Rock blocks.

W. Turner

March 14, 2006

Dedicated to the memory of Joe Silk.

Abstract

Consider representation theory associated to symmetric groups, or to Hecke algebras in type A, or to q-Schur algebras, or to finite general linear groups in non-describing characteristic. Rock blocks are certain combinatorially defined blocks appearing in such a representation theory, first observed by R. Rouquier. Rock blocks are much more symmetric than generic blocks, and every block is derived equivalent to a Rock block. Motivated by a theorem of J. Chuang and R. Kessar in the case of symmetric group blocks of abelian defect, we pursue a structure theorem for these blocks.

Contents.

Introduction. I. Highest weight categories, q-Schur algebras, Hecke algebras, and finite

general linear groups. II. Blocks of q-Schur algebras, Hecke algebras, and finite general linear

groups. III. Rock blocks of finite general linear groups and Hecke algebras, when

w < l. IV. Rock blocks of symmetric groups, in characteristic two. V. Schur-Weyl duality inside Rock blocks of symmetric groups. VI. Ringel duality inside Rock blocks of symmetric groups. VII. James adjustment algebras for Rock blocks of symmetric groups. VIII. Doubles, Schur super-bialgebras, and Rock blocks of Hecke algebras. IX. Power sums, and Schiver doubles of type A_{∞}.

References.

Introduction

Whilst the 20^{th} century was still in its infancy, an article by F. G. Frobenius was published in the Journal of the Berlin Science Academy, which contained a description of the irreducible complex characters of all symmetric groups [33]. Since then, representation theory has evolved into a deep and sophisticated art, to the point where most papers in the subject are incomprehensible to the multitude of mathematicians. After all this development however, some basic questions remain unanswered. Interrogate an expert on group representation theory over finite fields and you will quite soon witness a shrug of the shoulders, and a protestation of ignorance. The irreducible characters of symmetric groups, which Frobenius so casually exposed in characteristic zero, remain mysterious over fields of prime characteristic. This monograph comprises a sequence of reflections surrounding the modular representation theory of symmetric groups.

Our approach to the subject is homological, inspired by M. Broué’s abelian defect group conjecture [9], and encouraged by the proof of Broué’s conjecture for blocks of symmetric groups by J. Chuang, R. Kessar, and R. Rouquier.

The abelian defect group conjecture is the most homological of a menagerie of general conjectures in modular representation theory, each of which predicts a likeness between the representations of a finite group in characteristic l, and those of its l-local subgroups. It stakes that the derived category of any block A of a finite group is equivalent to the derived category of its Brauer correspondent B, so long as the blocks have abelian defect groups. Such an equivalence should respect the triangulated structure of the derived category, and therefore descend from a two sided tilting complex of A-B-bimodules, by a theorem of J. Rickard [57]. Some have postured to prove the conjecture by induction, and encountered the difficulty of lifting an equivalence of stable categories to an equivalence of derived categories [60]. Others have tried to prove the conjecture for particular examples, such as symmetric groups.

In 1991, R. Rouquier observed a certain class of blocks of symmetric groups, which he believed to possess a particularly simple structure. Indeed, Rouquier conjectured a beautiful structure theorem for such blocks of abelian defect, which was subsequently proved by J. Chuang, and R. Kessar [11]. A corollary was a proof of Broué’s conjecture for this class of blocks, whose defect groups could be arbitrarily large. In view of their history, these blocks should properly be called Rouquier, or Chuang-Kessar blocks. We use the curt abbreviation “RoCK blocks”. Such blocks can be defined in arbitrary defect, and in any species of type A representation theory.

Chuang and Rouquier proved in a later work that all symmetric group blocks of identical defect possess equivalent derived categories which, in conjunction with the previous study of Rock blocks, established the truth of Broué’s conjecture for all blocks of symmetric groups [12].

The will which motivated this text, was for a theorem like that of Chuang and Kessar, describing Rock blocks whose defect groups are not necessarily abelian. Such a result ought to be of broad interest since there is no known analogue of Broué’s conjecture in nonabelian defect. So far as symmetric groups are concerned, an algorithm of A. Lascoux, B. Leclerc and J-Y. Thibon gives a speculative description of all the decomposition numbers of blocks of abelian defect [48]. In nonabelian defect however, little is currently known about the decomposition numbers, even conjecturally. Symmetric group blocks of nonabelian defect are therefore beyond the influence of the general character theoretic predictions for algebraic groups made by G. Lusztig.

Over the length of this monograph, I hope to convince the Reader of the existence of a structure theorem for Rock blocks of arbitrary defect. Indeed, a conjectural description of an arbitrary Rock block of a Hecke algebra appears in chapter 8 (conjecture 162). A generalized conjecture for blocks of q-Schur algebras is given in chapter 9 (conjecture 176).

In formulating these conjectures, I produced more than a cupful of imaginative sweat, and earlier chapters of this booklet record theorems which point to the conjectures, and give evidence for them. For those readers with a fetish for decomposition numbers, the saltiest of these theorems is probably a formula for the decomposition matrix of a Rock block of a symmetric group, of arbitrary defect (theorem 129).

It would be polite of me to be a little more precise. Therefore, let us catalogue the more significant results of the article, and give some description of their character and logical intimacy, before plunging into the depths of the text.

In the first chapter, we recall E. Cline, B. Parshall, and L. Scott’s definition of a highest weight category. We discuss quasi-hereditary algebras, and summarise C. Ringel’s tilting theory for these algebras. We recognise the q-Schur algebra S_{q}(n) as the graded dual of the quantized coordinate ring of a matrix algebra, and recall S. Donkin’s tilting theory for S_{q}(n). Schur-Weyl duality relates q-Schur algebras, and Hecke algebras H_{q}(Σ_{r}) associated to symmetric groups Σ_{r}. We view this phenomenon in chapter one, and its relevance for the representation theory of finite general linear groups GL_{n}(𝔽_{q}) over a field k of characteristic l, coprime to q. Parametrizations of irreducible representations for all these algebras are given. We assemble a few facts concerning wreath products of algebras.

Chapter two opens with a description of the abacus presentation of partitions, due to G. James. We recall Nakayama’s parametrization of blocks kb_{τ,w}^{Σ} of symmetric groups by their l-core τ and their weight w, as well as parametrizations of blocks kb_{τ,w}^{S_{q}} of q-Schur algebras, blocks kb_{τ,w}^{H_{q}} of Hecke algebras, and unipotent blocks kb_{τ,w}^{G_{q}} of finite general linear groups. Blocks kb_{τ,w}^{Σ} have abelian defect groups if, and only if, w < l. Chuang and Rouquier’s general theory of 𝔰𝔩_{2}-categorification implies that kb_{τ,w}^{X} is derived equivalent to kb_{τ′,w}^{X}, for a fixed weight w, fixed X ∈ {Σ, S_{q}, H_{q}, G_{q}}, and various τ, τ′.

Blocks of symmetric groups are in one-one correspondence with weight spaces in the basic representation of 𝔰̂𝔩_{p}. Rock blocks are those blocks whose weight spaces lie on the boundary of this representation. For practical purposes, it is convenient for us to work with a combinatorial definition of the Rock blocks, given via an abacus presentation. When w < l, Chuang and Kessar’s structure theorem states that a Rock block kb_{ρ,w}^{Σ} of weight w is Morita equivalent to a wreath product kb_{∅,1}^{Σ} ≀ Σ_{w} of a cyclic defect block kb_{∅,1}^{Σ}, and a symmetric group Σ_{w} on w letters. A stated corollary of this theorem is a formula for the decomposition matrix of a symmetric group Rock block of abelian defect discovered by Chuang and K.M. Tan, and roughly simultaneously by B. Leclerc and H. Miyachi:

d_{\underline{λ}\underline{μ}} = ∑_{α∈Λ_{w}^{p+1},β∈Λ_{w}^{p}} ∏_{j=0}^{p−1} c(λ^{j}; α^{j}, β^{j})c(μ^{j}; β^{j}, (α^{j+1})′).

In this formula, \underline{λ} = [λ^{0}, ..., λ^{l−1}] is the p-quotient of a partition with pcore ρ, relative to a certain abacus presentation; the numbers c(λ; μ, ν) are Littlewood-Richardson coefficients.

We state R. Paget’s description of the Mullineux map on a Rock block of a Hecke algebra. We proceed to describe Brauer correspondence for blocks of finite general linear groups, and then for blocks of symmetric groups. This chapter ends with the statement of a criterion of Broué, for the lifting of a character correspondence between symmetric algebras to a Morita equivalence.

The first honest mathematics appears in chapter three. We give a proof, à la Chuang-Kessar, of a structure theorem for Rock blocks of finite general linear groups of abelian defect. More precisely, we show that a Rock block kb_{ρ,w}^{G_{q}} is Morita equivalent to the wreath product kb_{∅,1}^{G_{q}} ≀ Σ_{w}, when w < l (theorem 71). A simple corollary is the Morita equivalence of kb_{ρ,w}^{H_{q}} and kb_{∅,1}^{H_{q}} ≀ Σ_{w}, so long as w < l, and q ≠ 1 modulo l (theorem 78).

This structure theorem for Rock blocks of finite general linear groups of abelian defect appeared in a previous paper of mine, and was duplicated by Miyachi. An application given in my paper is the revelation of Morita equivalences between weight two blocks of finite general linear groups GL_{n}(𝔽_{q}), as q varies. Thanks to Chuang and Rouquier’s theory it must now be possible to generalise this result, and give comparisons between abelian defect blocks of GL_{n}(𝔽_{q}) of arbitrary weight, as q varies. We choose not to spend time chomping on this old pie, since we have become aware of dishes with a more exotic, and alluring aroma.

In chapter four, we turn to Rock blocks of symmetric groups, of nonabelian defect. We give a sweet proof that in characteristic two, once a Rock block of a symmetric group has been localised at some idempotent, Chuang and Kessar’s theorem generalises to non-abelian defect. To be precise, we prove that in characteristic two, ekb_{ρ,w}^{Σ}e is Morita equivalent to kΣ_{2} ≀ Σ_{w}, for some idempotent e ∈ kb_{ρ,w}^{Σ} (theorem 80). Our proof involves the Brauer morphism. This idea can be contorted and extended, to give a result in arbitrary characteristic. Indeed, to any Rock block kb_{ρ,w}^{Σ}, we associate a natural l-permutation module kM, whose endomorphism ring kE is Morita equivalent to kΣ_{w} (theorem 83).

Schur-Weyl duality for Σ_{w} is naturally visible inside a Rock block of weight w, via the Brauer homomorphism, and our fifth chapter is dedicated to establishing this truth. Formally, we prove that kb_{ρ,w}^{Σ}/Ann(kM) is Morita equivalent to the Schur algebra S(w, w), and that the S(w, w)-kΣ_{w}-bimodule corresponding to the kb_{ρ,w}^{Σ}-kE-bimodule kM is twisted tensor space (theorem 87).

Chapter six begins with a criterion for the lifting of a character correspondence between quasi-hereditary algebras to a Ringel duality (theorem 106), echoing Broué’s criterion for a Morita equivalence between symmetric algebras. We use this result to prove the existence of Ringel dualities between certain subquotients kA_{(0,a_{1},...,a_{p−2})} and kB_{(a_{1},....,a_{p−2},0)} of kb_{ρ,w}^{Σ} (theorem 120). We call this collection of Ringel dualities a “walk along the abacus”, because it is reminiscent of J. A. Green’s observations on the homological algebra of the Brauer tree [36].

In the seventh chapter, we introduce the James adjustment algebra of a Hecke algebra block. This is a quotient of the block by a nilpotent ideal, whose decomposition matrix is equal to the James adjustment matrix of the block. The principal result of this chapter is theorem 129, which states that the James adjustment algebra of kb_{ρ,w}^{Σ} is Morita equivalent to a direct sum,

⨁_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} (⨂_{i=1}^{p−1} S(a_{i}, a_{i})) ,

of tensor products of Schur algebras. The decomposition matrix of kb_{ρ,w}^{Σ} is a product of the matrix of sums of products of Littlewood Richardson coefficients defined by Chuang & Tan, and Leclerc & Miyachi, and the decomposition matrix of the James adjustment algebra.

At the entrance to chapter eight, we define a novel double construction. Indeed, given a bialgebra B equipped with an algebra anti-endomorphism σ, which is also a coalgebra anti-endomorphism, and a dual bialgebra B^{∗}, we project the structure of an associative algebra onto B ⊗ B^{∗} (theorem 135). Particular examples of these doubles show a remarkably close resemblance to Rock blocks.

If Q is a quiver, let P_{Q} be the path super-algebra of Q, modulo all quadratic relations. Let P_{Q}(n) be the super-algebra Morita equivalent to P_{Q}, all of whose irreducible modules have dimension n. The coordinate ring of P_{Q}(n) is a super-bialgebra, whose double we denote D_{Q}(n). We call such an algebra a Schur quiver double, or Schiver double. We prove that D_{Q}(n) is independent of the orientation of Q (theorem 154). Conjecture 162 predicts that the Rock block kb_{ρ,w}^{Σ} is Morita equivalent to a summand D_{A_{l−1}}(w, w) of the Schiver double D_{A_{l−1}}(w) associated to a quiver of type A_{l−1}.

The principal obstacle to a proof of this conjecture via the methods introduced here, has been my impotence in producing a suitable grading on a Rock block. The Schiver doubles are naturally ℤ_{+} × ℤ_{+}-graded algebras, the degree zero part being isomorphic to a tensor product

⨂_{v∈V (Q)} S_{v}(n)

of classical Schur algebras (remark 153). Such gradings on the Rock blocks remain elusive. Towards the end of chapter 8, we sketch a proof that the graded ring associated to a certain filtration on a Rock block, is Morita equivalent to an algebra summand of a Schiver double.

If the gradings proposed here on the Rock blocks do exist, then they will pass uncanonically via derived equivalences to ℤ × ℤ-gradings on arbitrary blocks. Is it true that every symmetric group block can be positively graded, so that the degree zero part is isomorphic to the James adjustment algebra of the block ?

The ninth, and final chapter of this article continues the study of Schiver doubles. Given any vertex v of the quiver Q with an arrow a emanating from it, we define a complex P_{r}(a) for D_{Q}(n, r) whose homology groups are all S_{v}(n, r) modules, and whose character is the power sum p_{r} (theorem 166).

We finally consider Schiver doubles associated to quivers of type A_{∞}, which enjoy a number of special homological properties. So long as n ≥ r, the module category of D_{A_{∞}}(n, r) is a highest weight category, and Ringel self-dual (theorem 173). We speculate that any Rock block kb_{ρ,w}^{S_{q}} of a q-Schur algebra is Morita equivalent to a certain subquotient of D_{A_{∞}}(w, w) (conjecture 176). We prove the existence of a long exact sequence of D_{A_{∞}}(n, r)- modules

......... → D_{A_{∞}}(n, r) → D_{A_{∞}}(n, r) → D_{A_{∞}}(n, r) → .........,

which generalises Green’s walk along the Brauer tree for an infinite Brauer line. (theorem 180)

Chapters one and two contain introductory material. Most of it should be familiar to students of symmetric groups, q-Schur algebras, or the like. General aspects of finite group representation theory such as Brauer correspondence are often omitted from presentations of type A representation theory, but we include a brief account of this correspondence here. I consider this to be important philosophically, as well as being necessary for some of our proofs. This article was conceived before the fire of modular representation theory laid by R. Brauer, and his vision of a sympathy between global and local representations is bred in its bones.

The third, fourth, and fifth chapters all make use of local representation theory, and should properly be read consecutively. The appearance of Schur algebras in chapter five should not be a great surprise to students of semisimple algebraic groups familiar with Steinberg’s tensor product theorem. However, I hope our approach via the Brauer morphism is at least provocative: it is so far unclear how to interpret J. Alperin’s conjecture homologically in nonabelian defect.

Chapter six can be read independently of chapters three to five, and rests on the theory of quasi-hereditary algebras. With R. Paget’s description of the Mullineux map on a Hecke algebra Rock block we cobble a pair of shiny black boots; wearing these we are able to comfortably walk along the abacus.

The description of the James adjustment algebra of a symmetric group Rock block in chapter seven relies on all the theory developed in earlier chapters. The results of chapter three allow one to understand some aspects of Rock blocks of Hecke algebras in characteristic zero, at a root of unity. The Schur algebra quotient of chapter five provides information which can be carried across the abacus using the Ringel dualities of chapter six.

Beyond this complex crescendo come chapters eight and nine. The conjectures made here concerning Rock blocks appear to be quite deep, and if proved, would envelop all the results of earlier chapters. However, their presentation is logically independent of chapters three to seven, and carries a lighter burden of notation.

The development of the article is in the direction of Time’s arrow, so that more recent ideas appear towards the end of the monograph.

I am most grateful to Joe Chuang, to Karin Erdmann, and to Rowena Paget, for encouraging me amongst these ideas, and to Steffen Koenig. Hannah Turner supported me financially (partly), and libidinously (entirely). The E.P.S.R.C. gave me some money, as well.

This work, its morality, and the wilful emotions which dominated its creation, are dedicated to Joe Silk. He was a dear, demonic friend to me, and I wish ... to wish him farewell.

Chapter I

Highest weight categories, q-Schur algebras, Hecke algebras, and finite general linear groups.

We brutally summarise the representation theory of the q-Schur algebra, of Hecke algebras of type A, and of finite general linear groups in nondescribing characteristic.

Although in later chapters, we will invoke such theory over more general commutative rings, for simplicity of presentation, in this chapter we only consider representation theory over a field k, of characteristic l.

Highest weight categories.

We state some of the principal definitions and results of E. Cline, B. Parshall, and L. Scott’s paper, [17].

Definition 1 ([17], 3.1) Let C be a locally Artinian, Abelian category over k, with enough injectives. Let Λ be an interval finite partially ordered set. The category C is a highest weight category with respect to Λ if,

(a) Λ indexes a complete collection {L(λ)}_{λ∈Λ} of non-isomorphic simple objects of C.

(b) Λ indexes a collection {∇(λ)}_{λ∈Λ} of “costandard objects” of C, for each of which there exists an embedding L(λ) ↪ ∇(λ), such that all composition factors L(μ) of ∇(λ)/L(λ) satisfy μ < λ.

(c) For λ, μ ∈ Λ, we have, dim_{k}Hom(∇(λ), ∇(μ)) < ∞, and in addition, [∇(λ) : L(μ)] < ∞.

(d) An injective envelope I(λ) ∈ C of L(λ) possesses a filtration

0 = F_{0}(λ) ⊂ F_{1}(λ) ⊂ ...,

such that,

(i) F_{1}(λ) ≅ ∇(λ).

(ii) For n > 1, we have F_{n}(λ)/F_{n−1}(λ) ≅ ∇(μ), for some μ = μ(n) > λ.

(iii) For μ ∈ Λ, we have μ(n) = μ for finitely many μ.

(iv) I(λ) = ⋃_{i} F_{i}(λ).

Definition 2 ([17], 3.6) Let S be a finite dimensional algebra over k. Then S is said to be quasi-hereditary if the category S − mod, of finitely generated left S-modules is a highest weight category.

For M ∈ C, and Γ ⊂ Λ, let M_{Γ} be the largest subobject of M, all of whose composition factors L(γ) correspond to elements γ ∈ Γ.

Theorem 3 ([17], 3.5) Let C be a highest weight category with respect to Λ. Let Γ ⊂ Λ be a finitely generated ideal, and let Ω ⊂ Λ be a finitely generated coideal. Suppose that Γ ∩ Ω is a finite set.

There exists a quasi-hereditary algebra S(Γ∩Ω) with poset Γ∩Ω, unique up to Morita equivalence, such that the derived category D^{b}(S(Γ∩Ω)−mod) may be identified as the full subcategory of D^{b}(C) represented as complexes of finite sums of modules I(ω)_{Γ}, with ω ∈ Γ ∩ Ω. □

Remark 4 If S is a quasi-hereditary algebra with module category C, under the hypotheses of theorem 3 we may choose S(Γ ∩ Ω) to be a subquotient i(S/SjS)i of S, where i, j are certain idempotents in S.

Theorem 5 ([17], 3.4, 3.11, [16], 4.3b) Let S be a quasi-hereditary algebra, with respect to a poset Λ. Then,

(a) Λ indexes a collection {Δ(λ)}_{λ∈Λ} of “standard objects” of C, for each of which there exists a surjection ϕ_{λ} : Δ(λ) → L(λ), such that all composition factors L(μ) of ker(ϕ_{λ}) satisfy μ < λ.

(b) The projective cover P (λ) of L(λ) possesses a filtration,

P (λ) = G_{0}(λ) ⊃ G_{1}(λ) ⊃ ... ⊃ G_{N}(λ) = 0,

such that,

(i) G_{0}(λ)/G_{1}(λ) ≅ Δ(λ).

(ii) For n > 0, we have G_{n}(λ)/G_{n+1}(λ) ≅ Δ(μ), for some μ > λ.

(c) For projective objects P in C, the number [P : Δ(λ)] of objects Δ(λ) appearing in a filtration by standard objects, is independent of filtration, for λ ∈ Λ.

For injective objects I in C, the number [I : ∇(λ)] of objects ∇(λ) appearing in a filtration by standard objects, is independent of filtration, for λ ∈ Λ.

(d) [I(μ) : ∇(λ)] = [Δ(λ) : L(μ)], for λ, μ ∈ Λ.

Dually, [P (μ) : Δ(λ)] = [∇(λ) : L(μ)], for λ, μ ∈ Λ. □

Remark 6 Let S be a finite dimensional algebra, whose simple modules {L(λ)}_{λ∈Λ} are parametrised by a poset Λ. Suppose that S satisfies conditions (a) and (b) of theorem 5. Then S is quasi-hereditary with respect to Λ, by a dual to theorem 5.

Definition 7 The decomposition matrix of a quasi-hereditary algebra S is the matrix (d_{λμ}) of composition multiplicities ([Δ(λ) : L(μ)]), whose rows amd columns are indexed by Λ.

Let S be a quasi-hereditary algebra with respect to Λ. We describe some elements of the theory of tilting modules for S, due to C. Ringel [58] (see also [26], A4).

Definition 8 A tilting module for S is a finite dimensional S-module, which may be filtered by standard modules, and may also be filtered by costandard modules.

Theorem 9 ([26], A4, theorem 1) For λ ∈ Λ, there is an indecomposable tilting module T (λ), unique up to isomorphism, such that [T (λ) : L(λ)] = 1, and all composition factors L(μ) of T (λ) satisfy μ ≤ λ.

Every tilting module for S is a direct sum of modules T (λ), λ ∈ Λ. □

Definition 10 A full tilting module for S is a tilting module, in which every T (λ) occurs as a direct summand.

A Ringel dual S′ of S is defined to be End_{S}(T)^{op}, where T is a full tilting module for S.

Remark 11 The Ringel dual of S is unique, up to Morita equivalence. We say the bimodule _{S}T_{S^{′op}} defines a Ringel duality between S, S′.

Theorem 12 The Ringel duals S′ of S are quasi-hereditary algebras, with respect to the poset Λ^{op}, opposite to Λ.

The standard S′-module corresponding to λ ∈ Λ^{op} is given by

Δ′(λ) = Hom_{S}(T, ∇(λ)).

The costandard S′-module corresponding to λ ∈ Λ^{op} is given by

∇′(λ) = Δ(λ)^{op} ⊗_{S} T.□

q-Schur algebras.

S. Donkin, and R. Dipper have associated, to a natural number n, and a non-zero element q ∈ k, a bialgebra A_{q}(n) [23]. This bialgebra is a qdeformation of the bialgebra A(n) = k[M] of regular functions on the associative algebra M = M_{n}(k), of n × n matrices over k. Whilst the undeformed bialgebra A(n) = A_{1}(n) is commutative, A_{q}(n) is noncommutative, for q ≠ 1.

In general, A_{q}(n) may be decomposed by degree as a direct sum,

A_{q}(n) = ⨁_{r≥0} A_{q}(n, r)

of finite dimensional coalgebras. Thus, upon writing S_{q}(n, r) for A_{q}(n, r)^{∗}, the bialgebra S_{q}(n), defined to be the graded dual of A_{q}(n), decomposes as a direct sum,

S_{q}(n) = ⨁_{r≥0} S_{q}(n, r)

of finite dimensional algebras. The algebras S_{q}(n, r) are the q-Schur algebras, first introduced by R. Dipper and G. James [25].

Let Λ(n, r) be the poset of partitions of r with n parts or fewer, dominantly ordered.

Theorem 13 ([26], 0.22) The q-Schur algebra S_{q}(n, r) is a highest weight category with respect to Λ(n, r). □

Let p be the least natural number such that 1 + q + ... + q^{p−1} = 0, if such exists. Otherwise, let p = ∞.

Theorem 14 ([26], 4.3(7)) If p = ∞, then S_{q}(n, r) is semisimple for all n, r. In this case, Δ(λ) ≅ L(λ), for all λ ∈ Λ(n, r).

Remark 15 The algebra S_{q}(n, r) possesses a natural anti-automorphism σ, inherited from the transpose operation on the matrix algebra M ([26], 4.1).

It is thus possible to twist a left/right S_{q}(n, r)-module by σ, to obtain a right/left S_{q}(n, r)-module. Composing this twist with the duality functor, we may associate to a left/right S_{q}(n, r)-module N, a left/right S_{q}(n, r)- module N^{∗}, its contragredient dual.

The contragredient dual of a standard module Δ(λ) is isomorphic to the costandard module, ∇(λ). The contragredient dual of a costandard module ∇(λ) is isomorphic to the standard module, Δ(λ).

Remark 16 In general, S_{q}(n, 1) is isomorphic to the matrix algebra M, and therefore has a unique irreducible left module E , and a unique irreducible right module E^{op}.

S. Donkin has defined the q-exterior powers ⋀_{q}^{r} E (respectively ⋀_{q}^{r} E^{op}) of E (respectively E^{op}), which are left (respectively right) S_{q}(n, r)-modules of

dimension ( \begin{matrix}n \\ r \end{matrix} ), exchanged under the anti-automorphism σ ([26], 1.2). He

has also defined q-exterior powers ⋀_{q}^{r} M of M, which are S_{q}(n, r)-S_{q}(n, r)-

bimodules of dimension ( \begin{matrix}n^{2} \\ r \end{matrix} ) ([26], 4.1).

For a sequence α = (α_{1}, ...., α_{m}) of natural numbers, whose sum is r, let us define

⋀_{q}^{α} E = ⋀_{q}^{α_{1}} E ⊗ ... ⊗ ⋀_{q}^{α_{m}} E.

⋀_{q}^{α} E^{op} = ⋀_{q}^{α_{1}} E^{op} ⊗ ... ⊗ ⋀_{q}^{α_{m}} E^{op}.

Theorem 17 (S. Donkin, [26], 1.2, 4.1)

(a) For τ ∈ Σ_{m} = Sym{1, ..., m}, there is an S_{q}(n, r)-module isomorphism between the exterior powers, ⋀_{q}^{(α_{1},...,α_{m})} E, and ⋀_{q}^{(α_{τ1},...,α_{τm})} E.

(b) There is a non-degenerate bilinear form,

<, >: ⋀_{q}^{α} E^{op} × ⋀_{q}^{α} E → k,

such that < x ∘ s, y >=< x, s ∘ y >. Therefore, ⋀_{q}^{α} E ≅ (⋀_{q}^{α} E)^{∗}, as left S_{q}(n, r)-modules, and ⋀_{q}^{α} E^{op} ≅ (⋀_{q}^{α} E^{op})^{∗}, as right S_{q}(n, r)-modules.

(c) Direct summands of q-exterior powers ⋀_{q}^{α} E are tilting modules for S_{q}(n, r). The restriction of ⋀_{q}^{r} M to a left S_{q}(n, r)-module is a full tilting module. The restriction of ⋀_{q}^{r} M to a right S_{q}(n, r)-module is also a full tilting module.

(d) Let n ≥ r. The S_{q}(n, r)-S_{q}(n, r)-bimodule ⋀_{q}^{r} M defines a Ringel duality between S_{q}(n, r) and S_{q}(n, r)^{op} = S_{q}(n, r).

Hecke algebras associated to symmetric groups.

Let Σ_{r} = Sym{1, 2, ..., r} be the symmetric group on r letters.

Definition 18 The Hecke algebra H_{q}(Σ_{r}) associated to Σ_{r}, is the associative k-algebra with generators

{T_{(i,i+1)}|i = 1, ..., r − 1},

subject to the relations,

T_{(i,i+1)}T_{(i+1,i+2)}T_{(i,i+1)} = T_{(i+1,i+2)}T_{(i,i+1)}T_{(i+1,i+2)}, i = 1, ..., r − 2,

T_{(i,i+1)}T_{(j,j+1)} = T_{(j,j+1)}T_{(i,i+1)}, |j − i| > 1,
(T_{(i,i+1)} − q)(T_{(i,i+1)} + 1) = 0, i = 1, ..., r − 1.

When q = 1, the Hecke algebra H_{q}(Σ_{r}) is isomorphic to the group algebra kΣ_{r}. Many of the properties of kΣ_{r} generalise to H_{q}(Σ_{r}). For example, we have the following theorem.

Theorem 19 (a) H_{q}(Σ_{r}) possesses an outer automorphism #, the “sign automorphism”, which takes T_{(i,i+1)} to −T_{(i,i+1)} + q − 1.

(b) Given w ∈ Σ_{r}, and a reduced expression w = s_{1}...s_{l(w)} as a product simple transpositions s_{j} ∈ {(i, i + 1), 1 ≤ i ≤ i − 1}, we may define an element T_{w} = T_{s_{1}}...T_{s_{l(w)}} ∈ H_{q}(Σ_{r}). The element T_{w} is independent of the choice of reduced expression.

(c) The set {T_{w}|w ∈ Σ_{r}} is a basis for H_{q}(Σ_{r}). □

Remark 20 There is a one dimensional “trivial module” for H_{q}(Σ_{r}), denoted k, on which T_{w} acts as q^{l(w)}. The twist of the trivial module by # is the “sign module”, denoted sgn, on which T_{w} acts as (−1)^{l(w)}.

There is an elementary relation between Hecke algebras, and q-Schur algebras, which is commonly exploited to describe the representation theory of the Hecke algebra. A proof, with references to various sources, is given in [46], 1.2.

Theorem 21 “Schur-Weyl duality”

(a) The S_{q}(n, r)-module E^{⊗r} is a tilting module.

(b) There is an algebra surjection H_{q}(Σ_{r}) ↠ End_{S_{q}(n,r)}(E^{⊗r}).

(c) S_{q}(n, r) = End_{H_{q}(Σ_{r})}(E^{⊗r}).

The S_{q}(n, r)-H_{q}(Σ_{r})-bimodule E^{⊗r} is often referred to as tensor space. If n ≥ r, then this bimodule is particularly regular.

Theorem 22 (J.A. Green [37], S. Donkin, [26], 2.1, 4.4) Let n ≥ r. (a) There is an idempotent ξ_{ω} ∈ S_{q}(n, r), such that

E^{⊗r} ≅ S_{q}(n, r)ξ_{ω},

as S_{q}(n, r)-modules.

(b) The S_{q}(n, r)-module E^{⊗r} is projective, and injective.

(c) H_{q}(Σ_{r}) = End_{S_{q}(n,r)}(E^{⊗r}) = ξ_{ω}S_{q}(n, r)ξ_{ω}.

(d) The Schur functor,

Hom_{S_{q}(n,r)}(E^{⊗r}, −) : S_{q}(n, r) − mod → H_{q}(Σ_{r}) − mod,

is exact, and takes simple modules either to simple modules, or to zero. We obtain all simple H_{q}(Σ_{r})-modules from the set of simple S_{q}(n, r)-modules, in this way.

(e) If λ is a p-nonrestricted partition, then Hom_{S_{q}(n,r)}(E^{⊗r}, L(λ)) = 0. Otherwise, if λ is a p-restricted partition, then D_{λ} = Hom_{S_{q}(n,r)}(E^{⊗r}, L(λ)) is a simple H_{q}(Σ_{r})-module.

(f) The set,

{D_{λ} | λ a p-restricted partition of r},

is a complete set of non-isomorphic irreducible H_{q}(Σ_{r})-modules. □

Remark 23 A partition λ is said to be p-regular if, and only if, λ does not have ≥ p identical parts.

A partition λ is said to be p-restricted if, and only if, λ′ is p-regular.

Remark 24 Let D^{λ′} = D_{λ}^{#}. It follows from theorem 22(f) that

{D^{λ} | λ a p-regular partition of r},

is a complete set of non-isomorphic irreducible H_{q}(Σ_{r})-modules.

Definition 25 The Specht module associated to a partition λ is the H_{q}(Σ_{r})- module,

S^{λ} = Hom_{S_{q}(r,r)}(E^{⊗r}, ∇(λ)).

The H_{q}(Σ_{r})-module S_{λ} is defined to be,

S_{λ} = Hom_{S_{q}(r,r)}(E^{⊗r}, Δ(λ)).

The decomposition matrix of H_{q}(Σ_{r}) is the matrix (d_{λμ}) of composition multiplicities [S^{λ} : D^{μ}], indexed by partitions λ of r, and p-regular partitions μ of r.

Remark 26 (a) We have S^{λ#} ≅ S_{λ′}.

(b) When q = 1, and k is a field of characteristic zero, the Specht modules S^{λ} form a complete set of non-isomorphic simple kΣ_{r}-modules. We write χ^{λ} for the irreducible character of Σ_{r} corresponding to S^{λ}.

Finite general linear groups.

Throughout this section, q is a prime power, coprime to l. We consider representations of the finite general linear group GL_{n}(q), over the field k.

Let V be an n dimensional vector space over 𝔽_{q}. Let \underline{n} = (n_{1}, …, n_{N}) be a sequence of natural numbers whose sum is n. Let us fix an 𝔽_{q}-basis of V. We define a collection of subgroups of GL(V) = GL_{n}(𝔽_{q}) = GL_{n}(q) relative to this basis:

A maximal torus T (q) - the subgroup of diagonal matrices.

A Borel subgroup B(q) - the subgroup of upper triangular matrices. A Levi subgroup L\underline{n}(q) - viewed as,

( \begin{matrix}GL_{n_{1}}(q) & 0 & … & 0 \\ 0 & ... & ... & ⋮ \\ ⋮ & ... & ... & 0 \\ 0 & … & 0 & GL_{n_{N}} (q) \end{matrix} )

A parabolic subgroup P\underline{n}(q) - viewed as,

( \begin{matrix}GL_{n_{1}}(q) & ∗ & … & ∗ \\ 0 & ... & ... & ⋮ \\ ⋮ & ... & ... & ∗ \\ 0 & … & 0 & GL_{n_{N}} (q) \end{matrix} )

A unipotent subgroup U\underline{n}(q) - viewed as,

( \begin{matrix}1_{n_{1}} & ∗ & … & ∗ \\ 0 & ... & ... & ⋮ \\ ⋮ & ... & ... & ∗ \\ 0 & … & 0 & 1_{n_{N}} \end{matrix} )

The Weyl group W ≅ Σ_{n} - permutation matrices. Lemma 27 (a) The normaliser of U\underline{n}(q) is P\underline{n}(q) = U\underline{n}(q) ⋊ L\underline{n}(q).

