ON COMPLEX REFLECTION GROUPS G(M,1,R)

ON COMPLEX REFLECTION

GROUPS G(m, 1, r)

AND THEIR HECKE ALGEBRAS

CHI KIN MAK

A thesis submitted in fulﬁlment Of the requirements for the degree of Doctor of Philosophy

School of Mathematics University of New South Wales

AUGUST, 2003 Abstract

We construct an algorithm for getting a reduced expression for any element in a complex reflection group G(m, 1, r) by sorting the element, which is in the form of a sequence of complex numbers, to the identity. Thus, the algorithm provides us a set of reduced expressions, one for each element. We establish a one-one correspondence between the set of all reduced expressions for an element and a set of certain sorting sequences which turn the element to the identity. In particular, this provides us with a combinatorial method to check whether an expression is reduced. We also prove analogues of the exchange condition and the strong exchange condition for elements in a G(m, 1, r). A Bruhat order on the groups is also defined and investigated. We generalize the Geck-Pfeiffer reducibility theorem for finite Coxeter groups to the groups G(m, 1, r). Based on this, we prove that a character value of any element in an Ariki-Koike algebra (the Hecke algebra of a G(m, 1, r)) can be determined by the character values of some special elements in the algebra. These special elements correspond to the reduced expressions, which are constructed by the algorithm, for some special conjugacy class representatives of minimal length, one in each class. Quasi-parabolic subgroups are introduced for investigating representations of Ariki- Koike algebras. We use n×n arrays of non-negative integer sequences to characterize double cosets of quasi-parabolic subgroups. We define an analogue of permutation modules, for Ariki-Koike algebras, corresponding to certain subgroups indexed by multicompositions. These subgroups are naturally corresponding, not necessarily one-one, to quasi-parabolic subgroups. We prove that each of these modules is free and has a basis indexed by right cosets of the corresponding quasi-parabolic subgroup. We also construct Murphy type bases, Specht series for these modules, and establish a Young’s rule in this case.

ii Acknowledgements

I thank my supervisor, Dr. Jie Du, for his time and patience. During the period of his supervision, his encouragement, positive remarks and constructive advice helped me struggle through difficult time and difficult points. I first came across the length formula 2.2.1 in [BM, 3.4] and I have used it in a published paper (see the appendix) to show that the Algorithm 2.1.7 outputs a reduced expression for an input element. I thank Professor Robert Proctor for pointing out that I can simply derive the length formula from the algorithm and prove that the product output is a reduced expression for the input element. Thanks to Dr. Andrew Francis for drawing my attention to the Geck-Pfeiffer reducibility theorem [GP1, Theorem 1.1] and the importance of studying the minimal length elements in conjugacy classes. I also thank Associate Professor Norman Wildberger and Dr. Hebing Rui for their time in listening to my presentations and their invaluable comments. Finally, I thank the examiners of my thesis for their precious comments and sug- gestions in improving the thesis.

iii Dedicated to Alvina, King and Kenny. Contents

Abstract ii

Acknowledgements iii

Introduction 1

Chapter 1. Symmetric Groups, Coxeter groups and their Hecke Algebras 5 1.1. Introduction 5 1.2. Symmetric Groups 6 1.3. Conjugacy Classes of Symmetric Groups 12 1.4. Young Subgroups 15 1.5. Iwahori-Hecke Algebras 19 1.6. Representations and Characters 21

Chapter 2. Complex Reﬂection Groups G(m, 1, r) 27 2.1. Characterizations of G(m, 1, r) 27 2.2. A Length Function and Reduced Expressions 37 2.3. Weak Exchange Condition 42 2.4. Bruhat Order 45

Chapter 3. Minimal Length Elements in Conjugacy Classes 50 3.1. Conjugacy Classes of G(m, 1, r) 50 3.2. Minimal Elements in Conjugacy Classes 55

Chapter 4. Characters of Ariki-Koike Algebras 67 4.1. Ariki-Koike Algebras 67 4.2. Characters of Ariki-Koike Algebras 70 v vi CONTENTS

Chapter 5. Quasi-Parabolic Subgroups 77 5.1. Quasi-parabolic Subgroups 77 5.2. Cosets of Quasi-parabolic Subgroups 79 5.3. Shortest and Longest Coset Representatives 81 5.4. Characterizing a Double Coset by an n × n Array 83

Chapter 6. q-Permutation Modules 86 6.1. Multicompositions and Multitableaux 87 6.2. q-Permutation Modules 91

6.3. Semistandard Basis of xλH 102

6.4. Specht Series and Young’s Rule for xλH 106

Chapter 7. Other Imprimitive Complex Reﬂection Groups 109 7.1. Presentation of G(m, p, r) 109 7.2. Reduced expressions for elements in G(2, 2, r) 111

Appendix. Published Paper – Quasi-Parabolic Subgroups of G(m, 1, r) 117

Bibliography 133 Introduction

We are interested in two well-known characterizations of a complex reﬂection group G(m, 1, r). The ﬁrst one is the presentation by generators and relations given in [She2]. Hence every element can be written as expressions in the generating set. The second one is a group of bijections. Suppose m, r are positive integers and ξ is a complex primitive mth root of unity. A generalized permutation is a bijection w on

{ξεa|0 6 ε 6 m − 1, 1 6 a 6 r}, where ε and a are integers such that the image of ξx, which is written as (ξx)w, is ξ(xw). The group of all generalized permutations is isomorphic to G(m, 1, r). Obviously, this group of bijections is also isomorphic to the group of r × r monomial matrices over the complex mth root of unity. An element in G(m, 1, r) is determined by the images on r = {1, . . . , r} only. Based on this, Read has introduced the two-line notation for the elements (see [Re]). Equivalently, if we just take the sequence of images on r in order, we have the one-line notation. An elementary but important observation is that the multiplication of a generator to the left of an element in one-line notation has a very strong combinatorial nature. It swaps terms or changes the exponent of ξ of the ﬁrst term. In the thesis, we make use of the one-line or two-line notations to get some results on the complex reﬂection groups and their Hecke algebras. Most of the results are analogues of those on symmetric groups. Reduced expressions are important to the study of Hecke algebras. We develop an algorithm for getting a reduced expression for any element based on sorting the one-line form of the element to the identity. The reduced expression is of the form d1d2 ··· dr, where dk is the distinguished right G(m, 1, k − 1)-coset in G(m, 1, k) and 1 2 INTRODUCTION

G(m, 1, 0) is the trivial group. We call this expression the distinguished reduced expression for the element. We also establish a one-one correspondence between the set of reduced expressions for an element with the set of length reduction sorts which turn the element to the identity. We have a simple combinatorial way to determine whether a sorting sequence is a length reduction sort. This provides us with an easy method to check whether an expression is reduced. Besides reduced expressions, we also study expressions in general. In Coxeter groups, elements obey the theorems of the exchange condition, deletion condition and the strong exchange condition [H, 1.7, 5.8]. We prove weaker analogues of the exchange conditions. The Bruhat order on a Coxeter group has a few characterizations [De, 1.1]. We attempt to study two of them. A character of an Iwahori-Hecke algebra (the Hecke algebra of a Coxeter group) is not constant on the set of elements corresponding to a conjugacy class in the corresponding Coxeter group. Starkey (see [Car, p.95]) and Ram (see [Ra]) have shown 0 independently that character values of Tw and Tw0 are equal if w and w are minimal length elements in the same conjugacy class in a type A Coxeter group. In [GP1], Geck and Pfeiﬀer have proved this for an Iwahori-Hecke algebra, in general, based on a reducibility theorem [GP1, 1.1]. Using the results in [GP1], Geck and Rouquier have constructed integral bases for centres of Iwahori-Hecke algebras in terms of irreducible characters of the algebras [GR]. Based on the reducibility theorem, Francis has constructed independently the same integral bases for centres of Iwahori-Hecke algebras in [F1, F2] by a purely combinatorial method. We generalize the reducibility theorem to one for G(m, 1, r). Based on this, we show that the character value of any element in an Ariki-Koike algebra (the Hecke algebra of a G(m, 1, r)) can be determined by the character values of elements corresponding to the distinguished reduced expressions for a complete set of minimal length conjugacy class representatives. We hope that in the future, we can use the reducibility theorem to construct integral bases for the centres of Ariki-Koike algebras similar to Francis’ construction. INTRODUCTION 3

Du and Scott have defined quasi-parabolic subgroups of type B Coxeter groups for the study of q-Schur2 algebras [DS]. The definition is generalized to the case of G(m, 1, r). We study the properties of cosets and double cosets of the quasi-parabolic subgroups. Also, we characterize double cosets by n×n arrays of non-negative integer sequences. Analogues of permutation modules for Hecke algebras of type A and type B have been introduced in [DJ1], [DS] and [DJM1]. All these permutation modules are corresponding to some subgroups. They are important to the study of the representation theory of Hecke algebras. A generalization of these modules is defined in [DJM2] for Ariki-Koike algebras, but they are not naturally corresponding to subgroups. We define another analogue of these modules which are naturally corresponding to certain subgroups indexed by multicomposition. These subgroups are naturally associated with quasi-parabolic subgroups. Each of these modules is free and has a basis indexed by right cosets of the associated quasi-parabolic subgroup. Also, we construct Murphy type bases, Specht series and a Young’s rule for these modules. The thesis is organized as follows. In Chapter 1, we collect some results of symmetric groups, Coxeter groups, and their Hecke algebras. We generalize these results in the subsequent chapters. In Chapter 2, we characterize elements in G(m, 1, r) by bijections and develop an algorithm to find reduced expressions for the elements. We define length reduction sort and establish a one-one correspondence between the set of length reduction sorts and the set of reduced expressions. Expositions of analogues of exchange conditions and Bruhat order are also included in this chapter. Geck and Pfeiffer prove their reducibility theorem by some block exchange lemmas. We generalize the exchange lemmas and the reducibility theorem in Chapter 3. In Chapter 4, we study characters of Ariki-Koike algebras. We include a counter example to show that characters of elements corresponding to distinguished reduced expressions of minimal length elements in a conjugacy class of G(m, 1, r) may not be the same. We prove that the character values of any element can be determined 4 INTRODUCTION by the character values of some special elements corresponding to minimal length elements in conjugacy classes. Definition of quasi-parabolic subgroups, properties of the coset and double cosets, and the characterization of double cosets by arrays are presented in Chapter 5 and in the published paper in the appendix. The q-permutation modules for Ariki-Koike algebras are defined and studied in Chapter 6. We prove that these modules are free. Using the Robinson-Schensted- Knuth correspondence, we construct Murphy type bases for our permutation modules from the bases of the cyclic modules in [DJM2]. Also based on their construction, we construct the Specht series and establish a Young’s rule for our modules. In the last chapter, we explore the possibility of generalizing some of the results to complex reflection groups of other imprimitive types. In particular, the algorithm for getting reduced expressions for elements in G(2, 2, r) is presented. CHAPTER 1

Symmetric Groups, Coxeter groups and their Hecke Algebras

1.1. Introduction

A real reflection is an invertible linear transformation on a finite dimensional real Euclidean space V such that this map fixes a hyperplane and sends a non-zero vector α which is orthogonal to the hyperplane to −α. Finite groups generated by reflections are finite reflection groups, for instance, symmetric groups. Each finite reflection group has a presentation as a group generated by a set of involutions S. They are Coxeter groups which are defined as follow.

