有限特殊線性群的特徵標

國立臺灣大學理學院數學系 碩士論文

Department of Mathematics College of Science

National Taiwan University Master Thesis

有限特殊線性群的特徵標

On characters of finite special linear groups in non-defining characteristic

陳鴻猷 Hung-You Chen

指導教授：于靖 博士 Advisor: Jing Yu, Ph.D.

中華民國 108 年 9 月 September 2019

致致致謝謝謝

首先我要感謝于靖教授。作為一個學生，毫無疑問我是不成熟且不合格的，但教 授從來沒有拋棄過我，僅論這點就令人難以回報。此外，教授無論是在課堂上、討

論會中，或是一般談話時，都展現了數學家的風格與思維模式，可以說教授本人就

是最良好的教材。最後，教授善於給予我們學習目標，並提供豐富的知識和各種論

文，成為最強大的後盾。能夠當上教授的學生，我感到非常榮幸。

其次我要感謝我的家人。他們時時刻刻在關心我，提供良好的環境，讓我在生活 上沒有後顧之憂，並在我遇到困難尋求幫忙時伸出援手。

最後感謝系辦各方面的支持，以及兩位口試委員的鼓勵。

中

中中文文文摘摘摘要要要

本文討論置換群與有限一般線性群的分解矩陣的各種性質，分析不同性質之間的 關係，並證明當特徵 p 不整除 q 時，有限特殊線性群亦擁有 (C, p)-性質。

關鍵詞：有限特殊線性群、模表現論、群論、分解矩陣、有限一般線性群

Abstract

In this thesis, we consider some properties of decomposition matrices of symmetric groups and finite general linear groups in non-defining characteristic, clarify the rela- tions among these properties, and show that SL_n(q) has an anologue property to S_n and GL_n(q) in non-defining characteristic, namely the (C, p)-property.

Keywords: finite special linear group, modular representation, group theory, decom- position matrix, finite general linear group

Contents

口 口

口試試試委委委員員員會會會審審審定定定書書書 i

致

致致謝謝謝 ii

中 中

中文文文摘摘摘要要要 iii

Abstract iv

0 Introduction 1

1 Preliminaries 6

1.1 Partitions . . . 6

1.2 Group Theory . . . 7

1.3 Conjugacy Classes in GL_n(q) . . . 9

1.4 Size of Conjugacy Classes of R_n . . . 12

1.5 Representations and Modules . . . 14

1.6 Characters . . . 17

1.7 The Decomposition Matrix . . . 20

1.8 Harish-Chandra Induction . . . 21

2 Representation Theory of GL_n(q) 22 2.1 Compositions, Tableaux and Permutations . . . 22

2.2 Subgroups . . . 24

2.3 Idempotents . . . 26

2.4 The Module MF(σ, (1)) . . . 27

2.5 The Module M_F(σ, (1^k)) . . . 31

2.6 The Module M_F(σ, λ), S_F(σ, λ) and D_F(σ, λ) . . . 34

2.7 The Module S_F(σ, λ) and D_F(σ, λ) . . . 39

3 Clifford Theory 42

3.1 Cyclic Quotient . . . 43

3.2 Direct Product . . . 45

3.3 Classification of Irreducible Representations of SLn . . . 46

3.4 G-tile and S-tile . . . 48

3.5 Representation Theory of G/S . . . 51

4 Field Theory 53 4.1 Basic Facts . . . 53

4.2 Elementary Number Theory . . . 54

4.3 Degree Extension Lemma . . . 56

4.4 Lemmas for Kleshchev-Tiep’s Theorem . . . 59

5 Kleshchev-Tiep’s Theorem 62 6 Main Theorem 68 6.1 The Canonical Composition Factor . . . 68

6.2 Main Results . . . 71

6.3 Relations Between Branching Numbers . . . 73

6.4 A Lower Unitriangular Submatrix . . . 74

7 Conclusion 77 A Appendix 79 A.1 The Original Problem . . . 79

A.2 The Implication Among Properties . . . 82

A.3 The Decomposition Matrix of GL₂ and SL₂ . . . 84

A.4 The Decomposition Matrix of Other Groups . . . 98

References 102

List of Tables

1 Decomposition matrix of S₆, p = 3 . . . 2

2 The property table for some families of groups . . . 77

3 Decomposition matrix of GL₂, p > 2, p | q − 1 . . . 90

4 Decomposition matrix of SL₂, p > 2, p | q − 1, q odd . . . 91

5 Decomposition matrix of SL₂, p > 2, p | q − 1, q even . . . 92

6 Decomposition matrix of GL₂, p > 2, p | q + 1 . . . 93

7 Decomposition matrix of SL₂, p > 2, p | q + 1, q odd . . . 94

8 Decomposition matrix of SL₂, p > 2, p | q + 1, q even . . . 95

9 Decomposition matrix of GL2, p = 2, q odd . . . 96

10 Decomposition matrix of SL₂, p = 2, q odd . . . 97

11 Decomposition matrix of GL₃(4), p = 3 . . . 99

12 Decomposition matrix of SL₃(4), p = 3 . . . 100

13 Decomposition matrix of A₆, p = 3 . . . 101

0 Introduction

One of the general problems of representation of finite groups is to describe the decom- position matrix of irreducible characters from the ordinary case to the p-modular case.

However, complete knowledge of such matrices is known only for few classes of groups, such as the symmetric groups and the general linear groups over finite fields. In this thesis, we are going to study properties of the decomposition matrices for the special linear groups, where most of the ingredients are from Kleshchev-Tiep [K].

Let Fq be the finite field of q elements. Let K = Q be the algebraic closure of the field of rational numbers, and F = Fp be the algebraic closure of the finite field of p elements, where p is a prime not dividing q.

Let G be a finite group. Denote irr_K(G) the set of irreducible ordinary characters, and irr_F(G) the set of irreducible Brauer (p-modular) characters. Let R⁺_K(G) be the set of all ordinary characters, and R_K(G) be the free Z-module generated by irrK(G), called the set of the virtual ordinary characters. Similarly, let R_F⁺(G) be the set of all Brauer characters, and R_F(G) be the free Z-module generated by irrF(G), called the set of the virtual Brauer characters.

For any finite generated KG-module V , with its character χV, we may take reduction modulo p to get a corresponding F G-module V . This process is not unique, but different reduction modulo p give the same Brauer character φ_V (cf. [S, Theorem 32].) Hence the map between characters χ_V 7→ φ_V is well-defined, and can be extended to a group homomorphism d : R_K(G) → R_F(G), with bases irr_K(G) and irr_F(G), respectively.