(b) We have Bruhat decomposition,

GL_{n}(q) = ∐_{w∈W} B(q)wB(q).□

Definition 28 Harish-Chandra induction is the functor, HCInd_{L\underline{n}(q)}^{GL_{n}(q)} : kL\underline{n}(q) − mod → kGL_{n}(q) − mod,

M ↦ k[GL_{n}(q)/U\underline{n}(q)] ⨂_{L\underline{n}(q)} M.

Harish-Chandra restriction is the functor, HCRes_{L\underline{n}(q)}^{GL_{n}(q)} : GL_{n}(q) − mod → L\underline{n}(q) − mod,

M ↦ k[U\underline{n}(q)\GL_{n}(q)] ⨂_{GL_{n}(q)} M.

Remark 29 The kGL_{n}(q)-kL\underline{n}(q) bimodule

k[GL_{n}(q)/U\underline{n}(q)],

is a summand of the ordinary induction bimodule _{GL_{n}(q)}k[GL_{n}(q)]_{L\underline{n}(q)}, and so Harish-Chandra induction is exact. Its left and right adjoint, Harish-Chandra restriction, is also exact.

Assuming the notation of J. Brundan, R. Dipper, and A. Kleshchev’s article [10], we introduce a set to parametrise conjugacy classes in GL_{n}(q).

For σ ∈ 𝔽̄_{q}^{∗} of degree d_{σ} over 𝔽_{q}, let (σ) be the companion matrix representing σ in GL_{d_{σ}}(q).

For a natural number k, let

(σ)^{k} = diag((σ), ..., (σ)) = ( \begin{matrix}(σ) & 0 & … & 0 \\ 0 & ... & ... & ⋮ \\ ⋮ & ... & ... & 0 \\ 0 & … & 0 & (σ) \end{matrix} )

be the block diagonal matrix embedding k copies of (σ) in GL_{d_{σ}k}(q).

For σ, τ ∈ 𝔽̄_{q}^{∗} of degree d over 𝔽_{q}, let us write σ ∼ τ if σ and τ have the same minimal polynomial over 𝔽_{q}.

Definition 30 (a) Let C_{ss}^{pre} be the set,

{((σ_{1})^{k_{1}}, ..., (σ_{a})^{k_{a}}) | σ_{i} ∈ 𝔽̄_{q}^{∗}, σ_{i} ≁ σ_{j} for i ≠ j, ∑ d_{σ_{i}}k_{i} = n}.

(b) Let ∼ denote an equivalence relation on C_{ss}^{pre}, given by

((σ_{1})^{k_{1}}, ..., (σ_{a})^{k_{a}}) ∼ ((τ_{1})^{m_{1}}, ..., (τ_{a})^{m_{b}})

so long as,

(i) a = b,

(ii) there exists w ∈ Σ_{a}, such that k_{wi} = m_{i}, and σ_{wi} ∼ τ_{i}, for i = 1, ..., a.

(c) Let C_{ss} = C_{ss}^{pre}/ ∼.

For s = ((σ_{1})^{k_{1}}, ..., (σ_{a})^{k_{a}}) ∈ C_{ss}, let κ(s) = (k_{1}, ..., k_{a}).

If \underline{λ} = (λ_{1}, ..., λ_{a}) is a sequence of partitions whose orders are described by the sequence κ, we write \underline{λ} ⊢ κ.

Theorem 31 (Jordan decomposition) The conjugacy classes of GL_{n}(q) are in one-one correspondence with the set,

\{(s, \underline{λ}) | s ∈ C_{ss}, \underline{λ} ⊢ κ(s)\}.□

Theorem 32 (J.A. Green, [35], [10], 2.3) Fix an embedding 𝔽̄_{q}^{∗} ↪ ℚ̄_{l}^{∗}. There is a one-one correspondence between the set of ordinary irreducible characters of GL_{n}(q) and the set,

\{(s, \underline{λ}) | s ∈ C_{ss}, \underline{λ} ⊢ κ(s)\}.□

Given an embedding 𝔽̄_{q}^{∗} ↪ ℚ̄_{l}^{∗}, and s ∈ C_{ss}, \underline{λ} ⊢ κ(s), we write χ(s, \underline{λ}) for the irreducible character of GL_{n}(q) corresponding to the pair (s, \underline{λ}) in the above theorem.

Remark 33 Let s = ((σ_{1})^{k_{1}}, ..., (σ_{a})^{k_{a}}), and \underline{λ} ⊢ κ(s). The irreducible character χ(s, \underline{λ}) corresponding to the pair (s, \underline{λ}), is equal to the Harish-Chandra induced character,

HCInd_{GL_{dσ_{1} k_{1}}(q)×...×GL_{dσ_{a} ka}(q)}^{GL_{n}(q)} (χ((σ_{1})^{k_{1}}, λ_{1}) ⊗ ... ⊗ χ((σ_{a})^{k_{a}}, λ_{a})) .

Furthermore, if μ, ν are partitions of m, n, and σ ∈ 𝔽̄_{q}^{∗} has degree d, then,

HCInd_{GL_{dm}(q)×GL_{dn(q)}}^{GL_{dm+dn}(q)} (χ((σ)^{m}, μ) ⊗ χ((σ)^{n}, ν))
= ∑ c(λ; μ, ν)χ((σ)^{m+n}, λ),

where c(λ; μ, ν) is the Littlewood-Richardson coefficient associated to λ, μ, ν.

Let C_{ss,l′} = {s ∈ C_{ss} | the order of s is coprime to l}.

Theorem 34 ([10], 2.4.6) There is a decomposition of the group algebra kGL_{n}(q) into a direct sum of two sided ideals,

kGL_{n}(q) = ⨁_{s∈C_{ss,l′}} kB_{s},

where the characters in B_{s} are given by the set,

\{χ(t, \underline{λ}) | \underline{λ} ⊢ κ(t), t ∈ C_{ss} has l-regular part conjugate to s\}.□

Definition 35 The characters {χ(1, λ) | λ ⊢ n} of GL_{n}(q) are the unipotent characters.

The indecomposable components of _{GL_{n}(q)}kB_{1}_{GL_{n}(q)} are the unipotent blocks.

Thanks to the following theorem, we may concentrate our curiosity on unipotent blocks.

Theorem 36 (C. Bonnafé, R. Rouquier, [5], 11.8) Every block of GL_{n}(q) is Morita equivalent to a unipotent block. □

Theorem 37 (N. Iwahori, E. Cline, B. Parshall, L. Scott, see [10], 3.2d, 3.5a) Let M = k[GL_{n}(q)/B(q)] denote the permutation module for kGL_{n}(q) on the coset space GL_{n}(q)/B(q).

(a) If q ≠ 1 modulo l, then M is a projective kGL_{n}(q)-module.

(b) M is a kB_{1}-module.

(c) End_{kB_{1}}(M) ≅ H_{q}(Σ_{n}). Under this isomorphism, the endomorphism defined by the double coset B(q)wB(q) maps to the element T_{w} of H_{q}(Σ_{n}), for w ∈ Σ_{n}.

(d) The annihilator ideal Ann_{kB_{1}}(M) is nilpotent.

(e) The quotient kB_{1}/Ann_{kB_{1}}(M) is Morita equivalent to S_{q}(n, n).

The standard S_{q}(n, n)-module Δ(λ) corresponds, under this Morita equivalence, to a kB_{1}-module, which is an l-modular reduction of the character χ(1, λ) of GL_{n}(q).

(f) The S_{q}(n, n)-H_{q}(Σ_{n})-bimodule corresponding to the kB_{1}-H_{q}(Σ_{n})-bimodule M under the Morita equivalence of (d), is isomorphic to twisted tensor space E^{⊗n#}. □

Definition 38 By theorem 37, simple kB_{1}-modules are in one-one correspondence with simple modules for the q-Schur algebra, which are in one-one correspondence with the set Λ(n) = Λ(n, n). We denote by D(λ) the simple kB_{1}-module corresponding to λ ∈ Λ(n).

Remark 39 By twisted tensor space E^{⊗n#}, we mean the tensor space of theorem 21, on which the right action of H_{q}(Σ_{n}) has been twisted by the signature automorphism, #.

Wreath products.

For a k-algebra A, and a natural number w, the wreath product A ≀ Σ_{w}, is defined to be the space A^{⊗w} ⊗ kΣ_{w}, with associative multiplication,

(x_{1} ⊗ ... ⊗ x_{w} ⊗ σ).(y_{1} ⊗ ... ⊗ y_{w} ⊗ τ) =
x_{1}.y_{σ^{−1}1} ⊗ ... ⊗ x_{w}.y_{σ^{−1}w} ⊗ σ.τ,

for x_{1}, ..., x_{w}, y_{1}, ..., y_{w} ∈ A, and σ, τ ∈ Σ_{w}. We collect a jumble of information concerning wreath products. One reference for such stuff is an article of J. Chuang and K.M. Tan [15].

For an A-module L, let us define the A ≀ Σ_{w}-module T^{(w)}L to be the w-fold tensor product L^{⊗w}, on which the subalgebra A^{⊗w} acts componentwise, and the symmetric group Σ_{w} acts by place permutations.

For an A ≀ Σ_{w}-module M, and a kΣ_{w}-module N, we define an A ≀ Σ_{w}- module M ⊗ N, by the action

(α ⊗ σ)(m ⊗ n) = (α ⊗ σ)m ⊗ σn,

for α ∈ A^{⊗w}, σ ∈ Σ_{w}, m ∈ M, n ∈ N.

Let I be a finite set. Given a set of natural numbers {w_{i}, i ∈ I}, whose sum is w, there is a Young subgroup of Σ_{w}, isomorphic to ×_{i∈I}Σ_{w_{i}}. Accordingly, there is a subalgebra of A ≀ Σ_{w}, isomorphic to ⨂_{i∈I} A ≀ Σ_{w_{i}}.

Let Λ_{l} be the set of all l-regular partitions. Let Λ_{l,w}^{I} be the set of I-tuples (λ^{i})_{i∈I} of l-regular partitions, whose orders (w_{i})_{i∈I} sum to w.

Theorem 40 (I. MacDonald, [51]) Let {S(i), i ∈ I} be a complete set of non-isomorphic simple A-modules. For \underline{λ} ∈ Λ_{l,w}^{I}, let

S(\underline{λ}) = Ind_{⊗_{i∈I}A≀Σ_{w_{i}}}^{A≀Σ_{w}} (⨂_{i∈I} T^{w_{i}}(S(i)) ⊗ D^{λ^{i}}) .

The set \{S(\underline{λ}) | λ ∈ Λ_{l,w}^{I}\}, is a complete set of non-isomorphic simple A ≀ Σ_{w}-modules. □

Let q be a prime power, coprime to l > 0. Let B(q) be a Borel subgroup of GL_{n}(q). Thus, the direct product ×^{w}B(q) of w copies of B(q) is a subgroup of the base group ×^{w}GL_{n}(q) inside GL_{n}(q) ≀ Σ_{w}.

The following theorem is a straightforward generalisation of theorem 37.

Theorem 41 Let M_{w} be the kB_{1} ≀ Σ_{w}-module k[GL_{n}(q) ≀ Σ_{w}/ ×^{w} B(q)].

(a) If q ≠ 1 modulo l, then M_{w} is a projective kGL_{n}(q) ≀ Σ_{w}-module.

(b) End_{GL_{n}(q)≀Σ_{w}}(M_{w}) ≅ H_{q}(Σ_{n}) ≀ Σ_{w}.

(c) The simple modules sent to zero by the functor,

Hom_{GL_{n}(q)≀Σ_{w}}(M_{w}, −),

are precisely those simples S(\underline{λ}) ∈ Λ_{l,w}^{Λ(n)}, for which some entry, λ^{μ} in \underline{λ}, is non-zero, for some p-nonrestricted partition μ. □

Proposition 42 The wreath product H_{q}(Σ_{n}) ≀ Σ_{w} is a symmetric algebra.

Proof:

Define a form <, > on H_{q}(Σ_{n}), extending the following form bilinearly:

< T_{u}, T_{v} >= \{ \begin{matrix}q^{l(u)} & if u = v^{−1} \\ 0 & otherwise. \end{matrix}

R. Dipper and G. James have proved ([24], (2.3)) that this is a symmetric, associative bilinear form. It is also non-degenerate, since q is a unit in k.

In case n = w, q = 1, one obtains a symmetric bilinear form <, >_{Σ} on kΣ_{w}. Now define a bilinear form on H_{q}(Σ_{n}) ≀ Σ_{w}, by the formula,

< x_{1} ⊗ ... ⊗ x_{w} ⊗ σ, y_{1} ⊗ ... ⊗ y_{w} ⊗ τ >_{≀}=
< x_{1}, y_{σ^{−1}1} > ... < x_{w}, y_{σ^{−1}w} >< σ, τ >_{Σ},

for x_{1}, ..., x_{w}, y_{1}, ..., y_{w} ∈ H_{q}(Σ_{n}), and σ, τ ∈ Σ_{w}. Then <, >_{≀} is an associative, symmetric, non-degenerate bilinear form on H_{q}(Σ_{n}) ≀ Σ_{w}. □

Chapter II

Blocks of q-Schur algebras, Hecke algebras, and finite general linear groups.

Let k be a field of characteristic l. We summarise some elements of block theory for the q-Schur algebras, the Hecke algebras, and the finite general linear groups in non-describing characteristic. We introduce the Rock blocks, and state some known results concerning them.

Abacus combinatorics.

Let p be a natural number. It is often convenient for us to picture partitions on an abacus with p runners, following G. James ([41], pg. 78).

We thus label the runners on an abacus 0, ..., p − 1, from left to right, and label its rows 0, 1, ..., from the top downwards. If λ = (λ_{1}, λ_{2}, ...) is a partition with m parts or fewer then we may represent λ on the abacus with m beads: for i = 1, ..., m, write λ_{i} + m − i = s + pt, with 0 ≤ s ≤ p − 1, and place a bead on the s^{th} runner in the t^{th} row. For example,

[figure: 0; 1; 2]

is an abacus representation of the partition (6, 4, 2^{2}, 1^{2}), when p = 3.

Sliding a bead one row upwards on its runner, into a vacant position, corresponds to removing a p-hook from the rim of the partition λ. Given an abacus representation of a partition, sliding all the beads up as far as possible produces an abacus representation of the p-core of that partition, a partition from which no further rim p-hooks can be removed. The pictured example (6, 4, 2^{2}, 1^{2}) is therefore a 3-core. The p-core of a partition is independent of the way in which p-hooks have been removed. The p-weight of a partition is the number of p-hooks removed to obtain the p-core.

Fix an abacus representation of a partition λ, and for i = 0, ..., p − 1, let λ_{1}^{i} be the number of unoccupied positions on the i^{th} runner which occur above the lowest bead on that runner. Let λ_{2}^{i} be the number of unoccupied positions on the i^{th} runner which occur above the second lowest bead on that runner, etc. etc.. Then λ^{i} = (λ_{1}^{i}, λ_{2}^{i}, ...) is a partition, and the p-tuple [λ^{0}, ..., λ^{p−1}] is named the p-quotient of λ. The p-quotient depends on the number of beads in the abacus representation of λ. The p-weight of λ is equal to the sum |λ^{0}| + ... + |λ^{p−1}|.

The partitions with a given p-core τ and p-weight w may be parametrized by p-quotients:

Fix m so that any such partition has m parts or fewer. Representing the partitions on an abacus with m beads, there is a p-quotient for each one. We thus introduce a bijection between the set of partitions with p-core τ and weight w, and the set of p-tuples [σ^{0}, ..., σ^{p−1}] such that |σ^{0}|+...+|σ^{p−1}| = w. It is a convention of this article, frequently to label a partition by its pquotient.

Whenever we represent partitions with a given p-core on an abacus, we assume that m is fixed as above.

Block parametrisation.

Let A_{S_{q}}(n) = S_{q}(n, n), let A_{H_{q}}(n) = H_{q}(Σ_{n}), and let A_{G_{q}}(n) = kB_{1}, a direct sum of the unipotent blocks of kGL_{n}(q).

We give parametrizations for the blocks of A_{X} (n) , for X ∈ {S_{q}, H_{q}, G_{q}}. The parametrizations depend on an invariant p.

Definition 43 If X ∈ {S_{q}, H_{q}}, then p = p(X) is defined to be the least natural number such that,

1 + q + ... + q^{p−1} = 0,

in k if such exists, and p = ∞, otherwise.

If X = G_{q}, for some prime power q, then we insist that l > 0. In this case, p = p(X) is defined to be the multiplicative order of q, modulo l.

Remark 44 Let l > 0. If q is a prime power such that q ≠ 1 modulo l, then p(G_{q}) = p(S_{q}).

However, if q = 1 modulo l, then p(G_{q}) = 1, while p(S_{q}) = l.

Theorem 45 (R. Brauer, P. Fong, B. Srinivasan [30], 7A, G. James, R. Dipper, [24], 4.13, S. Donkin, A. Cox, [20]) The blocks of A_{X} (n) are in one-one correspondence with pairs (w, τ), where w ∈ ℕ_{0}, and τ is a p-core of size n − wp.

We write b_{τ,w}^{X} for the block idempotent in A_{X} (n) corresponding to the pair (w, τ). We write kb_{τ,w}^{X} for the corresponding block algebra. We say that b_{τ,w}^{X} (respectively kb_{τ,w}^{X}) is a block of weight w, with core τ.

Remark 46 Here, and in the sequel, we write “block” in abbreviation of either “block idempotent”, or “block algebra”.

Theorem 47 Let τ be a partition of t, and a p-core. Let n = pw + t.

(a) A standard, costandard, or simple S_{q}(n, n)-module indexed by the partition λ lies in kb_{τ,w}^{S_{q}} if, and only if, λ has p-core τ, and p-weight w.

(b) A Specht, or simple H_{q}(Σ_{n})-module indexed by the partition λ lies in kb_{τ,w}^{H_{q}} if, and only if, λ has p-core τ, and p-weight w.

(c) The irreducible character,

χ (((1)^{k_{1}}, (σ_{2})^{k_{2}}, ..., (σ_{a})^{k_{a}}), (λ_{1}, ..., λ_{a})) ,

of GL_{n}(q) lies in kb_{τ,w}^{G_{q}} if, and only if, λ_{1} has p-core τ, and the order of σ_{i} is a power of l, for i ≥ 2. □

We write kb_{τ,w}^{S} for the block kb_{τ,w}^{S_{1}} of the Schur algebra S(n, n). We write kb_{τ,w}^{Σ} for the block kb_{τ,w}^{H_{1}} of the symmetric group Σ_{n}.

Theorem 48 (a) The blocks kb_{τ,w}^{Σ} and kb_{τ,w}^{G_{q}} have abelian defect groups if, and only if, w < l.

(b) If w < l, then a character χ (((1)^{k_{1}}, (σ_{2})^{k_{2}}, ..., (σ_{a})^{k_{a}}), (λ_{1}, ..., λ_{a})) in kb_{τ,w}^{G_{q}}, is constrained by the condition that σ_{i} is of degree p, for i ≥ 2. □

Derived equivalences, and Rock blocks.

Theorem 49 (J. Chuang, R. Rouquier, [12]) Let X be an element of the set {S_{q}, H_{q}, G_{q}, S, Σ}. Let τ, τ′ be p-cores. Then the bounded derived categories D^{b}(kb_{τ,w}^{X}), and D^{b}(kb_{τ′,w}^{X}), are equivalent. □

Definition 50 A p-core ρ = ρ(p, w) is said to be a Rouquier core if it has an abacus presentation, on which there are at least w − 1 more beads on runner i, than on runner i − 1, for i = 1, ..., p − 1.

Example 51 The partition (6, 4, 2^{2}, 1^{2}) pictured on the abacus at the beginning of the chapter is a Rouquier core, where p = w = 3.

Remark 52 Blocks of ⨁_{n≥0} H_{q}(Σ_{n}) are in one-one correspondence with weight spaces for the basic representation of 𝔰̂𝔩_{p}, as has been illucidated by A.

Lascoux, B. Leclerc, and J-Y. Thibon [48]. The Rouquier cores correspond to weight spaces on the boundary of this representation. In other words, the Rouquier cores correspond to those weight spaces which are sent to zero by some generator E_{i}, or some generator F_{i}, i ∈ ℤ/p in the Kac-Moody presentation for 𝔰̂𝔩_{p}.

R. Rouquier conjectured the following structure theorem [59].

Theorem 53 (J. Chuang, R. Kessar, [11]) Let w < l, and let ρ = ρ(p, w) be a Rouquier core. Any block kb_{ρ,w}^{Σ} of a symmetric group is Morita equivalent to the wreath product kb_{∅,1}^{Σ} ≀ Σ_{w}. □

Let X ∈ {S_{q}, H_{q}, G_{q}, S, Σ}.

Definition 54 A Rock block is any block b_{ρ,w}^{X} (respectively kb_{ρ,w}^{X}), where ρ = ρ(p, w) is a Rouquier core.

Remark 55 (a) “RoCK block” is slang for “Rouquier, or Chuang-Kessar block”.

(b) For fixed X , p, w, all Rock blocks are Morita equivalent (cf. lemma 107).

(c) Throughout this article, the letter ρ = ρ(p, w) represents a Rouquier core. Other p-cores will be represented by different letters, such as τ.

The next couple of results were deduced from theorem 53.

Theorem 56 (J. Chuang, K.M. Tan, [14]) Let w < l. The Rock block kb_{ρ,w}^{S} is Morita equivalent to kb_{∅,1}^{S} ≀ Σ_{w}. □

For a natural number a, let Λ_{w}^{a} be the set of a-tuples (α^{1}, ..., α^{a}) of partitions α^{i}, such that ∑_{i} |α^{i}| = w.

Let c(λ; μ, ν) be the Littlewood-Richardson coefficient corresponding to partitions λ, μ, ν, taken to be zero whenever |λ| ≠ |μ| + |ν|.

Theorem 57 (J. Chuang, K.M. Tan, [14], H. Miyachi, [52]) Let w < l. The decomposition matrix of the Rock block kb_{ρ,w}^{S} is equal to the decomposition matrix of kb_{∅,1}^{S} ≀ Σ_{w}. It is given by,

d_{\underline{λ}\underline{μ}} = ∑_{α∈Λ_{w}^{p+1},β∈Λ_{w}^{p}} ∏_{j=0}^{p−1} c(λ^{j}; α^{j}, β^{j})c(μ^{j}; β^{j}, (α^{j+1})′).□

The following theorem was also observed, upon viewing the canonical basis for the basic representation of 𝔰̂𝔩_{p}.

Theorem 58 (J. Chuang and K.M. Tan, [13], B. Leclerc, and H. Miyachi, [49]) Let l = 0, and let q be a p^{th} root of unity in k. The decomposition matrix of the Rock block kb_{ρ,w}^{S_{q}} is equal to the decomposition matrix of kb_{∅,1}^{S_{q}} ≀ Σ_{w}. It is given by,

d_{\underline{λ}\underline{μ}} = ∑_{α∈Λ_{w}^{p+1},β∈Λ_{w}^{p}} ∏_{j=0}^{p−1} c(λ^{j}; α^{j}, β^{j})c(μ^{j}; β^{j}, (α^{j+1})′).□

Lemma 59 The p-regular partitions [λ^{0}, λ^{1}..., λ^{p−1}] with p-core ρ, are those for which λ^{0} is empty. □

Theorem 60 (R. Paget, [54]) Let ρ be a Rouquier core. Then

D^{[∅,λ^{1},...,λ^{p−1}]} ≅ D_{[∅,λ^{p−1},...,λ^{1}]},

as kb_{ρ,w}^{H_{q}}-modules. □

Brauer correspondence for blocks of finite general linear groups.

In this book, when we are analysing the representations of a finite group G, it is convenient for us to work over an l-modular system (K, O, k) . Thus, O is a discrete valuation ring with field of fractions K, maximal ideal ℘, and residue field k = O/℘.

If a ∈ O, we denote the l-modular reduction of a by ā, an element of k. We insist that K is a splitting field for G.

Definition 61 (see [6]) For a finite group G, with an l-subgroup P, and an OG-module M, the Brauer homomorphism is defined to be the quotient map

Br_{P}^{G} : M^{P} → M(P) = M^{P} /( ∑_{Q<P} T r_{Q}^{P}(M^{Q}) + ℘M^{P}).

The Brauer quotient M(P) is the quotient of P -fixedpoints of M by relative traces from proper subgroups, reduced modulo l; M(P) is a kN_{G}(P)-module.

In case that G acts on M = OG by conjugation, without doubt OG(P) = kC_{G}(P), and the quotient map

Br_{P}^{G} : (OG)^{P} → kC_{G}(P)

is an algebra homomorphism, the classical Brauer homomorphism with respect to P, given by the rule

Br_{P}^{G}(∑_{g∈G} a_{g}g) = ∑_{g∈C_{G}(P)} ā_{g}g.

Theorem 62 (L. Scott) Let M be a kG-permutation module. Then M(P) ≠ 0 if, and only if, M has a direct summand with a vertex containing P. □

We present Brauer correspondence for unipotent blocks of finite general linear groups. Thus, let l > 0, and let q be a prime power, coprime to l.

Let t = n − wp ≥ 0. Let

L_{1} = ×^{w}GL_{p}(q) < GL_{wp}(q),
L_{2} = GL_{wp}(q) × GL_{t}(q) < GL_{n}(q),

be Levi subgroups. Let D be a Sylow l-subgroup of GL_{wp}(q).

Let τ be a partition of t, and a p-core. Let b_{τ,w}^{G_{q}} (respectively b_{τ,0}^{G_{q}}) be the unipotent block of GL_{n}(q) (respectively GL_{t}(q)) with p-core τ.

Theorem 63 (M. Broué, [7], 3.5) The block b_{τ,w}^{G_{q}} has D as a defect group, and

Br_{D}^{GL_{n}(q)}(b_{τ,w}^{G_{q}}) = 1_{kD} ⊗ b_{τ,0}^{G_{q}},

where Br_{D}^{GL_{n}(q)} is the classical Brauer morphism. □

We record some information on these defect groups D of unipotent blocks of GL_{n}(q), and on their centralizers and normalizers.

Let D_{1} be a Sylow l-subgroup of GL_{p}(q).

Lemma 64 ([30], 3D, 3E)

(a) ×^{w}D_{1} is a Sylow l-subgroup of L_{1}.

(b) D_{1} is isomorphic to a cyclic group of order l^{a}, where

a = max{i ∈ ℤ_{≥0}|l^{a} divides q^{p} − 1}.

(c) D ≅ ×^{w}D_{1} if, and only if, w < l. If w ≥ l, then D is non-abelian.

(d) The normalizer of D in GL_{wp}(q) is contained in the subgroup,

L_{1} ⋊ Σ_{w} ≅ GL_{p}(q) ≀ Σ_{w},

where Σ_{w} is the group of block permutation matrices, whose conjugation action on ×^{w}GL_{p}(q) permutes the GL_{p}(q)’s. □

The centralizer and normalizer of D in GL_{n}(q) are contained in L_{2}: Lemma 65 ([30], 1A, 3D, 3E)

C_{GL_{n}(q)}(D) = C_{GL_{wp}(q)}(D) × GL_{t}(q) ≤ L_{1} × GL_{t}(q),

and,

N_{GL_{n}(q)}(D) = N_{GL_{wp}(q)}(D) × GL_{t}(q) ≤ (GL_{p}(q) ≀ Σ_{w}) × GL_{t}(q).□

Brauer correspondence for blocks of symmetric groups.

We now describe Brauer correspondence for blocks of symmetric groups. Therefore, for the rest of this section, l = p > 0.

Let t = n − wp ≥ 0. Let

Y_{1} = ×^{w}Σ_{p} < Σ_{wp},
Y_{2} = Σ_{wp} × Σ_{t} < Σ_{n}

be Young subgroups. Let D be a Sylow p-subgroup of Σ_{wp}.

Let τ be a partition of t, and a p-core. Let b_{τ,w}^{Σ} (respectively b_{τ,0}^{Σ}) be the block of Σ_{n} (respectively Σ_{t}) with p-core τ. Theorem 66 (M. Broué, L. Puig, [7], 2.3) The block b_{τ,w}^{Σ} has D as a defect group, and

Br_{D′}^{Σ_{n}}(b_{τ,w}^{Σ}) = 1_{kD′} ⊗ b_{τ,0}^{Σ},

for any p-subgroup D′ of Σ_{wp} without fixed points in the set {1, 2, ..., wp}.

We offer some information on these defect groups D, and their centralizers and normalizers.

Let D_{1} be a Sylow p-subgroup of Σ_{p}. Lemma 67 (a) ×^{w}D_{1} is a Sylow p-subgroup of Y_{1}

(b) D_{1} ≅ C_{p}, a cyclic group of order p.

(c) D ≅ ×^{w}D_{1} if, and only if, w < p. If w ≥ p, then D is non-abelian.

(d) The normalizer of D in Σ_{wp} is contained in the subgroup

Y_{1} ⋊ Σ_{w} ≅ Σ_{p} ≀ Σ_{w},

where Σ_{w} is the group of block permutation matrices, whose conjugation action on ×^{w}Σ_{p} permutes the Σ_{p}’s. □

Lemma 68 ([41], 4.1.19,4.1.25)

C_{Σ_{n}}(D) = C_{Σ_{wp}}(D) × Σ_{t} ≤ Y_{1} × Σ_{t},

and, N_{Σ_{n}}(D) = N_{Σ_{wp}}(D) × Σ_{t} ≤ (Σ_{p} ≀ Σ_{w}) × Σ_{t}.□

Morita equivalence for symmetric algebras.

Definition 69 An O-order A, of finite rank, is said to be symmetric if there exists an O-linear form ϕ : A → O such that,

(i) ϕ(aa′) = ϕ(a′a) for all a, a′ ∈ A.

(ii) The map ϕ̂ : A → Hom_{O}(A, O) defined by ϕ̂(a)(a′) = ϕ(aa′), for a, a′ ∈ A, is an isomorphism of O-modules.

M. Broué has given a sufficient condition for two O-algebras to be Morita equivalent:

Theorem 70 ([8], 2.4) Let A and B be symmetric O-orders of finite rank, and let M be an A-B bimodule which is projective of finite rank at the same time as a left A-module and as a right B-module. Suppose that the functor

KM ⊗_{KB} − : KB − mod → KA − mod,

is an equivalence of categories. Then the functor

M ⊗_{B} − : B − mod → A − mod,

is an equivalence of categories. □

Chapter III Rock blocks of finite general linear groups and Hecke algebras, when w < l.

We prove the Morita equivalence of Rock blocks of finite general linear groups of abelian defect and their local blocks. Our proof of this theorem imitates Chuang and Kessar’s proof of theorem 53. We subsequently deduce an analogous result for Rock blocks of Hecke algebras.

Rock blocks of finite general linear groups.

Let (K, O, k) be an l-modular system. Let q be a prime power, coprime to l. Let p = p(G_{q}) be the multiplicative order of q, modulo l.

Theorem 71 (W. Turner [62], H. Miyachi [52]) Let w < l. The Rock block Ob_{ρ,w}^{G_{q}} is Morita equivalent to the principal block, Ob_{∅,1}^{G_{q}} ≀ Σ_{w}, of GL_{p}(q) ≀ Σ_{w}.

The proof is the length of this section.

Let the Rouquier core ρ = ρ(p, w) have size r. Let v = wp + r.

The principal block of GL_{p}(q) ≀ Σ_{w} is Morita equivalent to that block tensored with the defect 0 block Ob_{ρ,0}^{G_{q}}, a block of (GL_{p}(q)≀Σ_{w})×GL_{r}(q). We prove theorem 71 by showing that Green correspondence induces a Morita equivalence between this block of (GL_{p}(q) ≀ Σ_{w}) × GL_{r}(q), and the block Ob_{ρ,w}^{G_{q}} of GL_{v}(q), when w < l.

Let GL_{v}(q) = G = G_{0} > G_{1} > ... > G_{w} = L be a sequence of Levi subgroups of GL_{v}(q), where

G_{i} = GL_{p}(q)^{1} × ... × GL_{p}(q)^{i} × GL_{v−ip}(q).

Let

P_{1} > ... > P_{w}

be a sequence of parabolic subgroups of G_{0} > ... > G_{w−1} with Levi subgroups

G_{1} > ... > G_{w},

and unipotent complements

U_{1} > ... > U_{w},

such that P_{i} = U_{i} ⋊ G_{i} < G_{i−1}.

Let U_{i}^{+} = \frac{1}{|U_{i}|} ∑_{uϵU_{i}} u, a central idempotent in OU_{i}.

Note that |U_{i}| is a power of q^{p}, so equal to 1 (modulo l). Note also that G_{i} commutes with U_{i}^{+}.

Let a_{i} = b_{∅,1}^{G_{q}} be the principal block idempotent of GL_{p}(q)^{i}. Let

b_{i} = a_{1} ⊗ ... ⊗ a_{i} ⊗ b_{ρ,w−i}^{G_{q}},

a block idempotent of G_{i}, for 1 ≤ i ≤ w. We set G = G_{0}, b = b_{0}, and f = b_{w}.

Let Σ_{w} be the subgroup of permutation matrices of GL_{v}(q) whose conjugation action on L permutes the factors of GL_{p}(q)^{1} × ... × GL_{p}(q)^{w}.

We define N to be the semi-direct product of L and Σ_{w}, a subgroup of GL_{v}(q) isomorphic to (GL_{p}(q) ≀ Σ_{w}) × GL_{r}(q).

I’ll say it again. To prove theorem 71, we show that ONf and OGb are Morita equivalent, so long as w < l. It is not clear how to define the corresponding OGb-ONf-bimodule directly. However, we can describe the OGb-OLf-bimodule obtained by restriction.

Let Y =_{G}Y_{L} = OGb_{0}U_{1}^{+}b_{1}...U_{w}^{+}b_{w}, an (OGb, OLf)-bimodule. The functor Y ⊗_{L} − from L-mod to G-mod is

HCInd_{G_{1}b_{1}}^{G_{0}b_{o}}...HCInd_{G_{w}b_{w}}^{G_{w−1}b_{w−1}},

where HCInd is Harish-Chandra induction.

Proposition 72 The algebras ONf, and End_{G}(Y) have the same O-rank, equal to w!dim_{K}(KLF).

Proof:

The algebra End_{G}(Y) is O-free. It is therefore enough to compute the dimensions over K of End_{G}(KY) and KNf. One of these is straightforward - _{N}KNf is the induced module Ind_{L}^{N}(KLf), so certainly has dimension w!dim(KLf). The proposition will be proved when we have shown that End_{G}(KY) has the same dimension.

We calculate,

GKY =_{G} KY ⊗_{L} KLf = HCInd_{G_{1}b_{1}}^{G_{0}b_{0}}...HCInd_{G_{w}b_{w}}^{G_{w−1}b_{w−1}}(_{L}KLf).

This is to be done by first computing HCInd_{G_{1}b_{1}}^{G_{0}b_{0}}...HCInd_{G_{w}b_{w}}^{G_{w−1}b_{w−1}}(ψ), when ψ is an irreducible character of KLf, using the Littlewood-Richardson rule. The relevant combinatorics have already been described by Chuang and Kessar, and we record these below as lemma 73. Here, if λ = (λ_{1} ≥ λ_{2} ≥ ...) and μ = (μ_{1} ≥ μ_{2} ≥ ...) are partitions, we write μ ⊂ λ exactly when μ_{i} ≤ λ_{i} for i = 1, 2, .... An abacus is fixed so that the relevant p-quotients are well-defined.