Definition 1.1.1. A Coxeter group W is generated by a ﬁnite set S subject to the following relations,

mst (st) = e, for s, t ∈ S, s =6 t, 1 < mst < ∞ ;

s2 = e, for s ∈ S; where e is the identity. The pair (W, S) is called a Coxeter system.

The finite Coxeter groups are precisely the finite reflection groups, [H, Theorem 6.4]. Throughout the thesis, all reflection groups and Coxeter groups are assumed to be finite. In a Coxeter system (W, S), words in S are called expressions. Formally speaking, an element w in W is an equivalence class of expressions which can be convert to one another by the relations. Each expression in the equivalence class is called an expression for w or an expression of w. When it is clear from the context, we identify an element with any one of its expressions. The following length function and reduced expressions are widely used in the literature.

5 6 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS

Definition 1.1.2. Let (W, S) be a Coxeter system and w be an element in W .

Among all the expressions si1 ··· sip for w, where si1 , . . . , sip ∈ S, the expressions with minimal p are said to be reduced. This minimum value p is called the length of w and is denoted by `(w). The length of the identity is deﬁned to be 0.

We use the same symbol e to stand for both the identity and its reduced expression which should be an empty string. The notion of reflection is generalized to the case when V is a unitary space by Shephard, [She1, p.82]. A pseudo-reflection is an order m unitary transformation s on a unitary space which fixes a hyperplane. The finite groups generated by pseudo- reflections are called complex reflection groups. The complex reflection groups have been classified by Shephard and Todd in [ST]. We are going to investigate those of the imprimitive type, G(m, 1, r), where m, r are positive integers. A couple of features of the complex reflection groups G(m, 1, r) are very similar to those of symmetric groups. First, all complex reflection groups have a Coxeter group like presentation, although they are no longer generated by involutions, [BMR, Appendix 2]. Second, an element in G(m, 1, r) can also be characterized by a permutation, that is, a bijective function on a finite set of symbols. In the rest of this Chapter, we collect some well-known results in the symmetric groups, Coxeter groups and their Hecke algebras which can be found in [DJ1, GP2, JK, Mat, Mu3, NT, Sa]. For the consistency of notation, they are not word for word quotations. References are provided for those results which are not elementary. We omit most of the proofs which can be either found in the literature or are included in a more general case G(m, 1, r) given in subsequent chapters.

1.2. Symmetric Groups

Let r be a positive integer. The symmetric group Sr is deﬁned by presentation as follows. 1.2. SYMMETRIC GROUPS 7

Definition 1.2.1. The symmetric group Sr is generated by the set {s1, . . . , sr−1} subject to the following relations,

sisi+1si = si+1sisi+1, if 1 6 i 6 r − 2;

sisj = sjsi, if |i − j| > 2; and

2 2 s1 = ··· = sr−1 = e.

Here e is the identity.

The ﬁrst and second relations are known as braid relations, while the last one is called the order relation which tells us the orders of the generators of the group. From deﬁnition, the symmetric group is a Coxeter group with the set of generators

S = {s1, . . . , sr−1}. In the rest of this chapter, we use S to stand for the generating set of Sr except in the discussions of properties of a Coxeter system (W, S) in general.

It is well-known that an element in Sr can be considered as a permutation on r diﬀerent symbols, that is, a bijection on the set of these symbols. In particular, we can use the ﬁrst r positive integers as symbols.

Proposition 1.2.2. Let r be an integer and r = {1, 2, . . . , r}. The symmetric group Sr is the group of all permutations on r using the function composition as multiplication.

We write the image of a under a bijection σ ∈ Sr as aσ, and consequently, the multiplication of elements is done from left to right. For instance, the image of a under the permutation σπ is (aσ)π.

A permutation σ ∈ Sr is completely determined by the images of the elements in r. Naturally, we can present σ in a two-line form —

a1 a2 ··· ar σ = ,   a1σ a2σ ··· arσ   8 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS where {a1, a2, . . . , ar} = r. In particular, when ai = i for all 1 6 i 6 r, we can omit the ﬁrst line and get its one-line form as

σ = ((1σ, 2σ, . . . , rσ)).

Since the order of the terms in the one-line form is important, we call iσ the ith term or the ith entry of σ.

Example 1.2.3. In S5, let σ be the element given by

1σ = 4, 2σ = 5, 3σ = 2, 4σ = 1, 5σ = 3.

The one-line form for σ is ((4, 5, 2, 1, 3)), while in the two-line form, it can be written as 1 3 2 5 4 1 4 2 5 3 or . 4 2 5 3 1 4 1 5 3 2 The identity e in one-line form is ((1, 2, 3, 4, 5)). 

The multiplication of an element σ = ((a1, . . . , ar)) in the one-line form by a generator in S can be done simply by swapping entries of σ. It is straightforward to verify that the one-line form for siσ can be obtained by swapping the ith and the (i + 1)th terms of σ. The one-line form for σsi can be obtained by swapping the entries whose values are i and i + 1. Thus, we have a linear algorithm to write an element π ∈ Sr, which is given as an expression ω, in the one-line form. More precisely, let ω = si1 ··· sip with si1 , . . . , sip ∈ S. The number of steps used to write π in the one-line form is proportional to p.

Example 1.2.4. If σ = s2s4s1s3 ∈ S5, then

σ = s2s4s1s3((1, 2, 3, 4, 5)) σ = ((1, 2, 3, 4, 5))s2s4s1s3

= s2s4s1((1, 2, 4, 3, 5)) = ((1, 3, 2, 4, 5))s4s1s3

= s2s4((2, 1, 4, 3, 5)) and = ((1, 3, 2, 5, 4))s1s3

= s2((2, 1, 4, 5, 3)) = ((2, 3, 1, 5, 4))s3

= ((2, 4, 1, 5, 3)), = ((2, 4, 1, 5, 3)). 1.2. SYMMETRIC GROUPS 9

Since a symmetric group is a Coxeter group, the length and reduced expressions for an element in it are deﬁned in 1.1.2. The formula for the length of an element in

Sr can be derived from the one-line form of the element.

Definition 1.2.5. Let σ = ((a1, . . . , ar))be an element in Sr. The set of inversions of σ is deﬁned by

I(σ) = {(i, j)|i < j and ai > aj}, and the number of elements in I(σ) is denoted by n(σ).

Proposition 1.2.6. [ECHLPT, 9.1.5] If σ ∈ Sr, then `(σ) = n(σ).

The following lemma has been used in some other form to prove the above proposition in [ECHLPT]. Moreover, from it, we can derive an algorithm of getting reduced expressions for any given element.

Lemma 1.2.7. Let σ = ((a1, a2, . . . , ar)) be an element in Sr and 1 6 i 6 r − 1. If ai > ai+1 then n(σ) = n(siσ) + 1, otherwise n(σ) = n(siσ) − 1.

Let σ = ((a1, . . . , ar)) ∈ Sr. If σ =6 e, then there exists i = i1 such that ai > ai+1. Suppose that

0 σ = ((a1, . . . , ai−1, ai+1, ai, ai+2, . . . , ar)).

0 0 From the above lemma and proposition, we have `(σ) = `(σ )+1. Note that σ = siσ . 0 Then we can pick up another consecutive pair of terms in σ , say the i2th and the 00 (i2 + 1)th, such that the left one is greater than the right one. If σ is the element 0 00 0 formed after swapping these two terms we have σ = si1 σ = si1 si2 σ and `(σ ) = `(σ00) + 1. We can repeat this process until the element turns into the identity. Then we get a reduced expression si1 ··· sil for σ. We call this process a length reduction sort, because the element is changed to one with shorter length by each swap.

Remark 1.2.8. In the proof of [ECHLPT, 9.1.5], instead of getting i such that ai > ai+1 as in 1.2.7, they ﬁnd an inversion (i, j) of σ such that ai = aj + 1. Then they have n(σsi) + 1 = n(σ). A similar length reduction sort can then be easily 10 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS derived, though they have not done so. I discovered the length reduction sort for getting reduced expressions for an element in Sr and I generalized it to G(m, 1, r) (see the appendix of the thesis) long before I came across the proof in [ECHLPT]. To the best of my knowledge, using length reduction sort to get reduced expressions for an element in G(m, 1, r) is not in the literature.

Example 1.2.9. Below is a length reduction sort which turns σ = ((5, 1, 4, 2, 3)) to the identity.

((5, 1, 4, 2, 3)) = s1((1, 5, 4, 2, 3))

= s1s2((1, 4, 5, 2, 3))

= s1s2s3((1, 4, 2, 5, 3))

= s1s2s3s2((1, 2, 4, 5, 3))

= s1s2s3s2s4((1, 2, 4, 3, 5))

= s1s2s3s2s4s3((1, 2, 3, 4, 5))

Hence, σ = s1s2s3s2s4s3 is a reduced expression and `(σ) = 6.

In the rest of this section, we are going to state some well-known properties of expressions for elements in symmetric groups. Since all these properties are true for Coxeter groups in general, we state them in the more general setting that (W, S) is a Coxeter system.

Theorem . 1.2.10 (Deletion condition) [H, 1.7] Let si1 ··· sip be an expression of an element w ∈ W with sin ∈ S. If `(w) < p, then there exist 1 6 j < k 6 p such that

w = si1 ··· sˆij ··· sˆik ··· sip , where the hat denotes omission.

Theorem . 1.2.11 (Exchange condition) [H, 1.7] Suppose that si1 ··· sip is an expression (not necessarily reduced) of an element w ∈ W with sin ∈ S. If `(ws) < 1.2. SYMMETRIC GROUPS 11

`(w) for some generator s, then there exists an index k for which ws = si1 ··· sˆik ··· sip and thus w = si1 ··· sˆik ··· sip s, with a factor s exchanged for a factor sik . In particular, w has a reduced expression ending in s if and only if `(ws) < `(w).