Since for any prime p we have | irr_F(G)| ≤ | irr_K(G)| < ∞, we may write d into a matrix with respect to the bases and take its transpose, called the decomposition matrix of G. Each row of the matrix describes how an irreducible ordinary character decomposes into irreducible Brauer characters when passing from R_K(G) to R_F(G). For general group G, it is known that the map d is surjective [S, Theorem 33]. We know some finer properties of the map d for specific groups, like the family of symmetric

groups or finite general linear groups.

For example, Table 1 of [J1] is the decomposition matrix of G = S₆, the symmetric group of degree 6, for K = Q and F = F3 (p = 3). The index of rows and columns are partitions, which serves as labels for the irreducible ordinary characters of S6 over K and irreducible Brauer characters of S₆ over F respectively. The second row of the decomposition matrix means that the irreducible ordinary character χ_(5,1) maps to the Brauer character φ₍₆₎+ φ_(5,1).

(6) (5,1) (4,2) (32 ) (4,12 ) (3,2,1) (22 ,12 )

(6) 1 (5, 1) 1 1

(4, 2) 1

(3²) 1 1

(4, 1²) 1 1

(3, 2, 1) 1 1 1 1 1

(2², 1²) 1

(2³) 1 1

(3, 1³) 1 1

(2, 1⁴) 1 1

(1⁶) 1

Table 1: Decomposition matrix of S₆, p = 3

Observe that the upper square of this decomposition matrix is a lower unitriangular submatrix. This can be deduced from James’ Regularization Theorem for symmetric groups, see [J3, Theorem A]. With appropriate chosen order of labels, the decomposition matrix of the symmetric groups have the following properties:

• In each row, there exists an entry 1.

• In each row, the rightmost nonzero entry is 1, written in bold.

• In each column, there exists a bold 1.

These properties mainly come from the fact that the irreducible characters (and conjugacy classes) of the symmetric groups have a good way of labeling, via partitions.

When considering other related groups, like alternating groups and finite general linear groups, some of these properties remain hold, while some others do not. In this thesis, we concern about which of these nice properties hold for finite special linear groups of non-defining characteristic.

Fix an finite group G, a prime p, K = Q, F = Fp, and the corresponding decomposi- tion map d. For an ordinary character χ of G, write χ := d(χ). For a Brauer character φ of G, φ is said to be liftable if there exists some ordinary character χ of G satisfying χ = φ, and φ is almost liftable if there exists some ordinary character χ of G satisfying χ = aφ for some a ∈ N.

We say G has (R, p)-property, if property (R) (defined below) holds for G for prime p.

We use this terminology throughout the thesis, for properties listed in the introduction.

(R) All irreducible Brauer characters of G are liftable.

(QR) All irreducible Brauer characters of G are almost liftable.

Clearly property (R) implies property (QR). We are interesting about the following problem, which originally comes from an exercise of Serre’s (see Appendix, section A.1, for detail story.)

Problem 1. Find a finite group G and a prime p, such that G has (QR, p)-property, but not (R, p)-property.

This leads to the definition of the (L, p)-property.

(L) If G has (QR, p)-property, then G has (R, p)-property. That is, if every irreducible Brauer characters are almost liftable, then they are actually all liftable.

Note that if G does not have (QR, p)-property at all, then G automatically has (L, p)- property. Hence G is a solution to Problem 1 for some p, if and only if (L, p)-property fails for G. Therefore, to answer Problem 1, we move to the study of the (L, p)-property, starting from some of the common families of finite groups. Actually, (L, p)-property is a rather weak property, and often proven as a consequence of other stronger property.

By considering each irreducible ordinary character, we may strengthen (L) to properties

below.

(L⁰) For any χ ∈ irr_K(G), χ is either irreducible, or a sum of at least two distinct Brauer characters. In other words, if χ = aφ for some φ ∈ irr_F(G) and a ∈ N, then a = 1.

(L⁰⁰) For any χ ∈ irr_K(G), χ contains some φ ∈ irr_F(G) of multiplicity 1.

(C) There exists a partial order D on irrF(G), and a map irr_K(G) → irr_F(G), χ 7→ φ_χ, such that for each χ ∈ irr_K(G), χ contains φ_χ of multiplicity 1, and if χ contains φ ∈ irr_F(G), then φ D φχ.

It is clear that (C) =⇒ (L⁰⁰) =⇒ (L⁰) =⇒ (L). For example, James’ Regulariza- tion Theorem shows that the property (C) holds for symmetric groups for any prime p. Huang [H] proves that the property (L⁰) holds for alternating groups for any prime p , while (L⁰⁰) and (C) remains unknown.

We also have (R) =⇒ (L). The Fong-Swan Theorem [S, Theorem 38] shows that for a prime p, property (R) holds for all p-solvable groups, thus these groups have (L, p)- property as well, while any non-abelian p-group is a counterexample of (L⁰, p)-property.

There is a property (U ), looks similar to (C), considering each irreducible Brauer character instead. With suitable order of the bases of the decomposition matrix, we may find a lower unitriangular submatrix.

(U ) There exists a partial order D on irr^F(G), and a map irrF(G) → irrK(G), φ 7→ χφ, such that for each φ ∈ irr_F(G), χ_φcontains φ of multiplicity 1, and if χ_φcontains φ⁰ ∈ irr_F(G), then φ⁰D φ.

We have (R) =⇒ (U ), but (U ) does not imply (L) (theoretically.) There are no other trivial implication among these properties (See Appendix, section A.2, for details.) Kleshchev-Tiep [K, Proposition 6.3] proves that (U ) holds for both finite general and special linear groups in non-defining characteristic, but this is not enough to deduce property (L). Nevertheless, Kleshchev’s paper gives strong tools and rich ideas for analyzing properties of decomposition matrix of finite special linear groups, so we may

achieve our goal easily.

The main result of this thesis is to prove that the property (C) holds for finite special linear groups in non-defining characteristic (Theorem 6.8), and hence implies (L⁰⁰), (L⁰) and (L). In section 1.3, we start from the conjugacy classes of GLn(q), labeled as [(σ, λ)]. Then in section 2, we introduce L_F(σ, λ), modules of GL_n(q) over field F = K or F , which build up a complete set of non-isomorphic irreducible FGL_n(q)-modules.