Lemma 73 (J. Chuang, R. Kessar [11], Lemma 4) Let λ be a partition with p-core ρ and weight u ≤ w. Let μ ⊂ λ be a partition with p-core ρ and weight u − 1. Then there exists m with 0 ≤ m ≤ p − 1 such that μ^{l} = λ^{l} for l ≠ m and μ^{m} ⊂ λ^{m} with |μ^{m}| = |λ^{m}| − 1. Moreover the complement of the Young diagram of μ in that of λ is the Young diagram of the hook partition (m + 1, 1^{p−m−1}). □

In terms of character theory, by the Littlewood-Richardson rule, this means that Harish-Chandra induction from Kb_{∅,1}^{G_{q}} ⊗ Kb_{ρ,u−1}^{G_{q}} to Kb_{ρ,u}^{G_{q}} takes the character χ(1, (m + 1, 1^{p−m−1})) ⊗ χ(1, μ) to the sum of χ(1, λ)’s, such that λ is obtained from μ by moving a bead down the m^{th} runner.

Let us count the number of ways of sliding single beads down the e^{th} runner of a core j times, so that on the resulting runner the lowest bead has been lowered μ_{1}^{e} times, the second top bead has been lowered μ_{2}^{e} times, etc., so that μ_{1}^{e} ≥ μ_{2}^{e} ≥ ... and ∑_{i} μ_{i}^{e} = j. It is equal to the number of ways of writing the numbers 1, ..., j in the Young diagram of (μ_{1}^{e}, μ_{2}^{e}, ...) so that numbers increase across rows and down columns - that is, the degree of the character χ^{μ^{e}} of the symmetric group Σ_{j} (see [41], 7.2.7).

The characters in the block KLf are of the form,

ψ = χ(s_{1}, λ_{1}) ⊗ ... ⊗ χ(s_{w}, λ_{w}) ⊗ χ(1, ρ),

where either s_{i} = 1, and λ_{i} is a p-core, or else λ_{i} = (1), and s_{i} = (σ_{i}) is given by a degree p element σ_{i} ∈ 𝔽̄_{q}^{∗}, whose order is a power of l, for 1 ≤ i ≤ w.

A combinatorial description of the multiplicity of a given irreducible character of KGb in KY ⊗_{L} ψ is now visible:

Suppose that s_{1}, ..., s_{r_{0}} are all equal to 1, that λ_{i} = (1) for i ≥ r_{0} + 1, that σ_{r_{0}+1} ∼ ... ∼ σ_{r_{0}+r_{1}} =: θ_{1} are conjugate elements of 𝔽̄_{q}^{∗}, that σ_{r_{0}+r_{1}+1} ∼ ... ∼ σ_{r_{0}+r_{1}+r_{2}} =: θ_{2} are are conjugate elements of 𝔽̄_{q}^{∗}, not conjugate to θ_{1}, etc. etc. Also suppose that λ_{i}, for i = 1, ..., r_{0} is a p-hook, and that,

λ_{1} = ... = λ_{e_{0}} = (1^{p}),
λ_{e_{0}+1} = ... = λ_{e_{0}+e_{1}} = (2, 1^{p−1}),

...
λ_{e_{0}+..+e_{p−2}+1} = ... = λ_{e_{0}+..+e_{p−1}} = (p),

where e_{0} + .. + e_{p−1} = r_{0}. Then KY ⊗_{L} ψ is equal to the character sum,

∑(dimχ^{μ^{0}}...dimχ^{μ^{p−1}}.dimχ^{ν^{1}}.dimχ^{ν^{2}}...)
χ ((1, (θ_{1})^{r_{1}}, (θ_{2})^{r_{2}}, ...), (μ, ν^{1}, ν^{2}, ...))

Here, the summation is over partitions μ = [μ^{0}, .., μ^{p−1}] of |ρ| + r_{0}p with core ρ, such that (|μ^{0}|, ..., |μ^{p−1}|) = (e_{0}, e_{1}, ..., e_{p−1}); over partitions ν^{1} of r_{1}; over partitions ν^{2} of r_{2}, etc.

If, when we selected a character of L, we had permuted some of the (s_{i}, λ_{i})’s (there are (w!/e_{0}!..e_{p−1}!r_{1}!r_{2}!...) different arrangements), we would have seen the same character when we applied _{G}KY ⊗_{L} −. So the character of _{G}KY is the sum of characters,

∑_{|μ^{i}|=e_{i},|ν^{i}|=r_{i}} [(w!/e_{0}!..e_{p−1}!r_{1}!r_{2}!...)

×dimχ^{μ^{0}}...dimχ^{μ^{p−1}}.dimχ^{ν^{1}}.dimχ^{ν^{2}}...

×dimχ(1, (1^{p}))^{e_{0}}.dimχ(1, (2, 1^{p−2}))^{e_{1}}...dimχ(1, (p))^{e_{p−1}}
×dimχ(ρ_{X−1}) × dimχ(θ_{1}, (1))^{r_{1}}.dimχ(θ_{2}, (1))^{r_{2}}...
×χ ((1, (θ_{1})_{1}^{r}, (θ_{2})^{r_{2}}, ...), ([μ^{0}, ..., μ^{p−1}], ν^{1}, ν^{2}, ...)) ].

What is the dimension of the semisimple algebra End_{G}(KY) ? Remembering that ∑_{|σ|=m} |χ^{σ}|^{2} = m!, it is,

∑_{e_{0}+..+e_{p−1}+r_{1}+r_{2}+...=w} [(w!/e_{0}!..e_{p−1}!r_{1}!r_{2}!...)^{2}

×e_{0}!..e_{p−1}!r_{1}!r_{2}!...

× (dimχ(1, (1^{p})))^{2e_{1}} . (dimχ(1, (2, 1^{p−2})))^{2e_{2}} ... (dimχ(1, (p)))^{2e_{p}}

× (dimχ(ρ_{X−1}))^{2} × (dimχ(θ_{1}, (1)))^{2r_{1}} . (dimχ(θ_{2}, (1)))^{2r_{2}} ...]

= w! ∑_{e_{0}+..+e_{p−1}+r_{1}+r_{2}+...=w} [(w!/e_{0}!..e_{p−1}!r_{1}!r_{2}!...)

× (dimχ(1, (1^{p})))^{2e_{0}} . (dimχ(1, (2, 1^{p−2})))^{2e_{1}} ... (dimχ(1, (p)))^{2e_{p−1}}

× (dimχ(ρ_{X−1}))^{2} × (dimχ(θ_{1}, (1)))^{2r_{1}} . (dimχ(θ_{2}, (1)))^{2r_{2}} ...]
= w!.dim(KLf).□

Let D = D^{1} ×...×D^{w} be a Sylow l-subgroup of GL_{p}(q)^{1} ×...×GL_{p}(q)^{w}.

Lemma 74 Let w < l.

(a) D is a defect group of OG_{i}b_{i}, for i = 0, ..., p − 1.

(b) Br_{D}^{G}(b_{i}) = 1_{kD} ⊗ b_{ρ,0}^{G_{q}}, and Br_{D}^{G}(U_{i}^{+}) = 1.

(c) N stabilizes f, and as an O(N × L)-module, ONf is indecomposable with diagonal vertex ΔD . In particular, ONf is a block of N.

(d) OGb and ONf both have defect group D, and are Brauer correspondents.

Proof:

(a) D^{i+1} ×...×D^{w} is a defect group of Ob_{ρ,w−i}^{G_{q}}, and D^{j} is a defect group of OGL_{p}(q)^{j}a_{j}.

(b) G_{i} > C_{G}(D), so Br_{D}^{G}(b_{i}) = Br_{D}^{G_{i}}(b_{i}). Hence,

Br_{D}^{G}(b_{i}) = Br_{D}^{G_{i}}(b_{i})

= Br_{D^{1}}^{GL_{p}^{1}}(a_{1}) ⊗ ... ⊗ Br_{D^{i}}^{GL_{p}^{i}}(a_{i}) ⊗ Br_{D^{i+1}×...×D^{w}}^{GL_{v−ip}}(b_{ρ,w−i}^{G_{q}})

= 1_{kD} ⊗ b_{ρ,0}^{G_{q}},

where the last equality is by theorem 63.

In addition, P_{i} > C_{G}(D), and Br_{D}^{G}(U_{i}^{+}) = Br_{D}^{P_{i}}(U_{i}^{+}) = 1, because U_{i} ∩ C_{G}(D) = {1}.

(c) There can be little doubt that N stabilises f. By part (1), OLf has vertex ΔD. Since C_{G}(D) < L, the conjugate of ΔD by an element of N × L outside L × L is never conjugate to ΔD in L × L. But the vertex of the L × L-summand σOLf of ONf (where σ ∈ Σ_{w}) is equal to (ΔD)^{(σ,1)}. Consequently, the stabilizer of OLf in N × L is exactly L × L. So ONf = Ind_{L×L}^{N×L}(OLf) is indecomposable by Clifford’s theorem, and has vertex ΔD.

(d) Follows from (1) and (2) by Brauer’s First Main Theorem. □

By the Brauer correspondence, the O(G × G)-module OGb and the O(N × N)-module ONf both have vertex ΔD and are Green correspondents. Let X be the Green correspondent of OGb in G × N, an indecomposable summand of Res_{N×N}^{G×G}(OGb) with vertex ΔD. Because ONf is a direct summand of Res_{N×N}^{G×N}(X), we have Xf ≠ 0, so Xf = X and X is an (OGb, ONf)-bimodule.

We prove theorem 71 by verifying the following:

Proposition 75 Let w < l. There is a sequence of O-split monomorphisms of algebras,

ONf ↪ End_{G}(X) ↪ End_{G}(Y).

The algebras ONf and End_{G}(Y) have the same O-rank, so the above maps are isomorphisms.

The left OG-module _{G}X is a progenerator for OGb. Hence X ⊗_{N} − induces a Morita equivalence between ONf and OGb.

Proof:

Truly, _{G}OGb_{G} is isomorphic to a direct summand of Ind_{G×N}^{G×G}(X), since X and OGb are Green correspondents. Thus _{G}OGb is a direct summand of |G : N| copies of _{G}X, and _{G}X is a progenerator for OGb.

There is an O-split homomorphism of algebras

ONf → End_{G}(X),

born of the G × N-module structure of X. Since _{N}ONf_{N} is a direct summand of Res_{N×N}^{G×N}(X), this homomorphism is a monomorphism.

But Res_{G×L}^{G×N}(X) is indecomposable with vertex ΔD. For firstly, it is at least a direct sum of indecomposable modules whose vertices are conjugates of ΔD, by Mackey’s formula - X has vertex ΔD, and G × L is a normal subgroup of G × N containing ΔD. And secondly, Res_{G×L}^{G×N}(X) = _{G}X ⊗_{N} ONf_{L} is a summand of Ind_{N×L}^{G×L}(ONf), which by Green correspondence has exactly one summand with vertex ΔD (_{N}ONf_{N} has vertex ΔD and N_{G×L}(ΔD) < N × L).

So _{G}X_{L} is the only summand of _{G}OGb_{L} with vertex ΔD, by Mackey’s formula - for _{G}X_{N} is the only summand of _{G}OGb_{N} with vertex not strictly contained in ΔD.

Let _{G}Y_{L} = OGb_{0}U_{1}^{+}b_{1}...U_{w}^{+}b_{w}. Each b_{i} and each U_{i}^{+} is an idempotent in (OG)^{L}; these idempotents commute with each other, and their product is an idempotent contained in (OGb)^{L}. Hence, _{G}Y_{L} is also a direct summand of GOGb_{L}. We show that it has as direct summands all summands of _{G}OGb_{L} with vertex ΔD, using the Brauer morphism. So no doubt _{G}Y_{L} =_{G} X_{L} ⊕ ∗, revealing an O-split monomorphism End_{G}(X) ↪ End_{G}(Y). The calculation goes:

Y (ΔD) =
OGb_{0}U_{1}^{+}b_{1}...U_{w}^{+}b_{w}(ΔD) =
OC_{G}(D)Br_{D}^{G}(b_{0}U_{1}^{+}b_{1}...U_{w}^{+}b_{w}) =
OC_{G}(D)Br_{D}^{G}(b_{0})Br_{D}^{G}(U_{1}^{+})Br_{D}^{G}(b_{1})...Br_{D}^{G}(U_{w}^{+})Br_{D}^{G}(b_{w}) =
OC_{G}(D)Br_{D}^{G}(b_{0})Br_{D}^{G}(b_{1})...Br_{D}^{G}(b_{w}) =
OC_{G}(D)Br_{D}^{G}(b) =
OGb(ΔD).

Since _{G}OGb_{L} is an l-permutation module, the bimodule _{G}Y_{L} has indeed as summands all summands of _{G}OGb_{L} with vertex ΔD.

The O-ranks of ONf and End_{G}(Y) are equal, thanks to proposition 72 □

Theorem 71 is now proven. □

Remark 76 The correspondence between indecomposable modules of kNf and indecomposable modules of kGb given by Theorem 71 above is exactly Green correspondence between G and N. For if M is an indecomposable of kNf with vertex Q, the kG-module X ⊗_{kN} M cannot have a smaller vertex than Q, as then M = X^{∗} ⊗_{kG} X ⊗_{kN} M would have a smaller vertex than Q.

Remark 77 The proof of theorem 71 fails when w ≥ l, because the indecomposable module _{G}X_{N} does not restrict to an indecomposable module, GX_{L}.

Rock blocks of Hecke algebras, when w < l.

Let us persist with the notation of the last section. Thus, (K, O, k) is an l-modular system, and q is a prime power, coprime to l. Theorem 71 squashes to Hecke algebras as follows:

Theorem 78 Let w < l. Let p = p(G_{q}) = p(H_{q}) > 1. The Rock block Ob_{ρ,w}^{H_{q}} is Morita equivalent to Ob_{∅,1}^{H_{q}} ≀ Σ_{w}.

Proof:

Let B_{n}(q) be a Borel subgroup of GL_{n}(q) for n = p, v. Since p > 1, for n = p, v, the OGL_{n}(q)-module O[GL_{n}(q)/B_{n}(q)] is projective, having been induced from the projective trivial OB_{n}(q)-module.

We now construct a Morita bimodule from bimodules already available to us. Let G = GL_{v}(q) be as in theorem 71. Let H_{w} = ×^{w}GL_{p}(q). Let G_{w} = H_{w} ⋊ Σ_{w} = GL_{p}(q) ≀ Σ_{w}. Let _{G}Z_{G_{w}} be a bimodule inducing a Morita equivalence between Ob_{ρ,w}^{G_{q}} and Ob_{∅,1}^{G_{q}} ≀ Σ_{w}; such a bimodule exists by theorem 71. Let

η = \frac{1}{|B_{p}(q)|^{w}} ∑_{x∈B_{p}(q)×..×B_{p}(q)} x,

ξ = \frac{1}{|B_{v}(q)|} ∑_{b∈B_{v}(q)} b.

Then

ξZη

is an H_{q}(Σ_{v})-H_{q}(Σ_{p}) ≀ Σ_{w} bimodule. We show that this bimodule is left and right projective, and induces an isomorphism at the level of ordinary characters. Since Hecke algebras and their wreath products are symmetric algebras, an application of theorem 70 proves Morita equivalence.

We need to show that under the Morita equivalence from Ob_{ρ,w}^{G_{q}} to Ob_{∅,1}^{G_{q}} ≀ Σ_{w}; given by Z, simple G-modules (over K or k) killed by η become simple G_{w}-modules (over K or k) killed by ξ, and vice-versa. Then certainly the bimodule will induce an isomorphism on ordinary characters, and ξZη will be left and right projective.

The characters of G killed by ξ are the non-unipotent characters, by theorem 37. Under_{H_{w}}Z^{∗}⊗_{G}−, these become sums of characters χ^{1}⊗...⊗χ^{w}, such that one of the χ^{i}’s is a non-unipotent character of GL_{p}(q) (according to the proof of proposition 72). These are all killed by η. Conversely, by theorem 41, the characters of G_{w} killed by η are all composition factors of characters Ind_{H_{w}}^{G_{w}}(χ^{1} ⊗ ... ⊗ χ^{w}), where one of the χ^{i}’s is a non-unipotent character of GL_{p}(q). And _{G}Z ⊗_{G_{w}} − sends this induced character to a sum of non-unipotent characters of G, all of which are killed by ξ.

The simples for G which vanish under multiplication by ξ are those D(λ) indexed by p-singular partitions. These are simple composition factors of non-unipotent characters χ((1, (σ)^{|ν|}), (λ_{0}, ν)), where ν is a non-empty partition, and σ is an l-element of 𝔽̄_{q}^{∗} of degree p ([10], Theorem 4.4d). But these characters correspond (under the Morita equivalence) to characters for G_{w} which are killed by η. Conversely, simple modules for G_{w} which are killed by η are composition factors of induced characters Ind_{H_{w}}^{G_{w}}(χ^{1}⊗...⊗χ^{w}), where one of the χ^{i}’s is a non-unipotent character of GL_{p}(q). This becomes a character of G sent to zero by ξ on application of Z ⊗_{G_{w}} −. □

Chapter IV

Rock blocks of symmetric groups, in characteristic two.

We introduce notation, for the study of Rock blocks of symmetric groups, of arbitrary defect. We show that in characteristic two, a Rock block is isomorphic to the group algebra of its defect group, once it has been cut by a certain idempotent (theorem 80). Our proof involves an unusual application of the Brauer homomorphism. We generalise this method to obtain a result for Rock blocks of symmetric groups in arbitrary characteristic: we prove that the endomorphism ring of a certain l-permutation module M for kb_{ρ,w}^{Σ} is Morita equivalent to kΣ_{w}. (theorem 83)

Notation for Rock blocks of symmetric groups.

Throughout this chapter, we consider blocks of symmetric groups. Therefore, l = p.

Let (K, O, k) be a p-modular system. Let w be a natural number. Let ρ = ρ(p, w) be a Rouquier core, and a partition of r. Let v = wp + r.

Let Σ_{v} = Sym{1, 2, ..., v}. Let b_{ρ,w}^{Σ} be the Rock block of Σ_{v}. Let

L = Σ_{p}^{1} × ... × Σ_{p}^{w} × Σ_{r}^{0} ≤ Σ_{v},

where Σ_{p}^{i} = Sym{(i − 1)p + 1, ..., ip}, and Σ_{r}^{0} = Sym{wp + 1, ..., wp + r}. Let

f = b_{∅,1}^{Σ} ⊗ ... ⊗ b_{∅,1}^{Σ} ⊗ b_{ρ,w}^{Σ},

a block of L. Let

D = C_{p}^{1} × ... × C_{p}^{w} ≤ L,

where C_{p}^{i} is the group of order p generated by the single element ((i − 1)p + 1...ip). Let

N = L ⋊ Σ_{w} ≅ (Σ_{p} ≀ Σ_{w}) × Σ_{r}^{0},

the normaliser of L in Σ_{v}. Since the centralizer C_{N}(D) ≤ L, the idempotent f is also a block of N ([2] 15.1(5)).

Let e be an idempotent of kΣ_{v}, defined to the product e_{w}.e_{w−1}...e_{0} of block idempotents,

e_{i} = b_{∅,1}^{Σ} ⊗ ... ⊗ b_{∅,1}^{Σ} ⊗ b_{ρ,i}^{Σ},

of Σ_{p}^{1} × .. × Σ_{p}^{w−i} × Σ_{ip+r}.

Aping proposition 72, and its proof, we have, Proposition 79 The dimensions dim_{k}(ekΣ_{v}e) and dim_{k}(kNf) are both equal to w!.dim_{k}(kLf). □

Endomorphism rings.

Let P be the subgroup of D of order p generated by the element,

z = (1...p)(p + 1...2p)...((w − 1)p + 1...wp).

Let,

C = C_{Σ_{v}}(P) = D ⋊ Σ_{w} × Σ_{r}^{0} ≤ N.

We consider the classical Brauer homomorphism - the surjective algebra homomorphism from (kΣ_{v})^{P} to kC which truncates elements of the group algebra at C. The images of block idempotents under the Brauer morphism are given by theorem 66. In particular,

Br_{D}^{Σ_{v}}(e) = Br_{P}^{Σ_{v}}(e) = Br_{P}^{Σ_{v}}(f) = 1_{kD} ⊗ b_{ρ,0}^{Σ}.

Therefore,

Br_{P}^{Σ_{v}}((ekΣ_{v}e)^{P}) = Br_{P}^{Σ_{v}}(e).kC.Br_{P}^{Σ_{v}}(e) = kC.b_{ρ,0}^{Σ}.

If p = 2, then N = C, and f = Br_{P}^{Σ_{v}}(f) = Br_{P}^{N}(f). In this case, we can prove a one-sided version of theorem 53:

Theorem 80 If p = 2, then (ekΣ_{v}e)^{P} = ekΣ_{v}e. Furthermore, the Brauer homomorphism

Br_{P}^{Σ_{v}} : (ekΣ_{v}e)^{P} → kNf

is an isomorphism of algebras.

Proof:

From lemma 79, we know that dim_{k}(ekΣ_{v}e) = dim_{k}(kNf). Also N = C, so that

dim_{k}(ekΣ_{v}e)^{P} ≤ dim_{k}(ekΣ_{v}e) = dim_{k}(kNf) = dim_{k}(kCf).

But the Brauer homomorphism Br_{P}^{Σ_{v}} : (ekΣ_{v}e)^{P} → kCf is a surjection. So the dimensions above are all equal, and the theorem is proven. □

In [11], Chuang and Kessar prove that ekΣ_{v}e ≅ kNf so long as w < p.

Conjecture 81 ekΣ_{v}e ≅ kNf, for arbitrary w, p.

Remark 82 R. Paget [55] has computed the projective summands of Σ_{v}kΣ_{v}e, and shown that ekΣ_{v}e, and kNf have the same decomposition matrix. Her proof uses theorem 129, of this monograph.

If we cut down the module kΣ_{v}e, and the algebras that we are considering, it is possible to generalise the proof of theorem 80 to p ≥ 2. More precisely, let ζ_{w} = ∑_{x∈Σ_{p}^{1}×...×Σ_{p}^{w}} x.

Let M = OΣ_{v}eζ_{w}. Let E = End_{Σ_{v}}(M).

Since M is a p-permutation module, it is O-free, and End_{Σ_{v}}(RM) ≅ RE, for R ∈ {K, O, k} ([47], 1.14.5).

To complete this chapter, we present the following theorem, as the consequence of a triad of lemmas.

Theorem 83 Let R ∈ {K, k}. Then,

RE ≅ RΣ_{w} ⊗ Rb_{ρ,0}^{Σ}.

First, some notation. Let Θ_{u} be the set of partitions with core ρ of weight u obtained by moving beads only up the rightmost runner of the abacus representation of ρ. These correspond to partitions of u, via [∅, .., ∅, ν] ↔ ν. In the usual notation for partitions, this correspondence is

(ρ_{1} + pν_{1}, ρ_{2} + pν_{2}, ..., ρ_{u} + pν_{u}, ρ_{u+1}, ...) ↔ (ν_{1}, ν_{2}, ..., ν_{u}).

Lemma 84 The character of _{Σ_{v}}KM is equal to

dim(χ^{ρ}). ∑_{ν} dim(χ^{ν}).χ^{[∅,..,∅,ν]}.

Its endomorphism ring is isomorphic to KΣ_{w} ⊗ Kb_{ρ,0}^{Σ}.

Proof:

We have _{Σ_{v}}KM ≅ KΣ_{v}e ⊗_{L} KLfζ_{w}. This module is the KLf-module with character χ^{(p)} ⊗ ... ⊗ χ^{(p)} ⊗ Kb_{ρ,0}^{Σ}, induced to (⨂^{w−1} KΣ_{p}) ⊗ Kb_{ρ,1}^{Σ}, then induced to (⨂^{w−2} KΣ_{p}) ⊗ Kb_{ρ,2}^{Σ}, etc., etc.. That its character is as stated follows from a symmetric group analogue of proposition 72, and its proof.

The endomorphism ring is isomorphic to KΣ_{w} ⊗Kb_{ρ,0}^{Σ}, since all algebras concerned are semisimple. □

From now on, let us fix an isomorphism between the endomorphism ring of _{Σ_{v}}KM and KΣ_{w} ⊗ Kb_{ρ,0}^{Σ}, so that under the Morita equivalence, given by KM, between Kb_{ρ,w}^{Σ}/Ann(KM) and KΣ_{w} ⊗ Kb_{ρ,0}^{Σ}, the characters χ^{[∅,...,∅,λ]} and χ^{λ} ⊗ χ^{ρ} correspond.

Let ζ_{D} = ∑_{x∈D} x. The sum ζ_{D} is the image of ζ_{w} under Br_{P}. Note that C commutes with ζ_{D}, and so kC acts on the right of kC.b_{ρ,0}^{Σ}.ζ_{D}. Under this action, we have:

Lemma 85 We have an isomorphism of algebras,

kC.b_{ρ,0}^{Σ}/Ann(kC.b_{ρ,0}^{Σ}.ζ_{D}) ≅ kΣ_{w} ⊗ kb_{ρ,0}^{Σ}.

Proof:

It is true that C ≅ (D⋊Σ_{w})×Σ_{r}^{0}, so that the top subgroup Σ_{w} commutes with ζ_{D}. Thus,

kΣ_{w} ⊗ kb_{ρ,0}^{Σ} ≅ kC.b_{ρ,0}^{Σ}.ζ_{D}

as vector spaces, via x ↦ xζ_{D}. The annihilator Ann_{kC}(kC.b_{ρ,0}^{Σ}.ζ_{D}) is thus equal to I(D).kΣ_{w}.b_{ρ,0}^{Σ} as a right kC-module, where Σ_{w} is the top group, and where I(D) is the augmentation ideal of kD. Furthermore, the quotient of kC.b_{ρ,0}^{Σ} by the annihilator acts on kΣ_{w} ⊗ kb_{ρ,0}^{Σ} by the right regular action, and thus the quotient is actually isomorphic to kΣ_{w} ⊗ kb_{ρ,0}^{Σ}, as an algebra. □

We may prove a version of lemma 84 over k, using Br_{P}:

Lemma 86 We have an isomorphism of algebras,

kE ≅ kΣ_{w} ⊗ kb_{ρ,0}^{Σ}.

The implicit action of kΣ_{w} ⊗ kb_{ρ,0}^{Σ} on kM is,

m ∘ (x ⊗ y) = m.exye,

for x ∈ kΣ_{w}, y ∈ kb_{ρ,0}^{Σ}.

Proof:

First observe that _{L}kb_{ρ,wL}^{Σ} = _{L}ekΣ_{v}e_{L} ⊕ ∗, where ∗ has no summand with vertex containing ΔP, since,

ekΣ_{v}e(ΔP) = Br_{P} (e).kC.Br_{P} (e) =
kC.b_{ρ,0}^{Σ} = kb_{ρ,w}^{Σ}(ΔP).

Likewise, _{L}kNf_{L} is a summand of _{L}kb_{ρ,wL}^{Σ}, where the complement has summands whose vertices do not contain ΔP. However, a vertex of _{L}σkLf_{L} is ΔD^{(σ,1)} ≥ ΔP (for σ ∈ Σ_{w}). So _{L}kNf_{L} is the sum of summands of _{L}kb_{ρ,wL}^{Σ} with vertices containing ΔP. Hence, _{L}ekΣ_{v}e_{L} is a summand of _{L}kNf_{L}, and since their dimensions are equal by lemma 79, there are isomorphisms,

LekΣ_{v}e_{L} ≅_{L} kNf_{L} ≅ ⨁_{σ∈Σ_{w}}(_{L}σkLf_{L}).

It follows that dim_{k}(ekΣ_{v}eζ_{w}) = w!.dim_{k}(kb_{ρ,0}^{Σ}).

Let A be the subalgebra of (ekΣ_{v}e)^{P} generated by ekCe. Note that A acts on the right of kM = kΣ_{v}eζ_{w} by multiplication, thus commuting with the left action of kΣ_{v}. In fact, A acts on the right of (ekΣ_{v}e)^{P} ζ_{w} by multiplication. Applying the Brauer homomorphism Br_{P}, we realise an action of Br_{P} (A) = kC.b_{ρ,0}^{Σ} on the right of

Br_{P} ((ekΣ_{v}e)^{P} ζ_{w}) = Br_{P} ((ekΣ_{v}e)^{P}).Br_{P} (ζ_{w}) = kC.b_{ρ,0}^{Σ}ζ_{D}.

Thus there is an algebra homomorphism, given by the composition,

A → Br_{P} (A) → kC.b_{ρ,0}^{Σ}/Ann(kC.b_{ρ,0}^{Σ}.ζ_{D}),

in the kernel of which lies Ann_{A}((ekΣ_{v}e)^{P} ζ_{w}). Lemma 85 implies the existence of a surjection,

A/Ann_{A}((ekΣ_{v}e)^{P} ζ_{w}) → kΣ_{w} ⊗ kb_{ρ,0}^{Σ},

as well as the natural surjection,

A/Ann_{A}(kΣ_{v}eζ_{w}) → A/Ann_{A}((ekΣ_{v}e)^{P} ζ_{w}).

In addition, there is a sequence of natural injections,

A/Ann_{A}(kΣ_{v}eζ_{w}) → End_{Σ_{v}}(kΣ_{v}eζ_{w}) →
Hom_{Σ_{v}}(kΣ_{v}e, kΣ_{v}eζ_{w}) → ekΣ_{v}eζ_{w}.

But we have already agreed that ekΣ_{v}eζ_{w} has dimension equal to,

w!.dim_{k}kb_{ρ,0}^{Σ} = dim_{k}(kΣ_{w} ⊗ kb_{ρ,0}^{Σ}),

so in fact all of the above homomorphisms are isomorphisms. In particular,

End_{Σ_{v}}(kM) = End_{Σ_{v}}(kΣ_{v}eζ_{w}) ≅ kΣ_{w} ⊗ kb_{ρ,0}^{Σ}.□

Chapter V

Schur-Weyl duality inside Rock blocks of symmetric groups.

In this chapter, we strengthen theorem 80, and show that the Brauer homomorphism reveals Schur-Weyl duality between S(w, w), and kΣ_{w}, structurally inside the Rock block kb_{ρ,w}^{Σ} (theorem 87).

An alternative proof of the existence of a quotient of a symmetric group block, Morita equivalent to S(w, w) has been given by Cline, Parshall, and Scott, using Steinberg’s tensor product theorem for the algebraic group GL_{n} ([19], 5.3). If the Reader inclines towards an understanding of blocks of finite groups, the proof given here should be interesting, because it is independent of algebraic group theory.

A Schur algebra quotient.

We assume the notation of chapter 4. Let RI = Ann_{Rb_{ρ,w}^{Σ}}(RM), and let RB_{ρ,w}^{Σ} = Rb_{ρ,w}^{Σ}/RI, for R ∈ {K, O, k}.

Theorem 87 Let R ∈ {K, k}. Then RE = End_{Σ_{v}}(RM) is Morita equivalent to RΣ_{w}, and Rb_{ρ,w}^{Σ}/RI is Morita equivalent to the Schur algebra S(w, w), defined over R.

The S(w, w)-RΣ_{w}-bimodule corresponding to the Rb_{ρ,w}^{Σ}/RI-RE-bimodule RM is twisted tensor space, E^{⊗r#}.

Theorem 87 may be seen as a module theoretic interpretation of a theorem of Erdmann [29], which realises the decomposition matrix of the Schur algebra S(w, w) as a submatrix of the decomposition matrix of kb_{ρ,w}^{Σ}. Indeed, we have the following interesting corollary:

Corollary 88 (“Converse to Schur-Weyl duality”) Every block of polynomial GL_{n}(k)-modules is Morita equivalent to a quotient of some symmetric group algebra, localised at an idempotent. □

Theorem 87 is clearly analogous to theorem 37 concerning general linear groups in non-describing characteristic. However, our methods hardly resemble those of Brundan, Dipper, and Kleshchev.

The proof of theorem 87 is the length of this chapter. We first find summands of kM as a right kΣ_{w} × Σ_{r}^{0}-module which are twisted Young modules for kΣ_{w}, tensored with the block kb_{ρ,0}^{Σ}. We then observe that summands of these are all the indecomposable summands. We finally show that the map kb_{ρ,w}^{Σ} → End_{kE}(kM) is surjective by an explicit calculation.

Let us state and prove some combinatorial preliminaries.

Lemma 89 Let μ ∉ Θ_{w} be a partition of weight w with core ρ. Let θ ∈ Θ_{w}. Then μ ⋭ θ.

Proof: In an abacus representation of μ obtained from ρ, at least one bead must be moved up a runner other than the rightmost runner. This means that μ_{i} is greater than ρ_{i}, for some i > w. Thus, μ_{1} +μ_{2} +...+μ_{w} is at most ρ_{1} + ρ_{2} + ... + ρ_{w} + (wp − 1). But θ_{1} + θ_{2} + ... + θ_{w} = ρ_{1} + ρ_{2} + ... + ρ_{w} + wp. □

Lemma 90 Let θ ∈ Θ_{u} be equal to [∅, .., ∅, λ], where λ is a partition of u. Let t be such that u + t ≤ w. Then the character summand of

Ind_{(×^{p}Σ_{t})×Σ_{up+r}}^{Σ_{(t+u)p+r}}(χ^{(1^{t})} ⊗ ... ⊗ χ^{(1^{t})} ⊗ χ^{θ})

obtained by removing all character summands indexed by partitions outside Θ_{u+t} is given by

∑_{ν} l_{ν}χ^{[∅,...,∅,ν]},

where Ind_{Σ_{t}×Σ_{u}}^{Σ_{t+u}}(χ^{(1^{t})} ⊗ χ^{λ}) = ∑_{ν} l_{ν}χ^{ν}.

Proof:

We have θ = (ρ_{1}+pλ_{1}, ρ_{2}+pλ_{2}, ..., ρ_{u}+pλ_{u}, ρ_{u+1}, ...). By the Littlewood-Richardson rule, the only characters obtained by inducing χ^{(1^{t})} ⊗ χ^{θ} to Σ_{t+up+r} are obtained by adding nodes onto t different rows of θ. Repeating this process p times, the only way of obtaining a character indexed by a partition in Θ_{t+u} is by adding nodes onto the same t rows p times, in such a way that the resulting partitions lie in Θ_{t+u}. The ways of doing this correspond exactly to the ways of adding a node to t different rows of λ, so that the resulting composition is a strict partition. These correspond exactly to the character summands of Ind_{Σ_{t}×Σ_{u}}^{Σ_{t+u}}(χ^{(1^{t})} ⊗ χ^{λ}), by the Littlewood-Richardson rule. □

Permutation modules for Σ_{v}.