Denote the set of conjugates of S, w−1Sw, by T. The exchange condition can w∈W be generalized to a stronger one. S Theorem . 1.2.12 (Strong exchange condition) [H, 5.8] Suppose that si1 ··· sip is t an expression (not necessarily reduced) of an element w ∈ W with sin ∈ S. If `(w ) < t T t `(w) for some ∈ , then there exists an index k for which w = si1 ··· sˆik ··· sip .

Definition 1.2.13 (Bruhat order on a Coxeter group). For w, w0 ∈ W , deﬁne 0 0 w < w if there exist elements w = w0, w1, . . . , wk = w ∈ W such that, for all 1 6 i 6 k,

(a) `(wi−1) < `(wi), and −1 T (b) wiwi−1 ∈ .

We write w0 6 w if w0 < w or w0 = w. Obviously, 6 is a partial order on W . From deﬁnition, w0 < w implies `(w0) < `(w). Moreover, (b) can be replaced by

0 (b ) wi = twi−1, for some t ∈ T; or 00 (b ) wi = wi−1t, for some t ∈ T. (b0) is obviously equivalent to (b). To see (b0) and (b00) are equivalent, note that t −1 t −1 t T wi = wi−1 = wi−1(wi−1 wi−1) and wi−1 wi−1 ∈ . An important and useful characterization of Bruhat order in a Coxeter group is in terms of subword.

Definition 1.2.14. A subword of a word s ··· s is a word s ··· s such that i1 il ik1 ikp

1 6 k1 < ··· < kp 6 l.

The following theorem can be found in [De, 1.1(III)] or [H, 5.10].

Theorem . 0 1.2.15 Let w, w be elements in a Coxeter group. Let si1 ··· sil be a ﬁxed, but arbitrary, reduced expression for w. Then w0 6 w if and only if w0 can be obtained as a subword of this reduced expression. 12 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS

Thus, if an expression (not necessarily reduced) for w0 is a subword of a reduced expression for w, then w0 < w. As a result, for each reduced expression ω of w, there exists an expression of w0 which is a subword of ω.

1.3. Conjugacy Classes of Symmetric Groups

k Let σ be an element in Sr. The set A1 = {1σ |k ∈ N} is ﬁnite because it is a l subset of r. Let l be the minimal positive integer that 1σ = 1. An element σ1 is then deﬁned by

iσ when i ∈ A1, iσ1 =  i when i∈ / A1.

This element is called a cycle factorof length l or an l-cycle. In particular, a 2-cycle is known as a transposition. The cycle σ1 is denoted by

1 7→ 1σ 7→ 1σ2 7→ ... 7→ 1σl−1 7→ 1 . If A1 =6 r, we can take the minimum integer j ∈ r \ A1 and similarly deﬁne A2 and σ2. If r \ (A1 ∪ A2) is non-empty, we then take the minimum integer in it and deﬁne A3, σ3, and so on until we get A1 ∪ · · · ∪ An = r. The elements σ1, . . . , σn are called disjoint cycles of σ because the sets A1,...,An are disjoint. These factors are obviously mutually commutative. A not so obvious fact is that we do not have to take the minimum integer each time and still get the same set of disjoint cycles. We then write σ = σ1 ··· σn which is called the disjoint cycle decomposition for σ.

Example 1.3.1. The element ((4, 5, 2, 1, 3)) can be decomposed into the product of disjoint cycles as

(1 7→ 4 7→ 1)(2 7→ 5 7→ 3 7→ 2).

One of the advantages of the two-line notation is that we can easily write down the inverse of an element and hence its conjugates. Let σ, π be elements in Sr. We 1.3. CONJUGACY CLASSES OF SYMMETRIC GROUPS 13 have

1π 2π ··· rπ π−1 = ,  1 2 ··· r    1 2 ··· r σ = , and 1σ 2σ ··· rσ   1π 2π ··· rπ π−1σπ = . (1σ)π (2σ)π ··· (rσ)π   Since π is a bijection, immediately

i 7→ iσ 7→ iσ2 7→ · · · 7→ iσl−1 7→ i is an l-cycle of σ if and only if

iπ 7→ (iσ)π 7→ (iσ2)π 7→ · · · 7→ i(σl−1)π 7→ iπ is an l-cycle of π−1σπ. To see this, we put j = iπ and the second cycle above can be written as

j 7→ jπ−1σπ 7→ j(π−1σπ)2 7→ · · · 7→ j(π−1σπ)l−1 7→ j . Moreover, σ and π−1σπ have equal number of l-cycles for each l, where 1 6 l 6 r.

Definition 1.3.2 (cycle type). The cycle type of an element σ ∈ Sr is the expression

(1n1 , 2n2 , . . . , rnr ), where nl is the number of l-cycles in a disjoint cycle decomposition of σ.

Definition 1.3.3. A sequence of non-negative integers (λ1, λ2, . . . , λn) is called a n composition of r into n parts, if i=1 λi = r. In particular, if λ1 > λ2 > ··· > λn > 0, then λ is called a partition of r.P Furthermore, the conjugate of a composition λ is deﬁned to be the partition λ0 = 0 0 0 (λ1, λ2,... ) such that λi is the cardinality of {λj|λj > i}. 14 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS

Each cycle type (1n1 , 2n2 , . . . , rnr ) is associated with a partition in which there are 3 2 nl parts of value l. For instance, the cycle type (1, 2 , 4 ) will be associated with the partition (4, 4, 2, 2, 2, 1). Hence, there is a one-one correspondence between the set of partitions of r and the set of conjugacy classes.

Remark 1.3.4. The group S1 is the trivial group and S2 is the cyclic group of order 2. When r > 2, the group Sr can be presented as a group generated by two elements, [Co, 5.2]. More precisely, Sr is generated by {s, t} subjected to

tr = s2 = (ts)r−1 = (t−1sts)3 = (t−jstjs)2 = e,

1 where j = 2,..., [ 2 r]. Here [x] is the largest integer less than or equal to x for a real number x. We can identify s = ((2, 1, 3, . . . , r)) and t = ((2, 3, . . . , r, 1)). Note that s = (1 7→ 2 7→ 1) and t = (1 7→ 2 7→ 3 7→ · · · 7→ r 7→ 1) are elements with the smallest and largest order among the non-trivial cycles.

In a conjugacy class C of the symmetric group, usually there are more than one element of minimal length. For instance, both elements in the conjugacy class

{s1s2 = ((3, 1, 2)), s2s1 = ((2, 3, 1))} are of the same length and hence they are of minimal length. In general, in a Coxeter system (W, S), let Cmin be the set of all minimal length elements in the conjugacy class C of the group W .

Definition 1.3.5. [GP2, 3.2.3] The preorder relation → on W is deﬁned by 0 0 0 w → w for w, w ∈ W if there is a sequence of w = w0, w1, . . . , wn = w of elements

ski ski in W such that wi−1 −→ wi for some ski ∈ S for 1 6 i 6 n. Here, wi−1 −→ wi means wi = ski wi−1ski and `(wi) 6 `(wi−1).

Definition 1.3.6. [GP2, 3.2.4] The elements w, w0 ∈ W are said to be elementarily strongly conjugate in W , if `(w) = `(w0) and there is an element u ∈ W such that either wu = uw0 and `(wu) = `(w) + `(u) or uw = w0u and `(uw) = `(u) + `(w). 1.4. YOUNG SUBGROUPS 15

0 If there is a sequence w = w0, w1, . . . , wn = w of elements in W such that wi−1 and 0 wi are elementarily strongly conjugate for 1 6 i 6 n, then w and w are said to be strongly conjugate and this is denoted by w ∼ w0.

Theorem 1.3.7 (Geck-Pfeiﬀer reducibility theorem). [GP1, 1.1] Let C be a conjugacy class of a Coxeter group W . (a) For each w ∈ C there exists a minimal length element w0 ∈ C such that w → w0. (b) If w and w0 are both elements of minimal length in C, then w ∼ w0.

1.4. Young Subgroups

In this section we collect results of a special type of subgroups of the symmetric group Sr. These subgroups are important in the study of the representation theory of symmetric groups.

Definition 1.4.1. Corresponding to a composition λ = (λ1, . . . , λn) of r, the Young diagram of shape λ or λ-diagram is an array of n rows of boxes such that there are λi boxes in the ith row. For each σ = ((1σ, . . . , rσ)), the corresponding Young tableau or λ-tableau is an array obtained by ﬁlling in a λ-diagram by iσ in order along successive rows. If λ is a partition, a λ-tableau is said to be standard if the entries in each row and the entries in each column are increasing.

The correspondence, between the set of all λ-tableaux and Sr, in the above definition is one-one. It defines a bijective mapping δ between these two sets. The element in Sr corresponding to a λ-tableau t is denoted by δ(t). The λ-tableau corresponding λ to the identity in Sr is denoted by t . Furthermore, the λ-tableau corresponding to an element σ is denoted by tλσ and this defines a right action on the set of tableaux by elements in Sr.

Example 1.4.2. Suppose that λ = (4, 3, 1) is a composition of 8. Here are the λ-diagram and the λ-tableau corresponding to e. 16 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS

1 2 3 4 tλ = 5 6 7 8

Moreover if

5 2 6 8 1 2 4 7 s = 1 7 3 and t = 3 5 8 , 4 6 then t is standard but s is not. The elements corresponding to t and s are, respectively, δ(s) = ((5, 2, 6, 8, 1, 7, 3, 4)) and δ(t) = ((1, 2, 4, 7, 3, 5, 8, 6)).

The set r is naturally dissected into n disjoint subsets corresponding to the rows tλ tλ λ λ of . The set of entries of the ith row of is denoted by Ri . More precisely, Ri = ∅ when λi = 0. Otherwise

λ Ri = {λ1 + ··· + λi−1 + 1, . . . , λ1 + ··· + λi}.

It is clear that the set of entries in the ith row of t = tλσ is

λ λ Ri σ = {aσ|a ∈ Ri },

t which is denoted by Ri. Besides tableaux with distinct entries, a more general type of tableaux is also used to study symmetric groups in the literature.

Definition 1.4.3. Let λ be a composition. A λ-tableau t is an array formed by ﬁlling positive integers in a λ-diagram, one in each box and the same integer can be used more than once. The type of t is the composition µ = (µ1, µ2,... ) where the number of i’s in the tableau is µi.

Corresponding to a λ-tableaux t whose entries are distinct and in r, we can con- t µ struct a λ-tableau of type µ by replacing each i in by j, where i ∈ Rj . The so formed tableau is denoted by µ(t). 1.4. YOUNG SUBGROUPS 17

Definition 1.4.4. Let λ be a partition. A λ-tableau is semistandard if all rows are weakly increasing across and all columns are increasing down.