Next in section 3, we deduce some important lemmas from Clifford’s Theorem. Finally we prove the main result of this thesis in section 6. The proof is in fact independent of Kleshchev-Tiep’s theorem, the main theorem in [K], which is proved in section 5 for completeness, with the lemmas in part of section 3 and full of section 4. We make a table showing which property holds for which groups in Conclusion, section 7, and some other results in Appendix, section A.

1 Preliminaries

Let Fq be the finite field with q = p^f₀ elements, and Fq be the algebraic closure of F^q. Fixed a prime p not dividing q. In this thesis, K and F will always be a field of characteristic 0 and p > 0, respectively. If both K and F work for some result, we will put in the statement F = K or F .

1.1 Partitions

Given k ∈ Z^≥0, a partition λ ` k is a integer sequence (λ₁, λ₂, · · · ), where λ₁ ≥ λ₂ ≥

· · · ≥ 0 and k =P

iλ_i. For simplicity we may omit zeros and write in a compact form, e.g., (4, 2², 1³) instead of (4, 2, 2, 1, 1, 1, 0, · · · ). Let r_i := #{j ∈ N | λj = i}, then we write λ into (1^r1, 2^r2, · · · ) for expression of r_i.

Let λ, µ ` n and d ∈ N.

|λ| means λ₁+ λ₂+ · · · , that is, k;

λ + µ is (λ1+ µ1, λ2+ µ2, · · · ).

dλ is (dλ₁, dλ₂, · · · ).

λ⁰ is the transpose of λ, that is, λ⁰_i = #{j ∈ N | λj ≥ i}.

λ ^[+]µ = (λ⁰+ µ⁰)⁰, which combine and rearrange entries of λ and µ.

[d]λ = (dλ⁰)⁰, which combine and rearrange entries of d copies of λ.

λ D µ is the dominance order, Pi

j=1λ_j ≥Pi

j=1µ_j for every i.

∆(λ) is the greatest common divisor of λ_i. Let k be an a-tuple (k₁, · · · , k_a), k_i ∈ Z^≥0.

λ ` k is the multipartition (λ<sup>(1)</sup>, · · · , λ<sup>(a)</sup>), where λ<sup>(i)</sup> ` k_i; λ⁰ is (λ⁽¹⁾⁰, · · · , λ^(a)0);

λ D ν if λ⁽ⁱ⁾D ν⁽ⁱ⁾ for all i;

∆(λ) is the greatest common divisor of ∆(λ⁽ⁱ⁾).

Lemma 1.1. Let α⁽ⁱ⁾, β⁽ⁱ⁾ ` n_i with α⁽ⁱ⁾D β⁽ⁱ⁾ for i = 1, · · · , m.

(1) α⁽¹⁾+ · · · + α^(m)D β⁽¹⁾+ · · · + β^(m).

(2) If α⁽¹⁾+ · · · + α^(m) = β⁽¹⁾+ · · · + β^(m) then α⁽ⁱ⁾ = β⁽ⁱ⁾ for all i.

(3) α^{(1) [+]}· · ·^[+]α^(m)D β^{(1) [+]}· · ·^[+]β^(m).

(4) If α^{(1) [+]}· · ·^[+]α^(m) = β^{(1) [+]}· · ·^[+]β^(m) then α⁽ⁱ⁾ = β⁽ⁱ⁾ for all i.

Proof. By definition of the dominance order, Ps

j=1α⁽ⁱ⁾_j ≥ Ps

j=1β_j⁽ⁱ⁾ (∗) for any s ∈ N and i = 1, · · · , m. Thus Ps

j=1

Pm

i=1α⁽ⁱ⁾_j ≥ Ps j=1

Pm

i=1β_j⁽ⁱ⁾ (∗∗) and (1) holds. For (2), since the inequality of (∗∗) is actually equality, then (∗) as well and (2) follows. To prove (3), we start from α⁽ⁱ⁾D β⁽ⁱ⁾ and (α⁽ⁱ⁾)⁰E (β⁽ⁱ⁾)⁰, then by (1) Pm

i=1(α⁽ⁱ⁾)⁰EPm

i=1(β⁽ⁱ⁾)⁰ and hence^[+]m_i=1α⁽ⁱ⁾D^[+]mi=1β⁽ⁱ⁾. A similar argument to (2) gives (4).

1.2 Group Theory

Let G be any finite group, g ∈ G an element.

1_G or e is the identity element of G;

|G| is the cardinality of G;

|g| is the order of g, i.e. the smallest m ∈ N such that g^m = 1G; H ≤ G means H is a subgroup of G.

H E G means H is a normal subgroup of G.

O_p⁰(G) = {g ∈ G | |g| is prime to p}, the p⁰-part of G;

O_p(G) = {g ∈ G | |g| is a p-power}, the p-part of G;

Conventionally, when F = K, set O_p⁰(G) = G and O_p(G) = {1_G}.

GL_n denotes GL_n(Fq) or GL_n(q), if q is clear;

SL_n denotes SL_n(Fq) or SL_n(q);

R_n satisfies SL_n≤ R_n ≤ GL_n and R_n/SL_n = O_p⁰(GL_n/SL_n);

T_n satisfies SL_n≤ T_n ≤ GL_n and T_n/SL_n = O_p(GL_n/SL_n);

S_n denotes the symmetric group of degree n;

A_n denotes the alternating group of degree n.

Definition 1.2.

(1) A element g ∈ G is a p⁰-element if g ∈ O_p⁰(G), and is a p-element if g ∈ O_p(G). If F = K, then every g is a p⁰-element.

(2) A element g ∈ G is p-regular exactly if it is a p⁰-element, and is p-singular if it is not p-regular.

Proposition 1.3. For any g ∈ G, there is some p⁰-element g⁰ ∈ G and p-element g_p ∈ G, such that g = g⁰g_p. This decomposition is unique, both g⁰ and g_p is a power of g, and g⁰g_p = g_pg⁰.

Proof. Let |g| = p^cm with p6 | m. Then there is some a, b ∈ Z such that ap^c+ bm = 1.

Take g⁰ = g^apc and g_p = g^bm. Since the only element which is a p-element and a p⁰-element is 1_G, the uniqueness follows. The other statements are clear.

Definition 1.4.

(1) Given g ∈ G finite group and a prime p, the p⁰-part and p-part of g are g⁰, gp in the previous proposition, denoted by (g)_p⁰ and (g)_p, respectively. If F = K, set (g)_p⁰ = g and (g)_p = 1_G.