Suppose that λ = (λ_{1}, λ_{2}, ..., λ_{N}) is a partition of w. Let j_{λ}^{1} = 0, and let j_{λ}^{i} be the sum ∑_{r=1}^{i−1} λ_{r} (i = 2, 3, ..., N). Let J_{λ}^{i} (i = 1, 2, ..., N) be the subgroup,

Sym{pj_{λ}^{i} + 1, pj_{λ}^{i} + p + 1, ..., pj_{λ}^{i} + p(λ_{i} − 1) + 1}
×Sym{pj_{λ}^{i} + 2, pj_{λ}^{i} + p + 2, ..., pj_{λ}^{i} + p(λ_{i} − 1) + 2}
×... × Sym{pj_{λ}^{i} + p, pj_{λ}^{i} + p + p, ..., pj_{λ}^{i} + pλ_{i}}

of Σ_{v}, a product of p symmetric groups, each of which is isomorphic to S_{λ_{i}}. Note that J_{λ}^{i} has support {m ∈ ℤ|pj_{λ}^{i} + 1 ≤ m ≤ pj_{λ}^{i+1}}. Let

J_{λ} = J_{λ}^{1} × J_{λ}^{2} × ... × J_{λ}^{N},

a subgroup of Σ_{v} isomorphic to ×^{p}Σ_{λ}, a direct copy of p copies of the Young subgroup Σ_{λ} of Σ_{w}.

Recall that Σ_{r}^{0} is defined to be Sym{wp + 1, ..., wp + r}. We define

Σ_{r+pj_{λ}^{i}}^{0} = Sym{1, 2, 3, ..., pj_{λ}^{i}, wp + 1, wp + 2, ..., wp + r}.

This is the symmetric group whose support is equal to the support of the direct product J_{λ}^{1} × ... × J_{λ}^{i−1} × Σ_{r}^{0}.

Let ξ_{λ}^{i} = ∑_{y∈J_{λ}^{i}} y, for i = 1, 2, ..., N. Let

ξ_{λ} = (ξ_{λ}^{1}ξ_{λ}^{2}...ξ_{λ}^{N})b_{ρ,0}^{Σ}.b_{ρ,λ_{1}}^{Σ}.b_{ρ,λ_{1}+λ_{2}}^{Σ}...b_{ρ,w}^{Σ},

an element of OΣ_{v}. In this formula, we take b_{ρ,j_{λ}^{i}}^{Σ} to be an element of OΣ_{r+pj_{λ}^{i}}^{0}. Let η_{λ}^{i} = ∑_{y∈J_{λ}^{i}} sgn(y).y, for i = 1, 2, ..., N. Let

η_{λ} = (η_{λ}^{1}η_{λ}^{2}...η_{λ}^{N})b_{ρ,0}^{Σ}.b_{ρ,λ_{1}}^{Σ}.b_{ρ,λ_{1}+λ_{2}}^{Σ}...b_{ρ,w}^{Σ},

an element of OΣ_{v}.

Consider OΣ_{v}ξ_{λ}. This OΣ_{v}-module may be constructed as follows:

Take the projective module OΣ_{r}-module Ob_{ρ,0}^{Σ}. Tensor this module with p copies of the Young module Y^{(λ_{1})}, each of which is isomorphic to the trivial module for OΣ_{λ_{1}}. Cut the resulting p-permutation module off at the block of Σ_{r+pλ_{1}} with core ρ. Now tensor this module with p copies of Y^{(λ_{2})} and cut off at the block of Σ_{r+pλ_{1}+pλ_{2}} with core ρ. Repeat this process until a p-permutation module for OΣ_{v} is obtained, in the block with core ρ.

Why can OΣ_{v}ξ_{λ} be constucted in this way ? Because ξ_{λ}^{i} generates the trivial module for J_{λ}^{i}, which is isomorphic to p copies of the symmetric group Σ_{λ}^{i}. And because the idempotent b_{ρ,j_{λ}^{i}}^{Σ} commutes with the subgroup Σ_{r+pj_{λ}^{i}}^{0}.

We hava a similar construction of OΣ_{v}η_{λ}. This time, rather than tensoring with p copies of the trivial module each time before inducing, you should tensor with p copies of the alternating module.

When searching for summands of the right kΣ_{w} × Σ_{r}^{0}-module kM, we will be interested in the projective part of OΣ_{v}η_{λ} = (OΣ_{v}ξ_{λ})^{#}.

Pursuing this, let us first note that OΣ_{v}ξ_{λ} is a p-permutation module (a direct summand of a module induced from a trivial module tensored with a sum of Young modules, from a Young subgroup; the module Ob_{ρ,0}^{Σ} for OΣ_{r}^{0} is isomorphic to a sum of Young modules, since that block has defect zero).

Hence it is a direct sum of Young modules. It is known that a Young module Y^{μ} is projective if and only if μ is p-restricted. Also, a Young module Y^{μ} has a Specht filtration, and the Specht subquotients S^{γ} occuring satisfy γ ⊵ μ.

So any Young module summand of OΣ_{v}ξ_{λ} which is not projective only has Specht quotients S^{γ} where γ ⊵ μ for some p-restricted partition μ.

Now, OΣ_{v}η_{λ} = (OΣ_{v}ξ_{λ})^{#} ≅ OΣ_{v}ξ_{λ} ⊗ sgn, where sgn is the signature representation. And we know that S^{γ} ⊗ sgn ≅ (S^{γ′})^{∗}. Furthermore, tensoring with sgn preserves indecomposability and takes projectives to projectives and non-projectives to non-projectives.

Thanks to the above discussion, we know the following fact:

Lemma 91 Let OΣ_{v}η_{λ} = U ⊕ V, where U is projective and V has no projective summand. Then the character of KV has irreducible constituents χ^{δ} marked by δ ⊴ β where β is p-singular. □

Let K_{λ}^{i} (i = 1, 2, ..., N) be the diagonal subgroup of the direct product J_{λ}^{i} (here we fix isomorphisms between the factors of J_{λ}^{i} which preserve the ordering on {1, 2, .., wp}), a group isomorphic to Σ_{λ_{i}}. Let

K_{λ} = K_{λ}^{1} × K_{λ}^{2} × ... × K_{λ}^{N},

a subgroup of Σ_{v} isomorphic to a Young subgroup Σ_{λ} of Σ_{w}.

Note that N ≅ L ⋊ K_{(w)}, and K_{λ} < K_{(w)} normalises D. The following lemma is straightforward:

Lemma 92 (a) The direct product J_{λ}^{i} ≅ ×_{p}Σ_{λ_{i}} is normalised by P. The generator z of P acts on J_{λ}^{i} by circulating the p direct factors of J_{λ}^{i}.

(b) The direct product J_{λ} ≅ ×_{p}Σ_{λ} is normalised by P. The generator z of P acts on J_{λ} by circulating the p direct factors of J_{λ}.

(c) The diagonal subgroup K_{λ}^{i} of J_{λ}^{i} is equal to C_{G}(P)∩J_{λ}^{i}. The diagonal subgroup K_{λ} of J_{λ} is equal to C_{G}(P) ∩ J_{λ}.

(d) We have Br_{P} (ξ_{λ}^{1}ξ_{λ}^{2}...ξ_{λ}^{N}) = ∑_{K_{λ}} y, and Br_{P} (η_{λ}^{1}η_{λ}^{2}...η_{λ}^{N}) = ∑_{K_{λ}} sgn(y).y □

Summands of M_{E} as Young modules.

Recall from the proof of theorem 83 that the algebra A acts on the right of the module kM and its image is the endomorphism ring kE. The algebra A commutes with P, and thus preserves kM^{P} in its action. Hence, kE ≅ kΣ_{w} ⊗ kb_{ρ,0}^{Σ} preserves kM^{P} in its action.

Lemma 93 (a) We have η_{λ} ∈ (OΣ_{v})^{P}, and,

Br_{P} (η_{λ}) = (∑_{K_{λ}} sgn(y).y).b_{ρ,0}^{Σ}.

(b) Br_{P} (η_{λ}kM^{P}) ≅ Ind_{Σ_{λ}×Σ_{r}^{0}}^{Σ_{w}×Σ_{r}^{0}}(sgn ⊗ b_{ρ,0}^{Σ}), as a right Σ_{w} × Σ_{r}^{0}-module.

Proof:

Note that P commutes with b_{ρ,j_{λ}^{i}}^{Σ} for every i, and Br_{P} (b_{ρ,j_{λ}^{i}}^{Σ}) = b_{ρ,0}^{Σ}. Since the Brauer map is an algebra homomorphism, we compute the image under Br_{P} of the product η_{λ} to be

Br_{P} (η) = ( ∑_{C_{G}(P)∩J_{λ}} sgn(y).y).b_{ρ,0}^{Σ}.

This completes the proof of (a). Thus,

Br_{P} (η_{λ}.kM^{P})

= Br_{P} (η_{λ}(kΣ_{v})^{P} eζ_{w})

= ( ∑_{y∈K_{λ}} sgn(y)y).kC.b_{ρ,0}^{Σ}.ζ_{D}

= ζ_{D}.( ∑_{y∈K_{λ}} sgn(y)y).kC.b_{ρ,0}^{Σ},

which is isomorphic to Ind_{Σ_{λ}×Σ_{r}^{0}}^{Σ_{w}×Σ_{r}^{0}}(sgn ⊗ kb_{ρ,0}^{Σ}) as a right kΣ_{w} × Σ_{r}^{0}-module. □

Proposition 94 Let λ be a partition of w. There is an idempotent f_{λ} in OΣ_{v} such that,

(a) f_{λ}KM ≅ Ind_{Σ_{λ}×Σ_{r}^{0}}^{Σ_{w}×Σ_{r}^{0}}(sgn ⊗ Kb_{ρ,0}^{Σ}), as a right KΣ_{w} × Σ_{r}^{0}-module.

(b) f_{λ}kM is isomorphic to Ind_{Σ_{λ}×Σ_{r}^{0}}^{Σ_{w}×Σ_{r}^{0}}(sgn ⊗ kb_{ρ,0}^{Σ}), as a right kΣ_{w} × Σ_{r}^{0}- module.

(c) f_{λ}kM = η_{λ}kM = η_{λ}kM^{P} ≅ Br_{P} (η_{λ}kM^{P}).

Proof:

By lemma 91, we can write OΣ_{v}η_{λ} = U ⊕ V, where U is projective and V has no projective summand. And that the character of KV only has constituents labelled by δ ⊴ β where β is p-singular. Since OΣ_{v}η_{λ} is a quotient of OΣ_{v}, the submodule U may be generated by an idempotent. Let f_{λ} be such an idempotent. We prove (a) and (b) for this idempotent.

First we show that for γ ∈ Θ_{w}, we have

(χ^{γ}, char(KΣ_{v}η_{λ})) = (χ^{γ}, char(KΣ_{v}f_{λ})),

where char(X) denotes the character of a KΣ_{v}-module X.

To see this, only observe that for γ ∈ Θ_{w}, the formula (χ^{γ}, char(KV)) = 0 holds, for the complement V of U defined above. This is a consequence of lemma 89, for Θ_{w} contains no p-singular partitions.

As our second task, we show that η_{λ}KM = f_{λ}KM, so that (by pmodular reduction) η_{λ}kM = f_{λ}kM. Indeed,

η_{λ}KM ≅ Hom_{Σ_{v}}(KΣ_{v}η_{λ}, KM),

(note that the algebra KΣ_{v} is semisimple). And by lemma 84, char(KM) only has constituents in Θ_{w}. As has already been established, the part of KΣ_{v}η_{λ} in Θ_{w} is the same as the part of KΣ_{v}f_{λ} in Θ_{w}, so

Hom_{Σ_{v}}(KΣ_{v}η_{λ}, KM) ≅
Hom_{Σ_{v}}(KΣ_{v}f_{λ}, KM) ≅ f_{λ}KM.

As a third task, we should convince ourselves that the character component of KΣ_{v}η_{λ} which corresponds to Θ_{w} is equal to

(∗)  dim(χ^{ρ}). ∑_{ν⊢w} l_{ν}χ^{[∅,..,∅,ν]},

where l_{ν} is defined by Ind_{Σ_{λ}}^{Σ_{w}}(sgn ⊗ ... ⊗ sgn) = ∑_{ν} l_{ν}χ^{ν}.

From this formula, part (a) of this proposition follows directly - recall that −⊗_{Σ_{v}} KM matches the character χ^{[∅,...,∅,λ]} of KB_{ρ,w}^{Σ} with the character χ^{λ} ⊗ χ^{ρ} of KΣ_{w} ⊗ Kb_{ρ,0}^{Σ}.

To see (∗), consider our construction of OΣ_{v}η_{λ} as a module, tensored, induced and cut, tensored, induced and cut,.... At the same time meditate upon lemma 90. The formula is then visible, inductively.

Our final task is to prove part (b) of the proposition. The Brauer map gives (by lemma 85) a surjection

η_{λ}kM^{P} → Ind_{Σ_{λ}×Σ_{r}^{0}}^{Σ_{w}×Σ_{r}^{0}}(sgn ⊗ kb_{ρ,0}^{Σ}).

In addition, there is an injection

η_{λ}kM^{P} → η_{λ}kM = f_{λ}kM.

But part (a) of the proposition shows that the dimension of f_{λ}kM is equal to the dimension of Ind_{Σ_{λ}×Σ_{r}^{0}}^{Σ_{w}×Σ_{r}^{0}}(sgn ⊗ kb_{ρ,w}^{Σ}), so these two maps must be isomorphisms. This completes the proof of (b) and (c). □

Lemma 95 Every summand of kM as a right kΣ_{w} × Σ_{r}^{0}-module is isomorphic to a sum of summands of ⨁_{λ} f_{λ}kM.

Proof:

Let j be an idempotent such that _{Σ_{v}}OΣ_{v}j is the sum of ALL projective summands of _{Σ_{v}}OΣ_{v} with simple top in {D^{λ}|λ ∈ Θ_{w}}.

Note that OΣ_{v}f_{λ} has the projective cover of D^{[∅,...,∅,λ′]} as a summand - because its character is χ^{[∅,...,∅,λ′]} + ( a sum of χ^{μ}’s, μ ⋭ [∅, ..., ∅, λ]), by the formula (∗) and lemma 89. Thus, ⨁_{λ⊢w} f_{λ}M has every summand of jM as a summand.

But jM = M, since

iOΣ_{v}eΣ_{w} = Hom_{Σ_{v}}(OΣ_{v}i, M) = 0

for any projective indecomposable module OΣ_{v}i with simple top outside {D^{λ}|λ ∈ Θ_{w}} (recall KM has character summands corresponding to elements of Θ_{w}, and KΣ_{v}i has character summands corresponding to partitions outside Θ_{w} by lemma 89).

The lemma holds ! □

Let G be a finite group, with subgroups H and K. Let ζ_{H} = ∑_{h∈H} h and let ζ_{K} = ∑_{k∈K} k be corresponding sums in the group algebra kG.

Lemma 96 (a)As kG-modules, kG.ζ_{H} ≅ k[G/H] and kG.ζ_{K} ≅ k[G/K].

(b) As vector spaces, Hom_{G}(kGζ_{H}, kGζ_{K}) ≅ k[H\G/K]

(c) Any element of Hom_{G}(kG.ζ_{H}, kG.ζ_{K}) can be written as right multiplication by some element of kG.

Proof:

The map gζ_{H} ↦ gH, for g ∈ G defines a kG-module isomorphism kGζ_{H} ≅ k[G/H]. Furthermore,

k[H\G/K] ≅ Hom_{G}(k[G/H], k[G/K]),

HgK ↦ (H ↦ ∑_{x∈HgK/K} xK).

In other words, Hom_{G}(kGζ_{H}, kGζ_{K}) has a basis T_{s} indexed by elements s ∈ H\G/K, where

T_{HgK} : ζ_{H} ↦ ∑_{x∈HgK} x = ζ_{H}.g. ∑_{y∈T} y,

and T is a set of representatives for (g^{−1}Hg ∩ K)\K. So T_{HgK} can be defined as right multiplication by ∑_{y∈} y. □

Lemma 97 The natural map f_{λ}kΣ_{v}f_{μ} → Hom_{kE}(f_{μ}kM, f_{λ}kM) is surjective.

Proof:

Since f_{λ}kM = η_{λ}kM, left multiplication by x ∈ kΣ_{v} on f_{λ}kM ⊆ kΣ_{v} is equivalent to left multiplication by xf_{λ} ∈ kΣ_{v}. Similarly, if x.y ∈ η_{μ}kM = f_{μ}kM, then f_{μ}x.y = x.y. To prove the lemma then, it is sufficient to show that every element of Hom_{E}(η_{λ}kM, η_{μ}kM) is given (as left multiplication) by an element of kΣ_{v}.

Recall from lemma 94 and the proof of lemma 93 that,

η_{λ}kM = η_{λ}kM^{P} ≅ Br_{P} (η_{λ}kM^{P})

= ζ_{D}.( ∑_{y∈K_{λ}} sgn(y)y).kC.b_{ρ,0}^{Σ}

≅ ( ∑_{y∈K_{λ}} sgn(y).y)kK_{(w)} ⊗ kb_{ρ,0}^{Σ}.

Thanks to lemma 96, any homomorphism between,

( ∑_{y∈K_{μ}} sgn(y).y)kK_{(w)}, and

( ∑_{y∈K_{λ}} sgn(y).y)kK_{(w)},

is given as left multiplication by an element of kK_{(w)}. See this by applying the automorphism #. Thus, any homomorphism between,

η_{μ}kM ≅ ( ∑_{y∈K_{μ}} sgn(y).y)kK_{(w)} ⊗ kb_{ρ,0}^{Σ}, and

η_{λ}kM ≅ ( ∑_{y∈K_{λ}} sgn(y).y)kK_{(w)} ⊗ kb_{ρ,0}^{Σ},

is given as left multiplication by an element of kΣ_{v}. After this, the result follows on. □

Corollary 98 The natural algebra homomorphism from kb_{ρ,w}^{Σ} to the endomorphism ring End_{kE}(kM) is a surjection.

Proof:

The algebra kE is defined to be the endomorphism ring End_{Σ_{v}}(kM). So kΣ_{v} indeed maps to End_{kE}(kM). Thanks to lemma 95 and lemma 97, for any primitive idempotents i, j ∈ kb_{ρ,w}^{Σ}/kI, the natural map

i(kb_{ρ,w}^{Σ}/kI)j → Hom_{E}(jkM, ikM)

is surjective. On writing 1 ∈ kb_{ρ,w}^{Σ}/kI as a sum of primitive idempotents, the result follows. □

Remark 99 In the light of theorem 94 and lemma 95, theorem 87 is proven. □

Remarks and questions.

Recall that the set of irreducible characters for the Schur algebra KS(w, w) may be parametrized {χ(λ)|λ a partition of w}, and the set of simple kS(w, w)- modules {L(λ)|λ a partition of w}. In this way, χ(λ) corresponds under (non-twisted) Schur-Weyl duality to the character χ^{λ} of Σ_{w}. In addition, χ(λ) has a single composition factor isomorphic to L(λ), and for any other composition factor L(μ), we have λ ▷ μ.

Let A = End_{E}(M). Then A is O-free, and KA ≅ Kb_{ρ,w}^{Σ}/KI is Morita equivalent to the Schur algebra KS(w, w), by theorem 87. By convention, we match by this Morita equivalence the character χ(λ) of KS(w, w) with the character χ^{[∅,...,∅,λ′]} of KΣ_{v}.

Theorem 87 also informs us that kA = kb_{ρ,w}^{Σ}/kI is Morita equivalent to the Schur algebra kS(w, w). How do simple modules correspond ?

Proposition 100 Under the Morita equivalence between kS(w, w) and kb_{ρ,w}^{Σ}/kI of theorem 87, the simple module L(λ) corresponds to D^{[∅,...,∅,λ′]}.

Proof:

For λ, μ partitions of w, let the multiplicities m_{μ}^{λ} be defined by,

Ind_{Σ_{λ}}^{Σ_{w}}K = ∑_{μ} m_{μ}^{λ}.χ^{μ}.

We know for that m_{μ}^{λ} is zero for all partitions λ ▷ μ of w, and that m_{λ}^{λ} = 1 for all partitions λ of w.

The right KΣ_{w} × Σ_{r}-module f_{λ}KM has character,

dim(χ^{ρ}) ∑_{μ} m_{μ}^{λ}(χ^{μ′} ⊗ χ^{ρ}).

Consider Hom_{E}(M, f_{λ}M), a left A-module, which has character

dim(χ^{ρ}) ∑_{μ} m_{μ}^{λ}χ^{[∅,...,∅,μ′]},

Reduced modulo p, this is the kA-module,

Hom_{Σ_{w}×Σ_{r}}(kM, f_{λ}kM),

which via Morita equivalence, corresponds to to dim(χ^{ρ}) copies of the kS(w, w)- module,

E^{⊗r#}.( ∑_{σ∈Σ_{λ}} sgn(σ)σ).

In other words, we have dim(χ^{ρ}) copies of the kS(w, w)-module,

E^{⊗r}.( ∑_{σ∈Σ_{λ}} σ).

This module is isomorphic to a p-modular reduction of the KS(w, w)-module with character dim(χ^{ρ}) ∑_{μ} m_{μ}^{λ}.χ(μ).

Induction according to the dominance ordering now implies that the p-modular reduction of an KS(w, w)-module with character χ(λ), mapped under Morita equivalence to a kA-module, has the same composition factors as a p-modular reduction of a KΣ_{v}-module with character χ^{[∅,...,∅,λ′]}. The lower unitriangularity of the decomposition matrices of kS(w, w) and kΣ_{v} implies the result. □

Corollary 101 The decomposition matrix of kS(w, w) is a submatrix of the decomposition matrix of kb_{ρ,w}^{Σ}, where row and column λ corresponds to row and column [∅, ..., ∅, λ′]. □

Remark 102 It is not possible that kΣ_{v}/Ann(kΣ_{v}e) ≅ End_{kNf}(kΣ_{v}e) when p = 2.

Because on one hand (by [47], 1.14.5) the k-dimension of End_{ekΣ_{v}e}(kΣ_{v}e) is at least as great as the K-dimension of End_{KNf}(KΣ_{v}e) ≅ b_{ρ}(KΣ_{v}) (this isomorphism still holds when w ≥ p, by the character calculation in [11]). On the other hand, the dimension of kΣ_{v}/Ann(kΣ_{v}e) is strictly smaller than the dimension of Kb_{ρ,w}^{Σ}, because there are elements of kb_{ρ,w}^{Σ} which act on kΣ_{v}e as zero. For example, let xkΣ_{v} be a simple right b_{ρ,w}^{Σ}-module, not in the top of ekΣ_{v} (such exist, since ekΣ_{v}e and b_{ρ,w}^{Σ} are not Morita equivalent). In this situation, xkΣ_{v}e ≅ Hom(ekΣ_{v}, xkΣ_{v}) = 0.

Remark 103

(a) The modules in the top (and socle) of _{Σ_{v}}kM are those D^{[∅,...,∅,λ]}’s such that λ is p-regular, by theorem 87 and proposition 100.

(b) The Young module summands of _{Σ_{v}}kM are those Y^{[∅,...,∅,λ]}’s such that λ is p-regular, by theorem 87 and proposition 100.

Question 104 The following are open:

When p = 2, what are the summands of kΣ_{v}e_{N} ? What are their vertices and what are their sources ?

When p = 2, what is the vertex of _{Σ_{v}}kΣ_{v}e_{N} ? What is the source ?

Chapter VI

Ringel duality inside Rock blocks of symmetric groups.

We prove that a Rock block of a symmetric group, of arbitrary defect, possesses a family of internal symmetries, given as Ringel dualities between various subquotients (theorem 120). Since this sequence of symmetries resembles J.A. Green’s walk around the Brauer tree [36], we name it “a walk along the abacus”.

A criterion for Ringel duality.

Let (K, O, k) be an l-modular system. We prove a sufficient condition for Ringel duality between two split quasi-hereditary O-algebras, once they are viewed over k.

Let R be a commutative Noetherian ring. C.P.S. have defined split quasihereditary R-algebras ([17], 3.2). For example, the Schur algebra S(n, r), defined over R, is a split quasi-hereditary algebra, with respect to the poset Λ(n, r). (see [17], 3.7).

More generally, for Γ an ideal of Λ(n, r), and Ω a coideal of Λ(n, r), the generalised Schur algebra S(Γ ∩ Ω) is a split quasi-hereditary subquotient of S(n, r), with respect to the poset Γ ∩ Ω.

Definition 105 For a split quasi-hereditary O-algebra A, let us define a K-k- tilting module T to be a finitely generated A-order which is a tilting module over K, as well as a tilting module over k.

The following result resembles M. Broué’s theorem 70, which gives sufficient conditions for a Morita equivalence between symmetric O-algebras:

Theorem 106 Let A, B be split quasi-hereditary algebras over O with respect to posets Λ, Υ. Suppose that KA, KB are semisimple. Let T be an A-B-bimodule which is a K-k-tilting module at the same time as a left Amodule and as a right B-module. Suppose that the functors

KT ⊗_{KB} − : KB − mod → KA − mod
− ⊗_{KA} KT : mod − KA → mod − KB

are equivalences of categories, such that the resulting bijections of irreducible modules define order-reversing maps between Λ and Υ. Then kT defines a Ringel duality between kA and kB^{op}. Furthermore, B ≅ End_{A}(T), and A ≅ End_{B}(T).

Proof:

Let {P (μ), μ ∈ Υ} be the set of non-isomorphic principal indecomposable B-modules. Let Ψ be the order-reversing map from Υ to Λ defined by KT.

The A-module T has |Υ| distinct summands T ⊗_{B} P (μ). Viewed over K, such a summand has a single composition factor KΔ(Ψμ), and all other composition factors KΔ(λ), λ < Ψμ. This is because Ψ is order-reversing. Viewed over k, such a summand has a single composition factor L(Ψμ), and all other composition factors L(λ), λ < Ψμ. In other words, kT ⊗_{kB} P (μ) is the indecomposable tilting module T (Ψμ) for kA. Thus, kT is a full tilting module as a left kA-module.

Likewise, kT is a full tilting module as a right kB-module. It follows that kA and kB act faithfully on kT.

Note that there are natural isomorphisms KB ≅ End_{KA}(KT) and KA ≅ End_{KB}(KT) of algebras.

The endomorphism ring End_{A}(T) is an O-order, so that kEnd_{A}(T) injects into End_{kA}(kT), and such that KEnd_{A}(T) surjects onto End_{KA}(KT), which is isomorphic to KB. The dimension of End_{kA}(kT) is given ([26], A.2.2(ii)) by the formula,

∑_{ν}[kT : Δ_{k}(ν)][kT : ∇_{k}(ν)].

The dimension of End_{KA}(KT) is given by the formula,

∑_{ν}[KT : Δ_{K}(ν)][KT : ∇_{K}(ν)].

It follows from the equality of these formulae, that the rank of End_{A}(T) is equal to the rank of B, and kEnd_{A}(T) ≅ End_{kA}(kT). Indeed, kEnd_{A}(T) ≅ End_{kA}(kT) ≅ kB, since kB acts faithfully on kT.

We have now verified that kA and kB are in Ringel duality. Since the natural map from kB to kEnd_{A}(T) is an isomorphism, B is O-pure in End_{A}(T), and so B ≅ End_{A}(T). Likewise, A ≅ End_{B}(T). This completes the proof of theorem 106. □

Combinatorial preliminaries.

Let p be a prime number and w any natural number. Let ρ = ρ(p, w) be a minimal Rouquier core. Thus, in an abacus presentation, ρ has precisely w − 1 more beads on the i^{th} runner than on the i − 1^{th} runner.

Lemma 107 Let ρ̃ be an arbitrary Rouquier core. Then Ob_{ρ̃,w}^{Σ} is Morita equivalent to Ob_{ρ,w}^{Σ}. The resulting correspondence of partitions preserves p-quotients.

Proof:

Scopes’ isometries [61] correspond to the motion of k beads one runner leftwards on the abacus, where k ≥ w. Suppose that ρ̃ has N_{i} ≥ (w − 1) more beads on runner i than on runner i − 1. Suppose that N_{j} > (w − 1). Then there is a Scopes isometry which moves those N_{j} beads from runner j to runner j − 1, which may be followed by a Scopes isometry which moves N_{j} + N_{j+1} beads from runner j + 1 to runner j,....., which may be followed by a Scopes isometry which moves N_{j} +N_{j+1} +...+N_{p−1} beads from runner p−1 ro runner p−2. If j > 1, then the result is a p-core with N′_{j−1} > (w−1) more beads on runner j − 1 than on runner j − 2.

We may thus proceed with a further p − j Scopes isometries, moving beads from runners j − 1, ..., p − 2 leftwards. Following this procedure to its natural conclusion (always pushing beads leftwards), and at last circulating the ordering of runners on the abacus so that runner 0 becomes runner j, we obtain a p-core ρ̃_{0}, smaller than ρ̃ which still satisfies the condition that there are ≥ (w − 1) more beads on runner i than on runner i − 1. By induction, lemma 107 is proven. □

Let I = {λ | λ has core ρ and weight w}

be the indexing poset of kb_{ρ,w}^{S}. We wish to describe the dominance order on I. To λ ∈ I with p-quotient [λ^{0}, λ^{1}, ..., λ^{p−1}], let us associate an element < λ >∈ ℕ^{wp}, given by

(λ_{1}^{p−1}, λ_{2}^{p−1}, ..., λ_{w}^{p−1}, λ_{1}^{p−2}, λ_{2}^{p−1}, ..., λ_{w}^{p−2}, ..., λ_{1}^{0}, λ_{2}^{0}, ..., λ_{w}^{0}).

Let us place the dominance order on ℕ^{wp}. We then have:

Proposition 108 Let λ, μ ∈ I. In the dominance order, μ ⊴ λ if and only if < μ > ⊴ < λ >.

Proof:

Following lemma 107, and applying a series of Scopes isometries (which preserve the dominance order), we may replace ρ by ρ̃, where ρ̃ has at least N > 2w beads on runner i than runner i − 1 (for i = 1, ..., p − 1). Let

Ĩ = {λ | λ has core ρ̃, and weight w}.

Suppose that λ ◁ μ are neighbours in Ĩ. There is a sequence λ = λ_{0} ◁ λ_{1} ◁ ... ◁ λ_{m} = μ, where the Young diagram of λ_{j−1} is obtained from the Young diagram of λ_{j} by removing a box and placing it lower in the diagram (for j = 1, ..., m). We may assume that no box is removed from the core in this sequence of motions.

Thus, at each step we must remove a box from one of the p-hooks which has been added to ρ̃ to create μ. Since ρ̃ has N more beads on runner i than runner i − 1 (for i = 1, ..., p − 1), we must actually remove an entire p-hook if we are to end our sequence in Ĩ. Correspondingly, we must add an entire p-hook when we add boxes. The new additional p-hook must appear lower in the Young diagram than the old removed p-hook.

On an abacus, the corresponding motion looks as follows: move a single bead one place higher on its runner, move a second bead (necessarily above the first bead) one place lower on its runner.

This corresponds precisely to removing one box from the Young diagram of < λ >, and replacing it lower in the Young diagram to obtain < μ >. Thus, < λ > ◁ < μ >.

Working backwards, we find conversely that < λ > ◁ < μ > implies μ ◁ λ. □

We may deduce from the above proposition a number of combinatorial results concerning ideals and coideals of the poset I, ordered by the dominance ordering.

Definition 109 For natural numbers a_{1}, ..., a_{p−1} such that ∑_{i=0}^{p−1} a_{i} = w, let

I_{(a_{0},a_{1},..,a_{p−1})} = \left\{{λ ∈ I | λ ⊴ μ for some μ ∈ I with p-quotient \atop [μ^{0}, ..., μ^{p−1}], such that |μ^{i}| = a_{i}}\right\}

J_{(a_{0},a_{1},..,a_{p−1})} = \left\{{λ ∈ I | μ ⊴ λ for some μ ∈ I with p-quotient \atop [μ^{0}, ..., μ^{p−1}], such that |μ^{i}| = a_{i}}\right\}

For i = 0, ..., p − 1, let

I_{i} = I_{(0,...,0,w,0,...,0)}
J_{i} = I_{(0,...,0,w,0,...,0)},

where the w appears as the i^{th} entry in (0, ..., 0, w, 0, ..., 0). Let

I_{res} = I_{p−2}, I_{unres} = I − I_{res} I_{reg} = J_{1}, I_{sing} = I − I_{reg}.

Following these definitions, proposition 108 has a number of corollaries:

Corollary 110 These subsets of I are ideals:

I_{(a_{0},a_{1},..,a_{p−1})}, I_{i}, I_{res}, I_{sing}.

These subsets of I are coideals:

J_{(a_{0},a_{1},..,a_{p−1})}, J_{i}, I_{unres}, I_{reg}.□

Corollary 111 For i = 0, ..., p − 1,

I_{i} = \left\{{λ ∈ I with p-quotient [λ^{0}, ..., λ^{p−1}], such that \atop |λ^{i+1}| = |λ^{i+2}| = ... = |λ^{p−1}| = 0}\right\}

J_{i} = \left\{{λ ∈ I with p-quotient [λ^{0}, ..., λ^{p−1}], such that \atop |λ^{0}| = |λ^{1}| = ... = |λ^{i−1}| = 0}\right\} .□

Corollary 112

I_{res} = {λ ∈ I | λ is p-restricted }.
I_{unres} = {λ ∈ I | λ is p-nonrestricted }.
I_{reg} = {λ ∈ I | λ is p-regular }.
I_{sing} = {λ ∈ I | λ is p-singular }.□

Corollary 113 The intersection I_{(a_{0},a_{1},..,a_{p−1})}∩J_{(a_{0},a_{1},..,a_{p−1})} is equal to the set,

{λ ∈ I | λ has p-quotient [λ^{0}, ..., λ^{p−1}], where |λ^{i}| = a_{i}}.□

The intersection of ideals in the above lemma is analogous to the classical intersection {μ ⊴ λ} ∩ {μ ⊵ λ} = {λ}. Whilst this classical intersection may be used to index the characters of symmetric groups by partitions, we use the intersection of ideals above to study Rock blocks runner by runner.

Definition 114 For natural numbers a_{0}, ..., a_{p−1} such that ∑_{i=0}^{p−1} a_{i} = w, let

K_{(a_{0},...,a_{p−1})} = I_{(a_{0},...,a_{p−1})} ∩ J_{(a_{0},...,a_{p−1})} =
{λ ∈ I | λ has p-quotient [λ^{0}, ..., λ^{p−1}], where |λ^{i}| = a_{i}}.

For i = 0, ..., p − 1, let

K_{i} = I_{i} ∩ J_{i} =

\left\{{λ ∈ I | λ has p-quotient [λ^{0}, ..., λ^{p−1}], such that \atop |λ^{0}| = ... = |λ^{i−1}| = |λ^{i+1}| = ... = |λ^{p−1}| = 0, |λ^{i}| = w}\right\} .

Quasi-hereditary subquotients of Ob_{ρ,w}^{Σ}.

We introduce pairs of quasi-hereditary quotients of Ob_{ρ,w}^{Σ}, which are isomorphic under the signature automorphism.

Let ω = (1^{v}), a partition of v. Let S(v, v) be the Schur algebra associated to polynomial representations of GL_{v} of degree v, defined over O. Recall that ξ_{ω}S(v, v)ξ_{ω} is isomorphic to OΣ_{v} (theorem 21).

To an ideal Γ of I, let us associate the ideal X_{Γ} of S(v, v), the quotient by which, is the generalised Schur algebra, S(Γ). Let I_{Γ} = ξ_{ω}b_{ρ,w}^{S}X_{Γ}ξ_{ω} be the corresponding ideal of Ob_{ρ,w}^{Σ}.

Lemma 115 Suppose that Γ is an ideal of I_{res}. Then Ob_{ρ,w}^{Σ}/I_{Γ} is a quasihereditary algebra, with indexing poset Γ. The decomposition matrix of the algebra Ob_{ρ,w}^{Σ}/I_{Γ} is the square submatrix of the decomposition matrix of Ob_{ρ,w}^{Σ}, whose rows are indexed by elements of Γ.