Example 1.4.5. Continue from Example 1.4.2. Further let µ = (2, 3, 3). Then

2 1 3 3 1 1 2 3 µ(s) = 1 3 2 and µ(t) = 2 2 3 . 2 3

Furthermore, µ(s) is not semistandard while µ(t) is.

The tableaux are important tools for studying the following special type of subgroups of symmetric groups.

Definition . Sλ 1.4.6 Suppose λ is a composition of r. For each i ∈ n, let i be the S λ subgroup of r consisting of all λi! permutations which ﬁx the set r \ Ri element- SλSλ Sλ S wise. The subgroup 1 2 ··· n is denoted by λ and is called a (standard ) Young subgroup.

Remark 1.4.7. In a Coxeter system (W, S), subgroups of W generated by a subset of S are called standard parabolic, and parabolic subgroups are conjugate to the λ λ standard ones. In general, Young subgroups are similarly deﬁned when R1 ,...,Rn are replaced by disjoint subsets, the union of which is r. Hence, standard Young subgroups are standard parabolic and Young subgroups are parabolic. If we take S0 to be the trivial group, then a Young subgroup corresponding to λ is isomorphic to S S S the direct product λ1 × λ2 × · · · × λn . If α is the partition formed by shuﬄing the parts of λ, then Sλ is isomorphic to Sα. In the Thesis, we are only interested in the standard ones. Unless otherwise stated, we use Young subgroups and parabolic subgroups to mean standard Young subgroups and standard parabolic subgroups.

The following propositions are some well-known properties of cosets and double cosets of Young subgroups. They are generalized to some special subgroups of G(m, 1, r) in Chapter 5 and the appendix. 18 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS

Proposition 1.4.8. In each right coset of Sλ, there is a unique shortest element and there is a unique longest element.

The set of all the shortest elements in the right Sλ-cosets is denoted by Dλ. We call these elements the distinguished right coset representatives. The set of longest + elements in these right cosets is denoted by Dλ . −1 We also have analogous results for the left cosets. If σ = si1 ··· sip , then σ = Sλ −1 Sλ sip ··· si1 . Thus, the subgroup ( i ) is the same as i . As a result, the set of −1 distinguished (shortest) left coset representatives is Dλ .

Proposition 1.4.9. Let λ and µ be compositions of r. In each double coset in

Sλ \ Sr/Sµ, there are a unique shortest element and a unique longest element. The −1 set of shortest (distinguished) representatives of these double cosets is the set Dλ∩Dµ , + + −1 while the set of longest double coset representatives is the set Dλ ∩ (Dµ ) .

Proposition 1.4.10. [JK, 1.3.10] Let λ = (λ1, . . . , λn) and µ = (µ1, . . . , µn) be compositions of r. The set of double cosets Sλ\Sr/Sµ is in one-one correspondence with the set of n × n matrices (zij) of non-negative integer entries satisfying

n n

zij = λi and zij = µj. j=1 i=1 X X

A double coset SλσSµ is said to have the trivial intersection property if Sλσ ∩ −1 σSµ = {σ} or equivalently σ Sλσ ∩ Sµ = {e}.

Corollary 1.4.11. [JK, 1.3.13] There is a one-one correspondence between the set of double cosets with the trivial intersection property in Sλ \ Sr/Sµ and the set of n × n 0-1 matrices with row sums λi and column sums µj.

Corollary 1.4.12. [JK, 1.3.14] The number of double cosets in Sλ \ Sr/Sµ is

λ µ λ1 λn µ1 µn equal to the coeﬃcient of x y = x1 ··· xn y1 ··· yn in the formal power series

−1 (1 − xiyj) . 16i,j6n Y 1.5. IWAHORI-HECKE ALGEBRAS 19

The number of double cosets with the trivial intersection property is equal to the coeﬃcient of xλyµ in

(1 + xiyj). 16i,j6n Y Proposition 1.4.13. [D, 3.2, 3.4] The lengths of the shortest and the longest elements in the double coset corresponding to an n × n matrices (zij) of non-negative integer entries are respectively,

r z z and − z z . ij kl 2 ij kl 16i

1.5. Iwahori-Hecke Algebras

A Hecke algebra of a Coxeter group W can be defined as a deformation of the group algebra RW where R is a commutative algebra with 1. We first include a definition of the group algebra RG of a group G.

Definition 1.5.1. Let R be a commutative ring with 1 and G be a ﬁnite group. The group algebra RG is the set

rgg rg ∈ R ( ) g∈G X with addition

0 0 rgg + rgg = (rg + rg)g, g∈G g∈G g∈G X X X scalar multiplication

k rgg = krgg, and g∈G g∈G X X multiplication

0 0 rgg rgg = rhrh−1g g, ! ! ! Xg∈G Xg∈G Xg∈G Xh∈G 0 where k, rg, rg are elements in R. 20 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS

As in [GP2, p.7], we write

Prod (x, y; n) = xyx ··· , n factors | {z } and deﬁne the Iwahori-Hecke algebras, that is the Hecke algebras of Coxeter groups, as follows.

Definition 1.5.2. [GP2, p.123] Let R be a commutative ring with 1 and (W, S) 0 be a Coxeter system. Let Q = {qs, qs|s ∈ S} be a subset of R such that qs = 0 0 qt and qs = qt whenever s and t are conjugate in W . The Iwahori-Hecke algebra corresponding to (W, S) and Q, is the associative R-algebra with identity generated by {Ts|s ∈ S} subjected to the following relations

Prod (Ts,Tt; mst) = Prod (Tt,Ts; mst), for s =6 t, 1 < mst < ∞ ;

2 0 Ts = qs + qsTs, for s ∈ S.

Here mst denotes the order of st ∈ W for s, t ∈ S.

When the generating set S and the parameter set Q are understood, the Iwahori- 0 Hecke algebra is denoted by H(W ) or simply H. When qs takes the value 1 and qs takes the value 0, the algebra H(W ) is simply the group algebra RW . On the other hand, the ﬁrst of the two relations is the same as the ﬁrst one in the presentation of the Coxeter group. The second one is a deformation of the order relation. In this sense, the Hecke algebra is a q-deformation the group algebra.

Usually, we label the generators of W by integers, that is S = {s1, s2,... }. In order to save complicated subscripts, Tsi is denoted by Ti. The following theorem is fundamental in the study of Iwahori-Hecke algebras.

Theorem 1.5.3 (Matsumoto). [GP2, 1.2.2] Let (W, S) be a Coxeter system de- ﬁned by the presentation

2 mst W = hs, t ∈ S|s = e for s ∈ S;(st) = e for s =6 t, 1 < mst < ∞i. 1.6. REPRESENTATIONS AND CHARACTERS 21

Suppose (M, ·) is a monoid and f : S → M is a map such that

Prod (f(s), f(t); mst) = Prod (f(t), f(s); mst) for s, t ∈ S, s =6 t, mst < ∞ .

Then there exists a unique map F : W → M such that F (w) = f(si1 ) ··· f(sip ) whenever w = si1 ··· sip is a reduced expression with sik ∈ S.

Corollary 1.5.4. [GP2, 4.4.3(a)] Let H be an Iwahori-Hecke algebra corresponding to a Coxeter system (W, S). For each w ∈ W , if s ··· s and s 0 ··· s 0 are i1 i` i1 i` both reduced expressions for w, then T ··· T = T 0 ··· T 0 . This element is denoted i1 i` i1 i` by Tw. In particular, Te is the identity in H.

Theorem 1.5.5. [GP2, 4.4.6] The Iwahori-Hecke algebra over a commutative ring R with 1 corresponding to a Coxeter system (W, S) is free as an R-module with basis {Tw|w ∈ W }.

1.6. Representations and Characters

Let R be a commutative ring with 1 and A be an R-algebra. A matrix representation of A is an R-algebra homomorphism into the algebra of all n × n matrices over R for some positive integer n —

ρ : A → Mn(R).

The specially important ones are the faithful representations which are monomor- phisms. Representations provide tools for us to do calculations, in particular, when the algebra is deﬁned by presentation. Two representations ρ and ρ0 of A are said to be equivalent when there is an invertible matrix P such that

ρ0(a) = P −1ρ(a)P, for all a ∈ A.

Corresponding to a representation ρ, we can define a representation module which is an A-module Rn defined by av = ρ(a)v for all v ∈ Rn. Conversely, any finitely generated A-module which is free as an R-module defines a representation for A. 22 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS

A submodule W is said to be pure in V if the quotient module V/W is torsion-free. An irreducible representation is one whose representation module V does not have a proper pure submodule W .

If K ⊃ R are rings with common identity, any R-representation of ρ : A → Mn(R) K can obviously be treated as a K-representation ρ : A → Mn(K) because matrices over R can be treated as matrices over K. A representation over a field R is called absolute irreducible if it remains irreducible when regarding as any K-representation for any extension field K of R. From the representation theory point of view, it is interesting to find all non- equivalent irreducible representations for a given algebra. Further details can be found in [NT, Chapter 2]. The character corresponding to a representation ρ is the R-homomorphism defined by

χρ : A → R such that a 7→ trρ(a), where tr(M) is the trace of the matrix M. The representations and characters of a group G are simply the representations and characters of the group algebra RG. When R is a ﬁeld of characteristic 0 or p > 0 such that p is not a factor of the order of G, the number of irreducible representations is the same as the number of conjugacy classes. The representations over a ﬁeld of positive characteristic p which divides the order of G are far more complicated. The characters of G are class functions, in the sense that if w, w0 are conjugates in G then the character values on them are the same. More precisely, if w0 = g−1wg and χ is a character corresponding to the representation ρ, then

χ(w0) = χ(g−1wg) = tr(ρ(g−1)ρ(w)ρ(g)) = tr(ρ(g)ρ(g−1)ρ(w)) = tr(ρ(w)).

Note that tr(AB) = tr(BA) for any square matrices A, B of the same order. Fur- thermore, the group algebra RG has a basis {g|g ∈ G}, and so the values of all the characters on RG are determined by the values of the irreducible characters on the conjugacy classes of G. 1.6. REPRESENTATIONS AND CHARACTERS 23

However, in an Iwahori-Hecke algebra H(W ), we cannot expect and it is not true 0 that χ(Tw) = χ(Tw0 ) when w and w are conjugate in W . In Hecke algebras of type A, Starkey (see [Car, p.95]) and Ram (see [Ra]) have shown independently that the 0 character values of Tw and Tw0 are equal, when w and w are minimal length elements in the same conjugate class. Geck and Pfeiﬀer have proved in [GP1] that this is true in general for all Iwahori-Hecke algebras by an elementary and combinatorial method based on their reducibility theorem (1.3.7). They have further proved that based on the character values of elements in {Tw|w ∈ Cmin for some conjugacy class C}, the character value of Tw for any w ∈ W and hence the character value for an arbitrary element in the Iwahori-Hecke algebra can be determined [GP1, p.80]. In the rest of the section, we are going to state some results on the Iwahori-Hecke algebra of type A. We assume that R is a domain unless otherwise stated. Since all the generators of the symmetric group are conjugate to each other, we choose qs = q 0 and qs = q − 1 for all s ∈ S.