(2) Given r ∈ N and a prime p, write r = p^cm for non-negative integer c, m, p not dividing m. Then the p⁰-factor and p-factor of r are m, p^c, denoted by |r|_p⁰ and

|r|_p, respectively. If F = K, set |r|_p⁰ = r and |r|_p = 1. Do not confuse with p-adic norm, which does not appear in this thesis.

We emphasize that F^×q is the multiplication group of F^q, and we usually apply Defi- nition 1.2, 1.4 and Proposition 1.3 to the elements of F^×q or F^×_q^d.

1.3 Conjugacy Classes in GL

( q )

Given σ ∈ F^×q, deg(σ) = d, let B = B(σ) ∈ GL_d(q) be the companion matrix of minimal polynomial of σ over Fq. Then the corresponding Jordan block of size r is of the form

J_σ(r) =













 σ 1

σ 1 σ

. .. 1 σ















J_B(r) =















B I

B I

B

. .. I B















where Jσ(r) ∈ GLr(q^d) and JB(r) ∈ GLrd(q). With partition λ = (λ1, . . . , λm) given, we let J_σ(λ) = diag(J_σ(λ₁), · · · , J_σ(λ_m)) and J_B(λ) similarly.

For g ∈ GL_n(q), write the characteristic polynomial of g as f_g = f₁^k1· · · f_a^ka for some monic irreducible f_i. Then the Jordan canonical form of g over F^×q is

Jg = diag(Jσ1,1(λ⁽¹⁾), · · · , Jσ_1,d1(λ⁽¹⁾), · · · , Jσa,1(λ^(a)), · · · , Jσ_a,da(λ^(a)))

where σ_i,ji ∈ F^×q are roots of f_i for each j_i = 1, · · · , d_i with d_i = deg(f_i), and each λ⁽ⁱ⁾ ` k_i. And the rational canonical form of g over Fq is

R_g = diag(J_B1(λ⁽¹⁾), · · · , J_Ba(λ^(a)))

where B_i the companion matrix of f_i, λ⁽ⁱ⁾ ` k_i.

Definition 1.5. Let Mn(F^q) be the set of all n × n matrices over F^q.

(1) s ∈ Mn(F^q) is semisimple if it has an eigenbasis in (F^q)ⁿ. Equivalently, there is an x ∈ GL_n(Fq) such that xsx⁻¹ is a diagonal matrix.

(2) u ∈ M_n(Fq) is unipotent if (u − 1)^m = O for some m ∈ N. Equivalently, there is an x ∈ GL_n(Fq) such that xux⁻¹ is an upper unitrianglar matrix.

By Jordan decomposition [Sp1], every g ∈ GL_n(q) has a unique decomposition su, where s ∈ GL_n(q) is semisimple and u ∈ GL_n(q) is unipotent.

We need to construct an appropriate complete set of representatives for conjugacy classes in GLn(q).

Given σ ∈ F^×q, let [σ] be the set of all roots of the minimal polynomial of σ. Then σ₁ and σ₂ are (Galois) conjugate if and only if [σ₁] = [σ₂].

Definition 1.6. Let σ = (σ₁, · · · , σ_a), τ = (τ₁, · · · , τ_a) ∈ (F^×q)^a. In the following, for all i means for each i = 1, · · · , a.

(1) σ is p-regular if every σ_i is p-regular as a group element of F^×_qdi. (2) σ is non-repeated if for all i, [σ_i] are all different.

(3) σ is p-non-repeated if for all i, [(σ_i)_p⁰] are all different.

(4) σ and τ are (Galois) conjugate if deg(σi) = deg(τi) and [σi] = [τi] for all i.

(5) σ and τ are p-conjugate if deg(σ_i) = deg(τ_i) and [(σ_i)_p⁰] = [(τ_i)_p⁰] for all i.

Given σ ∈ F^×q, deg(σ) = d over Fq, then {1, σ, · · · , σ_d−1} is a Fq-basis of Fq(σ) = Fq^d, which produces a algebra embedding φ^σ : Fq^d → M_d(Fq) by φ^σ(σ) = B(σ), then restricts to a group embedding ι^σ : F^×_q^d → GL_d(q). Similarly, this produces a matrix algebra embedding φ^σ_k : M_k(Fq^d) → M_kd(Fq) by φ^σ_k(σE_ij) = B(σ) ⊗ E_ij, where ⊗ is the Kronecer product of matrices, and E_ij is the k × k matrix with (i, j)-entry 1 and other entries 0, and then restricts to a group embedding ι^σ_k : GLk(q^d) → GLkd(q). Note that ι^σ(σ) = B(σ), ι^σ_k(J_σ(k)) = J_B(k) and ι^σ_k(J₁(k)) = J_Id(k).

Let k = (k₁, · · · , k_a) ∈ N^a and λ ` k. Then π ∈ S_a acts naturally on each a-tuple, such as σ, k and λ. Write the action on the right.

Definition 1.7. Let σ = (σ₁, · · · , σ_a) ∈ (F^×q)^a, with deg(σ_i) = d_i for each i = 1, · · · , a.

Let λ = (λ⁽¹⁾, · · · , λ^(a)) ` k = (k₁, · · · , k_a) ∈ N^a.

(1) An n-admissible pair with a pairs is of the form (σ, λ), where both σ and λ are a-tuple for some a ∈ N, σ is non-repeated, and n =Pa

i=1kidi.

(2) We say (σ, λ) and (τ , ν) are equivalent if there exists some π ∈ S_a such that σ and τ π are conjugate, and λ = νπ. The equivalence class [(σ, λ)] is called an n-admissible symbol.

(3) We may write pairwisely the n-admissible pair and symbol with ◦ product, (σ, λ) = (σ₁, λ⁽¹⁾) ◦ · · · ◦ (σ_a, λ^(a))

[(σ, λ)] =([σ₁], λ⁽¹⁾) ◦ · · · ◦ ([σ_a], λ^(a))

(4) The dominance order of (multi)-partition naturally induces the partial order on n-admissible symbol. Denote [(σ, λ)] D [(τ , ν)] if there exists some π ∈ S^a such that σ and τ π are conjugate and λ D νπ.