Proof:

The generalised Schur algebra Ob_{ρ,w}^{S}/X_{Γ} is a quasi-hereditary algebra, whose indexing poset is Γ.

Over the field k, the idempotent ξ_{ω} sends to zero precisely those simple modules indexed by unrestricted partitions. Thus, (Ob_{ρ,w}^{S}/X_{Γ})ξ_{ω} is a progenerator for Ob_{ρ,w}^{S}/X_{Γ}. It follows that,

ξ_{ω}(Ob_{ρ,w}^{S}/X_{Γ})ξ_{ω} ≅ Ob_{ρ,w}^{Σ}/ξ_{ω}X_{Γ}ξ_{ω} = Ob_{ρ,w}^{Σ}/I_{Γ},

is Morita equivalent to Ob_{ρ,w}^{S}/X_{Γ}. □

Let a_{1}, ..., a_{p−1} be natural numbers such that ∑_{i=1}^{p−1} a_{i} = w.

We write I_{(a_{1},...,a_{p−1})} for I_{Γ}, where Γ is the ideal I_{(a_{1},...,a_{p−1},0)} in I.

Thus, Ob_{ρ,w}^{Σ}/I_{(a_{1},...,a_{p−1})} is a quasi-hereditary algebra whose poset is the ideal I_{(a_{1},...,a_{p−1},0)}, and {D_{λ}|λ ∈ I_{(a_{1},...,a_{p−1},0)}} is a complete set of nonisomorphic simple kb_{ρ,w}^{Σ}/I_{(a_{1},...,a_{p−1})}-modules.

So long as i = 1, ..., p − 1, let us write I_{i} for I_{(0,...,0,w,0,...,0)}, where w appears as the i − 1^{th} entry in (0, ..., 0, w, 0, ..., 0).

Thus, Ob_{ρ,w}^{Σ}/I_{i} is a quasi-hereditary algebra whose poset is I_{i−1}.

Let us write I_{unres} for I_{p−2}. Thus, Ob_{ρ,w}^{Σ}/I_{unres} is a quasi-hereditary algebra whose poset is I_{res}.

Let Ω ⊂ I be a coideal. Let x_{Ω} be an idempotent in OΣ_{v}, such that OΣ_{v}x_{Ω} is a maximal summand of OΣ_{v}, whose indecomposable summands have tops in {D_{λ}|λ ∈ I_{res} ∩ Ω}.

Lemma 116 Suppose that Ω is a coideal of I, and that Γ is an ideal of I_{res}. Let A_{Γ∩Ω} = x_{Ω}(Ob_{ρ,w}^{Σ}/I_{Γ})x_{Ω}. Then A_{Γ∩Ω} is a quasi-hereditary algebra, with indexing poset Γ ∩ Ω.

Proof:

A_{Γ∩Ω} is Morita equivalent to the generalised Schur algebra with indexing poset Γ ∩ Ω, via the bimodule ξ_{Ω}(S(v, v)/J_{Γ∩Ω})ξ_{ω}.x_{Ω}. □

Suppose that a_{1}, ..., a_{p−1} are natural numbers such that ∑_{i=1}^{p−1} a_{i} = w.

We write A_{(a_{1},...,a_{p−1})} for A_{Γ∩Ω}, where Ω = J_{(a_{1},...,a_{p−1},0)}, and Γ = I_{(a_{1},...,a_{p−1},0)}. Indeed, A_{(a_{1},...,a_{p−1})} is a quasi-hereditary algebra whose poset is K_{(a_{1},...,a_{p−1},0)}, by corollary 113.

So long as i = 1, ..., p−1, we write A_{i} for A_{(0,...,0,w,0,...,0)}, where w appears as the i − 1^{th} entry in (0, ..., 0, w, 0, ..., 0). Thus, A_{i} is a quasi-hereditary algebra whose poset is K_{i−1}.

For natural numbers a_{1}, ..., a_{p−1} such that ∑_{i=1}^{p−1} a_{i} = w, let J_{(a_{1},...,a_{p−1})} be the ideal I_{(a_{p−1},...,a_{1})}^{#}. Thus, Ob_{ρ,w}^{Σ}/J_{(a_{1},...,a_{p−1})} is a quasi-hereditary algebra whose poset is J_{(0,a_{1},...,a_{p−1})}^{op}.

We write {D^{λ}|λ ∈ J_{(0,a_{1},...,a_{p−1})}} for the set of simple kb_{ρ,w}^{Σ}/J_{(a_{1},...,a_{p−1})}- modules.

For i = 1, ..., p − 1, let J_{i} = I_{p−i}^{#}. Thus, Ob_{ρ,w}^{Σ}/J_{i} is a quasi-hereditary algebra whose poset is J_{i}^{op}.

Let J_{sing} = I_{unres}^{#}. Thus, Ob_{ρ,w}^{Σ}/J_{sing} is a quasi-hereditary algebra whose poset is I_{reg}^{op}.

Let y_{(a_{1},...,a_{p−1})} = x_{(a_{p−1},...,a_{1})}^{#}. Let

B_{(a_{1},...,a_{p−1})} = y_{(a_{1},...,a_{p−1})} (Ob_{ρ,w}^{Σ}/J_{(a_{1},...,a_{p−1})}) y_{(a_{1},...,a_{p−1})}.

Thus, B_{(a_{1},...,a_{p−1})} is a quasi-hereditary algebra whose poset is K_{(0,a_{1},...,a_{p−1})}^{op}.

So long as i = 1, ..., p − 1, let us write B_{i} for B_{(0,...,0,w,0,...,0)}, where w appears as the i^{th} entry in (0, ..., 0, w, 0, ..., 0). Thus, B_{i} is a quasi-hereditary algebra whose poset is K_{i}^{op}.

Remark 117 Simple kA_{(a_{1},...,a_{p−1})}-modules are in natural one-one correspondence with the set {D_{λ} | λ ∈ K_{(a_{1},...,a_{p−1},0)}}. By theorem 60, this set is equal to the set {D^{λ} | λ ∈ K_{(0,a_{1},...,a_{p−1})}}.

Simple kB_{(a_{1},...,a_{p−1})}-modules are also in natural one-one correspondence with the set {D^{λ} | λ ∈ K_{(0,a_{1},...,a_{p−1})}}.

By balancing the algebras Ob_{ρ,w}^{Σ}/I_{unres} and Ob_{ρ,w}^{Σ}/J_{sing} on the Mullineux map, we reveal Ringel dualities between different runners of Ob_{ρ,w}^{Σ}, in the following section.

Walking along the abacus.

Let p ≥ 3. For ∑_{i=1}^{p−2} a_{i} = w, consider the O-lattice

N_{(a_{1},...,a_{p−2})} = x_{(0,a_{1},...,a_{p−2})}OΣ_{v}y_{(a_{1},...,a_{p−2},0)}.

In this section we prove that N_{(a_{1},...,a_{p−2})} provides a Ringel duality between the quasi-hereditary subquotients A_{(0,a_{1},...,a_{p−2})} and B_{(a_{1},...,a_{p−2},0)} of Ob_{ρ,w}^{Σ}.

These Ringel dualities should be viewed as internal symmetries of the Rock block. For simple kA_{(0,a_{1},...,a_{p−2})}-modules are in natural correspondence with simple kb_{ρ,w}^{Σ}-modules D^{λ} indexed by elements of K_{(0,0,a_{1},...,a_{p−2})}. At the same time, simple kB_{(a_{1},...,a_{p−2},0)}-modules are in natural correspondence with simple kb_{ρ,w}^{Σ}-modules D^{μ} indexed by elements of K_{(0,a_{1},...,a_{p−2},0)}.

These symmetries therefore enable us to translate module-theoretic information along the abacus.

Here’s a technical lemma:

Lemma 118 Suppose that p ≥ 3. Let ∑_{i=1}^{p−2} a_{i} = w. Then

J_{sing}.x_{(0,a_{1},...,a_{p−2})} = 0,
J_{(a_{1},...,a_{p−2},0)}.x_{(0,a_{1},...,a_{p−2})} = x_{(0,a_{1},...,a_{p−2})}.J_{(a_{1},...,a_{p−2},0)} = 0,
I_{unres}.y_{(a_{1},...,a_{p−2},0)} = 0,
I_{(0,a_{1},...,a_{p−2})}.y_{(a_{1},...,a_{p−2},0)} = y_{(a_{1},...,a_{p−2},0)}.I_{(0,a_{1},...,a_{p−2})} = 0.

Proof:

The character of OΣ_{v}.x_{(0,a_{1},...,a_{p−2})} has irreducible components χ^{λ}, where λ lies in the coideal J_{(0,a_{1},...,a_{p−2},0)} of I. This character has no components which are p-singular, and no components which lie in I − J_{(0,a_{1},...,a_{p−2},0)}.

Over K, the ideal J_{sing} is equal to the Wedderburn component of Kb_{ρ,w}^{Σ} with p-singular components, whilst J_{(a_{1},...,a_{p−2},0)} is equal to the Wedderburn component with components in I −J_{(0,a_{1},...,a_{p−2},0)}. This, along with the fact that irreducible characters for symmetric groups are self-dual, proves the first two parts of the lemma. The third and fourth parts follow analogously. □

We now show that N_{(a_{1},...,a_{p−2})} is an A_{(0,a_{1},...,a_{p−2})}-B_{(a_{1},...,a_{p−2},0)}-bimodule:

Lemma 119 The kernel of the natural map from x_{(0,a_{1},...,a_{p−2})}OΣ_{v}x_{(0,a_{1},...,a_{p−2})} to End(N_{(a_{1},...,a_{p−2})}) contains the ideal x_{(0,a_{1},...,a_{p−2})}I_{(0,a_{1},...,a_{p−2})}x_{(0,a_{1},...,a_{p−2})}.

The kernel of the natural map from y_{(a_{1},...,a_{p−2},0)}OΣ_{v}y_{(a_{1},...,a_{p−2},0)} to the ring End(N_{(a_{1},...,a_{p−2})}), contains the ideal y_{(a_{1},...,a_{p−2},0)}J_{(a_{1},...,a_{p−2},0)}y_{(a_{1},...,a_{p−2},0)}.

Thus, N_{(a_{1},...,a_{p−2})} is an A_{(0,a_{1},...,a_{p−2})}-B_{(a_{1},...,a_{p−2},0)}-bimodule.

Proof:

x_{(0,a_{1},...,a_{p−2})}I_{(0,a_{1},...,a_{p−2})}x_{(0,a_{1},...,a_{p−2})}OΣ_{v}y_{(a_{1},...,a_{p−2},0)}
⊆ x_{(0,a_{1},...,a_{p−2})}I_{(0,a_{1},...,a_{p−2})}y_{(a_{1},...,a_{p−2},0)} = 0,

by lemma 118. Likewise,

x_{(0,a_{1},...,a_{p−2})}OΣ_{v}y_{(a_{1},...,a_{p−2},0)}J_{(a_{1},...,a_{p−2},0)}y_{(a_{1},...,a_{p−2},0)}
⊆ x_{(0,a_{1},...,a_{p−2})}J_{(a_{1},...,a_{p−2},0)}y_{(a_{1},...,a_{p−2},0)} = 0.□

Theorem 120 (“Walking along the abacus”) The bimodule N_{(a_{1},...,a_{p−2})} defines a Ringel duality between kA_{(0,a_{1},...,a_{p−2})} and kB_{(a_{1},...,a_{p−2},0)}^{op}.

Proof:

We first show that N_{(a_{1},...,a_{p−2})} is a K-k-tilting module both as a left A_{(0,a_{1},...,a_{p−2})}-module, and as a right B_{(a_{1},...,a_{p−2},0)}-module.

Let R ∈ {K, k}. Recall that under Schur-Weyl duality, Specht modules correspond to costandard modules. Therefore, the costandard modules for Rb_{ρ,w}^{Σ}/I_{unres} are those Specht modules indexed by restricted partitions. The costandard modules for Rb_{ρ,w}^{Σ}/I_{(0,a_{1},...,a_{p−2})} are those Specht modules indexed by elements of I_{(0,a_{1},...,a_{p−2},0)}. The costandard modules for A_{(0,a_{1},...,a_{p−2})} are those modules x_{(0,a_{1},...,a_{p−2})}S, where S is a Specht module indexed by an element of K_{(0,a_{1},...,a_{p−2},0)}.

Since RΣ_{v}y_{(a_{1},...,a_{p−2})} is a projective module, it has a filtration by Specht modules. Suppose that S is a Specht module in this filtration. Then S is a costandard module for Rb_{ρ,w}^{Σ}/I_{(0,a_{1},...,a_{p−2})}, by lemma 118. Thus, x_{(0,a_{1},...,a_{p−2})}S is a costandard module for A_{(0,a_{1},...,a_{p−2})}. So N_{(a_{1},...,a_{p−2})} has a filtration by costandard modules.

Since RΣ_{v}y_{(a_{1},...,a_{p−2})} is self-dual, it may also be filtered by dual Specht modules. The same argument as above now shows that N_{(a_{1},...,a_{p−2})} has a filtration by standard modules.

Thus, N_{(a_{1},...,a_{p−2})} is a left K-k-tilting module for A_{(0,a_{1},...,a_{p−2})}. In the same way, N_{(a_{1},...,a_{p−2})} is a right K-k-tilting module for B_{(a_{1},...,a_{p−2},0)}.

We would like to apply theorem 106.

Recall that the KΣ_{v}-KΣ_{v}- bimodule KΣ_{v} has character, ⨁_{λ} χ^{λ}⊗χ^{λ}. In this way, the x_{(0,a_{1},...,a_{p−2})}KΣ_{v}x_{(0,a_{1},...,a_{p−2})}- y_{(a_{1},...,a_{p−2},0)}KΣ_{v}y_{(a_{1},...,a_{p−2},0)}- module KN_{(a_{1},...,a_{p−2})} has character,

⨁_{λ∈K_{(0,a_{1},...,a_{p−2},0)}} x_{(0,a_{1},...,a_{p−2})}.χ^{λ} ⊗ χ^{λ}.y_{(a_{1},...,a_{p−2},0)}.

Note that,

{x_{(0,a_{1},...,a_{p−2})}.χ^{λ} | λ ∈ K_{(0,a_{1},...,a_{p−2},0)}},

is a complete set of irreducible left KA_{(0,a_{1},...,a_{p−2})}-modules. And that,

{χ^{λ}.y_{(a_{1},...,a_{p−2},0)} | λ ∈ K_{(0,a_{1},...,a_{p−2},0)}^{op}},

is a complete set of irreducible right KB_{(a_{1},...,a_{p−2},0)}-modules.

Thus, KN_{(a_{1},...,a_{p−2})} induces a Morita equivalence between KA_{(0,a_{1},...,a_{p−2})} and KB_{(a_{1},...,a_{p−2},0)}, which reverses order on the indexing posets. It is now a consequence of theorem 106, that kA_{(0,a_{1},...,a_{p−2})}, and kB_{(a_{1},...,a_{p−2},0)}^{op} are in Ringel duality. □

Remark 121 Under the Morita equivalence provided by KN_{(a_{1},...,a_{p−2})}, a simple KA_{(0,a_{1},...,a_{p−2})}-module S_{A}^{λ} corresponds to a simple KB_{(a_{1},...,a_{p−2},0)}- module S_{B}^{λ}.

Note 122 Both A_{(a_{1},...,a_{p−1})} and B_{(a_{1},...,a_{p−1})} have simple modules in natural correspondence with {D^{λ}|λ ∈ K_{(0,a_{1},...,a_{p−1})}}. Let L(a_{1}, ..., a_{p−1}) be the Serre subcategory of kΣ_{v} − mod generated by {D^{λ}|λ ∈ K_{(0,a_{1},...,a_{p−1})}}.

Chapter VII

James adjustment algebras for Rock blocks of symmetric groups.

Let kb_{ρ,w}^{Σ} be a Rock block of a symmetric group, whose weight is w. We show that there is a nilpotent ideal N of kb_{ρ,w}^{Σ}, such that kb_{ρ,w}^{Σ}/N is Morita equivalent to a direct sum,

⨁_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} (⨂_{i=1}^{p−1} S(a_{i}, a_{i})) ,

of tensor products of Schur algebras (theorem 129). The decomposition matrix of this quotient is equal to the James adjustment matrix of kb_{ρ,w}^{Σ}.

In chapter five, we proved the existence of a quotient of kb_{ρ,w}^{Σ}, equivalent to S(w, w). In this chapter, we show that the Ringel dualities of chapter VI, may be applied simultaneously with the ideas of chapter V, to set up an induction, proving the existence of a quotient kb_{ρ,w}^{Σ}/N, described above.

Although we choose not explicitly to describe its proof here, the there exists a generalisation of the main result of this chapter to arbitrary Hecke algebras of type A. Indeed, there exists a nilpotent ideal N of kb_{ρ,w}^{H_{q}}, such that kb_{ρ,w}^{H_{q}}/N is Morita equivalent to a direct sum,

⨁_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} (⨂_{i=1}^{p−1} S(a_{i}, a_{i})) ,

of tensor products of unquantized Schur algebras. The decomposition matrix of this quotient is equal to the James adjustment matrix of kb_{ρ,w}^{H_{q}}.

The James adjustment algebra of a Hecke algebra.

Let (K, O, k) be an l-modular system. Let ℘ be the maximal ideal of O, so that O/℘ ≅ k. Let q ∈ O be a primitive p^{th} root of unity, whose image in k, is non-zero.

This section is devoted to the Hecke algebra H_{q}(Σ_{n}), We define a quotient of this algebra, whose representation theory controls the James adjustment matrix of H_{q}(Σ_{n}) (see [34]).

Let us label the simple KH_{q}(Σ_{n})-modules,

{D_{q}^{λ} | λ a p-regular partition of n},

in contrast with the simple kH_{q}(Σ_{n})-modules, which we label,

{D^{λ} | λ a p-regular partition of n}.

The algebra KH_{q}(Σ_{n}) is far from semisimple in general, and thus has a non-trivial radical. Let ON_{q} be the intersection of this radical with the subalgebra OH_{q}(Σ_{n}) of KH_{q}(Σ_{n}). Thus, ON_{q} is equal to the annihilator in OH_{q}(Σ_{n}) of all simple KH_{q}(Σ_{n})-modules. The ideal ON_{q} is an O-pure sublattice of OH_{q}(Σ_{n}).

The algebra G_{q}(Σ_{n}) = OH_{q}(Σ_{n})/ON_{q} is an O-free algebra, whose square decomposition matrix, ([D_{q}^{λ} : D^{μ}]) is equal to the James adjustment matrix of H_{q}(Σ_{n}). We call it the James adjustment algebra of H_{q}(Σ_{n}). Indeed, we have (cf. [34], 2.3),

Dec_{℘}(OH_{q}(Σ_{n})) = Dec_{<t−q>}(K[t]_{(t−q)}H_{t}(Σ_{n})) × Dec_{℘}(G_{q}(Σ_{n})).

Here, we write Dec_{J} (A) for the decomposition matrix of an algebra A, defined over a ring R, relative to a maximal ideal J.

Cutting G_{q}(Σ_{n}) at a block b_{τ,w}^{H_{q}} of H_{q}(Σ_{n}), we obtain the James adjustment algebra of the block, which we denote Gb_{τ,w}^{H_{q}}. If q = 1, we label this block Gb_{τ,w}^{Σ}.

Preliminaries on adjustment algebras for Rock blocks.

We now concentrate on Rock blocks of symmetric groups. Thus, we assume that the image of q in k, is equal to 1, and k has characteristic l = p. And we adopt the notation of chapters 4-6.

Let x_{(a_{1},...,a_{p−1}),q} be the q-analogue of x_{(a_{1},...,a_{p−1})}, an idempotent of Kb_{ρ,w}^{H_{q}}. Let A_{(a_{1},...,a_{p−1}),q} be the q-analogue of A_{(a_{1},...,a_{p−1})}, and let B_{(a_{1},...,a_{p−1}),q} be the q-analogue of B_{(a_{1},...,a_{p−1})},. These are subquotients of Kb_{ρ,w}^{H_{q}}.

For natural numbers a_{i} such that ∑_{i=1}^{p−1} a_{i} = w, let L_{q}(a_{1}, ..., a_{p−1}) be the Serre subcategory of Kb_{ρ,w}^{H_{q}} − mod generated by simple modules

{D_{q}^{λ} | λ ∈ K_{(0,a_{1},...,a_{p−1})}}.

Let N_{a_{1},...,a_{p−1},q} be the ideal of Kb_{ρ,w}^{H_{q}}, the quotient by which, is equal to the Wedderburn component of the quotient KG_{q}(Σ_{v}) with components in L_{q}(a_{1}, ..., a_{p−1}).

Proposition 123 There are natural isomorphisms of algebras,

Kb_{ρ,w}^{H_{q}}/KN_{a_{1},...,a_{p−1},q} ≅ KA_{(a_{1},...,a_{p−1}),q} ≅ KB_{(a_{1},...,a_{p−1}),q}.

Proof:

Kb_{ρ,w}^{H_{q}}/KN_{a_{1},...,a_{p−1},q}
= x_{(a_{1},...,a_{p−1}),q} (Kb_{ρ,w}^{H_{q}}/KN_{a_{1},...,a_{p−1},q}) x_{(a_{1},...,a_{p−1}),q},

which is equal to the Wedderburn component of,

x_{(a_{1},...,a_{p−1}),q} (Kb_{ρ,w}^{H_{q}}/Radical) x_{(a_{1},...,a_{p−1}),q},

whose simple components correspond to simple objects of L_{q}(a_{1}, ..., a_{p−1}).

But the simple KA_{(a_{1},...,a_{p−1}),q}-modules are in one-one correspondence with simple objects of L_{q}(a_{1}, ..., a_{p−1}). Thus, KA_{(a_{1},...,a_{p−1}),q} surjects onto the quotient

Kb_{ρ,w}^{H_{q}}/KN_{a_{1},...,a_{p−1},q}.

However, KA_{(a_{1},...,a_{p−1}),q} is semisimple by proposition 134(1), so this surjection is an isomorphism.

Applying # to this isomorphism, we discover that in addition,

Kb_{ρ,w}^{H_{q}}/KN_{a_{p−1},...,a_{1},q} ≅ KB_{(a_{p−1},...,a_{1}),q}.□
Quotients of kb_{ρ,w}^{Σ}.

The algebras A_{(a_{1},..,a_{p−1})} and B_{(a_{1},...,a_{p−1})} may be realised as quotients of kb_{ρ,w}^{Σ}, and not merely as subquotients. This will be shown in general in proposition 132, and is crucial to the proof of our main result, theorem 129. As a preliminary, in this section we prove that A_{(0,a_{2},...,a_{p−2},0)} and B_{(0,a_{2},...,a_{p−2},0)} may be realised as quotients of kb_{ρ,w}^{Σ}.

We also prove here a baby version of theorem 129, so you may see how these ideas fit together with those of chapter 6 to provide information on kb_{ρ,w}^{Σ} − mod.

Lemma 124 Suppose that p ≥ 3. Let ∑_{i=1}^{p−2} a_{i} = w.

Then

kb_{ρ,w}^{Σ}/I_{(a_{1},...,a_{p−2},0)} − mod ≅

{M ∈ kΣ_{v} − mod|M has composition factors D^{λ}, λ ∈ I_{(0,a_{1},...,a_{p−2},0)} ∩ I_{reg}}.

Let i = 2, ..., p − 1. Then

kb_{ρ,w}^{Σ}/J_{(0,a_{1},...,a_{p−2})} − mod ≅

{M ∈ kΣ_{v} − mod | M has composition factors D^{λ}, λ ∈ J_{(0,0,a_{1},...,a_{p−2})}}.

Proof:

Let p ≥ 3, and ∑_{i=1}^{p−2} a_{i} = w. Then,

{M ∈ kΣ_{v} − mod|M has simple factors D^{λ}, λ ∈ I_{(0,a_{1},...,a_{p−2},0)} ∩ I_{reg}}

⊆ {M ∈ kΣ_{v} − mod | M is generated by kΣ_{v}y_{(a_{1},...,a_{p−2},0)}} ≅

{M ∈ kb_{ρ,w}^{Σ}/I_{unres} − mod | M is generated by kΣ_{v}y_{(a_{1},...,a_{p−2},0)}},

where the latter isomorphism is by lemma 118. Thus,

{M ∈ kΣ_{v} − mod|M has simple factors D^{λ}, λ ∈ I_{(0,a_{1},...,a_{p−2},0)} ∩ I_{reg}}

= \left\{{M ∈ kb_{ρ,w}^{Σ}/I_{unres} − mod | M has simple factors D^{λ}, \atop λ ∈ I_{(0,a_{1},...,a_{p−2},0)} ∩ I_{reg}}\right\}

= \left\{{M ∈ kb_{ρ,w}^{Σ}/I_{unres} − mod | M has simple factors D_{λ}, \atop λ ∈ I_{(a_{1},...,a_{p−2},0,0)}}\right\}

≅ \left\{{M ∈ kb_{ρ,w}^{Σ}/I_{(a_{1},...,a_{p−2},0)} − mod | M has simple factors D_{λ} \atop λ ∈ I_{(a_{1},...,a_{p−2},0,0)}}\right\} ,

where the latter isomorphism holds thanks to the quasi-heredity of the quotient kb_{ρ,w}^{Σ}/I_{unres}. We deduce that, for i = 1, ..., p − 2,

kb_{ρ,w}^{Σ}/I_{(a_{1},...,a_{p−2},0)} − mod ≅

{M ∈ kΣ_{v} − mod|M has composition factors D^{λ}, λ ∈ I_{(0,a_{1},...,a_{p−2},0)} ∩ I_{reg}}.

The second part of the lemma follows on an application of #. □

Note the following fact:

Lemma 125 Suppose that I, J are ideals in an algebra A. Then

A/(I + J) − mod ≅ (A/I − mod) ∩ (A/J − mod).□

Note that A_{i} and B_{i} both have simple modules in natural correspondence with the set {D^{λ} | λ ∈ K_{i}}. Let L(i) = L(0, ..., 0, w, 0, ..., 0) be a “single runner subcategory” of kΣ_{v} −mod, associated to runner i, for i = 1, ..., p−1. Thus, C(i) is the Serre subcategory of kΣ_{v} − mod generated by

{D^{λ}|λ ∈ K_{i}}.

We may now produce “single runner quotients” of kb_{ρ,w}^{Σ}, for p ≥ 3.

Proposition 126 Let p ≥ 3.

For i = 1, ..., p − 1, there exists an ideal α_{i} of kb_{ρ,w}^{Σ}, such that

kb_{ρ,w}^{Σ}/α_{i} − mod ≅ L(i).

There are natural algebra isomorphisms,

kA_{i} ≅ kB_{i} ≅ kb_{ρ,w}^{Σ}/α_{i}.

Proof:

For ideals α_{i}, we take I_{i} + J_{i}, so long as 2 ≤ i ≤ p − 2. We take α_{1} = I_{1}, and α_{p−1} = J_{p−1}. From lemmas 124 and 125 above it is clear that kb_{ρ,w}^{Σ}/α_{i} − mod ≅ L(i).

Let i = 1, ..., p − 1. We have kb_{ρ,w}^{Σ}/α_{i} = y_{i} (kb_{ρ,w}^{Σ}/α_{i}) y_{i}, since any projective summand of kb_{ρ,w}^{Σ}, maximal subject to the restriction that its top lies in C(i), must be isomorphic to a summand of kΣ_{v}y_{i}.

Note that α_{i} = I_{i,q} + J_{i,q}(mod p), where I_{i,q} (resp. J_{i,q}, α_{i,q}) is the q-analogue of I_{i} (resp. J_{i}, α_{i}), an ideal of Ob_{ρ,w}^{H_{q}}.

By proposition 123, and lemma 125, we know that Ob_{ρ,w}^{H_{q}}/α_{i,q} = KB_{i,q}. We therefore witness the inclusion, y_{i,q}I_{i,q}y_{i,q} ⊆ y_{i,q}J_{i,q}y_{i,q}, over the field K. Taking intersections with the natural O-form for KH_{q}(Σ_{v}), we reveal the inclusion y_{i,q}I_{i,q}y_{i,q} ⊆ y_{i,q}J_{i,q}y_{i,q}, over O.

Reducing modulo p, we find that y_{i}J_{i}y_{i} contains y_{i}I_{i}y_{i}. In conclusion,

kb_{ρ,w}^{Σ}/α_{i} = y_{i} (kb_{ρ,w}^{Σ}/α_{i}) y_{i} ≅ y_{i} (kb_{ρ,w}^{Σ}/J_{i}) y_{i} = kB_{i},

for i = 1, ..., p − 1. Likewise, kb_{ρ,w}^{Σ}/α_{i} ≅ kA_{i}, for i = 1, ..., p − 1. □

Generalising the above proposition and its proof, we have,

Proposition 127 Let p > 3, and let ∑_{i=2}^{p−2} a_{i} = w.

There exists an ideal N_{0,a_{2},...,a_{p−2},0} of kb_{ρ,w}^{Σ} such that,

kb_{ρ,w}^{Σ}/N_{0,a_{2},...,a_{p−2},0} − mod ≅ L(0, a_{2}, ..., a_{p−2}, 0).

There are algebra isomorphisms,

kA_{(0,a_{2},...,K_{p−2},0)≅kB_{(0,a_{2},...,a_{p−2},0)}≅kb_{ρ,w}^{Σ}/N_{0,a_{2},...,a_{p−2},0}.□}

To complete this section, we bear an infant theorem 129.

Proposition 128 (a) Let p = 2. There is a nilpotent ideal α_{1} of kb_{ρ,w}^{Σ}, such that

kb_{ρ,w}^{Σ}/α_{1} − mod ≅ S(w, w) − mod.

(b) Let p ≥ 3, and let i = 1, ..., p − 1. There are equivalences of abelian categories,

L(i) ≅ kb_{ρ,w}^{Σ}/α_{i} − mod ≅ S(w, w) − mod.

Proof:

(a) This is theorem 87, in case p = 2.

(b) Let i = p − 1. The first equivalence is then a particular case of proposition 126. To see the second equivalence recall from theorem 87, that there is a kb_{ρ,w}^{Σ}-module kM, with composition factors in L(p − 1), such that kb_{ρ,w}^{Σ}/Ann(kM) is Morita equivalent to the Schur algebra S(w, w).

We have α_{p−1} ⊆ Ann(kM), by the first part of the proposition. A dimension count yields an isomorphism between kb_{ρ,w}^{Σ}/Ann(kM) and the quotient kb_{ρ,w}^{Σ}/α_{p−1}.

Let i = 2, ..., p − 1. Theorem 120 provides a Ringel duality between kA_{i} and and kA_{i−1}. In the light of proposition 126, and the knowledge that S(w, w) is Ringel self-dual, we discover that L(i) ≅ S(w, w) − mod, for i = 1, ..., p − 1. □

A global-local theorem, and the James adjustment matrix of a Rock block.

In the final sections of this chapter, we prove the following:

Theorem 129 There is a Morita equivalence between kGb_{ρ,w}^{Σ} and a direct sum,

⨁_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} (⨂_{i=1}^{p−1} S(a_{i}, a_{i})) ,

of tensor products of Schur algebras.

Under this Morita equivalence,

L(a_{1}, ..., a_{p−1}) ≅ (⨂_{i=1}^{p−1} S(a_{i}, a_{i})) − mod,

and the correspondence of simple modules is:

D^{[∅,...λ_{p−3},λ_{p−2},λ_{p−1}]} ↔ ... ⊗ L(λ′_{p−3}) ⊗ L(λ_{p−2}) ⊗ L(λ′_{p−1}).

The James adjustment matrix of kb_{ρ,w}^{Σ} is equal to the decomposition matrix of,

⨁_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} (⨂_{i=1}^{p−1} S(a_{i}, a_{i})) .

That is to write,

[D_{q}^{[∅,...,λ′_{p−3},λ_{p−2},λ′_{p−1}]} : D^{[∅,...,μ′_{p−3},μ_{p−2},μ′_{p−1}]}] =

= \{ \begin{matrix}∏_{i=1}^{p−1}[Δ(λ_{i}) : L(μ_{i})], & if |λ_{i}| = |μ_{i}|, i = 1, ..., p − 1 \\ 0, & otherwise \end{matrix}

Here, we set S(0, 0) ≅ k.

Remark 130 When w < p, the quotient kGb_{ρ,w}^{Σ} is merely the quotient of kb_{ρ,w}^{Σ} by its radical. We are therefore far from the strength of theorem 53.

Our proof of theorem 129 is inductive, on one hand applying the Ringel dualities of chapter 6, and on the other generalising the theory of chapter 5.

There falls an elegant description of the decomposition matrix of kb_{ρ,w}^{Σ}, in terms of Littlewood-Richardson coefficients, and decomposition matrices of Schur algebras which are bounded in degree by w.

Corollary 131 The decomposition matrix of kb_{ρ,w}^{Σ} is equal to the matrix product,

Dec_{<t−q>} (K[t]_{(t−q)}b_{∅,1}^{H_{t}} ≀ Σ_{w}) ×

Dec ( ⨁_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} (⨂_{i=1}^{p−1} S(a_{i}, a_{i}))) .

Here, Dec_{<t−q>} (K[t]_{(t−q)}b_{∅,1}^{H_{t}} ≀ Σ_{w})) is the decomposition matrix of a wreath product of the principal block of K[t]_{(t−q)}H_{t}(Σ_{p}), with Σ_{w}. Formulae for the entries in this matrix are given in terms of Littlewood-Richardson coefficients (theorem 58). □

Note that kA_{(a_{1},...,a_{p−1})} is the reduction modulo p of OA_{(a_{1},...,a_{p−1}),q}. The proof of theorem 129 rests upon the following proposition:

Proposition 132 Let a_{i} be natural numbers such that ∑_{i=1}^{p−1} a_{i} = w. Then there exists an ideal N_{a_{1},...,a_{p−1}} in kb_{ρ,w}^{Σ}, such that,

(a) N_{a_{1},...,a_{p−1}} is equal to the reduction modulo p of ON_{a_{1},...,a_{p−1},q}.

(b) There are isomorphisms,

kb_{ρ,w}^{Σ}/N_{a_{1},...,a_{p−1}} ≅ A_{(a_{1},...,a_{p−1})} ≅ B_{(a_{1},...,a_{p−1})}.

(c) kb_{ρ,w}^{Σ}/N_{a_{1},...,a_{p−1}} is Morita equivalent to

⨂_{i=1}^{p−1} S(a_{i}, a_{i}).

(d) Under the Morita equivalence of (3), the simple module

... ⊗ L(λ′_{p−3}) ⊗ L(λ_{p−2}) ⊗ L(λ′_{p−1})

for ⨂_{i=1}^{p−1} S(a_{i}, a_{i}), corresponds to the simple kb_{ρ,w}^{Σ}-module D^{[∅,...,λ_{p−3},λ_{p−2},λ_{p−1}]} in L(a_{1}, ..., a_{p−1}).

(e) The decomposition matrix of the O-algebra A_{(a_{1},...,a_{p−1})} is equal to the decomposition matrix of the quasi-hereditary algebra, ⨂_{i=1}^{p−1} S(a_{i}, a_{i}).