One of the ways to construct all irreducible representations for H = H(Sr) is to construct q-Specht modules which are submodules of q-permutation modules. All irreducible H-modules (i.e. irreducible H-representations) are quotient modules of these Specht modules, [DJ1, Section 4]. Recall that corresponding to a composition

λ of r, there is a Young subgroup Sλ. Let

−`(σ) (1.6.1) xλ = Tσ, and yλ = (−q) Tσ. S S σX∈ λ σX∈ λ

Definition 1.6.2. Let λ be a composition of r. The cyclic right H-module xλH is called a q-permutation module.

The module xλH is called a q-permutation module because when q = 1 the module is isomorphic to the coset space with basis {Sλd|d ∈ D}, where D is a complete set of coset representatives. The right action of an element in Sr on the coset space permutes the basis elements. 24 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS

Lemma 1.6.3. [DJ1, 3.2(i)] Let λ be a composition of r and Dλ be the set of all minimal length coset representatives of Sλ in Sr. The permutation module xλH is free with basis {xλTd|d ∈ Dλ}.

Recall that λ0 is the conjugate of the composition λ deﬁned in 1.3.3.

Definition . λ 0 1.6.4 A q-Specht module S is the cyclic module xλTwλ yλ H, where

λ is a partition and wλ is the unique longest element in the Young subgroup Sλ.

The q-Specht modules are R-free.

We then deﬁne an R-bilinear form h , i [DJ1, p.34] on xλH such that

q`(u) if u = v, hxλTu, xλTvi =  0 otherwise, for u, v ∈ Dλ. Extending this linearly to xλH, we get a symmetric bilinear form. With respect to this bilinear form, we deﬁne

λ⊥ λ S = {h ∈ xλH|hh, yi = 0 for all y ∈ S }.

Denote the module Sλ/(Sλ ∩ Sλ⊥) by Dλ.

Theorem 1.6.5. [DJ1, p.36] Let R be a ﬁeld and q be a non-zero element in it. Then Dλ is zero or an absolutely irreducible H-module, and

{Dλ|λ is a partition and Dλ =6 0} is a complete set of non-equivalent irreducible H-modules.

Theorem 1.6.6. [DJ1, 4.15] When R is a ﬁeld of characteristic zero and q ∈ Q, q > 0, we have Sλ = Dλ for any partition λ.

In [DJ1], Sλ and Dλ are deﬁned for a composition λ. In the paper, they have shown that Dλ ∼= Dµ if and only if λ0 = µ0, where λ0 and µ0 are conjugates of λ and µ. 1.6. REPRESENTATIONS AND CHARACTERS 25

Murphy has constructed another basis for H in [Mu2, Mu3]. Based on this basis, another version of Specht modules has been constructed. We describe the construction brieﬂy as follows.

Definition 1.6.7. Let λ = (λ1, . . . , λl) and µ = (µ1, . . . , µn) be compositions of r. Then λ dominates µ, denoted by λ D µ, if

k k

λi > µi, for all k. i=1 i=1 X X We write λ . µ, if λ D µ and λ =6 µ.

Definition 1.6.8. Let λ be a composition of r, and s and t be row-standard λ-tableaux. Deﬁne

xst = Tδ(s)−1 xλTδ(t).

Theorem 1.6.9 (Murphy basis). [Mu3, 4.17] The following set is an R-basis for

H(Sr).

{xst|s and t are standard λ-tableaux for some λ a partition of r}.

A pair of tableau (s, t) is called a standard µ-pair if both s and t are standard µ-tableaux for some partition µ. For any partition λ, the R-submodules spanned by

{xst|(s, t) is a standard µ-pair, µ is a partition of r, µ . λ}

{xst|(s, t) is a standard µ-pair, µ is a partition of r, µ D λ} are two-sided ideals of H and are denoted by H.λ and HDλ, respectively.

Definition 1.6.10 (Another version of Specht modules). Let λ be a partition, .λ Dλ .λ and zλ be the element xλ + H in the quotient algebra H /H . The submodule λ Dλ .λ S˜ generated by zλ in H /H is also called a Specht module.

Theorem 1.6.11. [Mu3, 5.3] The module S˜λ is H-isomorphic to the dual of Sλ. 26 1. SYMMETRIC GROUPS, COXETER GROUPS AND THEIR HECKE ALGEBRAS

Theorem 1.6.12 (Young’s rule). [Mu3, 7.2] Let λ be a composition. The q- permutation module, xλH, has a ﬁltration

xλH = M1 > M2 > ··· > Mk+1 = 0 such that for each i the quotient Mi/Mi+1 is isomorphic to a Specht module. Further- more, the number of such factors isomorphic to S˜µ (or Sµ) is the same as the number of semistandard µ-tableaux of type λ. CHAPTER 2

Complex Reﬂection Groups G(m, 1, r)

I have published the algorithm in Section 2.1 for getting a reduced expression for an element in G(m, 1, r) in a paper (see the appendix). In the paper, the length formula in [BM] is used to prove that the output is a reduced expression, but we shall give an alternate proof of the fact that the output is reduced and use it to prove the length formula here.

2.1. Characterizations of G(m, 1, r)

Let m and r be arbitrary positive integers. As mentioned in Chapter 1, we are going to investigate the imprimitive complex reﬂection groups G(m, 1, r). In [Co], Coxeter gave a presentation of these groups in terms of two generators for r > 2. He also found that there are mrr! elements in each of them. We start from the deﬁnition of the groups as a presentation in terms of generators and relations given in [She2, 4.12].

Definition 2.1.1. The group generated freely by S = {s0, s1, . . . , sr−1} subject to the following relations

s0s1s0s1 = s1s0s1s0;

sisi+1si = si+1sisi+1, if 1 6 i 6 r − 2;

sisj = sjsi, if |i − j| > 2; and

m 2 2 s0 = s1 = ··· = sr−1 = e, where e is the identity, is a complex reﬂection group G(m, 1, r).

From now on, we denote G(m, 1, r) by W and denote the generator set by S, unless otherwise stated. 27 28 2. COMPLEX REFLECTION GROUPS G(m, 1, r)

Let ξ be a primitive mth root of unity. The multiplicative subgroup of C \{0} ε generated by ξ is denoted by Cm. We set Cmr = {ξ a | a ∈ r, 0 6 ε 6 m − 1}. We ? ? also write Cm = Cm \{1} and so Cmr carries the obvious meaning. In the rest of the thesis, unless otherwise stated, we assume a, b, ai, bi, k, p to be in r or n; x, xi to be in Cmr or Cmn; and 0 6 ε, εi 6 m − 1. We also define a mapping : Cmr → r by ξεa = a. Determining the equality of two expressions as elements in a group defined by a presentation is not easy. For this, we treat W as a generalized symmetric group which contains some special bijections on Cmr as in [Re]. We are going to define a generalized symmetric group and give an alternate proof of the fact that it is the same as W .

ε Definition 2.1.2. A bijection w : Cmr → Cmr satisfying the condition (ξ i)w = ξε(iw), for all i ∈ r, is called a generalized permutation. The group of all generalized permutations with composition of functions as multiplication is called the generalized symmetric group, and is denoted by G, or more precisely by Gm,r.

As in symmetric groups, we write the image of i under w ∈ G as iw and hence the group multiplication is evaluated from left to right. For any w ∈ G, let iw = ξεk, jw = ξνk for some i, j, k ∈ r and 0 6 ε, ν 6 m − 1. By deﬁnition, we have iw = ξε−ν(ξνk) = (ξε−νj)w. Since w is a bijection, we have i = j and hence {1w, . . . , rw} = r. Thus G has the order of mrr!. Furthermore, any element w is uniquely determined by the images 1w, 2w, . . . , rw. Thus we can write an element w in G either in a one-line notation as ((1w, 2w, . . . , rw)) or in two-line notation (see [Re, p.145]) as

1 2 ··· r . 1w 2w ··· rw In general, we can deﬁne an element w by 

ε1 ε2 εr ξ a1 ξ a2 ··· ξ ar ,  ε1 ε2 εr  (ξ a1)w (ξ a2)w ··· (ξ ar)w   2.1. CHARACTERIZATIONS OF G(m, 1, r) 29 as long as ((a1, a2, . . . , ar)) ∈ Sr. Using one-line notation, the underlying set of the group G is

ε1 εr {((ξ a1, . . . , ξ ar)) | ((a1, . . . , ar)) ∈ Sr, 0 6 εi 6 m − 1, ∀i ∈ r}.

The product of elements in G can be easily evaluated as the composition of functions. For instance, take u = ((ξ3, ξ21, 2, 4))and v = ((ξ22, ξ23, ξ4, 1))in G(3, 1, 4). We have 1(uv) = (1u)v = ξ(3v) = ξ24 and so on. Hence uv = ((ξ24, ξ2, ξ23, 1)).

Definition 2.1.3. As elements in G, deﬁne

((ξ1, 2, . . . , r)) when i = 0, fi =  ((1, . . . , i − 1, i + 1, i, i + 2, . . . , r)) when 1 6 i 6 r − 1.

It is straightforward to verify the following product formulae.

ε1 εr Lemma 2.1.4. If w = ((ξ a1, . . . , ξ ar)) is in G, then

ε1 εi+1 εi εr (a) fiw = ((ξ a1, . . . , ξ ai+1, ξ ai, . . . , ξ ar)) for any 1 6 i 6 r − 1 and

ε1+1 ε2 εr (b) f0w = ((ξ a1, ξ a2, . . . , ξ ar)).

Further if ap = 1, aj = i and ak = i + 1, then

εj εk (c) wfi is the element formed by replacing the terms ξ i and ξ (i + 1) of w by ξεj (i + 1) and ξεk i respectively and

ε1 εp+1 εr (d) wf0 = ((ξ a1, . . . , ξ 1, . . . , ξ ar)).

