

i∈I n

then



ch ei M =

ci1 in−1 i ei 

i∈I n−1

Proof This follows from Lemma 22.3.8 and Corollary 19.2.2. r

r

There are also divided power operators ei and fi . Again we just state a lemma characterizing them uniquely, rather than giving their explicit definition: Lemma 22.3.15 Let D be an irreducible n -supermodule, and i ∈ I. r

(i) There is an n−r -supermodule ei D, that ⎧ r ⊕r! ⎪ ⎨ ei D )r/2* r resi r D  ei D⊕2 r! ⎪ ⎩ r ⊕2)r+1/2* r! ei D r

(ii) There is an n+r -supermodule fi D, that ⎧ r ⊕r! ⎪ ⎨ fi D )r/2* r indi r D  fi D⊕2 r! ⎪ ⎩ r ⊕2)r+1/2* r! fi D

unique up to isomorphism, such if i = 0, if i = 0 and D is of type M, if i = 0 and D is of type Q. unique up to isomorphism, such if i = 0, if i = 0 and D is of type M, if i = 0 and D is of type Q.

Proof Use Lemma 19.3.1. Details omitted. Note comparing Lemmas 22.3.12 and 22.3.15, we see that r

eir = r!ei 

r

fir = r!fi

(22.19)

at the level of characters. Finally, we “desuperize”, that is deduce results about the usual (ungraded) n -modules from our theory of supermodules. At the same time we get results on n -modules, where n =  n 0¯ is the twisted group algebra of the alternating group An , see Section 13.1. Using Theorem 22.3.1, Corollary 12.2.10

22.3 Spin representations of Sn

267

and Proposition 12.2.11, we get the following. If a is odd then D decomposes as an ungraded module as D = D + ⊕ D − for two non-isomorphic irreducible n -modules D + and D −. More over, the restrictions res nn D + and res nn D − are irreducible and isomorphic to each other. We denote

E 0 = res nn D ± If a is even, then D is irreducible viewed as an ungraded n -module, but we denote it instead by D 0 to make it clear that we are no longer considering a 2 -grading. Moreover,

res nn D 0  E + ⊕ E − for two non-isomorphic irreducible

n -modules

E + and E −. Finally:

Theorem 22.3.16 & D 0   ∈ p n a even

D + D −   ∈ p n a odd

is a complete set of pairwise non-isomorphic irreducible n -modules, and & E 0   ∈ p n a odd

E + E −   ∈ p n a even

is a complete set of pairwise non-isomorphic irreducible

n -modules.

Remark 22.3.17 We would like to emphasize that, as it is usual for spin representations of symmetric groups even in characteristic 0, the parametrizations obtained in Theorem 22.3.16 are “defective” in the sense that we cannot effectively distinguish between D + and D −. So we believe that when working with spin representations of Sn and An , it is more natural to work with the category of supermodules for as long as possible, and then to “desuperize” in the last moment. Examples of that are given by Theorem 22.3.16, as well as Theorems 22.3.18, 22.3.19, and 22.3.20 below. As another illustration of “desuperization” procedure, we give the solution to a problem important for group theory, namely description of irreducible restrictions from n to n−1 and from n to n−1 .

Double covers

268

Theorem 22.3.18 Let  ∈ p n. Then:

(i) If a is even, res nn−1 D 0 is irreducible if and only if 
0  =
i  = 1 i∈I

(ii) If a is odd, res nn−1 D ± is irreducible if and only if 
i  = 1 i∈I

Proof (i) D 0 is just D considered as an ungraded module. So, by Theorem 22.3.4(i), the restriction to res nn−1 D 0 is irreducible only if
i  = 1 for some i ∈ I and
j  = 0 for all j = i. Moreover, in this case the restriction is ei D, considered as an ungraded module. In view of Theorem 22.3.4(vi), ei D is an irreducible supermodule of type M if i = 0 and type Q otherwise, see (22.14) and Theorem 22.3.1. So the restriction res nn−1 D 0 is irreducible if and only if i = 0. (ii) In view of Corollary 12.2.10, we have D +  D − ⊗ sgn where sgn is the 1-dimensional sign representation of n . It follows that res nn−1 D + is irreducible if and only if res nn−1 D − is irreducible. So n n res n−1 D + is irreducible if and only if res n−1 D has two composition factors when considered as an ungraded module. It is now easy to see using  Theorems 22.3.4 and 22.3.1 that this happens if and only if i∈I
i  = 1

Theorem 22.3.19 Let  ∈ p n. Then: (i) If a is even, res

n n−1

E ± is irreducible if and only if 
i  = 1 i∈I

(ii) If a is odd, res

n n−1

E 0 is irreducible if and only if 
0  =
i  = 1 i∈I

Proof Similar to the proof of Theorem 22.3.18. Finally, we give a description of the ordinary (ungraded) blocks of n . This does not quite follow from the description of the “superblocks” in