It is the concern of the last section of this chapter to prove proposition 132. First, let us give a proof of theorem 129 from proposition 132:

Let

N = ⋂_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} N_{a_{1},...,a_{p−1}},

Ω = {(a_{i})_{i=1,...,p−1}|a_{i} ∈ ℤ_{≥0}, ∑_{i=1}^{p−1} a_{i} = w}.

Then, for α ∈ Ω, we know that kb_{ρ,w}^{Σ}/((⋂_{ω∈Ω−α} N_{ω}) + N_{α}) − mod is empty, by theorem 132(4) and lemma 125. We deduce that kb_{ρ,w}^{Σ} = ((⋂_{ω∈Ω−α} N_{ω})+ N_{α}). By linear algebra, it follows that,

kb_{ρ,w}^{Σ}/N ≅ ⨁_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} kb_{ρ,w}^{Σ}/N_{a_{1},...,a_{p−1}}.

Note that proposition 132(1) implies that N is equal to the p-modular reduction of

⋂_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} ON_{a_{1},...,a_{p−1},q} = b_{ρ,w}^{H_{q}}.ON_{q}.

Thus, kb_{ρ,w}^{Σ}/N is isomorphic to kGb_{ρ,w}^{Σ}.

These isomorphisms, along with proposition 132(3), complete the proof of the Morita equivalence of theorem 129, and the correspondence between simple modules under this Morita equivalence.

To see the correspondence between decomposition numbers, first note that the decomposition matrix of Gb_{ρ,w}^{Σ} is equal to the decomposition matrix of

⨁_{\begin{matrix}a_{1},...,a_{p−1}∈ℤ_{≥0} \\ ∑ a_{i}=w \end{matrix}} OA_{(a_{1},...,a_{p−1}),q},

by proposition 123. Secondly, note that the decomposition matrix of OA_{(a_{1},...,a_{p−1}),q} is equal to the decomposition matrix of OA_{(a_{1},...,a_{p−1})}, since both of these algebras are semisimple over K. Thirdly, note that the decomposition matrix of OA_{(a_{1},...,a_{p−1})} is equal to the decomposition matrix of ⨂_{i=1}^{p−1} S(a_{i}, a_{i}), by proposition 132(5). □

Induction.

The intent of this section is to convince the Logos of the truth of proposition 132. Let w be a natural number. We assume proposition 132 is proven for Rock blocks of weight strictly less than w, and deduce the same result for a Rock block kb_{ρ,w}^{Σ} of weight w.

Let a_{p−1} be a natural number, 0 < a_{p−1} ≤ w. Let u = n − a_{p−1}p.

Let L_{a_{p−1}} = Σ_{p}^{1} × ... × Σ_{p}^{a_{p−1}} × Σ_{u}^{0} ≤ Σ_{v}, where Σ_{p}^{i} = Sym{(i − 1)p + 1, ..., ip}, and Σ_{u}^{0} = Sym{a_{p−1}p + 1, ..., wp + r} .

Let e_{a_{p−1}} be an idempotent of kΣ_{v}, defined to be the product of block idempotents of L_{i} with cores ∅, ..., ∅, ρ, for i = 0, .., a_{p−1}.

Let ζ_{a_{p−1}} = ∑_{x∈Σ_{p}^{1}×...×Σ_{p}^{a_{p−1}}} x. Let kM_{a_{p−1}} = OΣ_{v}e_{a_{p−1}}ζ_{a_{p−1}}.

Suppose that a_{1}, ..., a_{p−1} are natural numbers, whose sum is w. By our inductive assumption, there is an ideal N_{a_{1},...,a_{p−2},0} of kb_{ρ,w−a_{p−1}}^{Σ}, such that kb_{ρ,w−a_{p−1}}^{Σ}/N_{a_{1},...,a_{p−2},0} is Morita equivalent to ⨂_{i=1}^{p−2} S(a_{i}, a_{i}), and whose simple modules are the simple objects of L(a_{1}, ..., a_{p−2}, 0).

Let kM_{(a_{1},....,a_{p−1})} be equal to the quotient (kM_{a_{p−1}}/kM_{a_{p−1}}N_{a_{1},...,a_{p−2},0}). Since kM_{a_{p−1}} is projective as a right kΣ_{u}^{0}-module, we have

kM_{(a_{1},...,a_{p−1})} ≅ kM_{a_{p−1}} ⨂_{Σ_{u}^{0}} (kb_{ρ,w−a_{p−1}}^{Σ}/N_{a_{1},...,a_{p−2},0}) .

Therefore kM_{(a_{1},...,a_{p−1})} is a kΣ_{v}-kΣ_{u}^{0}-bimodule.

The following proposition generalises theorem 87.

Proposition 133 (a) Consider the kb_{ρ,w}^{Σ}-module kM_{(a_{1},...,a_{p−1})}. Its endomorphism ring kE_{(a_{1},...,a_{p−1})} is isomorphic to kb_{ρ,w−a_{p−1}}^{Σ}/N_{a_{1},...,a_{p−2},0} ⊗ kΣ_{a_{p−1}}.

(b) The quotient of kb_{ρ,w}^{Σ}, by the annihilator kI_{(a_{1},...,a_{p−1})} of kM_{(a_{1},...,a_{p−1})}, is Morita equivalent to the tensor product,

⨂_{i=1}^{p−1} S(a_{i}, a_{i}),

of Schur algebras.

(c) The (⨂_{i=1}^{p−1} S(a_{i}, a_{i}))- (⨂_{i=1}^{p−2} S(a_{i}, a_{i})) ⊗ kΣ_{a_{p−1}}- bimodule which corresponds via Morita equivalence to the kb_{ρ,w}^{Σ}/kI_{(a_{1},...,a_{p−1})}- kE_{(a_{1},...,a_{p−1})} bimodule kM_{(a_{1},...,a_{p−1})} is isomorphic to,

(⨂_{i=1}^{p−2} S(a_{i}, a_{i})) ⊗ E^{⊗a_{p−1}#}.

(d) Under the Morita equivalence between kb_{ρ,w}^{Σ}/kI_{(a_{1},...,a_{p−1})} and the tensor product ⨂_{i=1}^{p−1} S(a_{i}, a_{i}), the correspondence between simple modules is:

D^{[∅,...λ_{p−3},λ_{p−2},λ_{p−1}]} ↔ ... ⊗ L(λ′_{p−3}) ⊗ L(λ_{p−2}) ⊗ L(λ′_{p−1}).□

Let KM_{(a_{1},...,a_{p−1}),q} be the q-analogue of kM_{(a_{1},...,a_{p−1})}.

The proposition below is a q-analogue of proposition 87, valid in characteristic zero.

Proposition 134 Let K be a splitting field for H_{q}(Σ_{v}). Let a_{1}, ..., a_{p−1} be natural numbers whose sum is w.

(a) KA_{(a_{1},...,a_{p−1}),q} is a semisimple algebra.

(b) KM_{(a_{1},...,a_{p−2},a_{p−1}),q} is a semisimple Kb_{ρ,w}^{H_{q}}-module. Its endomorphism ring KE_{(a_{1},...,a_{p−1}),q} is isomorphic to (Kb_{ρ,w−a_{p−1}}^{H_{q}}/KN_{a_{1},...,a_{p−2},0,q}) ⊗ KΣ_{a_{p−1}}.

(c) The quotient of Kb_{ρ,w}^{H_{q}}, by the annihilator KI_{(a_{1},...,a_{p−1}),q} of M_{(a_{1},...,a_{p−1}),q} is Morita equivalent to the tensor product, ⨂_{i=1}^{p−1} S(a_{i}, a_{i}) of semisimple Schur algebras.

(d) The (⨂_{i=1}^{p−1} S(a_{i}, a_{i}))- (⨂_{i=1}^{p−2} S(a_{i}, a_{i})) ⊗ KΣ_{a_{p−1}}-bimodule, which corresponds (under Morita equivalence) to KM_{(a_{1},...,a_{p−1}),q} is isomorphic to

(⨂_{i=1}^{p−2} S(a_{i}, a_{i})) ⊗ E^{⊗a_{p−1}#}.

(e) The annihilator KI_{(a_{1},...,a_{p−1}),q} is precisely N_{a_{1},...,a_{p−1},q}. □

Proof:

Dirichlet’s theorem guarantees the existence of infinitely many prime numbers l′, such that l′ = 1 (modulo p) (i.e. such that 𝔽_{l′} contains primitive p^{th} roots of unity). Let us choose such a prime, such that w < l′.

Let q be a primitive p^{th} root of unity. A second application of Dirichlet’s theorem provides a prime number q′, such that q′ = q (modulo l′).

Let (K_{l′}, O_{l′}, k_{l′}) be an l′-modular system, such that K_{l′} is a splitting field for GL_{v}(q′), and such that q ∈ O_{l′}.

By theorem 78, there is an equivalence between k_{l′}b_{ρ,w}^{H_{q}}, and k_{l′}b_{∅,1}^{H_{q}} ≀ Σ_{w}. The decomposition matrix of this algebra is equal to the decomposition matrix of K_{l′}b_{ρ,w}^{H_{q}}, by theorem 58. Therefore, the James adjustment algebra KGb_{ρ,w}^{H_{q}} is semisimple, over ANY splitting field K.

The proposition is now visible, by induction on w. □

Here goes the induction. Proof of proposition 132:

How to define the ideals N_{a_{1},...,a_{p−1}} ?

Case 1: if a_{1} = a_{p−1} = 0, the ideal of proposition 127 suffices.

Case 2: If a_{p−1} ≠ 0, proposition 87 above provides the ideal : set N_{a_{1},...,a_{p−1}} = kI_{(a_{1},...,a_{p−1})}.

Case 3: If a_{p−1} = 0, and a_{1} ≠ 0, set N_{a_{1},....,a_{p−1}} = N_{a_{p−1},...,a_{1}}^{#}.

(a) Case 1: Note that, by a q-analogue of proposition 127, we know that I_{(0,a_{2},...,a_{p−2},0),q} +J_{(0,a_{2},...,a_{p−2},0),q} is contained in N_{0,a_{2},...,a_{p−2},0,q}, over the field K, and hence also over O. Thus,

N_{0,a_{2},...,a_{p−2},0} ⊆ N_{0,a_{2},...,a_{p−2},0,q} (modulo p).

However, proposition 123 and proposition 127 imply that both quotients, Kb_{ρ,w}^{H_{q}}/N_{0,a_{2},...,a_{p−2},0,q} and kb_{ρ,w}^{Σ}/N_{0,a_{2},...,a_{p−2},0} have the same dimension, equal to the dimension of A_{(a_{1},...,a_{p−1})}.

Case 2: Let KM_{(a_{1},...,a_{p−1}),q} be the q-analogue of kM_{(a_{1},...,a_{p−1})}. By proposition 134, KM_{(a_{1},...,a_{p−1}),q} is a semisimple KH_{q}(Σ_{v})-module, and the quotient Ob_{ρ,w}^{H_{q}}/ON_{a_{1},...,a_{p−1},q} surjects onto kb_{ρ,w}^{Σ}/N_{a_{1},...,a_{p−1}}.

Furthermore, on writing kE_{(a_{1},...,a_{p−1})} (respectively KE_{(a_{1},...,a_{p−1}),q}) for the endomorphism ring of kM_{(a_{1},...,a_{p−1})} (respectively KM_{(a_{1},...,a_{p−1}),q}), we have

kb_{ρ,w}^{Σ}/N_{a_{1},...,a_{p−1}} = End_{kE_{(a_{1},...,a_{p−1})}}(kM_{(a_{1},...,a_{p−1})}),
Kb_{ρ,w}^{H_{q}}/KN_{a_{1},...,a_{p−1},q} = End_{KE_{(a_{1},...,a_{p−1}),q}}(KM_{(a_{1},...,a_{p−1}),q}).

By propositions 133 and 134, these two endomorphism rings have the same dimensions. Thus, N_{a_{1},...,a_{p−1}} = ON_{a_{1},...,a_{p−1},q}(modulo p).

Case 3: Note that

N_{a_{1},...,a_{p−1}} = N_{a_{p−1},...,a_{1}}^{#} =
N_{a_{p−1},...,a_{1},q}^{#}(modulo p) = N_{a_{1},...,a_{p−1},q}(modulo p).

(b) follows from proposition 123 and (a), by p-modular reduction.

(c) What is the Morita type of the quotient kb_{ρ,w}^{Σ}/N_{a_{1},...,a_{p−1}} ?

In case 2, a_{p−1} ≠ 0, we know that kb_{ρ,w}^{Σ}/N_{a_{1},....,a_{p−1}} is Morita equivalent to ⨂_{i=1}^{p−1} S(a_{i}, a_{i}), by proposition 133. Recall that S(n, n) is Ringel selfdual for any n (theorem 17). The Ringel dualities of theorem 120, and the isomorphisms of part (b), show that kb_{ρ,w}^{Σ}/N_{a_{1},....,a_{p−1}} is Morita equivalent to ⨂_{i=1}^{p−1} S(a_{i}, a_{i}) in general.

(d) Tracing back through proposition 133, along remark 121, past theorem 120 to proposition 60, the correspondence between simple modules is visible.

(e) The decomposition matrix of A_{(a_{1},...,a_{p−1})} as an O-algebra is equal to the decomposition matrix ([Δ(λ) : L(μ)]) of kA_{(a_{1},...,a_{p−1})} as a quasihereditary algebra. (e) is now clear from (b) and (c). □

Chapter VIII

Doubles, Schur super-bialgebras, and Rock blocks of Hecke algebras.

J. Alperin’s weight conjecture [1], now along with a number of examples (eg. theorem 37, theorem 129, [22]), suggests that the category of modules for a finite group algebra, over a field of prime characteristic, should resemble a highest weight category. However, group algebras are symmetric algebras, and far from quasi-hereditary. This chapter presents a conjectural resolution to this problem, for symmetric groups.

Indeed, we associate symmetric associative algebras to certain bialgebras, via a double construction (theorem 135). To any super-algebra, we then assign a “Schur super-bialgebra”. From the algebra of n × n matrices, concentrated in parity zero, we thus recover the classical Schur bialgebra, S(n). Applying the aforementioned double construction to certain Schur super-bialgebras, which correspond to quivers of type A, we reveal symmetric algebras which should be Morita equivalent to Rock blocks for Hecke algebras (conjecture 162).

A double construction.

Let k be a field. Let B be an bialgebra over k, endowed with a kendomorphism σ, which is an algebra anti-homomorphism, and a coalgebra anti-homomorphism. Let B^{∗} be a bialgebra, dual to B.

Let us write comultiplication as Δ(x) = ∑ x_{(1)} ⊗ x_{(2)}.

Theorem 135 The tensor product D(B) = B ⊗ B^{∗} is a k-algebra, with associative product given by,

(a ⊗ α).(b ⊗ β) = ∑ a_{(2)}b_{(1)} ⊗ β_{(2)}α_{(1)} < a_{(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(2)}^{σ} > .

Furthermore, D(B) possesses a symmetric associative bilinear form,

< a ⊗ α, b ⊗ β >=< a^{σ}, β >< α, b^{σ} > .

Therefore, if σ is invertible, then D(B) is a symmetric algebra.

So long as B is cocommutative, there are algebra homomorphisms,

Δ_{l} : D(B) → D(B) ⊗ B,

Δ_{l} : a ⊗ α ↦ ∑ a_{(1)} ⊗ α ⊗ a_{(2)},

Δ_{r} : D(B) → B ⊗ D(B)
Δ_{r} : a ⊗ α ↦ ∑ a_{(1)} ⊗ a_{(2)} ⊗ α.

Proof:

We first check associativity:

((a ⊗ α).(b ⊗ β)).(c ⊗ γ) = (∑ a_{(2)}b_{(1)} ⊗ β_{(2)}α_{(1)} < a_{(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(2)}^{σ} >).(c ⊗ γ) = ∑(a_{(2)}b_{(1)})_{(2)}c_{(1)} ⊗ (β_{(2)}α_{(1)})_{(1)}γ_{(2)} < a_{(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(2)}^{σ} >

< (a_{(2)}b_{(1)})_{(1)}^{σ}, γ_{(1)} >< (β_{(2)}α_{(1)})_{(2)}, c_{(2)}^{σ} >

= ∑ a_{(3)}b_{(2)}c_{(1)} ⊗ γ_{(2)}β_{(2)}α_{(1)} < a_{(1)}^{σ}, β_{(1)} >< α_{(3)}, b_{(3)}^{σ} >

< b_{(1)}^{σ}a_{(2)}^{σ}, γ_{(1)} >< β_{(3)}α_{(2)}, c_{(2)}^{σ} >

= ∑ a_{(3)}b_{(2)}c_{(1)} ⊗ γ_{(2)}β_{(2)}α_{(1)} < a_{(1)}^{σ}, β_{(1)} >< α_{(3)}, b_{(3)}^{σ} >

< b_{(1)}^{σ}, γ_{(1)(1)} >< a_{(2)}^{σ}, γ_{(1)(2)} >< α_{(2)}, c_{(2)(1)}^{σ} >< β_{(3)}, c_{(2)(2)}^{σ} >

= ∑ a_{(3)}b_{(2)}c_{(1)} ⊗ γ_{(3)}β_{(2)}α_{(1)} < a_{(1)}^{σ}, β_{(1)} >< α_{(3)}, b_{(3)}^{σ} > < b_{(1)}^{σ}, γ_{(1)} >< a_{(2)}^{σ}, γ_{(2)} >< α_{(2)}, c_{(2)}^{σ} >< β_{(3)}, c_{(3)}^{σ} >.

This final symmetric expression may similarly be shown to equal (a ⊗ α).((b ⊗ β).(c ⊗ γ)). Associativity is proven !

Now suppose that σ is invertible. There can be little doubt of the symmetry, nor the non-degeneracy of the bilinear form we have defined on D(B). What about associativity ?

< (a ⊗ α).(b ⊗ β), (c ⊗ γ) > = ∑ < a_{(2)}b_{(1)} ⊗ β_{(2)}α_{(1)}, c ⊗ γ >< a_{(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(2)}^{σ} > = ∑ < b_{(1)}^{σ}a_{(2)}^{σ}, γ >< β_{(2)}α_{(1)}, c^{σ} >< a_{(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(2)}^{σ} > = ∑ < b_{(1)}^{σ}, γ_{(1)} >< a_{(2)}^{σ}, γ_{(2)} >< α_{(1)}, c_{(1)}^{σ} >

< β_{(2)}, c_{(2)}^{σ} >< a_{(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(2)}^{σ} >.

This final symmetric expression may similarly be shown to equal < (a ⊗ α), (b ⊗ β).(c ⊗ γ) >. Associativity of <, > is proven !

The last check we make is that Δ_{r} is an algebra homomorphism, so long as B is cocommutative (a similar calculation can be written down for Δ_{l}):

Δ_{r}((a ⊗ α).(b ⊗ β)) = ∑(a_{(2)}b_{(1)})_{(1)} ⊗ (a_{(2)}b_{(1)})_{(2)} ⊗ β_{(2)}α_{(1)} < a_{(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(2)}^{σ} > = ∑ a_{(2)}b_{(1)} ⊗ a_{(3)}b_{(2)} ⊗ β_{(2)}α_{(1)} < a_{(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(3)}^{σ} > = ∑ a_{(1)}b_{(1)} ⊗ a_{(3)}b_{(2)} ⊗ β_{(2)}α_{(1)} < a_{(2)}^{σ}, β_{(1)} >< α_{(2)}, b_{(3)}^{σ} > = ∑ a_{(1)}b_{(1)} ⊗ a_{(2)(2)}b_{(2)(1)} ⊗ β_{(2)}α_{(1)} < a_{(2)(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(2)(2)}^{σ} > = (a_{(1)} ⊗ a_{(2)} ⊗ α).(b_{(1)} ⊗ 1 ⊗ b_{(2)} ⊗ β) = Δ_{r}(a ⊗ α).Δ_{r}(b ⊗ β). □

Remark 136 Since B possesses an algebra anti-automorphism σ, the dual of a left/right B-module may be given the structure of a left/right B-module as well. The left/right regular action of B on itself, may thus be dualised to define a left/right action of B on B^{∗}. We obtain a simpler expression,

(a ⊗ α).(b ⊗ β) = ∑ a_{(2)}b_{(1)} ⊗ (a_{(1)} ∘ β)(α ∘ b_{(2)}),

for the associative product on D(B).

Remark 137 The coproducts Δ_{l} and Δ_{r} satisfy the following property:

Let M be a B-module, and let N be a D(B)-module. The D(B)-module M ⊗ N (formed via Δ_{r}) is isomorphic to the D(B)-module N ⊗ M (formed via Δ_{l}).

In the examples of this article, we find ourselves in the following situation:

Lemma 138 Suppose that B = ⊕_{r∈ℤ_{+}}B(r) is a bialgebra, which is direct sum of finite dimensional pieces, B(r). Suppose that B possesses a degree preserving k-endomorphism σ, which is an algebra antiautomorphism, and a coalgebra automorphism. We write B^{∗} for the graded dual, ⊕_{r∈ℤ_{+}}B(r)^{∗}, of B. Suppose further, that

i. B(r) is a subalgebra of B, for r ∈ ℤ_{+}.

ii. B(0) ≅ k.

iii. B = ⊕_{r∈ℤ_{+}}B(r) is a graded coalgebra, with a counit in B(0). Thus,

Δ : B(r) → ⨁_{d=0}^{r} B(r − d) ⊗ B(d).

Then the degree r part of D = D(B),

D(r) = ⨁_{d=0}^{r} B(r − d) ⊗ B^{∗}(d),

is a finite-dimensional, graded, symmetric algebra summand of D, where B(r − d) ⊗ B^{∗}(d) is given degree d.

The ideal,

N (r) = ⨁_{d=1}^{r} B(r − d) ⊗ B^{∗}(d),

of D(r) is nilpotent.

The quotient D(r)/N (r) is isomorphic to the degree zero part D^{0}(r) = B(r), of D(r).

Irreducible D(r)-modules are in natural correspondence with irreducible B(r)-modules.

In this way, D is a graded associative algebra, whose degree zero part is isomorphic to B, as an algebra.

On writing N for ideal ⊕_{r∈ℤ_{+}}N (r) of D, a splitting of the natural algebra monomorphism B → D becomes visible:

D → D/N ≅ B.

The degree d part B(r − d) ⊗ B^{∗}(d) of D(r) inherits a D^{0}(r)-D^{0}(r)- bimodule structure from D(r), for d = 0, ..., r. This is nothing but the natural B(r)-B(r)-bimodule structure on B(r − d) ⊗ B^{∗}(d). □

We require a super- generalisation of theorem 135.

Suppose that B is a super-bialgebra. Thus, B is ℤ/2-graded algebra and coalgebra, so that the product m, and the coproduct Δ, preserve the grading:

m : B^{i} ⊗ B^{j} → B^{i+j},

Δ : B^{k} → ⨁_{i+j=k} B^{i} ⊗ B^{j}, for i, j, k ∈ ℤ/2.

In addition,

Δ(a.b) = ∑(−1)^{|a_{(2)}||b_{(1)}|}a_{(1)}.b_{(1)} ⊗ a_{(2)}.b_{(2)}.

Suppose that B is endowed with a parity preserving endomorphism σ, which is a coalgebra anti-automorphism, and an algebra anti-automorphism.

Theorem 139 The tensor product D(B) = B ⊗ B^{∗} is a super-algebra, with associative product given by,

(a ⊗ α).(b ⊗ β) =
∑(−1)^{s(a,α,b,β)}a_{(2)}b_{(1)} ⊗ β_{(2)}α_{(1)} < a_{(1)}^{σ}, β_{(1)} >< α_{(2)}, b_{(2)}^{σ} >,

where

s(a, α, b, β) = |a_{(1)}|(|a_{(2)}| + |b_{(1)}|) + |b_{(1)}||α| + |α_{(1)}||β|,

and ℤ/2-grading, given by

|a ⊗ α| = |a| + |α|.

In fact, D(B) is endowed with a symmetric associative bilinear form,

< a ⊗ α, b ⊗ β >=< a^{σ}, β >< α, b^{σ} > .

Therefore, if σ is invertible, D(B) is a symmetric super-algebra.

Proof:

Write out the proof of theorem 135 diagrammatically, rather than algebraically (thus, a variable should be represented by a string, a product by the fraying of a string into two parts, a coproduct by the joining of two strings together, etc. etc.). To generalise this proof to the super- situation, we need only introduce the sign (−1)^{|a||b|} whenever two strings (corresponding to variables a,b, in degrees |a|, |b|) cross. Here is a picture of the product a ⊗ α and b ⊗ β in D(B):

a  α  b  β

The sign allocated to this diagram is (−1), raised to the power

|a_{(1)}|(|a_{(2)}| + |b_{(1)}|) + |b_{(1)}|(|α_{(1)}| + |α_{(2)}|) + |α_{(1)}(|β_{(1)}| + |β_{(2)}|).

A slightly simpler expression is (−1)^{s(a,α,b,β)}. □

Remark 140 The ℤ_{+}-grading on B of lemma 138 is necessarily a different grading from the ℤ/2-grading on B of theorem 139. Indeed, lemma 138 generalises to apply to a bialgebra which is ℤ_{+} × ℤ/2-graded, so that the ℤgrading is compatible with lemma 138, whilst the ℤ/2-grading is compatible with theorem 139.

Examples.

Example 141 Let B = S(1) be the Schur bialgebra associated to GL_{1}(k), with trivial antiautomorphism. As a coalgebra, S(1) is the graded dual of the polynomial ring k[X] in one variable. Each homogeneous component S(1, r) is isomorphic (as an algebra) to a copy of the field k.

Then we are in the context of lemma 138, and D(r) is isomorphic to the uniserial algebra k[Y]/(Y^{r}). □

Example 142 Let B = B(0) ⊕ B(1) = k ⊕ T_{n}(k), be the direct sum of a copy of the field k (in degree zero), and the algebra of n×n upper triangular matrices (in degree one). On writing ϵ for the unit in B(0) (not a unit for B), we see that B is a cocommutative bialgebra via the coproduct,

Δ : x ↦ x ⊗ ϵ + ϵ ⊗ x, x ∈ B(1)
Δ : ϵ ↦ ϵ ⊗ ϵ.

The bialgebra B possesses an algebra anti-automorphism σ, acting trivially on B(0), but non-trivially on B(1) taking E_{i,j} to E_{n−j+1,n−i+1}. This map σ is a coalgebra anti-automorphism.

We are in the setup of lemma 138, and B(1) is isomorphic to the circular quiver algebra, modulo the ideal of paths of length ≥ n + 1. This is a uniserial algebra, otherwise known as the circular Brauer tree algebra, with multiplicity one. □

Example 143 Let Q be a quiver, without loops or multiple edges, equipped with an orientation reversing automorphism. Let B = k ⊕ kQ/I_{2} be the direct sum of a copy of the field k (in degree zero), and the path algebra kQ, modulo the ideal of paths of length ≥ 2 (in degree one). Just as in example 142, B may be given the structure of a bialgebra, equipped with an algebra anti-automorphism which is a coalgebra anti-automorphism.

This time, D(1) is isomorphic to the zigzag algebra (see [43]), whose graph is the underlying graph of Q. If the underlying graph is an ordinary Dynkin diagram of type A, the zigzag algebra is otherwise known as a linear Brauer tree algebra, with multiplicity one. □

Brauer tree algebras appear naturally in the block theory of finite groups with cyclic defect groups. There appear to be more mysterious instances of doubles appearing in finite group theory:

Example 144 Let k be a field of characteristic two. Let B = S(2) be the Schur bialgebra associated to GL_{2}(k), with transpose antiautomorphism. Then D(2) is Morita equivalent to the Rock block kb_{ρ,2}^{Σ}, otherwise known as the principal block of kΣ_{5}. This follows from Erdmann’s description of the basic algebras for tame blocks of group algebras. □

Example 145 Let k be a field of characteristic two. Let B = S(2) be the Schur bialgebra associated to GL_{2}(k), with transpose antiautomorphism. Let

C = (0 → D(2)ξ_{(2)} ⊗ ξ_{(2)}D(2)ξ_{(1^{2})} → D(2)ξ_{(1^{2})} → 0).

Then C is a tilting complex for D(2), and its endomorphism ring E in the homotopy category is Morita equivalent to kΣ_{4}. This follows from Holm’s description [42] of derived equivalences between tame blocks of group algebras. □

Remark 146 The equivalences of examples 144 and 145 both lift to Hecke algebras at −1, over fields of arbitrary characteristic.

Schur super-bialgebras.

Let A, B be super-algebras (i.e. ℤ/2-graded associative algebras). Their tensor product, A ⊗ B, becomes a super-algebra, with parity,

|a ⊗ b| = |a| + |b|,

and super-product,

(a ⊗ b).(a′ ⊗ b′) = (−1)^{|b||a′|}(aa′ ⊗ bb′).

For a super-algebra A, we define the super-algebra A^{⊗r} inductively, to be A^{⊗r} = A^{⊗r−1} ⊗ A, with super-product as above.

The symmetric group Σ_{r} acts naturally as parity-preserving automorphisms on A^{⊗r}. A simple reflection (i, i + 1) ∈ Σ_{r} acts as:

(a_{1} ⊗ ... ⊗ a_{r})^{(i,i+1)} = (−1)^{|a_{i}||a_{i+1}|}a_{1} ⊗ ... ⊗ a_{i−1} ⊗ a_{i+1} ⊗ a_{i} ⊗ a_{i+2} ⊗ ... ⊗ a_{r}.

Let A be a super-algebra. Let T (A) be the direct sum,

T (A) = ⨁_{d≥0} A^{⊗d},

of super-algebras. This algebra becomes a super-bialgebra, with parity,

|a_{1} ⊗ ... ⊗ a_{r}| = |a_{1}| + ... + |a_{r}|,

and coassociative comultiplication,

Δ(a_{1} ⊗ ... ⊗ a_{r}) = ∑_{i=0}^{r}(a_{1} ⊗ ... ⊗ a_{i}) ⊗ (a_{i+1} ⊗ ... ⊗ a_{r}).

Definition 147 Let A be a super-algebra. The Schur super-bialgebra associated to A is the graded sub-super-bialgebra,

S(A) = ⨁_{r≥0} S(A)(r) = ⨁_{r≥0}(A^{⊗r})^{Σ_{r}},

of Σ_{r}-fixpoints on T (A).

Remark 148 In case A = M = M_{n}(k) is the algebra of n × n matrices, concentrated in parity zero, we recover the classical Schur bialgebra S(n) in this way.

Remark 149 If A is a super-algebra, equipped with a parity-preserving anti-automorphism σ, then A^{⊗r} may also be equipped with a parity-preserving anti-automorphism,

σ : a_{1} ⊗ ... ⊗ a_{r} ↦ σ(a_{r}) ⊗ ... ⊗ σ(a_{1}).

Indeed, this map acts as a coalgebra anti-automorphism on T (A), and restricts to a coalgebra anti-automorphism on S(A).

Therefore, if A is a super-algebra, equipped with a parity-preserving anti-automorphism σ, then S(A) is a super-bialgebra, equipped with an endomorphism σ, which is an algebra and coalgebra anti-automorphism. Under such circumstances, we may apply theorem 139, and form the double, D(S(A)).

Schiver super-bialgebras.

The example which concerns us most in this booklet, is the special case when A is Morita equivalent to the path algebra of a quiver Q, modulo the ideal of paths of length > 1. We are particularly interested in this when Q is a Dynkin quiver of type A.

Let Q be a quiver (that is, a locally finite, oriented graph). We note its vertex set V, and its set of edges E. For any edge e ∈ E, we denote its source s(e) ∈ V, and its tail t(e) ∈ V.

Let P_{Q} be the path algebra of Q, modulo the ideal of paths of length > 1. To any natural number n, let us assign the algebra P_{Q}(n), which is Morita equivalent to P_{Q}, and whose simple modules all have dimension n. Thus, P_{Q}(n) = End_{P_{Q}}(P_{Q}^{⊕n}), and as vector spaces we have

P_{Q}(n) ≅ M^{⊕V} ⊕ M^{⊕E},

where M is the algebra of n×n matrices over k. The space P_{Q}(n) is naturally an algebraic affine super-variety, where paths in P_{Q}(n) of length 0 and 1 are given parities 0 and 1 respectively.

Definition 150 The Schur quiver super-bialgebra, or Schiver super-bialgebra associated to (Q, n), is the Schur super-bialgebra, S_{Q}(n) = S(P_{Q}(n)), associated to P_{Q}(n). Its graded dual, the ring of functions,

A_{Q}(n) ≅ (A(M^{∗}))^{⊗V} ⊗ (⋀(M^{∗}))^{⊗E} ,

on P_{Q}(n), is isomorphic as an algebra, to a tensor product of symmetric and exterior algebras.

Our sole motivation for this definition is the apparent emergence of such structure in type A representation theory. But the alternating structure also bears encouraging homological consequences. For example, the supersymmetric aspect of this definition looks rather becoming, when one considers Koszul duality for these super-algebras - see remark 181.

Remark 151 When Q has merely one vertex, and no arrows, we recover the classical Schur bialgebras S(n) according to this construction. In general the tensor product,

S_{V} (n) = (S(n))^{⊗V} ,

of Schur algebras, is naturally a sub-bialgebra of

S_{Q}(n) ≅ (S(n))^{⊗V} ⊗ (⋁(n))^{⊗E} .

The inclusion map splits as an algebra homomorphism, via

S_{Q}(n) → S_{Q}(n)/J ≅ (S(n))^{⊗V} ,

where J is the direct sum of subspaces,

(⨂_{v∈V} S(n, a_{v})) ⊗ (⨂_{e∈E} ⋁(n, b_{e}))

of S_{Q}(n), such that b_{e} > 0 for some e ∈ E. Thus, J is a nilpotent ideal of the algebra S_{Q}(n).

If Q is a quiver, then the disjoint union Q ∐ Q^{op} of quivers possesses an obvious orientation-reversing automorphism, exchanging Q and Q^{op}. Thus, S_{Q ∐ Q^{op}}(n) may be equipped with a k-endomorphism σ, which is an algebra and coalgebra anti-automorphism. Under such circumstances, we may apply theorem 139, and form the double, D(S_{Q ∐ Q^{op}}(n)).

Definition 152 The Schiver double associated to (Q, n), is the natural algebra summand,

D_{Q}(n) = S_{Q}(n) ⊗ A_{Q^{op}}(n),

of the double D(S_{Q ∐ Q^{op}}(n)) corresponding to Q ∐ Q^{op}. Thus, D_{Q}(n) is a direct sum, ⨁_{r≥0} D_{Q}(n, r) of algebras, where

D_{Q}(n, r) = ⨁_{r_{1}+r_{2}=r} S_{Q}(n, r_{1}) ⊗ A_{Q^{op}}(n, r_{2}).

Remark 153 Each algebra summand D_{Q}(n, r) is ℤ_{+} × ℤ_{+}-graded, where the component,

(⨂_{v∈V} S(n, a_{v})) ⊗ (⨂_{e∈E} ⋁(n, b_{e}))

⊗ (⨂_{v∈V} A(n, c_{v})) ⊗ (⨂_{e∈E} ⋀(n, d_{e})) ,

is given degree (∑_{E} b_{e} + ∑_{V} c_{v}, ∑_{V} c_{v} + ∑_{E} d_{e}).