2 2 Example 2.1.5. Let w = ((ξ 4, ξ2, ξ 3, 1)) ∈ G3,4. We have

2 2 2 2 f1w = ((ξ2, ξ 4, ξ 3, 1)), wf1 = ((ξ 4, ξ1, ξ 3, 2)),

2 2 2 f0w = ((4, ξ2, ξ 3, 1)) and wf0 = ((ξ 4, ξ2, ξ 3, ξ1)).

Note that when f1 multiplies on the left of w, it swaps the first and second terms of w. When f1 multiplies on the right, it swaps only the numerals 1 and 2 but not the powers of ξ. When f0 multiplies on the left of w, the exponent of ξ of the first term is increased by 1. But when f0 multiplies on the right, the change is made to the term ξε1 instead. When an element in G is given as a product of elements 30 2. COMPLEX REFLECTION GROUPS G(m, 1, r) in F = {f0, f1, . . . , fr−1}, we can effectively write the element in one-line form by function composition. We shall derive an algorithm to factorize any element in G as a product of elements in F . Consequently, the set F generates G. To establish the algorithm, we need the following order.

0 Definition 2.1.6. Deﬁne a total order 4 on Cmr by x 4 x if and only if either x = x0 or x ≺ x0, where

ξm−1r ≺ ξm−2r ≺ · · · ≺ ξr ≺ ξm−1(r − 1) ≺ · · · ≺ ξ(r − 1) ≺ · · ·

≺ ξm−11 ≺ · · · ≺ ξ1 ≺ 1 ≺ · · · ≺ r.

When m = 1 or 2, we have Cm = {1} or {1} and hence the order 4 is simply the usual order 6 on r or r ∪ −r. The following algorithm is motivated by a well-known sorting algorithm — the bubble sort. Each of the following operations on an element w = ((1w, . . . , rw)) is treated as a single sorting step — (1) reduce the exponent ε of ξ in the ﬁrst term by one if 1 6 ε 6 m − 1, (2) swap two adjacent terms iw and (i + 1)w if (i + 1)w ≺ iw.

0 0 00 At any stage if w is sorted to w by (1), we have w = f0w . If w is sorted to w by 00 0 00 (2), then w = fiw . For iteration, we rename w or w as w. Briefly, the algorithm starts from the first term and it removes the power of ξ of the first term iteratively by (1) if necessary. After the first p terms have been changed to positive integers and sorted in ascending order, we then handle the (p + 1)th term. We have two cases. Case 1 — if x = (p + 1)w ≺ 1, then swap the (p + 1)th term with the pth term, (p − 1)th term and so on until x appears as the first term. Remove the power of ξ and then sort the first (p + 1) terms of the resulting element in ascending order. Case 2 — if (p + 1)w ∈ r, then just sorts the first (p + 1) terms of w in ascending order. If we keep track of all the elements in F involved, we can factorize w as an expression in elements of F . The algorithm is written in pseudo-code as follows.

ε1 εr Algorithm 2.1.7. Given any element w = ((ξ a1, . . . , ξ ar)) ∈ G. The algorithm outputs an expression u = fi1 ··· fin for w and the number n = n(w). 2.1. CHARACTERIZATIONS OF G(m, 1, r) 31

(a) Set n := 0, k := 1, and u := e.

(b) (Note that 0 6 ε1 6 m − 1.) Set j := 1. Deﬁne 0 6 ε 6 m − 1 and a ∈ r by 1w = ξεa. If ε =6 0, perform (i) set 1w := a, ε (ii) set u := u · f0 , (iii) increase n by ε, and (c) If j < k and (j + 1)w ≺ jw, perform (i) swap the jth and the (j + 1)th terms in w,

(ii) set u := u · fj, and (iii) increase n by one, (iv) increase j by one, repeat step (c). (d) At this stage, the terms 1w, . . . , kw are positive integers and they are in ascending order. If k = r then go to step (g), else set j := k and increase k by one. (e) If j > 0 and (j + 1)w ≺ jw, perform (i) swap the jth and the (j + 1)th terms in w,

(ii) set u := u · fj, and (iii) increase n by one, (iv) decrease j by one, repeat step (e). (f) If j := 0, then go to step (b) else go to step (d). (g) Output n = n(w) and u.

In 1.2.5, we define n(σ) to be the order of the set of inversions I(σ) for σ ∈ Sr. It turns out that n(σ) is the same as `(σ). The algorithm 2.1.7 defines a function n : G → N. More precisely, n(w) is the number of sorting steps used by the algorithm for turning w to e. In the next section, we can see that the function n defined by 2.1.7 is an extension of the one in 1.2.5. In the algorithm, w is used iteratively as in a computer programme. In the following example, for clarity, we start from w = w0 and use a new subscript whenever 32 2. COMPLEX REFLECTION GROUPS G(m, 1, r) swap or change in exponent takes place. In each stage, we are handling the term(s) on the left of the vertical line.

2 2 Example 2.1.8. Take an element w = w0 = ((ξ 2, ξ 3, 4, ξ1))in G3,4. By the above algorithm, u is sorted to ((1, 2, 3, 4)) as follows.

i Term(s) Step wi Relationship between

considered 2.1.7 wi−1 and wi 2 2 1 1 (b) ((2|ξ 3, 4, ξ1)) w0 = f0 w1, n(w0) = n(w1) + 2 2 2 1,2 (e) ((ξ 3, 2|4, ξ1)) w1 = f1w2, n(w1) = n(w2) + 1 2 3 1 (b) ((3, 2|4, ξ1)) w2 = f0 w3, n(w2) = n(w3) + 2

4 1, 2 (c) ((2, 3|4, ξ1)) w3 = f1w4, n(w3) = n(w4) + 1

5 3, 4 (e) ((2, 3, ξ1, 4)) w4 = f3w5, n(w4) = n(w5) + 1

6 2, 3 (e) ((2, ξ1, 3, 4)) w5 = f2w6, n(w5) = n(w6) + 1

7 1, 2 (e) ((ξ1, 2, 3, 4)) w6 = f1w7, n(w6) = n(w7) + 1

8 1 (b) ((1, 2, 3, 4)) w7 = f0w8, n(w7) = n(w8) + 1

2 2 Hence, w = f0 f1f0 f1f3f2f1f0 and n(w) = 10.

Proposition 2.1.9. The algorithm terminates and the expression u output from

ε1 εr the above is equal to the input element ((ξ a1, . . . , ξ ar)).

Proof. When k = 1, step 2.1.7(b) turns the ﬁrst term to a positive integer if necessary, and step 2.1.7(c) will not be performed. Thus, when step 2.1.7(d) is performed for the ﬁrst time, the subsequence 1w is a single positive integer. In general, suppose that at some stage, when step 2.1.7(d) is performed, k has a value p.

Then 1w, . . . , pw is a sequence of increasing integers. If εp+1 = 0, then step 2.1.7(e) will arrange 1w, . . . , pw, ap+1 in ascending order. If εp+1 =6 0, step 2.1.7(e) will bring this term to the beginning and step 2.1.7(b) will change it to a positive integer. Step 2.1.7(c) will then arrange the subsequence of p + 1 positive integers in ascending order. Hence, the algorithm terminates and w becomes ((1, . . . , r)). Also, at any stage,

ε1 εr the element uw is ((ξ a1, . . . , ξ ar)). Thus, when the algorithm terminates, u is an

ε1 εr expression for the element ((ξ a1, . . . , ξ ar)). 2.1. CHARACTERIZATIONS OF G(m, 1, r) 33

Corollary 2.1.10. The subset F = {f0, f1, . . . , fr−1} generates G.

Remark 2.1.11. We shall prove that the expression output in the algorithm is reduced in the next section. Our algorithm is not a generalization of Algorithm A in [GP2, p.9] for Coxeter groups. The latter generates a reduced expression for an element in a Coxeter group. The input element for Algorithm A has to be in the form of a matrix with respect to a basis corresponding to the Coxeter generators.

Proposition 2.1.12. The groups W and G are isomorphic.

Proof. Deﬁne a mapping φ : S → F by φ(si) = fi. By composition of functions, we check that all the relations in Deﬁnition 2.1.1 hold when si are replaced by fi. Hence we can lift φ to a group homomorphism W → G. By Corollary 2.1.10, this homomorphism is surjective. Hence, card G 6 card W . From [Co, 5.4], the number of elements in W is mrr! There are also mrr! elements in G. Hence, φ is a group isomorphism.

Instead of using the number of elements in W as in the proof above, we can prove that card G > card W by the same argument for symmetric groups given in [B, note C]. To establish the alternate proof, we need the following lemmas which are well-known and easy to check.

Lemma 2.1.13. Let t1 = s0 and ti+1 = sitisi for 1 6 i 6 r − 1 be elements in W . Then

(a) tisi−1 = si−1ti−1 when i =6 1,

(b) tisi = siti+1 when i =6 r, and

Proof. For parts (a) and (b),

tisi−1 = si−1ti−1si−1si−1 = si−1ti−1,

tisi = sisitisi = siti+1. 34 2. COMPLEX REFLECTION GROUPS G(m, 1, r)

For part (c), if j > i, then for any factor sp of ti the diﬀerence of j and p is greater than 1 and the formula follows. When j 6 i − 2

tisj = si−1 ··· sjtjsj ··· si−1sj

= si−1 ··· sj+2sj+1sjtj(sjsj+1sj)sj+2 ··· si−1

= si−1 ··· sj+2sj+1sjtj(sj+1sjsj+1)sj+2 ··· si−1

= si−1 ··· sj+2(sj+1sjsj+1)tjsjsj+1sj+2 ··· si−1

= si−1 ··· sj+2(sjsj+1sj)tjsjsj+1sj+2 ··· si−1

= sjti.

Lemma 2.1.14. Let 1 6 i 6 r − 1 and j > 0. We have

sj · sr−1 ··· si when j < i − 1,  s ··· s when j = i − 1,  r−1 i−1 sr−1 ··· si · sj =   sr−1 ··· si+1 when j = i, s · s ··· s when j > i.  j−1 r−1 i   Proof. It is straightforward for the ﬁrst three cases. When j > i, we have

sr−1 ··· si · sj = sr−1 ··· sj+1(sjsj−1sj)sj−2 ··· si

= sr−1 ··· sj+1(sj−1sjsj−1)sjsj−2 ··· si = sj−1 · sr−1 ··· si.

Alternate proof of card G > card W . We are going to prove it by induction on r. When r = 1, W is a cyclic group of order m and hence card W = m = card G.

Let L be the subgroup of W generated by S \{sr−1}. Consider the set of elements m−1 D = ε=0 Dε in W , where

S ε ε ε Dε = {sr−1 ··· s1t1, sr−1 ··· s2t2, . . . , tr}. 2.1. CHARACTERIZATIONS OF G(m, 1, r) 35

We want to show that W = d∈D Ld. The superset part is obvious, but we need to check the subset part. Since theS identity is in D0, we only need to show, for any s ∈ S and d ∈ D, there exist d0 ∈ D such that Lds = Ld0. We verify this case by case.