(5) If (σ, λ) is an n-admissible pair, then we associate an element g := su ∈ GL_n(q), where s is semisimple and u is unipotent,

s = s(σ, k) := diag

(ι^σ1(σ₁))^(k1), · · · , (ι^σa(σ_a))^(ka)

= diag



B₁^(k1), · · · , B^(k_a^a)

 u = u(λ, k) := diag ι^σ_k¹

1 J₁(λ⁽¹⁾)
, · · · , ι^σ_k^a

a J₁(λ^(a))

= diag



JI_d1(λ⁽¹⁾), · · · , JI_da(λ^(a))



where B_i is the companion matrix of the minimal polynomial of σ_i over Fq. Note that diag(B^(k)) means k copies of B on diagonal, not k power of B.

Then it is not hard to see that Proposition 1.8.

(1) (σ, λ) and (τ , ν) are equivalent if and only if their corresponding su are conjugate to each other.

(2) The set Σ_K := { [(σ, λ)] | (σ, λ) is an n-admissible pair } is the complete set of representatives of conjugacy classes of GL_n(q).

(3) The set Σ_F := { [(σ, λ)] | (σ, λ) is an n-admissible pair, σ is p-regular } is the com- plete set of representatives of p-regular conjugacy classes of GL_n(q).

Proof. (1) Note that π means changing the order of blocks of s and u, and if σ₁ is conjugate to τ1 over F^q, they produce the same companion matrix of their common minimal polynomial.

(2) We show that every g ∈ GL_n(q) is conjugate to some s(σ, k)u(λ, k). The rational canonical form of g is

R_g = diag(J_B1(λ⁽¹⁾), · · · , J_Ba(λ^(a)))

so it suffices to prove the case R_g = J_B(k) for some d × d matirx B = B(σ) and k ∈ N. Let S = diag(B^(k)), U = J_Id(k), N = U − I_d, then R_g = S + N . Now take D = diag(Id, B, · · · , B^k−1), then we have D⁻¹SD = S and D⁻¹N D = SN . Hence D⁻¹R_gD = S + SN = SU and we are done.

(3) Since u is always p-regular (p does not divide q), su is p-regular if and only if s is p-regular, which is equivalent to σ is p-regular.

Some other properties of the n-admissible symbols are put in chapter 4.

Given su = s(σ, k)u(λ, k), if g ∈ GL_n(q) centralize su, then it centralize both s and u by the uniqueness of the decomposition. That is, the centralizer

C_GLn(q)(su) = C_GLn(q)(s) ∩ C_GLn(q)(u) The centralizer of s is,

C_GLn(q)(s) = (ι^σ_k¹

1 × · · · × ι^σ_k¹

1) GL_k1(q^d1) × · · · × GL_ka(q^da)

The centralizer of u is more complicated. Nevertheless, by [Sp2, I, 2.2], the size of the centralizer of u is,

C_GLn(q)(u)

= q^NY

i≥1

|GL_ri(q)| (1)

where N = N (λ) is defined as follows. Write λ =^[+]a_i=1[d_i]λ⁽ⁱ⁾ into (1^r1, 2^r2, · · · ). Then N =P

i≥1(λ⁰_i)²− r_i². Note that λ⁰_i =P

j≥ir_j, hence N ≥ 0.

1.4 Size of Conjugacy Classes of R

Recall that Rn is a subgroup of GLn satisfying SLn ≤ Rn ≤ GLn and Rn/SLn = Op⁰(GLn/SLn). In the later proof, we need to find the size of conjugacy classes of Rn.

For g ∈ G, let Conj_G(g) be the conjugacy class of g in G. The following is the general lemma we use.

Lemma 1.9. Let S E G, G/S cyclic, and S ≤ R ≤ G. For any g ∈ R, let c = (G : CG(g)S), d = (G : R). Then | Conj_G(g)|/| Conj_R(g)| = gcd(c, d).

Proof. Denote C = C_G(g) and D = C_R(g) = C ∩ R. Then

| Conj_G(g)|/| Conj_R(g)| = (G : C)/(R : D)

The key step is to drag C, D to CS, DS, where we can count their index in G. Now C ∩ S = C ∩ R ∩ S = D ∩ S, thus |DS|/|CS| = |D|/|C|. Consider π : G → G/S = hxi, then π(CS) = hx^ci, π(R) = hx^di, and π(CS ∩ R) = hx^lcm(c,d)i. Since CS ∩ R = DS, we have (G : DS) = lcm(c, d). Therefore

| Conj_G(g)|

| Conj_R(g)| = |G|

|C|

|D|

|R| = |G|

|CS|

|DS|

|G|

|G|

|R| = c · d

lcm(c, d) = gcd(c, d)

Lemma 1.10. Let u = J₁(λ) be a Jordan block of GL_k(q), λ ` k. Then det maps C_GLk(q)(u) onto hε^∆(λ)i ≤ F^×q, where ε is a generator of F^×q.

Proof. Denote C = C_GLn(q)(u) and write λ into (1^r1, 2^r2, · · · ). Then u is similar to L

i:ri>0Iri⊗ J1(i), hence D =L

i:ri>0GLri(q) ⊗ Ii commutes with u and is a subgroup of C. Consider P₀ a Sylow p₀-subgroup of C. By equation (1), (C : D) = q^N, thus we have C = P₀D. Now for any element a ∈ P₀, a^qr = I_n for some r ∈ N, while det^q(a) = det(a) since it’s an element of F^×q. Hence det(P₀) = {1}, det(C) = det(D), and det(GL_ri(q) ⊗ I_i) = hεⁱi gives det(D) = hε^∆(λ)i, as desired.

Proposition 1.11. Let g = su = s(σ, k)u(λ, k) be a representative of a p-regular conjugacy class in GLn(q) corresponding to [(σ, λ)] ∈ ΣF. Then g ∈ Rn and

| Conj_GLn(g)|

| Conj_Rn(g)| = gcd{(GLn: Rn), ∆(λ)}

Proof. Let π⁰ : GLn → GLn/Rn. Note that GLn/Rn is a p-power, thus π⁰(g) must be identity, hence g ∈ Rn.