Each algebra summand D_{Q}(n, r) is ℤ_{+}-graded, where the component above is given degree (∑_{E} b_{e} + 2 ∑_{V} c_{v} + ∑_{E} d_{e}). We write D_{Q}^{i}(n, r) for the degree i part with respect to this grading. Thus,

D_{Q}(n, r) = ⨁_{i=0}^{2r} D_{Q}^{i}(n, r),

as a direct sum of graded pieces. In degree zero, we have

D_{Q}^{0}(n, r) = S_{V (Q)}(n, r).

Schiver doubles: independence of quiver orientation.

This section is devoted to a proof of the following result...

Theorem 154 The Schiver double D_{Q}(n) is independent of the orientation of Q, and as such, is an invariant of the underlying graph of Q. □

For a locally finite graph Γ, we thus write D_{Γ}(n) for the Schiver double D_{Q}(n), where Q is any orientation of Γ.

We give a proof of theorem 154, in case Q = A_{1} is the quiver with two vertices, and one arrow connecting those two vertices. To say that the corresponding double is independent of orientation, is to say that there is an algebra isomorphism between the double corresponding to the quiver, ∘————-▷∘, and the double corresponding to the quiver, ∘◁————-∘. We therefore reveal an algebra automorphism of D_{A_{1}}(n).

Theorem 154 follows for a general quiver from the case Q = A_{1}. To see this, first observe that distinct arrows do not interact with one another when multiplied in D_{Q}(n). Therefore, if Q′ is obtained from Q, by the reversing of an arrow, we have D_{Q′}(n) ≅ D_{Q}(n). Secondly note that we may obtain one orientation of Q from another by reversing a collection of arrows, and so D_{Q}(n) is indeed independent of the orientation of Q.

We present a triad of preliminary lemmas.

Let ⋁(n) = ⋁(M), and let ⋀(n) = ⋀(M^{∗}). Theorem 17 implies the following lemma.

Lemma 155 There is an S(n)-S(n)-bimodule isomorphism, which exchanges ⋁(n) and ⋀(n), for n ≥ 0.

When n = 1, this isomorphism is defined by the structure of a symmetric algebra on M. □

We write ∗ for either of the inverse homomorphisms which describe the isomorphism of lemma 155. In this way, (x ∧ y)^{∗} = (x^{∗} ∨ y^{∗}), and (x ∨ y)^{∗} = (y^{∗} ∧ x^{∗}).

The tranpose anti-automorphism σ of S_{A_{1}}(n) maps s ⊗ λ ⊗ s′ to s^{′T} ⊗ λ^{T} ⊗ s^{T}.

Lemma 156 The left action of S_{A_{1}}(n) on A_{A_{1}}(n), is given by,

(s ⊗ λ ⊗ t) ∘ (a ⊗ μ ⊗ b) =

∑(−1)^{|μ_{(2)}||λ|}(t_{(2)} ∘ a) ⊗ (t_{(1)} ∘ μ_{(2)}) ⊗ (μ_{(1)(2)}.(s ∘ b)) < μ_{(1)(1)}, λ^{T} > .

The right action of S_{A_{1}}(n) on A_{A_{1}}(n), is given by,

(a ⊗ μ ⊗ b) ∘ (s ⊗ λ ⊗ t) =

∑(−1)^{|μ_{(1)}||μ_{(2)}|}((a ∘ t).μ_{(2)(1)}) ⊗ (μ_{(1)} ∘ s_{(2)}) ⊗ (b ∘ s_{(1)}) < μ_{(2)(2)}, λ^{T} > .

Proof:

We record a calculation for the left action:

< (s ⊗ λ ⊗ t) ∘ (a ⊗ μ ⊗ b), (s′ ⊗ λ′ ⊗ t′) > =< a ⊗ μ ⊗ b, (s ⊗ λ ⊗ t)^{σ} ∘ (s′ ⊗ λ′ ⊗ t′) > =< a ⊗ μ ⊗ b, (t^{T} ⊗ λ^{T} ⊗ s^{T}) ∘ (s′ ⊗ λ′ ⊗ t′) > = ∑ < a ⊗ μ ⊗ b, (t^{T})_{(1)}s′ ⊗ (λ^{T} ∘ t′_{(1)}) ∨ ((t^{T})_{(2)} ∘ λ′) ⊗ s^{T} t′_{(2)} > = ∑(−1)^{|μ_{(2)}||λ|} < a, t_{(2)}^{T}s′ >< μ_{(1)}, (λ^{T} ∘ t′_{(1)}) >

< μ_{(2)}, (t_{(1)}^{T} ∘ λ′) >< b, s^{T} t′_{(2)} >

= ∑(−1)^{|μ_{(2)}||λ|} < t_{(2)} ∘ a, s′ >< μ_{(1)(1)}, λ^{T} >< μ_{(1)(2)}, t′_{(1)} >

< t_{(1)} ∘ μ_{(2)}, λ′ >< s ∘ b, t′_{(2)} >

= ∑(−1)^{|μ_{(2)}||λ|} < (t_{(2)} ∘ a ⊗ t_{(1)} ∘ μ_{(2)} ⊗ μ_{(1)(2)}.(s ∘ b)), (s′ ⊗ λ′ ⊗ t′) > < μ_{(1)(1)}, λ^{T} > □

Dual to the bilinear form

ϵ : ⋀(n) ⊗ ⋁(n) →→ k,

there is a natural map

ϵ^{∗} : k → ⋀(n) ⊗ ⋁(n).

Squeezing the identity map on S(n) inside ϵ, we obtain a map

ϕ : S(n) → ⋀(n) ⊗ S(n) ⊗ ⋁(n).

The right action of S(n) on ⋀(n) may be formulated as a map,

m_{1} : ⋀(n) ⊗ S(n) → ⋀(n).

The left action of S(n) on ⋁(n) may be formulated as a map,

m_{2} : S(n) ⊗ ⋁(n) → ⋁(n).

Lemma 157 The diagram

\xymatrix{S(n) \ar[d]^{ϕ} \ar[r]^{ϕ} & ⋀(n) ⊗ S(n) ⊗ ⋁(n) \ar[d]^{T ⊗m_{2}} \\ ⋀(n) ⊗ S(n) ⊗ ⋁(n) \ar[r]^{m_{1}⊗T} & ⋀(n) ⊗ ⋁(n)}

commutes.

Proof:

Let {ξ_{ij} | i, j = 1, ..., n} be a basis for M. Let {X_{ij}} be the basis for M^{∗} which is identified with {ξ_{ij}} via the symmetric structure on M. The Schur algebra may be given basis, whose elements have the form ξ_{a_{1}b_{1}}....ξ_{a_{r}b_{r}}. Let Σ(a, b) be the stabilizer in Σ_{n} of the sequence (a_{i}, b_{i})_{i=1}^{n}. We have,

(m_{1} ⊗ T) ∘ ϕ(ξ_{a_{1}b_{1}}....ξ_{a_{r}b_{r}})

= (m_{1} ⊗ 1) (∑_{i_{k},j_{k}} X_{i_{1}j_{1}} ∧ ... ∧ X_{i_{n}j_{n}} ⊗ ξ_{a_{1}b_{1}}....ξ_{a_{r}b_{r}} ⊗ ξ_{j_{1}i_{1}} ∨ .... ∨ ξ_{j_{r}i_{r}} )

= (m_{1} ⊗ 1) ( \begin{matrix}∑_{i_{k},j_{k},σ∈Σ(a,b)} X_{i_{1}j_{1}} ∧ ... ∧ X_{i_{n}j_{n}} ⊗ ξ_{a_{σ1}b_{σ1}} ⊗ .... ⊗ ξ_{a_{σr}b_{σr}}⊗ \\ ξ_{j_{1}i_{1}} ∨ .... ∨ ξ_{j_{r}i_{r}} \end{matrix} )

= ∑_{i_{k},σ∈Σ(a,b)} X_{i_{1}b_{σ1}} ∧ ... ∧ X_{i_{n}b_{σn}} ⊗ ξ_{a_{σ1}i_{1}} ∨ .... ∨ ξ_{a_{σr}i_{r}}.

A similar computation shows that (T ⊗ m_{2}) ∘ ϕ is equal to the same sum. □

The duals of m_{1}, m_{2} are maps,

m_{1}^{∗} : ⋁(n) → ⋁(n) ⊗ A(n).
m_{2}^{∗} : ⋀(n) → A(n) ⊗ ⋀(n).

We write m_{1}^{∗}(γ) = ∑ γ_{(1)} ⊗ γ_{(2)} for γ ∈ ⋁(n), and m_{2}^{∗}(δ) = ∑ δ_{(1)} ⊗ δ_{(2)} for δ ∈ ⋀(n).

Lemma 158 For α ∈ ⋁(n), β ∈ ⋀(n), we have,

∑ α_{(2)} < α_{(1)}, β^{T} >= ∑ < α^{∗T} , β_{(2)}^{∗}, > β_{(1)}^{∗}

Proof:

By lemma 157, the following diagram commutes:

\xymatrix{ & S(n) \ar[dl]^{ϕ} \ar[dd]_{ϕ} \ar[dr]^{(∗⊗1⊗∗)ϕ} \\ ⋀(n) ⊗ S(n) ⊗ ⋁(n) \ar[dd]_{m_{1}⊗T} &  & ⋁(n) ⊗ S(n) ⊗ ⋀(n) \ar[dd]^{T ⊗m_{2}} \ar@{-->}[dl]_{∗⊗1⊗∗} \\  & ⋀(n) ⊗ S(n) ⊗ ⋁(n) \ar@{-->}[dl]^{T ⊗m_{2}} \\ ⋀(n) ⊗ ⋁(n) \ar[rr]_{∗⊗∗} &  & ⋁(n) ⊗ ⋀(n)}

Therefore, the diagram dual to this one commutes. The two passages from ⋁(n) ⊗ ⋀(n) to A(n) around the boundary of this dual diagram describe the two sides to the formula of the lemma. □

Theorem 159 There is an involutory algebra automorphism θ of D_{A_{1}}(n), given by,

θ(s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b) = (−1)^{|λ||ξ|}(t ⊗ ξ^{∗} ⊗ s ⊗ b ⊗ λ^{∗} ⊗ a).

Proof:

[s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b].[u ⊗ μ ⊗ v ⊗ c ⊗ η ⊗ d] = ∑ ±(s ⊗ λ ⊗ t)_{(2)}.(u ⊗ μ ⊗ v)_{(1)} ⊗ [(s ⊗ λ ⊗ t)_{(1)} ∘ (c ⊗ η ⊗ d)]⊗

[(a ⊗ ξ ⊗ b) ∘ (u ⊗ μ ⊗ v)_{(2)}]

= ∑ ±(s_{(2)} ⊗ λ_{(2)} ⊗ t_{(2)}).(u_{(1)} ⊗ μ_{(1)} ⊗ v_{(1)})⊗ [(s_{(1)} ⊗ λ_{(1)} ⊗ t_{(1)}) ∘ (c ⊗ η ⊗ d)]⊗ [(a ⊗ ξ ⊗ b) ∘ (u_{(2)} ⊗ μ_{(2)} ⊗ v_{(2)})]

= ∑ ±s_{(2)(1)}u_{(1)} ⊗ (λ_{(2)} ∘ v_{(1)(1)}) ∨ (s_{(2)(2)} ∘ μ_{(1)}) ⊗ t_{(2)}v_{(1)(2)}⊗ [t_{(1)(2)} ∘ c ⊗ t_{(1)(1)} ∘ η_{(2)} ⊗ η_{(1)(2)}.(s_{(1)} ∘ d)]. [(a ∘ v_{(2)}).ξ_{(2)(1)} ⊗ ξ_{(1)} ∘ u_{(2)(2)} ⊗ c ∘ u_{(2)(1)}] < η_{(1)(1)}, λ_{(1)}^{T} >< ξ_{(2)(2)}, μ_{(2)}^{T} >

= ∑ ±s_{(2)}u_{(1)} ⊗ (λ_{(2)} ∘ v_{(1)}) ∨ (s_{(3)} ∘ μ_{(1)}) ⊗ t_{(3)}v_{(2)}⊗ [t_{(2)} ∘ c ⊗ t_{(1)} ∘ η_{(3)} ⊗ η_{(2)}.(s_{(1)} ∘ d)]. [(a ∘ v_{(3)}).ξ_{(2)} ⊗ ξ_{(1)} ∘ u_{(3)} ⊗ c ∘ u_{(2)}] < η_{(1)}, λ_{(1)}^{T} >< ξ_{(3)}, μ_{(2)}^{T} >

= ∑ ±s_{(2)}u_{(1)} ⊗ (λ_{(2)} ∘ v_{(1)}) ∨ (s_{(3)} ∘ μ_{(1)}) ⊗ t_{(3)}v_{(2)}⊗ (t_{(2)} ∘ c).(a ∘ v_{(3)}).ξ_{(2)} ⊗ (t_{(1)} ∘ η_{(3)}) ∧ (ξ_{(1)} ∘ u_{(3)})⊗ η_{(2)}.(s_{(1)} ∘ d).(c ∘ u_{(2)}) < η_{(1)}, λ_{(1)}^{T} >< ξ_{(3)}, μ_{(2)}^{T} > .

Thus,

θ([s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b]).θ([u ⊗ μ ⊗ v ⊗ c ⊗ η ⊗ d]) = [t ⊗ ξ^{∗} ⊗ s ⊗ b ⊗ λ^{∗} ⊗ a].[v ⊗ η^{∗} ⊗ u ⊗ d ⊗ μ^{∗} ⊗ c] = ∑ ±t_{(2)}v_{(1)} ⊗ (ξ_{(2)}^{∗} ∘ u_{(1)}) ∨ (t_{(3)} ∘ η_{(1)}^{∗}) ⊗ s_{(3)}u_{(2)}⊗

(s_{(2)} ∘ d).(b ∘ u_{(3)}).λ_{(2)}^{∗} ⊗ (s_{(1)} ∘ μ_{(3)}^{∗}) ∧ (λ_{(1)}^{∗} ∘ v_{(3)})⊗ μ_{(2)}^{∗}.(t_{(1)} ∘ c).(d ∘ v_{(2)}) < μ_{(1)}^{∗}, ξ_{(1)}^{∗T} >< λ_{(3)}^{∗}, η_{(2)}^{∗T} > .

On the other hand,

θ([s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b].[u ⊗ μ ⊗ v ⊗ c ⊗ η ⊗ d]) = ∑ ±t_{(3)}v_{(2)} ⊗ [(t_{(1)} ∘ η_{(3)}) ∧ (ξ_{(1)} ∘ u_{(3)})]^{∗} ⊗ s_{(2)}u_{(1)}⊗

η_{(2)}.(s_{(1)} ∘ d).(c ∘ u_{(2)}) ⊗ [(λ_{(2)} ∘ v_{(1)}) ∨ (s_{(3)} ∘ μ_{(1)})]^{∗}⊗ (t_{(2)} ∘ c).(a ∘ v_{(3)}).ξ_{(2)} < η_{(1)}, λ_{(1)}^{T} >< ξ_{(3)}, μ_{(2)}^{T} > .

= ∑ ±t_{(3)}v_{(2)} ⊗ (ξ_{(1)}^{∗} ∘ u_{(3)}) ∨ (t_{(1)} ∘ η_{(3)}^{∗}) ⊗ s_{(2)}u_{(1)}⊗ η_{(2)}.(s_{(1)} ∘ d).(c ∘ u_{(2)}) ⊗ (s_{(3)} ∘ μ_{(1)}^{∗}) ∧ (λ_{(2)}^{∗} ∘ v_{(1)})⊗

(t_{(2)} ∘ c).(a ∘ v_{(3)}).ξ_{(2)} < η_{(1)}, λ_{(1)}^{T} >< ξ_{(3)}, μ_{(2)}^{T} > .

Given the cocommutativity of the classical Schur algebra, it is clear from lemma 158, that up to a sign, the terms in our expression for

θ([s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b].[u ⊗ μ ⊗ v ⊗ c ⊗ η ⊗ d])

agree with the terms in our expression for

θ([s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b]).θ([u ⊗ μ ⊗ v ⊗ c ⊗ η ⊗ d]).

The difference in sign of each term is precisely

(−1)^{|λ_{(1)}||λ_{(2)}|+|μ_{(1)}||μ_{(2)}||ξ_{(1)}||ξ_{(2)}|+|η_{(1)}||η_{(2)}|}.

Diagrammatically comparing a term of

θ(θ([s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b]).θ([u ⊗ μ ⊗ v ⊗ c ⊗ η ⊗ d]))

with the relevant term of

[s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b].[u ⊗ μ ⊗ v ⊗ c ⊗ η ⊗ d],

one sees that their difference is also

(−1)^{|λ_{(1)}||λ_{(2)}|+|μ_{(1)}||μ_{(2)}||ξ_{(1)}||ξ_{(2)}|+|η_{(1)}||η_{(2)}|}.

Therefore, our expressions for

θ([s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b].[u ⊗ μ ⊗ v ⊗ c ⊗ η ⊗ d]),

and

θ([s ⊗ λ ⊗ t ⊗ a ⊗ ξ ⊗ b]).θ([u ⊗ μ ⊗ v ⊗ c ⊗ η ⊗ d])

agree. This completes the proof of theorem 159. □

Schiver doubles and wreath products.

Schiver bialgebras and their doubles may be understood to be generalisations of certain wreath products, as is illustrated by the following example:

Let S(n) be the Schur bialgebra associated to GL_{n}(k), and let the Schur algebra S(n, r) be the subalgebra of degree r. Let the double of S(n) be denoted D(n), and let its degree r part be written D(n, r).

Let n ≥ r, and let ω = (1^{r}) be the partition of r with r parts. According to Green’s presentation of Schur-Weyl duality (theorem 21),

ξ_{ω}S(n, r)ξ_{ω} ≅ kΣ_{r}.

We have the following generalisation (for a further generalisation, see [63], theorem 3):

Proposition 160 (a) The endomorphism ring ξ_{ω}D(n, r)ξ_{ω} is isomorphic to the wreath product k[x]/(x^{2}) ≀ Σ_{r}.

(b) If char(k) = 2, then ξ_{ω}D(n, r)ξ_{ω} is isomorphic to the wreath product kΣ_{2} ≀ Σ_{r}.

(c) If char(k) = 0, or w < char(k), then D(n, r) is Morita equivalent to k[x]/(x^{2}) ≀ Σ_{r}.

Proof:

Let Ω = {1, ..., r}. Let us identify a subset Π ⊂ Ω of size d with the d-tuple (π_{1}, ..., π_{d}) ∈ I(n, d), where Π is ordered just as Ω is ordered.

Thus, ξ_{Ω,Ω} = ξ_{ω}. The set

{ξ_{Πσ,Π} ⊗ X_{(Ω−Π)σ,(Ω−Π)}|Π ⊂ Ω, σ ∈ Σ_{r}}

is a basis for ξ_{ω}D(n, r)ξ_{ω}. The size of this basis set is 2^{r}.r!.

The subspace spanned by {ξ_{Ωσ,Ω}|σ ∈ Σ_{r}} is a subalgebra, naturally isomorphic to kΣ_{r}.

For Π ⊂ Ω, let ξ_{Π} = ξ_{Π,Π}. We show that the subspace spanned by {ξ_{Π} ⊗ X_{Ω−Π}|Π ⊂ Ω} is also a subalgebra, isomorphic to (k[x]/(x^{2}))^{⊗r} via

Φ : ξ_{Π} ⊗ X_{Ω−Π} ↦ α_{1} ⊗ ... ⊗ z_{r},

where z_{i} = 1 if i ∈ Π, and z_{i} = 0 otherwise.

The isomorphism of the lemma is then quite plain from the formulas,

ξ_{Π} ⊗ X_{Ω−Π}.ξ_{Ωσ,Ω} = ξ_{Πσ,Π} ⊗ X_{Ω−Πσ,Ω−Π},
ξ_{Ωσ^{−1},Ω}.ξ_{Π} ⊗ X_{Ω−Π}.ξ_{Ωσ,Ω} = ξ_{Π} ⊗ X_{(Ω−Π)σ}.

Let Γ, Π be subsets of Ω. Let s = ξ_{Π}, and let a = X_{Ω−Π}. Let t = ξ_{Γ}, and let b = X_{Ω−Γ}. We compute the product of s ⊗ a and t ⊗ b.

Δ(ξ_{Π}) = ∑_{α⊂Π} ξ_{α} ⊗ ξ_{Π−α}

Δ(ξ_{Γ}) = ∑_{β⊂Π} ξ_{β} ⊗ ξ_{Π−β}.

Thus, non-zero terms of (s ⊗ a)(t ⊗ b) only appear when b_{(1)} = X_{α}, s_{(2)} = ξ_{Π−α}, a_{(2)} = X_{Γ−β}, and t_{(1)} = ξ_{β}.

This implies, in turn, that only b = b_{(1)} = b_{(2)} = X_{α} gives possible non-zero terms in (s ⊗ a).(t ⊗ b). Thus, α = Ω − Γ.

In addition, only a = a_{(1)} = a_{(2)} = X_{Γ−β} gives possible non-zero terms in (s ⊗ a).(t ⊗ b). Thus, Γ − β = Ω − Π.

For the product s_{(2)}t_{(1)} to be non-zero, we now require β = Π−α. Thus, β = Γ ∩ Π and α = Ω − Γ.

From all this, we conclude that (s ⊗ a).(t ⊗ b) is zero unless Π ∪ Γ = Ω, and that in this case (s ⊗ a).(t ⊗ b) is equal to ξ_{Π∩Γ} ⊗ X_{Ω−Π∩Γ}. It follows that Φ is indeed an algebra isomorphism, and (a) is proven !

(b) is immediate from (a), since kΣ_{2} ≅ k[x]/(x^{2}), in characteristic two.

(c) also follows immediately from (a), because kΣ_{r} is semisimple so long as r! is invertible in k. Thus, D(n, r) and k[x]/(x^{2}) ≀ Σ_{r} have the same number of simple modules. □

Schiver doubles and blocks of Hecke algebras.

Here is a summary of our education concerning Rock blocks of symmetric groups, in characteristic two, in chapters 4 and 5.

Theorem 161 Let k be a field of characteristic two. Let kb_{ρ,w}^{Σ} be a Rock block of a symmetric group, whose weight is w. Then kb_{ρ,w}^{Σ} contains a nilpotent ideal N, such that kb_{ρ,w}^{Σ}/N is Morita equivalent to S(w, w). Let e be an idempotent in kb_{ρ,w}^{Σ}, such that the indecomposable summands of kb_{ρ,w}^{Σ}e are precisely those indecomposable summands with tops in the set,

{D^{[∅,λ]}|λ is p-regular }.

Then ekb_{ρ,w}^{Σ}e is Morita equivalent to kΣ_{2} ≀ Σ_{w}. □

This theorem, proposition 160, and indeed example 144 of this paper, give evidence that Rock blocks kb_{ρ,w}^{Σ}, are in fact Morita equivalent to the Schur doubles D(n, w), for n ≥ w, when the field k has characteristic two. But the Ringel dualities of chapter 6 impose a more general conjecture on us:

Let A_{p−1} be the ordinary Dynkin graph with p − 1 vertices:

∘————–∘————–∘.............∘————–∘————–∘

Let q ∈ k^{×}, and let p be the least natural number such that,

1 + q + ... + q^{p−1} = 0.

Let n, w be natural numbers, such that n ≥ w. Let D_{A_{p−1}}(n) be the Schiver double associated to the graph A_{p−1}.

Conjecture 162 The degree w part D_{A_{p−1}}(n, w) of D_{A_{p−1}}(n), is Morita equivalent to the Rock block kb_{ρ,w}^{H_{q}} of a Hecke algebra, whose weight is w.

Indeed, D_{A_{p−1}}(n, w) is derived equivalent to any block kb_{τ,w}^{H_{q}} of a Hecke algebra, whose weight is w.

Since any two blocks of the same weight are derived equivalent, the conjectured derived equivalences would follow immediately from the conjectured Morita equivalences.

Remark 163 When q = 1, the Hecke algebra H_{q}(Σ_{n}) is isomorphic to the symmetric group algebra kΣ_{n}, and p is merely the characteristic of the field k. In this case, conjecture 162 could be viewed as relating to a non-abelian generalisation of Broué’s abelian defect group conjecture [9] for symmetric groups, as proved by Chuang, Kessar, and Rouquier [11], [12].

So far, however, I can see no natural interpretation of the Schiver double D_{A_{p−1}}(n) in terms of p-local group theoretic information, weights for symmetric groups, etc. etc.

Doubles, and graded rings associated to Rock blocks of Hecke algebras.

I am unable to prove conjecture 162. The best I can do, is describe a filtration,

kb_{ρ,w}^{H_{q}} = N [0] ⊃ N [1] ⊃ N [2] ⊃ ... ⊃ N [2w] ⊃ N [2w + 1] = 0,

on the Rock block of a Hecke algebra of weight w, such that N [i].N [j] ⊆ N [i + j], and the associated graded ring,

gr_{ρ,w}^{H_{q}} = ⨁_{i=0}^{2w} N [i]/N [i + 1],

is Morita equivalent to the Schiver double D_{A_{p−1}}(w, w).

In this section, we explain a strategy to prove this result, in case p ≥ 4. We thus explain how conjecture 162, concerning Rock blocks of Hecke algebras, should be reduced to the conjecture that kb_{ρ,w}^{H_{q}} is graded, in a certain way.

Step 1. If K is a certain field of characteristic zero, and q ∈ K is a primitive p^{th} root of unity, then Kb_{ρ,w}^{H_{q}} is Morita equivalent to Kb_{∅,1}^{H_{q}} ≀ Σ_{w}.

Proof:

By Dirichlet’s theorem, there exists a prime number l, such that l = 1 (modulo p). Similarly, there exists a prime number q̄, such that q̄ = q (modulo l). By theorem 78, there is an l-modular system (K, O, k), such that kb_{ρ,w}^{Hq̄} is Morita equivalent to kb_{∅,1}^{Hq̄} ≀ Σ_{w}. We can lift this equivalence by the following argument, due to Joe Chuang.

The bimodule inducing Morita equivalence is a summand of the kb_{ρ,w}^{Hq̄}- kb_{∅,1}^{Hq̄} ≀ Σ_{w}-bimodule,

kT = kb_{ρ,w}^{Hq̄} ⨂_{⊗^{w}kb_{∅,1}^{Hq̄}} kb_{∅,1}^{Hq̄} ≀ Σ_{w}.

We would like to define a summand of the Ob_{ρ,w}^{H_{q}}- Ob_{∅,1}^{H_{q}} ≀ Σ_{w}-bimodule,

T = Ob_{ρ,w}^{H_{q}} ⨂_{⊗^{w}Kb_{∅,1}^{Hq}} Ob_{∅,1}^{H_{q}} ≀ Σ_{w},

which induces a Morita equivalence. It is enough lift idempotents from End(kT) to End(T). Algebraically, this translates to the problem of lifting centralizers of parabolic subalgebras from characteristic l to characteristic zero. The arguments of A. Francis ([31], 3.6, 3.8) show that this is possible.

Step 2. Use the approach of C.P.S. ([19], 5.3) to generalise theorem 87 from blocks of symmetric groups to blocks of Hecke algebras. Generalise the results of chapters 6 and 7 from blocks of symmetric groups, to blocks of Hecke algebras. Thus produce a nilpotent ideal N of kb_{ρ,w}^{H_{q}}, such that kb_{ρ,w}^{H_{q}}/N is Morita equivalent to S_{V (A_{p−1})}(w, w).

Step 3. Define the filtration N [i] on kb_{ρ,w}^{H_{q}}, using good idempotents for the q-Schur algebra, as well as the signature automorphism on the Hecke algebra.

The method of definition generalises that of the bimodule N, of chapter 6. Whilst N is defined to be xkb_{ρ,w}^{H_{q}}y, for fixed idempotents x, y, the ideal N [i] contains sums of terms xH_{q}y, yH_{q}x, xH_{q}yH_{q}x, and yH_{q}xH_{q}y, for various idempotents x, y.

By comparison with the characteristic zero case (Step 1), it can be seen that the filtration satisfies N [i].N [j] ⊆ N [i + j], and N = N [1].

Step 4. Show that the kb_{ρ,w}^{H_{q}}/N - kb_{ρ,w}^{H_{q}}/N -bimodule, N [i]/N [i + 1] corresponds, via Morita equivalence, to the D_{A_{p−1}}^{0}(w, w)-D_{A_{p−1}}^{0}(w, w)-bimodule, D_{A_{p−1}}^{i}(w, w). Prove this by induction on w.

Step 5. Show that for compositions (a_{1}, ..., a_{p−1}), (c_{1}, .., c_{p−1}) of w, for which there exists a composition (b_{1}, ..., b_{p−1}) of w, such that,

(a_{1}, ..., a_{p−1}) ◁ (b_{1}, ..., b_{p−1}) ◁ (c_{1}, ..., c_{p−1}),

we have,

Ext^{1}(L_{q}(a_{1}, ..., a_{p−1}), L_{q}(c_{1}, ..., c_{p−1})) =
Ext^{1}(L_{q}(c_{1}, ..., c_{p−1}), L_{q}(a_{1}, ..., a_{p−1})) = 0.

Prove this by induction on w, using duality and Frobenius reciprocity, with base case w = 1.

Step 6. Show (using Step 5) that the multiplication morphism,

ϕ : N [i]/N [i + 1] ⨂_{kb_{ρ,w}^{Hq}} N [j]/N [j + 1] → N [i + j]/N [i + j + 1],

of kb_{ρ,w}^{H_{q}}/N - kb_{ρ,w}^{H_{q}}/N -bimodules, is a surjection. In other words, gr_{ρ,w}^{H_{q}} is generated in degrees 0 and 1.

Step 7 The generalized Koszulity of D_{A_{∞}}(w, w) (remark 181) implies that D_{A_{∞}}(w, w) is quadratic. When p ≥ 4, D_{A_{p−1}}(w, w) is also quadratic.

The correspondences of Step 4 may be fixed so that in degree (1, 1), ϕ corresponds to the morphism

D_{A_{p−1}}^{1}(w, w) ⨂_{D_{A_{p−1}}^{0}(w,w)} D_{A_{p−1}}^{1}(w, w)

→ D_{A_{p−1}}^{2}(w, w).

of D_{A_{p−1}}^{0}(w, w)-D_{A_{p−1}}^{0}(w, w)-bimodules.

Step 8. Conclude from Steps 6 and 7 that D_{A_{p−1}}(w, w) surjects onto an algebra Morita equivalent to gr_{ρ,w}^{H_{q}}. By a dimension count, this surjection is an isomorphism. □

Remark 164 Since in case w < l and in case l = 0, the Rock blocks kb_{ρ,w}^{H_{q}} are suitably graded (by theorem 53, theorem 78, and Step 1 above), conjecture 162 holds in these situations.

Chapter IX Power sums, and Schiver doubles of type A_{∞}.

At the beginning of this chapter, we define certain chain complexes for the Schiver doubles, whose Grothendieck character describes the symmetric functions p_{r} = x_{1}^{r} + x_{2}^{r} + ... (theorem 166).

The existence of such complexes is consistent with the apparent affinity between derived categories of Schiver doubles, and those of blocks of Hecke algebras. Indeed, whilst blocks of Hecke algebras define a category lifting the Fock space realization of the basic representation of 𝔰̂𝔩_{p}, power sums play a defining role in the combinatorial formulation of the principal homogeneous realization of the basic representation for 𝔰̂𝔩_{p} by I. Frenkel, N. Jing and W. Wang.

We then consider the infinite Dynkin quiver A_{∞}:

..........∘————–∘————–∘————–∘..........

We prove that the module category of D_{A_{∞}}(w, w) is a highest weight category, and then conjecture that Rock blocks kb_{ρ,w}^{S_{q}}, of q-Schur algebras, are Morita equivalent to certain subquotients of the algebra D_{A_{∞}}(w, w), defined in chapter 8 (conjecture 176).

We also describe a walk along D_{A_{∞}}(n), analogous to J.A. Green’s walk along the Brauer tree (theorem 180).

Complexes for Schiver doubles, and power sums.

Most of the bases for the ring of symmetric functions given in I. Mac-Donald’s book [50] have natural interpretations as characters of modules for the Schur algebra. Elementary symmetric functions correspond to exterior powers of the natural module for M_{n}(k), complete symmetric functions correspond to symmetric powers of the natural module, and Schur functions correspond to Weyl modules. However, the power sums p_{r} = x_{1}^{r} + x_{2}^{r} + ... have no such interpretation.

In this section, we describe complexes P_{r} for Schiver doubles D_{Γ}(n, r), whose homology describes the power sum p_{r}. Indeed, we may define one such complex P_{r}(a), for every pair (γ, a), where γ is an vertex of Γ, and a an edge emanating from γ.

Lemma 165 Let n ≥ r. Let Q be a quiver. Suppose that γ_{1}, γ_{2} are vertices in Q. Suppose that a is an arrow in Q, whose source is γ_{1}, and tail is γ_{2}, and that a is the only such arrow. Let r_{1}, r_{2} be natural numbers, whose sum is r. There is a graded module M_{r_{1},r_{2}} = M_{r_{1},r_{2}}(a) for S_{Q}(n, r), whose graded pieces are the S_{V (Q)}(n, r)-modules,

Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{2}}^{⊗r_{2}}
Δ_{γ_{1}}(r_{1}, 1) ⊗ E_{γ_{2}}^{⊗r_{2}−1}
...
Δ_{γ_{1}}(r_{1}, 1^{r_{2}}),

in degrees 0, 1, ..., r_{2}.

Proof:

Let ξ_{r_{1},r_{2}} be the unit of the algebra S_{γ_{1}}(n, r_{1})⊗S_{γ_{2}}(n, r_{2}), an idempotent in S_{Q}(n, r). Then,

N_{r_{1},r_{2}}(a) = S_{Q}(n, r)ξ_{r_{1},r_{2}} ⨂_{S_{γ_{1}}(n,r_{1})⊗S_{γ_{2}}(n,r_{2})} Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{2}}^{⊗r_{2}},

is a graded module for S_{Q}(n, r), whose graded pieces are,

Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{2}}^{⊗r_{2}}
Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{1}} ⊗ E_{γ_{2}}^{⊗r_{2}−1}
...
Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{1}}^{⊗r_{2}},

in degrees 0, 1, ..., r_{2}.

For a partition λ of t, let ξ_{>λ} = ∑_{μ>λ} ξ_{μ} be the sum of Green’s idempotents ξ_{μ} [37], corresponding to partitions μ, greater than λ, with respect to the dominance ordering.

Let i_{r_{1},r_{2}}^{j} be the idempotent ξ_{>(r_{1},1^{j})} ⊗ξ_{(1^{r_{2}−j})}, an element of the algebra,

S_{γ_{1}}(n, r_{1} + j) ⊗ S_{γ_{2}}(n, r_{2} − j).

Let i_{r_{1},r_{2}} = ∑_{j=1}^{r_{2}} i_{r_{1},r_{2}}^{j}, an idempotent in the algebra,

⨁_{j=1}^{r_{2}} S_{γ_{1}}(n, r_{1} + j) ⊗ S_{γ_{2}}(n, r_{2} − j).

Let us define,

M_{r_{1},r_{2}}(a) = N_{r_{1},r_{2}}(a)/S_{Q}(n, r)i_{r_{1},r_{2}}N_{r_{1},r_{2}}(a),

to be the quotient of N_{r_{1},r_{2}}(a), relative to a trace from the projective module S_{Q}(n, r)i_{r_{1},r_{2}}.