22.3 Spin representations of Sn

269

Theorem 22.3.1(iii), unless we invoke the work of Humphreys [H]; in fact, all we need from [H] is to know the number of ordinary blocks. Theorem 22.3.20 Let D
 and D   be ungraded irreducible n -modules, see Theorem 22.3.16. Then, with one exception, D
 and D   lie in the same block if and only if  and have the same p-bar core. The exception is if  = is a p-bar core, a is odd and
= −, when D
 and D   are in different blocks. Proof If the p-bar core of  is different from the p-bar core of , then, by Theorem 22.3.1(iii), the supermodules D and D  belong to different blocks of the superalgebra n . This means that they correspond to two mutually orthogonal even central idempotents in n , whence D
 and D   belong to different ungraded blocks. Next assume that  = is a p-bar core and a is odd. By Theorem 22.3.1(iv), D is a projective supermodule, whence D + and D − are projective ungraded modules. So they are in different blocks. Thus, the combinatorial conditions in the statement of the theorem separate the irreducible n -modules into classes which are unions of blocks. In view of [H, 1.1], the amount of these classes is equal to the number of blocks of n . So each class must comprise exactly one block. Remark 22.3.21 Using the same method as the one explained in Remark 11.2.29, we can prove the following (see [BK6 ] for details): if B is a superblock of p-bar weight w of n , then the determinant of the Cartan matrix of B is pN where N equals  p−3  p−3    2r1 + 2r3 + 2r5 +    p−3 + r1 + r2 + r3 2 2 2  p−1 r1 r2 r3 the sum being over all partitions  = 1r1 2r2     of w. In order to “desuperize” this result, note that for blocks of type M this same formula gives the Cartan determinant of the corresponding ungraded block of n . We have conjectured in [BK5 ] that the same is true for blocks of type Q (in this case the ungraded block has twice as many irreducibles as in the corresponding superblock). Recently Bessenrodt and Olsson confirmed this conjecture.

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Index

M , 37, 83, 193 Mi , 36, 138, 192 1
, 90, 229
, 3 i , 37, 193  , 37, 83, 193 , 163 , 3, 157 , 156 1, 41, 51 1Sn , 27 a, 259 i , 67, 183 n , 174 admissible transposition, 15 affine Hecke algebra, 25 affine Sergeev superalgebra, 175 B, 86, 224 B, 86, 224 basic spin supermodule, 258, 260 block, 4, 38, 83, 194, 264 of cyclotomic Hecke algebra, 118 of cyclotomic Sergeev algebra, 247 branching graph, 9 branching problem, x branching rules, 16, 139, 141, 256, 260 c, 67, 183 n , 154 central character, 3 ch , 36, 192 Clifford superalgebra, 154 Clifford supermodule, 158 Clifford–Grassman superalgebra, 155 completely splittable module, 143 conti, 83, 193 cont, 5, 251

content of a tuple, 83, 193 residue of a partition, 5, 251 p-core, 134 p-bar core, 254 crystal, 123 cyclotomic crystal operators, 86, 224 cyclotomic Hecke algebra, 68 cyclotomic Sergeev superalgebra, 184 D  , 29 a , 44, 200 , 67, 183 V , 151 am , 44, 200 decomposition numbers problem, x ei r , 87, 90, 225–227 ei , 46, 67, 139, 184, 222 ei , 222 r ei , 67, 87, 90, 139, 184, 225, 226 ei , 83
i , 133, 253
i , 90, 229 e˜ a , 47 e˜ i∗ , 69, 229 e˜ i , 86, 224 e˜ i , 133, 204, 253
i , 45, 142, 201
∗i , 69, 229 extremal weight, 135, 144 fi r , 89, 90, 226, 227 fi r , 225 fi , 67, 184 r fi , 67, 139, 184 fi , 84, 223 !n , 168 f˜a , 47

275

276 f˜i∗ , 69, 229 f˜i , 86, 224 i , 34, 179 i , 133, 253 i , 90, 229 f˜i , 133, 203, 253 i , 125, 142, 248 Fock space, 22 formal character, 36, 138, 192, 263 n , 25 "n , 168 #n , 83, 193 , 66, 67, 183 Gelfand–Zetlin basis, 9 Grassman superalgebra, 154 hi , 67, 183, 184 hi , 113, 244
n , 25
n , 68
, 28 hi  j , 67, 183
n -mod , 83
n -mod , 38 head, 3 I, 3, 181  , 68, 184 indi, 54, 206 indn , 28, 177 indi , 219 infl , 69, 184 integrable module, 115 integral module, 65, 181 integral representation, 65, 181 intertwining elements, 34, 179 Jucys–Murphy elements, 7, 167, 173 K, 104, 238 K, 105, 238 kab , 58 Kato module, 38 Kato supermodule, 194 Li, 36, 191 Li1      in , 51, 205 Lk , 7, 173 Lm in , 40, 197 , 66, 67, 181 w, 5 i , 67, 183 Mk , 167 Mackey Theorem, 30

Index Mackey theorem, 178 cyclotomic, 73, 187 node, 4 i-addable, 5, 251 addable, 5, 252 i-cogood, 132, 252 i-conormal, 131, 251 cogood, 132, 252 conormal, 132, 252 i-good, 132, 252 good, 132, 252 i-normal, 131, 251 normal, 132, 252 i-removable, 5, 251 removable, 5, 252 P, 67, 183 P+ , 67, 183 , 4 p , 4 ", 156 n , 5, 25 n, 4 p n, 4 parabolic subalgebra, 28, 177 partition, 4 p-regular, 4 p-strict, 250 p-restricted, 250 strict, 250 pr  , 69, 184 qi, 181 RepI , 65, 182 RepI n  , 194 RepI
n  , 83 resn , 28, 177 resnn−1 , 29, 177 resi , 202 resi , 219 residue, 4, 251 residue content, 5, 251 p , 250 p n, 250 , 31, 178 sgn, 41 Scopes Morita equivalence, 147 Sergeev superalgebra, 168 Shapovalov–Jantzen form, 115 Shuffle Lemma, 37, 192 socle, 3 Specht module, 137 superalgebra, 152

Index supermodule, 155 irreducible, 157 superspace, 151

w, 134, 255 weight space, 11, 138, 262 wt, 123, 125, 133, 248, 253

-tableau, 15 n , 166 , 31, 178 twisted group algebra, 166 type M, 157 type Q, 157

n , 175 n , 184  , 177 n -smod , 194

U , 67, 184 U , 67, 184 U− , 67, 184 n , 166

n , 168 Young diagram, 4 Young graph, 14

n , 5

277