Denote C = C_GLn(q)(g). To apply Lemma 1.9, we need to find the index c = (GL_n : C · SL_n), which leads to finding det(C · SL_n) = det(C). Then

C = C_GLn(q)(su) = C_GLn(q)(s) ∩ C_GLn(q)(u) = C_CGLn(q)(s)(u)

=

a

Y

i=1

C_ισi

ki(GL_k1(q^d1))(ι^σ_kⁱ

i(J₁(λ⁽ⁱ⁾))) =

a

Y

i=1

ι^σ_kⁱ

i

 C_GL

k1(qd1)J₁(λ⁽ⁱ⁾) Now for k, d ∈ N, σ ∈ F^×q, deg(σ) = d, consider the following commute diagram

GL_k(q^d) GL_kd(q)

F^×_q^d F^×q ι^σ_k

det det

N_F

qd/ Fq

For each d, take a generator εd ∈ F^×_q^d such that ε = N_F

qd/ Fq(εd). Then det(C) =

a

Y

i=1

det ◦ ι^σ_kⁱ

i

 C_GL

k1(q^d1)J₁(λ⁽ⁱ⁾)

=

a

Y

i=1

N_F

qdi/ Fq ◦ det C_GL

k1(q^d1)J₁(λ⁽ⁱ⁾)

=

a

Y

i=1

N_F

qdi/ Fq

hε^∆(λ_d ⁽ⁱ⁾⁾i

=

a

Y

i=1

hε^∆(λ(i))i = hε^∆(λ)i

Hence c = ∆(λ). Applying Lemma 1.9 with d = (GL_n: R_n) yields the result.

1.5 Representations and Modules

Let G be a finite group, R a commutative ring with 1, and V an R-module. We call (V, ρ) a representation of G over R if ρ : G → GL(V ) is a group homomorphism, where GL(V ) is the group of R-module automorphisms of V . In this thesis R will always be a field F, and V is a finite vector space over F.

Given a representation ρ : G → GL(V ), it can be extended to an FG-module, also denoted V . In contrast, given an FG-module V , it defines a representation ρ of G in V over F. We will also call V a representation, although it is the underlying module of ρ.

Definition 1.12. Let V be an FG-module.

(1) An FG-module W is a submodule (subrepresentation) of V , denoted W ⊂ V , if W ⊂ V as vector space over F, and stable under the action of G.

(2) A submodule W of V is a (direct) summand of V , denoted W | V , if there is another submodule W⁰ of V such that V = W ⊕ W⁰ as vector space.

(3) V is an irreducible representation, or a simple FG-module, if {0} and V are the only submodules of V .

(4) Denote Irr_F(G) to be the set of all isomorphic types of non-isomorphic irreducible representations of G over F.

(5) V is a trivial representation if V ∼= F and ρ acts trivially, denoted idG.

(6) V is semi-simple or complete reducible, if every submodule of V is a summand of V . Hence V =L W_i for some irreducible submodules W_i of V .

(7) The dual of V , denoted V^∗, is the F-module HomF(V, F), equipped with the action of G, (gf )(v) = f (g⁻¹v), becoming an FG-module. It is known that Hom_F(V₁, V₂) ∼= V₁^∗⊗ V₂.

Let H ≤ G. We may construct some FH-module from an FG-module, or vise versa.

Definition 1.13. Let V be an FG-module, and W be an FH-module.

(1) The restriction of V from G to H, denoted as Res^G_H(V ) or V ↓^G_H, simply restrict the action of FG to FH. If G is clear, we may also write V ↓_H for simplicity.

(2) The induction of W from H to G, denoted as Ind^G_H(W ) or W ↑^G_H, is defined to be FG ⊗_FH W . If H is clear, we may also write W ↑^G for simplicity.

(3) Assume F = K or F = F with p6 | |G|. For FG-modules V₁, V₂, define hV₁, V₂i_G = dim_FHom_FG(V₁, V₂). Similarly for hW₁, W₂i_H.

(4) For g ∈ G, let^gH := gHg⁻¹. Then^gW := g ⊗ W is naturally an F(^gH)-module.

(5) The kernel of V , denoted ker V , is the unique maximal subgroup H ≤ G such that H acts trivially on the FH-module V ↓_H. It is known that ker V E G.

The basic properties of restriction and induction is on, for example, [F, II], so we omit them here. We only list some important theorems here.

Theorem 1.14. Let V be an FG-module, and W be an FH-module.

(1) (Frobenius Reciprocity) Assume F = K or F = F with p6 | |G|.

hV, W ↑^Gi_G = hV ↓_H, W i_H

hW ↑^G, V i_G = hW, V ↓_Hi_H (2) (Mackey Decomposition) Let A ≤ G. Then

(W ↑^G_H)↓^G_A∼=M

x

x(W ↓^G_H∩Ax)
↑^Ax_H∩A

summing over the complete set of double coset representative x ∈ [A\G/H].

Let S E G. We may construct some F(G/S)-module from an FG-module, or vise versa.

Definition 1.15. Let V be a FG-module, and W be a F(G/S)-module.

(1) The S-fixed point of V , denoted as V^S, is the abelian group {v ∈ V | sv = v for all s ∈ S} equipped with the action of G, which can be viewed as an F(G/S)- module.

(2) The inflation of W , denoted as infl^G_G/S(W ), has the same underlying space W , equipped with the action g · w = π(g)w for any g ∈ G, w ∈ W and the canonial homomorphism π : G → G/S.

Their basic properties is on, for example, [GR, Chap 4]. It is known that the fixed point construction is adjoint to inflation, so they has an analogue to the Frobenius reciprocity. We list the following relations here for later usage.

Proposition 1.16. Let S E G, A any other subgroup of G.

(1) (Restriction commutes with inflation) Let V be an F(G/S)-module. Then

infl^A_A/A∩S(V ↓^G/S_AS/S) ∼= (infl^G_G/S(V ))↓^G_A

Note that we identify A/A ∩ S ∼= AS/S.

(2) (Induction commutes with inflation) Assume additionally S E A. Let W be a F(A/S)-module. Then

infl^G_G/S(W ↑^G/S_A/S) ∼= (infl^A_A/S(W ))↑^G_A

1.6 Characters

Given K ⊂ Q, let OK be the ring of integer of K. Pick any prime p and any prime ideal p of OK containing p, there is a corresponding p-adic valuation νp. Take the ring A = {α ∈ K | ν_p(α) ≥ 0}, which has a unique maximal ideal m = {α ∈ K | ν_p(α) > 0}.

The residue field F = A/ m is of characteristic p. If K = Q is algebraically closed, then F = Fp is also algebraically closed.

Definition 1.17. The triple (K, A, F ) is called a p-modular system.