ε 0 0 Case (1), s = s0. If d = sr−1 ··· s1t1 for some ε > 0, then ds ∈ Dε , where ε ≡ ε+1 (mod m). Otherwise, we have ds = sd and so Lds = Ld.

Case (2), s = sj with 1 6 j 6 r − 1 and d ∈ D0. If d = e, then

L when 1 6 j < r − 1, Lds = Ls =  Lsr−1 when j = r − 1.

Suppose d = sr−1 ··· si for some 1 6 i 6 r − 1. By Lemma 2.1.14,

Ld when j =6 i − 1, i,  Lds = Lsr−1 ··· si−1 when j = i − 1,   Lsr−1 ··· si+1 when j = i =6 r − 1,   and obviously Lsr−1sr−1 =L.

Case (3), s = sj with 1 6 j 6 r − 1 and d ∈ Dε for some 1 6 ε 6 m − 1. Let ε d = sr−1 ··· siti and s = sj. By Lemmas 2.1.13 and 2.1.14, we have

sjd when j < i − 1,  s ··· s tε when j = i − 1,  r−1 i−1 i−1 ds =   ε sr−1 ··· si+1ti+1 when j = i, s d when j > i.  j−1  0  0 We also have Lds = Ld for some d ∈ D. We complete the veriﬁcation. By the above and card D = mr, we have card W 6 mr · card L. By induction, we have card W 6 mrr! = card G.

∼ Since W = G, from now on we identify W and G, and identify si with fi for all 1 6 i 6 r − 1.

Let C denote the kernel of the group epimorphism : W → Sr deﬁned by

w = ((1w, . . . , rw))= ((a1, . . . , ar)), 36 2. COMPLEX REFLECTION GROUPS G(m, 1, r)

ε1 εr where w = ((ξ a1, . . . , ξ ar)). Hence,

ε1 εr C = {((ξ 1, . . . , ξ r))|0 6 εi 6 m − 1 ∀i}, and it is an abelian subgroup of W .

In one-line form, ti = ((1, . . . , i − 1, ξi, i + 1, . . . , r)). Thus, the subgroup C is generated by {t1, . . . , tr}. It is not diﬃcult to see that any element w ∈ W can be 0 0 written as cσ or σc , where c, c ∈ C and σ ∈ Sr. For instance,

((ξ24, ξ2, ξ23, 1))= ((ξ21, ξ2, ξ23, 4))((4, 2, 3, 1))= ((4, 2, 3, 1))((1, ξ2, ξ23, ξ24)).

Furthermore, the power of ξ of ic is the same as that of iw, while the power of ξ of ic0 is the same as that of the term ξεi in w.

Lemma 2.1.15. The group W is the product of its subgroups C and Sr, and that is W = CSr = SrC.

Lemma . ι 2.1.16 Let w = si1 ··· sip , where si1 , . . . , sip ∈ S. Denote sip ··· si1 by w . Then ι : w 7→ wι is an anti-involution of W .

Proof. Corresponding to any relation u = u0 in Deﬁnition 2.1.1, uι = (u0)ι is also a relation. Then the lemma follows from (wι)ι = w.

We can easily write down the one-line form for w−1 from w by the fact that ε −1 m−ε ι −1 iw = ξ a if and only if aw = ξ i. For an element σ ∈ Sr we have σ = σ , but this is not generally true in W .

Lemma 2.1.17. For any i ∈ r and w ∈ W , if iw = ξεa then awι = ξεi.

ε0 ε0 Proof. Let w = cσ for some c ∈ C and σ ∈ Sr. If ic = ξ i, then iw = ξ (iσ). Since iw = ξεa, we have ε0 = ε and iσ = a. Hence awι = aσιcι = (aσ−1)c = ic = ξεi. 2.2. A LENGTH FUNCTION AND REDUCED EXPRESSIONS 37

Example 2.1.18. Let w = ((ξ24, ξ2, ξ23, 1)) ∈ G(3, 1, 4). Then

ξ24 ξ2 ξ23 1 1 2 3 4 w−1 = =  1 2 3 4 4 ξ22 ξ3 ξ1     1 2 3 4 wι = 4 ξ2 ξ23 ξ21  

2.2. A Length Function and Reduced Expressions

Based on Deﬁnition 2.1.1, a length function ` : W → N and reduced expressions can be deﬁned as in 1.1.2. Instead of using the length formula [BM, 3.4] to show that the expression output by the Algorithm 2.1.7 is reduced, we take Proctor’s advice and prove the length formula based on the algorithm. First we show that n(w) has the same formula provided by [BM, 3.4].

ε1 εr Lemma 2.2.1. Let w = ((ξ a1, . . . , ξ ar)) ∈ W . Then

n(w) = εi + 2 · card{(i, j)|i < j, ai < aj, εj =6 0} + card{(i, j)|i < j, ai > aj}, i=1 X where n(w) is the number output by Algorithm 2.1.7 for w.

Proof. Let nj be the number of steps used to sort the ﬁrst j terms of w to an increasing order of positive integers in Algorithm 2.1.7. Obviously, the value of

εp εr n1 is ε1. Suppose w is sorted to ((b1, . . . , bp−1, ξ ap, . . . , ξ ar)) by np−1 steps where

0 < b1 < b2 < ··· < bp−1 and

p−1

np−1 = εi + 2 · card{(i, j)|i < j 6 p − 1, ai < aj, εj =6 0} i=1 X + card{(i, j)|i < j 6 p − 1, ai > aj}.

εp If ap < b1, then we need to swap ξ ap to the ﬁrst term and remove the power of ξ when necessary and

np = np−1 + (p − 1) + εp. 38 2. COMPLEX REFLECTION GROUPS G(m, 1, r)

If ap > bp−1, we then have

np−1 + 2(p − 1) + εp when εp =6 0, np =  np−1 when εp = 0.

If b1 < ··· < bi < ap < bi−1< ··· < bp−1, then

np−1 + (p − 1) + i + εp when εp =6 0, np =  np−1 + p − (i + 1) when εp = 0.

In all the above cases,  p

np = εi + 2 · card{(i, j)|i < j 6 p, ai < aj, εj =6 0} i=1 X + card{(i, j)|i < j 6 p, ai > aj}.

By induction, the lemma follows.

ε1 εr Lemma 2.2.2. Let w = ((ξ a1, . . . , ξ ar)), and

0 ε1−1 ε2 εr w = ((ξ a1, ξ a2, . . . , ξ ar)),

00 ε1 εi−1 εi+1 εi εi+2 εr w = ((ξ a1, . . . , ξ ai−1, ξ ai+1, ξ ai, ξ ai+2, . . . , ξ ar)).

Then

0 (a) if 1 6 ε1 6 m − 1, then n(w) = 1 + n(w ); 0 (b) if ε1 = 0, then n(w ) = n(w) + (m − 1); and (c) if iw (i + 1)w, then n(w) = 1 + n(w00).

Proof. For the element w, we deﬁne

A(w) = {(i, j)|i < j, ai > aj}, and

B(w) = {(i, j)|i < j, ai < aj, εj =6 0}.

The sets A(w0),A(w00),B(w0) and B(w00) are similarly deﬁned. Parts (a) and (b) follow from A(w) = A(w0),B(w) = B(w0) and the formula for n(w) in 2.2.1. 2.2. A LENGTH FUNCTION AND REDUCED EXPRESSIONS 39

For part (c), if iw (i + 1)w, we have the following cases to analyse. 00 Case 1, εi = εi+1 = 0 and ai > ai+1. We have A(w) = A(w ) ∪ {(i, j)} and B(w) = B(w00). 00 Case 2, εi = 0, εi+1 =6 0 and ai > ai+1. We have A(w) = A(w ) ∪ {(i, j)} and B(w) = B(w00). 00 Case 3, εi = 0, εi+1 =6 0 and ai < ai+1. We have A(w ) = A(w) ∪ {(i, j)} and B(w) = B(w00) ∪ {(i, j)}. 00 Case 4, εi =6 0, εi+1 =6 0 and ai < ai+1. We have A(w ) = A(w) ∪ {(i, j)} and B(w) = B(w00) ∪ {(i, j)}. In all four cases, we have n(w) = 1 + n(w00).

Corollary 2.2.3. If s ∈ S and w ∈ W , then n(sw) 6 n(w) + 1.

Proof. By Lemma 2.2.2, n(sw) = n(w)1 when s =6 s0; while n(s0w) = n(w)+1 or n(s0w) = n(w) − (m − 1). The corollary then follows.

Proposition 2.2.4. For any w ∈ W , the value of n(w) output in Algorithm 2.1.7 equals the length `(w). If the factors fi are replaced by si for all i, the output u is a reduced expression for w.

Proof. By Algorithm 2.1.7 and Proposition 2.1.12, the element w can be written as si1 ··· sin where n = n(w). Hence `(w) 6 n(w). On the other hand, if sj1 ··· sj` is a reduced expression of w with ` = `(w), we have

n(sj1 ··· sj` ) 6 n(sj2 ··· sj` ) + 1.

Hence, we have n(w) 6 ` + n(e) = `. Hence, n(w) = `(w) and the proposition follows.

From now on, we identify `(w) with n(w) for all w ∈ W . Thus the formula for n(w) is the length formula and the expression output from the algorithm is reduced. The following generalization of 2.2.2(a) is useful.

ε1 εr Lemma 2.2.5. Let w = ((ξ a1, . . . , ξ ar)) ∈ W . If 1 6 εi 6 m − 1 for some i ∈ r −1 and m > 1, then `(w) > `(ti w). 40 2. COMPLEX REFLECTION GROUPS G(m, 1, r)

Proof. For easy reference, we write `(w) = `1(w) + `2(w) + `3(w) with

`1(w) = εi, `2(w) = 2 · cardB(w), and `3(w) = cardA(w). i=1 X In one-line notation,

−1 ε1 εi−1 εi−1 εi+1 εr ti w = ((ξ a1, . . . , ξ ai−1, ξ ai, ξ ai+1, . . . , ξ ar)).

−1 −1 −1 When 2 6 εi 6 m−1, we have `1(ti w) = `1(w)−1, `2(ti w) = `2(w) and `3(ti w) = −1 −1 −1 `3(w). When εi = 1, we have `1(ti w) = `1(w) − 1, `2(ti w) 6 `2(w) and `3(ti w) =

`3(w). The lemma hence follows.