Since the tensor product of a projective S_{γ_{1}}(n, r_{1} + j)-module with the natural representation E_{γ_{1}} is projective, with Δ-composition factors given by the branching rule, we may compute the composition factors of M_{r_{1},r_{2}}(a), and find them to be,

Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{2}}^{⊗r_{2}}
Δ_{γ_{1}}(r_{1}, 1) ⊗ E_{γ_{2}}^{⊗r_{2}−1}
...
Δ_{γ_{1}}(r_{1}, 1^{r_{2}}),

in degrees 0, 1, ..., r_{2}. □

The following theorem generalises the above lemma:

Theorem 166 Let n ≥ r. Let Q be a quiver. Suppose that γ_{1}, γ_{2} are vertices in Q. Suppose that a is an arrow in Q, whose source is γ_{1}, and tail is γ_{2}, and that a is the only such arrow. Let r_{1}, r_{2} be natural numbers, whose sum is r. There is a graded module C_{r_{1},r_{2}} = C_{r_{1},r_{2}}(a) for D_{Q}(n, r), whose graded pieces are the S_{V (Q)}(n, r)-modules,

Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{2}}^{⊗r_{2}}
Δ_{γ_{1}}(r_{1} − 1) ⊗ E_{γ_{2}}^{⊗r_{2}+1} + Δ_{γ_{1}}(r_{1}, 1) ⊗ E_{γ_{2}}^{⊗r_{2}−1}
Δ_{γ_{1}}(r_{1} − 1, 1) ⊗ E_{γ_{2}}^{⊗r_{2}} + Δ_{γ_{1}}(r_{1}, 1^{2}) ⊗ E_{γ_{2}}^{⊗r_{2}−2}
...
Δ_{γ_{1}}(r_{1} − 1, 1^{r_{2}−1}) ⊗ E_{γ_{2}}^{⊗2} + Δ_{γ_{1}}(r_{1}, 1^{r_{2}})
Δ_{γ_{1}}(r_{1} − 1, 1^{r_{2}}) ⊗ E_{γ_{2}},

in degrees 0, 1, ..., r_{2} + 1.

There is a natural homomorphism d_{r_{1},r_{2}} = d_{r_{1},r_{2}}(a) of degree one, from C_{r_{1}−1,r_{2}+1} to C_{r_{1},r_{2}}. The sequence of maps {d_{r_{1},r_{2}}|r_{1} + r_{2} = r} defines a chain complex, P_{r}(a), given by,

... → 0 → C_{1,r−1} → C_{2,r−2} → ... → C_{r−1,1} → C_{r,0} → 0 → ...

The homology of this complex at term C_{r_{1},r_{2}} is isomorphic as a D_{Q}(n, r)- module, to the standard module Δ_{γ_{1}}(r_{1}, 1^{r_{2}}) for S_{γ_{1}}(n, r), concentrated in degree r_{1} − 1.

Proof:

Consider the D_{Q}(n, r) − S_{γ_{1}}(n, r_{1}) ⊗ S_{γ_{2}}(n, r_{2})-bimodule,

X_{r_{1},r_{2}} = S_{Q}(n, r)ξ_{r_{1},r_{2}} ⊕ (S_{Q}(n, r − 1)ξ_{r_{1}−1,r_{2}} ⊗ ⋀(n, 1)_{γ_{2},γ_{1}}),

given as a quotient of the D_{Q}(n, r)- S_{γ_{1}}(n, r_{1}) ⊗ S_{γ_{2}}(n, r_{2})-bimodule,

D_{Q}(n, r)ξ_{r_{1},r_{2}},

modulo terms of higher degree. Note that

Y_{r_{1},r_{2}} = (S_{Q}(n, r − 1)ξ_{r_{1}−1,r_{2}} ⊗ ⋀_{γ_{2},γ_{1}}(n, 1)),

is a sub-bimodule of X_{r_{1},r_{2}}. Let us define the D_{Q}(n, r)-module,

U_{r_{1},r_{2}} = X_{r_{1},r_{2}} ⨂_{S_{γ_{1}}(n,r_{1})⊗S_{γ_{2}}(n,r_{2})}(Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{2}}^{⊗r_{2}}),

which contains the submodule,

V_{r_{1},r_{2}} = Y_{r_{1},r_{2}} ⨂_{S_{γ_{1}}(n,r_{1})⊗S_{γ_{2}}(n,r_{2})}(Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{2}}^{⊗r_{2}}).

Thus, U_{r_{1},r_{2}} is a graded module, whose graded pieces are the S_{V (Q)}(n, r)- modules,

Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{2}}^{⊗r_{2}}
Δ_{γ_{1}}(r_{1} − 1) ⊗ E_{γ_{2}}^{⊗r_{2}+1} + Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{1}} ⊗ E_{γ_{2}}^{⊗r_{2}−1}
Δ_{γ_{1}}(r_{1} − 1) ⊗ E_{γ_{1}} ⊗ E_{γ_{2}}^{⊗r_{2}} + Δ_{γ_{1}}(r_{1}) ⊗ E_{γ_{1}}^{⊗2} ⊗ E_{γ_{2}}^{⊗r_{2}−2}
...
Δ_{γ_{1}}(r_{1} − 1) ⊗ E_{γ_{1}}^{r_{2}−1} ⊗ E_{γ_{2}}^{⊗2} + Δ_{γ_{1}}(r_{1}, 1^{r_{2}})
Δ_{γ_{1}}(r_{1} − 1) ⊗ E_{γ_{1}}^{r_{2}} ⊗ E_{γ_{2}},

in degrees 0, 1, ..., r_{2} + 1. Let us define,

O_{r_{1},r_{2}} = D_{Q}(n, r)i_{r_{1}−1,r_{2}+1}V_{r_{1},r_{2}},

a submodule of U_{r_{1},r_{2}}. As in lemma 165, D_{r_{1},r_{2}} = V_{r_{1},r_{2}}/O_{r_{1},r_{2}} is a graded module, whose graded pieces are the S_{V (Q)}(n, r)-modules,

Δ_{γ_{1}}(r_{1} − 1) ⊗ E_{γ_{2}}^{⊗r_{2}+1}
Δ_{γ_{1}}(r_{1} − 1, 1) ⊗ E_{γ_{2}}^{⊗r_{2}}

...
Δ_{γ_{1}}(r_{1} − 1, 1^{r_{2}−2}) ⊗ ⊗E_{γ_{2}}^{⊗2}
Δ_{γ_{1}}(r_{1} − 1, 1^{r_{2}}) ⊗ E_{γ_{2}},

in degrees 0, 1, ..., r_{2}. Let us define,

P_{r_{1},r_{2}} = D_{Q}(n, r)i_{r_{1},r_{2}}(U_{r_{1},r_{2}}/O_{r_{1},r_{2}}),

a submodule of M_{r_{1},r_{2}}/O_{r_{1},r_{2}}. We now define,

C_{r_{1},r_{2}} = U_{r_{1},r_{2}}/(O_{r_{1},r_{2}} + P_{r_{1},r_{2}}),

which contains D_{r_{1},r_{2}} as a submodule. The graded pieces of C_{r_{1},r_{2}} are visibly those described in the statement of the theorem.

Let η be the element, 1_{S_{γ_{1}}(n,r_{1}−1)} ⊗1_{M_{n}(k)} ⊗1_{S_{γ_{2}}(n,r_{2}+1)} of the S_{γ_{1}}(n, r_{1} − 1) ⊗ S_{γ_{2}}(n, r_{2} + 1)- S_{γ_{1}}(n, r_{1}) ⊗ S_{γ_{2}}(n, r_{2})- bimodule,

S_{γ_{1}}(n, r_{1} − 1) ⊗ ⋀_{γ_{2},γ_{1}}(n, 1) ⊗ S_{γ_{2}}(n, r_{2}).

Then multiplication by η defines a map from X_{r_{1}−1,r_{2}+1} to Y_{r_{1},r_{2}}. By restriction, there is a map,

d_{r_{1},r_{2}} : C_{r_{1}−1,r_{2}+1} → C_{r_{1},r_{2}}.

The kernel of d_{r_{1}+1,r_{2}−1} is equal to the submodule D_{r_{1}−1,r_{2}+1} + Δ_{γ_{1}}(r_{1}, 1^{r_{2}}) of C_{r_{1},r_{2}}, and the image of d_{r_{1},r_{2}} is isomorphic to D_{r_{1},r_{2}}. The chain complex P_{r}(a) defined by the sequence of maps, thus has homology Δ_{γ_{1}}(r_{1}, 1^{r_{2}}), at term C_{r_{1},r_{2}}. □

Theorem 167 The Grothendieck character of the complex P_{r}(a) of D_{Q}(n, r)- modules describes the power sum p_{r}, at γ_{1}.

Proof:

The Grothendieck character of a standard module Δ(λ) is given by the Schur function s_{λ}. There is a formula for the power sum p_{r}, given by ([50], I.4, example 10),

p_{r} = s_{(r)} − s_{(r−1,1)} + s_{(r−2,1^{2})} − ... ± s_{(1^{r})}.

The Grothendieck character of P_{r}(a) thus describes the power sum p_{r}. □

I. Frenkel, N. Jing and W. Wang have given a description of the homogeneous vertex operator construction of the basic representation of an affine Lie algebra of type ADE, via wreath products of finite group algebras [32]. The characters of symmetric groups which are used in that paper correspond in symmetric function theory to elementary symmetric functions, to complete symmetric functions, and to power sums.

We have observed here that there are objects in the derived category of a Schiver double which correspond to all these functions.

In fact, upon studying Frenkel, Jing and Wang’s construction more carefully, one realises that the Schiver double afforded to a simply-laced Dynkin diagram Γ, underlies a category for a vertex representation for the affinization of the Kac-Moody Lie algebra defined by Γ, at least when Γ is ordinary/affine, of type ADE.

By this, we only mean that we can describe a category whose complexified Grothendieck group is the vertex representation, and we can describe functors which correspond to the vertex operators, upon passing to the Grothendieck group. The category is a direct product of derived categories for Schiver doubles, and the functors are described by certain complexes of bimodules. Note that we have not explored the extent to which relations in the affine Lie algebra lift to relations between functors.

Conjecture 162 is comfortingly consistent with this categorical perspective. The blocks of Hecke algebras are already well known to be categories which describe the basic representation for 𝔰̂𝔩_{p}, as has most elegantly been described by I. Grojnowski [39], following theory of A. Lascoux, B. Leclerc, and J-Y. Thibon [48], as well as S. Ariki [3], and A. Kleshchev [44], [45]. In Grojnowski’s article, relations in the affine Lie algebra do lift to relations between functors.

In general, we expect an equivalence of 𝔰̂𝔩_{p}-categorifications,

D^{b} ( ⨁_{w≥0,s∈Ŵ/W} D_{A_{p−1}}(w, w)) ≅ D^{b} (⨁_{r≥0} H_{q}(Σ_{r})) ,

although it is unclear to me what this means precisely; so far, only 𝔰𝔩_{2}- categorifications possess an axiomatic definition [12]. Above, W is the Weyl group of 𝔰𝔩_{p}, and Ŵ is the corresponding affine Weyl group.

Schiver bialgebras for A_{∞}.

Consider the infinite Dynkin quiver A_{∞}, which has vertex set

V_{∞} = {v_{i}, i ∈ ℤ},

and edge set E_{∞} = {e_{i}, i ∈ ℤ}. The source of any edge e_{i} is the vertex v_{i+1}, and its tail is v_{i}.

We present the Schiver bialgebra corresponding to A_{∞}. This bialgebra amplifies the category of chain complexes over k, as the classical Schur bialgebra amplifies the category of vector spaces over k.

Definition 168 Let \underline{Λ}(n, r) (respectively \underline{Λ}′(n, r)) be the set \{\underline{λ} = (λ_{i})_{i∈ℤ}\} of sequences of partitions, with n parts or fewer, whose sizes sum to r.

For two elements \underline{λ}, \underline{μ} of \underline{Λ}(n, r) (respectively \underline{Λ}′(n, r)), let \underline{λ} ⊴ \underline{μ} if and only if, the sequence (λ_{i})_{i∈ℤ} can be obtained from the sequence (μ_{i})_{i∈ℤ} in finitely many steps, by repeatedly either,

(1) removing a box from the Young diagram of μ_{i}, and replacing it lower down on the same Young diagram, to create a new partition, or

(2) removing a box from the Young diagram of μ_{i}, and adding it on to the Young diagram of μ_{i−1} (respectively μ_{i+1}), to create a new partition.

The posets \underline{Λ}(n, r), \underline{Λ}′(n, r) generalise the dominantly ordered poset of partitions of r with n parts or fewer.

Recall that the the degree zero part of the algebra S_{A_{∞}}(n, r) is equal to,

S_{V_{∞}}(n, r) ≅ ⨁ ; (r_{i})_{i∈ℤ},∑ r_{i}=r (⨂_{i∈ℤ}S(n, r_{i})) .

Recall further, that the module category S(n, r) − mod for the classical Schur algebra, is a highest weight category, whose standard modules Δ(λ) are named Weyl modules, indexed by partitions of r with n parts or fewer. We write ∇(λ) for the costandard S(n, r)-module corresponding to a partition λ of r.

Definition 169 Let \underline{λ} = (λ_{i})_{i∈ℤ} be a sequence of partitions, finitely many of which are non-empty, whose sizes are given by the sequence (r_{i})_{i∈ℤ}. Let r = ∑_{i∈ℤ} r_{i}.

The standard module, Δ_{1}(\underline{λ}) for S_{A_{∞}}(n, r), is given by,

S_{A_{∞}}(n, r) ⨂_{S_{V∞}(n,r)} (⨂_{i∈ℤ}Δ(λ_{i})) .

The costandard module, ∇_{1}(\underline{λ}) for S_{A_{∞}}(n, r), is the S_{V_{∞}}(n, r)-module,

⨂_{i∈ℤ}∇(λ_{i}).

The costandard module, ∇_{2}(\underline{λ}) for S_{A_{∞}}(n, r), is given by,

Hom_{S_{V∞}(n,r)} (S_{A_{∞}}(n, r), ⨂_{i∈ℤ}∇(λ_{i})) .

The standard module, Δ_{2}(\underline{λ}) for S_{A_{∞}}(n, r), is the S_{V_{∞}}(n, r)-module,

⨂_{i∈ℤ}Δ_{2}(λ_{i}).

Theorem 170 Let n ≥ r be natural numbers.

The module category S_{A_{∞}}(n, r) − mod, is a highest weight category with respect to the poset \underline{Λ}(n, r). Given an element \underline{λ} of \underline{Λ}(n, r), the corresponding standard module is Δ_{1}(\underline{λ}), and the corresponding costandard module is ∇_{1}(\underline{λ}).

The module category S_{A_{∞}}(n, r) − mod, is also a highest weight category with respect to the poset \underline{Λ}′(n, r). Given an element \underline{λ} of \underline{Λ}′(n, r), the corresponding standard module is Δ_{2}(\underline{λ}), and the corresponding costandard module is ∇_{2}(\underline{λ}).

Proof:

We describe only the quasi-hereditary structure with respect to \underline{Λ}(n, r). The quasi-hereditary structure with respect to \underline{Λ}′(n, r) may be understood similarly.

Throughout this proof, we indiscriminately use the Ringel self-duality of S_{.}(n, t), for n ≥ t (theorem 17).

We first show that the projective cover of a simple module L(\underline{λ}) is filtered by standard modules Δ_{1}(\underline{μ}), with \underline{λ} ⊴ \underline{μ}. Note that if P (\underline{λ}) is the projective S_{V_{∞}}(n, r) module, with top L(\underline{λ}), then S_{A_{∞}}(n, r) ⊗_{S_{V∞}(n,r)} P (\underline{λ}) is the projective cover of L(\underline{λ}) as a S_{A_{∞}}(n, r)-module. Since P (\underline{λ}) is filtered by Δ(\underline{μ}), with \underline{λ} ⊴ \underline{μ}, and for n ≥ t, the functor ⊗_{S_{V∞}(n,t)} ⋁(n, t) is exact on the category of Δ-filtered S_{V_{∞}}(n)-modules, we deduce that S_{A_{∞}}(n) ⊗_{S_{V∞}(n)} P (\underline{λ}) is filtered by Δ_{1}(\underline{μ}), with \underline{λ} ⊴ \underline{μ}.

We secondly remark that the simple composition factors of Δ_{1}(\underline{λ}) are indexed by elements \underline{μ} of \underline{Λ}(n, r), such that \underline{μ} ⊴ \underline{λ}. This is a consequence of the branching rule for classical Schur algebras, as well as the quasi-heredity of S_{V_{∞}}(n).

Thirdly, that the costandard modules relevant to this highest weight structure are the ∇_{1}(\underline{λ})’s is now visible. By duality, we need only observe that Δ_{2}(\underline{λ}) is the largest quotient of the projective cover of L(\underline{λ}) for whose composition factors L(\underline{μ}) (excepting the top L(\underline{λ})), the multipartition \underline{μ} is strictly smaller than \underline{λ}, with respect to the ordering on \underline{Λ}′(n, r). This is apparent from the structure we have already described on the projective cover of L(\underline{λ}). □

We consider the Schiver double, D_{A_{∞}}(n). An immediate corollary of theorem 154 is,

Theorem 171 The action of the infinite dihedral group D_{∞} as graph automorphisms of A_{∞} lifts to an action of D_{∞} as algebra automorphisms on D_{A_{∞}}(n). □

Let

′[1] : \underline{Λ}(n, r) → \underline{Λ}(n, r),
(\underline{λ}′[1])_{i} = λ′_{i−1},

be the map on \underline{Λ}(n, r), which shifts a sequence by 1, and then conjugates each entry in the sequence.

Lemma 172 Let n ≥ r be natural numbers.

There is an isomorphism, Δ_{1}(\underline{λ}) ≅ ∇_{2}(\underline{λ}′[1]).

Proof:

The natural sequence of homomorphisms,

Δ_{1}(\underline{λ})^{op} ⊗ Δ_{1}(\underline{λ}′[1]) → Δ_{1}(\underline{λ})^{op} ⊗_{S_{A∞}(n,r)} Δ_{1}(\underline{λ}′[1])

≅ (⨂_{i∈ℤ}Δ(λ_{i})^{op}) ⊗_{S_{V∞}(n,r)} (⨂_{i∈ℤ}⋁(n, r_{i})) ⊗_{S_{V∞}(n,r)} (⨂_{i∈ℤ}∇(λ′_{i}))

≅ (⨂_{i∈ℤ}Δ(λ_{i})^{op}) ⊗_{S_{V∞}(n,r)} (⨂_{i∈ℤ}Δ(λ_{i})) ≅ k,

defines a non-degenerate bilinear form,

<, >: Δ_{1}(\underline{λ})^{op} × Δ_{1}(\underline{λ}′[1]) → k,

such that < x ∘ s, y >=< x, s ∘ y >, for s ∈ S_{A_{∞}}(n, r).

Since the dual of the S_{A_{∞}}(n, r)-module Δ_{1}(\underline{λ}), is isomorphic to ∇_{2}(\underline{λ}), the existence of such a bilinear form seals the proof of the lemma. □

Theorem 173 Let n ≥ r be natural numbers.

The Schiver algebra S_{A_{∞}}(n, r) is Ringel self-dual. Indeed, Ringel duality exchanges the two highest weight structures we have introduced on S_{A_{∞}}(n, r) − mod.

The module category D_{A_{∞}}(n, r) − mod is a highest weight category, with respect to the poset \underline{Λ}(n, r). Given an element \underline{λ} of \underline{Λ}(n, r), the corresponding standard module is Δ_{1}(\underline{λ}).

Furthermore, D_{A_{∞}}(n, r) − mod is a highest weight category, with respect to the poset \underline{Λ}′(n, r). Given an element \underline{λ} of \underline{Λ}′(n, r), the corresponding costandard module is ∇_{2}(\underline{λ}).

Indeed, D_{A_{∞}}(n, r) is Ringel self-dual, and Ringel duality exchanges these two highest weight structures on D_{A_{∞}}(n, r) − mod.

Proof:

As a consequence of lemma 172, and theorem 170, the regular representation of S_{A_{∞}}(n, r) can be filtered by ∇_{2}’s, as well as filtered by Δ_{2}’s.

Thus, the regular representation is a full tilting module for S_{A_{∞}}(n, r) with respect to \underline{Λ}′(n, r), and indeed S_{A_{∞}}(n, r) is Ringel self-dual. Ringel duality exchanges the two highest weight structures we have defined on S_{A_{∞}}(n, r), because the functor Hom_{S_{A∞}(n,r)}(S_{A_{∞}}(n, r), −) must affect costandard modules to become standard modules. □

Remark 174 Let C be a highest weight category, with poset Λ, and let Π = Γ ∩ Ω be the intersection of an ideal Γ ⊂ Λ and a coideal Ω ∈ Λ. Then there is a canonically defined highest weight category C(Π), whose poset is Π (see theorem 3). So long as Π is a finite set, C is the module category of a quasi-hereditary algebra.

Let p be a natural number. We define Γ_{p}(n, r) (respectively Γ′_{p}(n, r)) to be the ideal (respectively coideal), of sequences (λ_{i}) ∈ \underline{Λ}(n, r) (respectively \underline{Λ}′(n, r)), all of whose entries λ_{i} are zero, for i > p − 1.

Let us define Ω_{p}(n, r) (respectively Ω′_{p}(n, r)), to be the coideal (respectively ideal), of sequences (λ_{i}) ∈ \underline{Λ}(n, r) (respectively \underline{Λ}′(n, r)), all of whose entries λ_{i} are zero, for i < 0.

Let Π_{p}(n, r) = Γ_{p}(n, r) ∩ Ω_{p}(n, r) (respectively Π′_{p}(n, r) = Γ′_{p}(n, r) ∩ Ω′_{p}(n, r) ), be the set of sequences (λ_{i}) ∈ \underline{Λ}(n, r) (respectively \underline{Λ}′(n, r)), all of whose entries λ_{i} are zero, for i < 0, and i > p − 1.

For n ≥ r, let Q_{p}(n, r) (respectively Q′_{p}(n, r)), be the quasi-hereditary subquotient of D_{A_{∞}}(n, r), whose poset is Π_{p}(n, r) (respectively Π′_{p}(n, r)), and whose module category is the highest weight category, (D_{A_{∞}}(n, r) − mod)(Π_{p}(n, r)) (respectively (D_{A_{∞}}(n, r) − mod)(Π′_{p}(n, r))).

Remark 175 By theorem 173, Ringel duality exchanges the quasi-hereditary algebras, Q_{p}(n, r) and Q′_{p}(n, r).

The quiver A_{∞} possesses an orientation-reversing automorphism, which exchanges vertex v_{0}, and vertex v_{p−1}. By theorem 171, this automorphism lifts to an automorphism Θ of D_{A_{∞}}(n, r).

The automorphism Θ provides an isomorphism between Q_{p}(n, w) and Q′_{p}(n, w). Therefore, Q_{p}(n, w) is Ringel self-dual.

We may now formulate a generalisation of conjecture 162 to q-Schur algebras.

Let q ∈ k^{×}, and let p be the least natural number such that,

1 + q + ... + q^{p−1} = 0.

Let n, w be natural numbers, such that n ≥ w.

Conjecture 176 The quasi-hereditary algebra Q_{p}(n, w) is Morita equivalent to any Rock block kb_{ρ,w}^{S_{q}} of a q-Schur algebra, whose weight is w.

Indeed, Q_{p}(n, w) is derived equivalent to any block kb_{τ,w}^{S_{q}} of a q-Schur algebra, whose weight is w.

How far does this conjecture generalise ?

Question 177 Can all blocks of q-Schur algebras of weight w be ℤ_{+}-graded, so that the degree zero part is Morita equivalent to the James adjustment algebra ?

Remark 178 Conjecture 162 and conjecture 176 are related as follows:

For n ≥ w, the double D_{A_{p−1}}(n, w) is obtained from Q_{p}(n, w) by cutting at the idempotent j_{p} ∈ Q_{p}(n, r) corresponding to the subset Π_{p−1}(n, w) ⊂ Π_{p}(n, w).

We should thus define the Specht modules for D_{A_{p−1}}(n, w), to be those modules j_{p}.Δ, where Δ is a standard module for Q_{p}(n, w).

It is now possible to deduce the following result from theorem 129, the definition of standard modules for D_{A_{∞}}(w, w), and formula 58, for the decomposition matrix of Kb_{ρ,w}^{H_{q}}, where q is a p^{th} root of unity.

Corollary 179 Let k be a field of characteristic p. Then the symmetric group Rock block kb_{ρ,w}^{Σ} has the same decomposition matrix as D_{A_{p−1}}(w, w). □

One proof uses the Littlewood-Richardson rule, concerning tensor products of modules for the Schur algebra. Conjecture 162 thus structurally clarifies formula 58 of Chuang-Tan, and Leclerc-Miyachi.

Walking along A_{∞}.

The super-algebra P_{A_{∞}}, is endowed with a natural differential d of degree 1, given by the infinite sum, ∑_{i∈ℤ} e_{i}, of all edges. Indeed, the complex,

......... → P_{A_{∞}} → P_{A_{∞}} → P_{A_{∞}} → .........,

with differential given by right multiplication by d, is a linear exact sequence of left P_{A_{∞}}-modules.

In the last passage of this letter, we lift this elegant differential structure on P_{A_{∞}}, to the super-bialgebra S_{A_{∞}}(n), and its double D_{A_{∞}}(n). We call the resulting chain complex a “walk along A_{∞}”, since it generalises a homological structure discovered by J.A. Green on blocks of finite groups of cyclic defect: the “walk around the Brauer tree”.

Let d be the differential on P_{A_{∞}}(n) of degree 1, given by

d = (0^{×V}) × (1^{×E}) ∈ (End_{k}(k^{n}))^{×V} × (End_{k}(k^{n}))^{×E} .

Let d_{r} be the differential on P_{A_{∞}}(n)^{⊗r} of degree 1, given by

d ⊗ 1 ⊗ ... ⊗ 1 + 1 ⊗ d ⊗ 1 ⊗ ... ⊗ 1 + .... + 1 ⊗ ... ⊗ 1 ⊗ d.

Note that d_{r} is invariant under the action of the symmetric group Σ_{r}, and so d_{r} is a differential on the Schiver super-bialgebra S_{A_{∞}}(n, r).

Theorem 180 (“Walk along A_{∞}”) The chain complex,

......... → S_{A_{∞}}(n, r) → S_{A_{∞}}(n, r) → S_{A_{∞}}(n, r) → .........,

with differential given by right multiplication by d_{r}, is a linear exact sequence of left S_{A_{∞}}(n, r)-modules. The chain complex,

......... → D_{A_{∞}}(n, r) → D_{A_{∞}}(n, r) → D_{A_{∞}}(n, r) → .........,

with differential given by right multiplication by d_{r}, is a linear exact sequence of left D_{A_{∞}}(n, r)-modules.

Proof:

The classical Koszul complex on End(k^{n})^{∗} is the acyclic chain complex,

A(n) ⊗ ⋀(n),

whose differential is given by,

d : A(n, l) ⊗ ⋀(n, m) → A(n, l + 1) ⊗ ⋀(n, m − 1),

y_{1}...y_{l} ⊗ x_{1} ∧ ... ∧ x_{m} ↦

∑_{i=1}^{m}(−1)^{i−1}y_{1}...y_{l}x_{i} ⊗ x_{1} ∧ ... ∧ x_{i−1} ∧ x_{i+1} ∧ ... ∧ x_{m}.

Its dual is an acyclic chain complex,

S(n) ⊗ ⋁(n).

Tensoring together ℤ copies of this dual Koszul complex, and forming the total complex, in degree r we obtain an acyclic chain complex which corresponds precisely to the first exact sequence of theorem 180.

Tensoring together ℤ copies of the Koszul complex, along with ℤ copies of the dual Koszul complex, in degree r we obtain an exact sequence which corresponds precisely to the second exact sequence of theorem 180. □

Remark 181 The super-algebra, P_{A_{∞}}, is a Koszul algebra [4]. Its Koszul dual is the path algebra kA_{∞}^{op} on the quiver A_{∞}, with opposite orientation.

The Schiver super-algebra S_{A_{∞}}(n) is not a Koszul algebra (unless k is a field of characteristic zero). Its degree zero part,

S_{V_{∞}}(n) = ⨂_{v∈ℤ}S(n),

is not semisimple. However, it does possess a linear resolution, and the algebra,

Ext_{S_{A∞}(n)}^{∗}(S_{V_{∞}}(n), S_{V_{∞}}(n)),

is isomorphic to S(kA_{∞}^{op})(n), while the path algebra kA_{∞}^{op} is concentrated in parity zero.

The algebra,

Ext_{D_{A∞}(n)}^{∗}(S_{V_{∞}}(n), S_{V_{∞}}(n)),

is isomorphic to the algebra S(Π_{A_{∞}})(n), where Π_{A_{∞}} is the preprojective algebra on the graph A_{∞}, concentrated in parity zero.

A similar statement is true, relating DÃ_{p−1}(n) and S(ΠÃ_{p−1})(n). I prove this, along with various stronger results, in my paper “On seven families of algebras” [63]. Sending p to infinity, one obtains theorems for the algebras associated to A_{∞}.

Index

(K, O, k), 28 ; (σ)^{k}, 19 ; (d_{λμ}), 12, 17 ; A ≀ Σ_{w}, 21 ; A_{∞}, 100, 105 ; A_{i}, 63 ; A_{(a_{1},...,a_{p−1}),q}, 68 ; A_{(a_{1},...,a_{p−1})}, 63 ; A_{X} (n), 25 ; B, B^{∗}, 79 ; B(q), 17 ; B_{i}, 63 ; B_{(a_{1},...,a_{p−1}),q}, 68 ; B_{(a_{1},...,a_{p−1})}, 63 ; Br_{P}^{G}, 28 ; C, 42 ; D, 41 ; D(B), 79 ; D(λ), 21 ; D(r), 81 ; D^{λ}, 16 ; D_{∞}, 108 ; D_{λ}, 16 ; D_{q}^{λ}, 68 ; E, 14, 86 ; G, 32 ; GL_{n}(q), 17 ; G_{i}, 32 ; HCInd_{L\underline{n}(q)}^{GL_{n}(q)}, 18 ; HCRes_{L\underline{n}(q)}^{GL_{n}(q)}, 18 ; I(λ), 10 ; I_{i}, 62 ; I_{(a_{1},...,a_{p−1})}, 62 ; I_{unres}, 62 ; J_{λ}^{i}, 47 ; J_{λ}, 48 ; J_{i}, 63  J_{(a_{1},...,a_{p−1})}, 63 ; J_{sing}, 63 ; KI_{(a_{1},...,a_{p−1}),q}, 76 ; KM_{(a_{1},...,a_{p−1}),q}, 76 ; K_{λ}^{i}, 49 ; K_{λ}, 49 ; KE_{(a_{1},...,a_{p−1}),q}, 76 ; L, 32, 41 ; L(λ), 10 ; L\underline{n}(q), 17 ; L_{a_{p−1}}, 75 ; M, 13, 43 ; M(P), 28 ; M^{P}, 28 ; M_{Γ}, 11 ; N, 33, 41 ; N^{∗}, 13 ; N_{(a_{1},...,a_{p−2})}, 64 ; P (λ), 11 ; P_{Q}, P_{Q}(n), 86 ; P\underline{n}(q), 17 ; Q, 86 ; RI, 46 ; S(Γ ∩ Ω), 11 ; S(\underline{λ}), 22 ; S^{λ}, 17 ; S_{λ}, 17 ; T (A), 85 ; T (q), 17 ; T^{(w)}L, 22 ; U\underline{n}(q), 18 ; V, 86 ; W, 18 ; X, 37 ; Y, 33 ; Z, 39 ; [λ^{0}, ..., λ^{p−1}], 24

#, 15 ΔD, 36 Δ(x), 79 Δ_{1}(\underline{λ}), ∇_{1}(\underline{λ}), 106 Δ_{2}(\underline{λ}), ∇_{2}(\underline{λ}), 107 Δ_{l}, 79 Δ_{r}, 80 Γ_{p}(n, r), 109 Λ(n), 21 Λ(n, r), 13 Λ_{w}^{a}, 27 Λ_{l,w}^{I}, 22 Ω_{p}(n, r), 109 Π_{p}(n, r), 109 Σ_{r}, 15 Σ_{u}^{0}, 75 Θ_{u}, 43 ⋁(n), ⋀(n), 89 ⋀_{q}^{r} M, 14 ⋀_{q}^{α} E, 14 χ(λ), 54 χ(s, \underline{λ}), 20 χ^{λ}, 17 η_{λ}, η_{λ}^{i}, 48 κ(s), 19 λ, 59 ∇(λ), 10 ρ, 26 τ, 25 \underline{Λ}(n, r), 106 ξ_{λ}, ξ_{λ}^{i}, 48 ξ_{ω}, 16 ζ_{D}, 44 ζ_{H}, ζ_{K}, 52 ζ_{w}, 43 ζ_{a_{p−1}}, 75 b, 33 b_{i}, 33 b_{τ,w}^{X}, 25

c(λ; μ, ν), 20 ; d_{σ}, 19 ; e, 41 ; e_{a_{p−1}}, 75 ; f, 33, 41 ; f_{λ}, 50 ; gr_{ρ,w}^{H_{q}}, 96 ; j_{λ}^{i}, 47 ; k, 10 ; kB_{s}, 20 ; kI_{(a_{1},...,a_{p−1})}, 76 ; kM_{(a_{1},....,a_{p−1})}, 75 ; kM_{a_{p−1}}, 75 ; kT, 97 ; kb_{τ,w}^{X}, 25 ; kE_{(a_{1},...,a_{p−1})}, 76 ; l, 10 ; l′, 77 ; m_{μ}^{λ}, 54 ; p, 25 ; q, 13 ; r, 32 ; sgn, 15 ; v, 32 ; w, 25 ; x_{Ω}, 63 ; x_{(a_{1},...,a_{p−1}),q}, 68 ; y_{(a_{1},...,a_{p−1})}, 63 ; A, 54 ; A_{Q}(n), 87 ; A_{q}(n), A_{q}(n, r), 13 ; C_{ss}^{pre}, 19 ; C_{ss,l′}, 20 ; C_{ss}, 19 ; D_{Q}(n), D_{Q}(n, r), 88 ; D_{Q}^{i}(n, r), 88 ; E, 43 ; G_{q}(Σ_{n}), 68 ; H_{q}(Σ_{r}), 15

I, 59 I_{i}, 60 I_{(a_{0},a_{1},..,a_{p−1})}, 60 I_{res}, I_{unres}, I_{reg}, I_{sing}, 60 J_{i}, 60 J_{(a_{0},a_{1},..,a_{p−1})}, 60 K_{i}, 61 K_{(a_{0},...,a_{p−1})}, 61 L(a_{1}, ..., a_{p−1}), 66 L_{q}(a_{1}, ..., a_{p−1}), 68 M, 21 M_{w}, 22 N, 74, 82 N (r), 82 N [i], 96 N_{a_{1},...,a_{p−1},q}, 68 N_{a_{1},...,a_{p−1}}, 74 ON_{q}, 68 Q_{p}(n, r), 109 S(A), 86 S_{Q}(n), 87 S_{q}(n), S_{q}(n, r), 13

References

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