Let G be a finite group, ρ : G → GL(V ) an representation over K. Then the (ordinary) character χ_V : G → K of G corresponding to V , is defined to be χ_V(g) = Tr(ρ(g)), with ρ(g) : v 7→ ρ(g)v written as an invertible matrix with a chosen basis of V . It is clear that the definition of character does not depend on the basis, by the property of the trace.

The properties of characters can be founded in the textbook of Serre [S].

Definition 1.18.

(1) We say χ_V is irreducible if V is.

(2) Write irr_K(G) the set of all irreducible ordinary characters.

(3) A class function f : G → K, is a function satisfied f (xgx⁻¹) = f (g) for any g, x ∈ G.

By the property of the trace, it is clear that characters are class functions.

(4) For two characters χ, φ of G, define hχ, φi = |G|⁻¹P

g∈Gχ(g)φ(g⁻¹).

(5) Let R⁺_K(G) be the set of all characters of G over K, and R_K(G) be the additive group generated by the characters of G over K. The elements of R_K(G) are called the virtual characters.

Proposition 1.19. Let K = Q.

(1) There are only finite irreducible characters of G, written as χ₁, · · · , χ_h.

(2) R_K(G) is a Z-module with basis {χ1, · · · , χ_h}, and χ_i are mutually orthogonal with the inner product h·, ·i

(3) Every class functions are virtual characters, hence the set of all class functions coincides R_K(G). By considering that any class function is constant on a conjugacy class of G, the number of irreducible characters of G is exactly the same as the number of conjugacy classes of G.

(4) V and W are isomorphic representations of G if and only if χ_V = χ_W. (5) If V = Lh

i=1n_iW_i for W_i ∈ Irr_K(G), n_i means W_i appears n_i times. Then χ_V = Ph

i=1n_iχ_i, where χ_i are characters corresponding to W_i.

In general, if K is a field of characteristic 0 with algebraic closure K, one can view any KG-module V as a KG-module V_K by scalar extension, then define χ_V = χ_V

K. However, some irreducible KG-module cannot be realized over K, hence R_K(G) may be a proper subgroup of R_K(G). To ensure R_K(G) = R_K(G), a sufficient condition is that K contains the mth root of unity [S, Theorem 24], where m = lcm{|g| | g ∈ G}.

In this case we say K is sufficiently large for G. In other words, R_K(G) is independent of K as long as K is sufficiently large for G, and we may replace K = Q by any K sufficiently large for G in Proposition 1.19.

Now consider G_reg := {g ∈ G | g is p-regular}. Let (K, A, F ) be a p-modular system, with K, F sufficiently large for G_reg, and F = A/ m. Pick ζ a m⁰th root of unity of F , where m⁰ = lcm{|g| | g ∈ G_reg}, and ˜ζ a m⁰th root of unity of K which passing from K to F is ζ.

Let V be an F G-module of dimension n. For g ∈ G_reg, let ρ(g) : v 7→ gv for v ∈ V . Then ρ(g) is diagonalizable, and its eigenvalues t₁, · · · , t_n are all powers of ζ, hence

˜t1, · · · , ˜tn are corresponding powers of ˜ζ. Define φV(g) = Pn

i=1˜ti. Then φV : Greg → K is called the Brauer character of V .

The properties of Brauer characters are also in [S]. It shares many properties with ordinary characters, but without orthogonality.

Definition 1.20.

(1) We say φV is irreducible if V is.

(2) Write irr_F(G) the set of all irreducible ordinary characters.

(3) Let R⁺_F(G) be the set of all Brauer characters of G over F , and R_F(G) be the additive group generated by the Brauer characters of G over F . The elements of R_F(G) are called the virtual Brauer characters.

Proposition 1.21. Let F = Fp.

(1) If 0 → V₁ → V → V₂ → 0 is an exact sequence of F G-modules, then φ_V = φ_V1+φ_V2. (2) RF(G) is a Z-module with basis {φ¹, · · · , φh⁰}.

(3) The number of irreducible Brauer characters of G is exactly the same as the number of conjugacy classes of G_reg. Hence the number of irreducible Brauer characters is equal to or less than the number of irreducible ordinary characters.

(4) V and W have the same composition factors if and only if φ_V = φ_W.

(5) If for each W_i ∈ Irr_F(G), V has n_i composition factors isomorphic to W_i, then φ_V =Ph

i=1n_iφ_i, where φ_i are Brauer characters corresponding to W_i.

Note that in the case of characteristic 0, every KG-module is complete reducible.

Hence two modules have common composition factors are actually isomorphic.

We may say F is sufficiently large for G with the same definition to K, replacing F = Fp by any F sufficiently large for G in Proposition 1.21.

Given any ring R, let C be a category of R-modules. Then the Grothendieck group of C is the abelian group defined by the generators [V ] for any V ∈ C, and relations [V ] = [V₁] + [V₂] if 0 → V₁ → V → V₂ → 0 is an exact sequence.

Then for any field F = K or F sufficiently large over G, the Grothendieck group of FG-modules has a canonical group isomorphism to R_F(G) by [V ] 7→ χ_V or φ_V.

Hereafter, we identify R_F(G) with the Grothendieck group of FG-modules, and omit the bracket. That is, when we say V = V₁ + V₂ in the Grothendieck group of FG- modules, we actually means [V ] = [V₁] + [V₂], hence χ_V = χ_V1 + χ_V2 or φ_V = φ_V1+ φ_V2. We use this terminology because the name of the representation is often quite long (e.g.

L_K(σ, λ)), which is not suitable to write in character form.

1.7 The Decomposition Matrix

Let K ⊂ Q be a field of characteristic 0, sufficiently large with respect to G, and F = A/ m be the field of characteristic p defined in the first paragraph of section 1.6, so (K, A, F ) forms a p-modular system.

For a KG-module V , pick a lattice V₁, a finite generated A-submodule of V , gener- ating V as a K-module. Let V₂ be the sum of the image of V₁ under elements of G, hence V₂ is also a lattice of V which is stable under G. Define V = V₂/mV₂. Then V is an F G-module, called a reduction modulo p of V , written as V = V mod p (although it is actually mod m.)

The following proposition are from [S].

Proposition 1.22. Let V be an KG-module.

(1) V is not unique, but they share common composition factors, hence having the same Brauer characters φ_V.

(2) We have φ_V = χ_V|_Greg.

Hence the reduction modulo p of an ordinary (virtual) character χ may be defined as χ := χ|Greg, and the group homomorphism d : RK(G) → RF(G) defined by reduc- tion modulo p is well-defined, and send characters R⁺_K(G) to characters R⁺_F(G). The transpose of the matrix form of d is a | Irr_K(G)| × | Irr_F(G)| matrix with non-negative integer entries, called the decompostion matrix. Each row of the decompostion matrix shows how χ decompose into sum of Brauer characters.

A theorem [S] shows that d is surjective (so the decomposition matrix has full rank), but little else properties are known. Finding out the properties of decomposition matrix is a main objective in the study of representation theory.

We list here some properties of reduction modulo p.

Proposition 1.23. Reduction modulo p commutes with conjugation, restriction, induc- tion and inflation, in the sense of common composition factor.

Proof. These are all simple with character and using Proposition 1.22(2).

1.8 Harish-Chandra Induction

In this thesis, we are concerning about Harish-Chandra induction, a construction of FGL_n-module from FGL_r-module and FGL_s-module with n = r + s.

Let L, U, P ≤ GL_n be the group of matrices of the form:

L = g_r g_s



U =  I_r ∗ I_s



P = g_r ∗ g_s



with g_r ∈ GL_r, g_s ∈ GL_s, and I_r, I_sidentity. It is easy to check that L ∼= GLr×GL_s, U E P , and L ∼= P/U canonically.

Let W_r, W_s be FGL_r-module and FGL_s-module, respectively. Then W_r⊗ W_s gives an FL-module. The Harish-Chandra induction is defined as

W_r◦ W_s = infl^P_L(W_r⊗ W_s)
↑^GL_P ⁿ

Proposition 1.24. Let W∗ be FGL∗-module for ∗ = r, s, t.

(1) W_r◦ W_s∼= Ws◦ W_r.

(2) (W_r◦ W_s) ◦ W_t ∼= Ws◦ (W_r◦ W_t).

(3) W_r◦ W_s∼= Ws◦ W_r.

The commutativity and associativity of this Harish-Chandra induction can be easily extended to the case n =Pa

i=1n_i for any a ∈ N.

2 Representation Theory of GL_n

(

)

Here we follow [J] to gives a construction of irreducible representations of GL_n= GL_n(q) over F = K or F with characteristic not dividing q. F needs to contain p0-root of unity, and be sufficiently large for all the groups we have considered in this section.

2.1 Compositions, Tableaux and Permutations

Although we have defined patitions of k in section 1.1, it is natural to consider compo- sitions, an unordered version of partition, when talking about the FGL_n-modules.

A composition λ of a non-negative k, denoted λ |= k, is a non-negative integer sequence (λ₁, λ₂, · · · ) where P

iλ_i = k. A composition is a partition if λ₁ ≥ λ₂ ≥ · · · , written as λ ` k. The transpose λ⁰ := (λ⁰₁, λ⁰₂, · · · ) is defined as λ⁰_i = #{λ_j | λ_j ≥ i}.

Note that λ⁰ must be a partition, and λ⁰⁰ is the partition rearranging each terms of λ by order. If λ, µ |= k, the dominance order λ D µ is defined by that of partitions λ⁰E µ⁰, that is,Pj

i=1λ⁰_i ≤Pj

i=1µ⁰_i for all j ∈ N.

Fixed some r ∈ N, a partition λ ` k is r-singular if for some i, λi = λ_i+1 = · · · = λ_i+r−1> 0, otherwise it is r-regular. Hence every nonempty partition is 1-singular, and conventionally every partition is ∞-regular.

For λ |= k, a λ-tableau is a bijection from [λ] := {(i, j) | 1 ≤ i, 1 ≤ j ≤ λ_i} to {1, 2, · · · , k}. For a λ-tableau t, we may draw it by embedding [λ] into N × N, with x-axis point to south and y-axis point to east, and put the number on its corresponding coordinate. The following are examples of tableaux.

A (2², 3)-tableau t₁ A (4, 3, 2)-tableau t₂ A (4, 3, 2)-tableau t₃ 2 4

1 7 3 5 6

1 2 3 4 5 6 7 8 9

1 4 7 9 2 5 8 3 6

A tableau t is said to be row standard if in each row, the number increase as the y-coordinate increase. The t₁, t₂, t₃ above are all row standard.

Let t^λ be the unique λ-tableau, put the number in lexicographical order of the coor- dinate, comparing x-coordinate first. Similarly, let t^λw_λ be the unique λ-tableau, put the number in lexicographical order of the coordinate, comparing y-coordinate first.

For example, if λ = (4, 3, 2), then t^λ = t2 and t^λwλ = t3 above.

For λ |= k, let t be a λ-tableau, (i, j) ∈ [λ] a node of [λ], m ≤ k.

(i, j)t is the number corresponding to the node.

row_t(m) := i, if (i, j)t = m. That is, m is on the ith-row of t.

col_t(m) := j, if (i, j)t = m. That is, m is on the jth-column of t.

row_λ(m) := row_t^λ(m), an abbreviation for tableau t^λ.

For w ∈ S_k, let tw be the λ-tableau such that (i, j)(tw) = ((i, j)t)w. w_λ ∈ S_k is the permutation consistent with the notation t^λwλ above. For example, if λ = (4, 3, 2), then w_λ = (2 4 9 6 5)(3 7 8). Let R(t) := {w ∈ S_k | row_t(i) = row_t(iw) for all i}, the row stabilizer of t, and C(t) := {w ∈ S_k | col_t(i) = col_t(iw) for all i}, the column stabilizer of t.

Let d, k ∈ N and n = dk. If λ |= k, define dλ |= n by (dλ)i = d(λ_i).

For w ∈ S_k, define π_w ∈ S_n by

π_w : ad − b 7→ (aw)d − b, 1 ≤ a ≤ k, 0 ≤ b ≤ d − 1

That is, if we divide {1, 2, · · · , n} into k packs v1 = (1, 2, · · · , d), v2 = (d + 1, d + 2, · · · , 2d), · · · , v_k = ((k − 1)d + 1, (k − 1)d + 2, · · · , kd), then π_w permutes the index of {v_i} while not changing its order inside.

Let W_n ≤ GL_n(q) be the permutation group. We identify W_n with S_n via π 7→

(e_i 7→ e_iπ), where {e_i} is the standard basis of (Fq)ⁿ. Then W_k→n := {π_w | w ∈ S_k} is a subgroup of W_n.

For example, if λ = (4, 3, 2), w = (3 5 4 7 8), d = 2, then we have