The following Proposition, which is a direct consequence of Algorithm 2.1.7, tells us more about the output reduced expression.

Proposition 2.2.6. Any element w ∈ W can be decomposed into the product i w = d1d2 ··· dr with `(w) = `(d1) + ··· + `(dr) and di ∈ D , where

i ε ε ε D = {si−1 ··· s1s0, si−1 ··· s1s0s1, . . . , si−1 ··· s1s0s1 ··· si−1|1 6 ε 6 m − 1}

∪ {e, si−1, si−1si−2, . . . , si−1 ··· s1}.

In particular, di is a coset representative of G(m, 1, i − 1) in G(m, 1, i), where G(m, 1, 0) is regarded as the trivial group.

The element di corresponds to the sorting sequence which put the ith term of w in position when all the terms before the ith are sorted to positive integers in ascending order. Each element in Di has a unique reduced expression. If we identify i the elements in D with their reduced expressions, then as an expression, d1d2 ··· dr is reduced. When m = 1 or 2, the elements d1, . . . , dr are all distinguished coset representatives. Thus the expression deserves to be called distinguished.

Definition 2.2.7. Let w ∈ W and ω = d1d2 ··· dr be the unique reduced expression of the decomposition in Proposition 2.2.6. Then ω is called the distinguished reduced expression for w. 2.2. A LENGTH FUNCTION AND REDUCED EXPRESSIONS 41

Remark 2.2.8. The set Dr is basically the same as the set R in the remark after [BM, 3.3]. They called a reduced expression an R-expression when it is a product with an expression in R and a reduced expression in G(m, 1, r − 1).

The algorithm gives us just one reduced expression for an element. However, any process of sorting w = ((x1, . . . , xr)) to the identity iteratively by the following operations gives us an expression for w, not necessarily reduced.

ε ε−1 (op0) If the ﬁrst entry x1 is ξ a with a ∈ r, then turn it to ξ a.

(opi) Swap the ith entry xi and the (i + 1)th entry xi+1.

More precisely, if we can sort w by the sequence of operations opi1 ··· opi` performed from left to right, then si1 ··· si` is an expression for w. Furthermore, if op0 is performed when 1 6 ε 6 m − 1 and opi is done under the condition that xi+1 ≺ xi, the length of the element is decreased by 1. If w is sorted to an element u by operations satisfying these extra conditions, the sorting sequence is called a length reduction sort. The expression corresponding to a length reduction sort which turns w to e is a reduced expression for w. This generalizes the length reduction sort for symmetric group elements on page 9.

Example 2.2.9. For the element w = ((2, 5, 4, ξ33, 6, ξ21)) in G(4, 1, 6), we can sort it to the identity by

2 2 op3op2op3op1op0op5op4op3op0op1op2op1op0.

2 2 We then have a reduced expression s3s2s3s1s0s5s4s3s0s1s2s1s0.

On the other hand, if si1 ··· si` is a reduced expression for w, we can sort w to the identity by opi1 ··· opi` .

Proposition 2.2.10. There is a one-one correspondence between the set of all reduced expressions for an element w =6 e and the set of length reduction sorts which turn w to the identity.

Proof. Suppose the length reduction sort opi1 ··· opi` turns w to the identity, and that w is sorted to w1 by opi1 . If w1 = e, we have w = si1 . If w1 =6 e, then 42 2. COMPLEX REFLECTION GROUPS G(m, 1, r)

w = si1 w1 and `(w) = 1 + `(w1). By induction si1 ··· si` is a reduced expression for w.

ε1 εr For the converse, let si1 ··· si` be a reduced expression for w = ((ξ a1, . . . , ξ ar)), and u = si2 ··· si` . Thus, u is reduced and `(w) = 1 + `(u). If i1 = 0, then

−1 ε1−1 ε2 εr u = s0 w = ((ξ a1, ξ a2, . . . , ξ ar)).

By formulae for `(w) and `(u), we have 1 6 ε1 6 m − 1. Hence w turns to u by the length reduction sort op0.

When 1 6 i1 6 r − 1, if i1w ≺ (i1 + 1)w then i1u (i1 + 1)u. By Lemma 2.2.2(c), this contradicts the fact that `(w) = 1 + `(u). Thus, w turns to u by the length reduction sort opi1 . Iteratively, opi1 ··· opi` is a length reduction sort which turns w to the identity. The proposition then follows.

The above proposition implies that ω is a reduced expression if and only if it corresponds to a length reduction sort. This provides us with a simple and combinatorial way to check whether an expression is reduced.

2.3. Weak Exchange Condition

If we take W = G(m, 1, r) and S = {s0, s1, . . . , sr−1}, the deletion condition 1.2.10 and the exchange condition 1.2.11 are not true in general. When m > 3, deleting two factors of s0 from an expression of an element, we cannot expect to get another expression for the same element. Hence, we cannot expect the exchange condition to be true when s = s0. The following example shows that, in general, the deletion condition is not true even if we are allowed to delete any number of factors and the exchange condition is not true even when s ∈ S \{s0}.

Example 2.3.1. Using the notation in 2.1.1, we let t = s0 and s = s1 be the generators for G(3, 1, 2). Let u = ((ξ2, ξ21)) which has only one reduced expression tst2. Now, consider the expression st2sts for u. We cannot get an expression for u by deleting one or more factors from st2sts. Thus, the deletion condition fails. 2.3. WEAK EXCHANGE CONDITION 43

Let w = ((ξ1, ξ22)) which has an expression st2st. We have ws = tst2 and `(ws) < `(w). We cannot get an expression for w by exchanging any factor of st2st with a right factor s. In this sense, the exchange condition does not hold.

However, the second paragraph of the exchange condition theorem does hold for W .

Proposition 2.3.2 (weak exchange conditions). Let w ∈ W and s ∈ S. Then (a) w has a reduced expression starting with s if and only if `(s−1w) < `(w), (b) w has a reduced expression starting with s−1 if and only if `(sw) < `(w), (c) w has a reduced expression ending with s if and only if `(ws−1) < `(w), (d) w has a reduced expression ending with s−1 if and only if `(ws) < `(w).

ε1 εr Proof. “Only if” parts are obvious. For “if” parts, let w = ((ξ a1, . . . , ξ ar)).

First we consider the case that s = s0. In part (a), we see that ε1 =6 0, otherwise `(s−1w) = `(w) + m − 1 by the length formula. By Algorithm 2.1.7, w has a reduced expression starting with s0. In part (b), we have ε1 = m − 1. By the algorithm again, m−1 −1 w has a reduced expression starting with s0 = s0 .

When s = si ∈ S \{s0}, parts (a) and (b) are identical. Since `(sw) < `(w), by 2.2.2(c) we have iw (i + 1)w and `(sw) = `(w) + 1. If ω is a reduced expression for sw, then sω is a reduced expression for w. Parts (c) and (d) of the lemma follow from applying ι to the ﬁrst two.

When w and s satisfy one of the weak exchange conditions, although we cannot exchange one of the factors of any expression for w by s or s−1, there is a reduced expression for w that one or m − 1 factors are exchanged by s or s−1. If w is an element in a Coxeter group and s is a generator such that `(s−1w) = `(ws) and `(s−1ws) = `(w), then s−1ws = w (see [GP2, 1.2.6]). This is not true in

ε1 εr G(m, 1, r) when m > 2 and r > 1. Let s = si ∈ S \{s0} and w = ((ξ a1, . . . , ξ ar)). −1 If {ai, ai+1} = {i, i + 1}, εi, εi+1 =6 0, and εi =6 εi+1, then we have `(s w) = `(ws) and `(s−1ws) = `(w), but

−1 ε1 εi+1 εi εr s ws = ((ξ a1, . . . , ξ ai, ξ ai+1, . . . , ξ ar)) =6 w. 44 2. COMPLEX REFLECTION GROUPS G(m, 1, r)

We are going to show that this is the only exception.

ε1 εr −1 Lemma 2.3.3. For m > 2, let s ∈ S and let w = ((ξ a1, . . . , ξ ar)). If `(s w) = `(ws) and `(s−1ws) = `(w), then either

(a) s−1ws = w, or

(b) s = si =6 s0, {ai, ai+1} = {i, i + 1}, εi, εi+1 =6 0, and εi =6 εi+1.

Proof. We need to show if `(s−1w) = `(ws), `(s−1ws) = `(w) and s−1ws =6 w, then (b) is true. −1 Suppose that s = s0. When a1 = 1, we have w = s ws. When a1 =6 1, let iw = ξν1. Therefore,

ε1 ν w = ((ξ a1, . . . , ξ 1,... )), and

−1 ε1−1 ν+1 s ws = ((ξ a1, . . . , ξ 1,... )).

−1 Since `(s ws) = `(w), we have either (i) ε1 = 0 and ν = m−1; or (ii) 1 6 ε1 6 m−1 and 0 6 ν 6 m − 2. In case (i), `(s−1w) = `(w) + m − 1 but `(ws) = `(w) − m + 1. In case (ii), `(s−1w) = `(w) − 1 but `(ws) = `(w) + 1. In both cases, `(s−1w) =6 `(w).

εi εi+1 Suppose s = si for some 1 6 i 6 r −1. If `(siw) > `(w), we have ξ ai ≺ ξ ai+1.

If {ai, ai+1}= 6 {i, i + 1}, we claim that

εi εi+1 εi εi+1 ξ ai ≺ ξ ai+1 ⇒ (ξ ai)si ≺ (ξ ai+1)si, for all εi, εi+1.

For the case {ai, ai+1} ∩ {i, i + 1} = ∅, the implication follows from the fact that if ε ε a∈ / {i, i + 1} then ξ asi = ξ a. In case of {ai, ai+1} ∩ {i, i + 1} is a singleton, one of the possibilities is ai = i but ai+1 =6 i + 1. Then we have

εi εi εi εi+1 εi+1 (ξ ai)si = (ξ i)si = ξ (i + 1) ≺ ξ ai+1 = (ξ ai+1)si, because i and i + 1 are consecutive integers. The other possibilities follow similarly.

Thus iwsi ≺ (i + 1)wsi, and so `(siwsi) = 1 + `(wsi) =6 `(w) if `(wsi) = `(siw).

Similar arguments apply in the case that `(siw) < `(w), and hence {ai, ai+1} = −1 {i, i + 1}. From s ws =6 w, we have εi =6 εi+1. If only one of εi, εi+1 is zero, we can 2.4. BRUHAT ORDER 45 check by the length formula and the elements in one-line form that

`(siw) =6 `(wsi) when ai = i + 1, ai+1 = i,

`(siwsi) =6 `(w) when ai = i, ai+1 = i + 1.

The lemma hence follows.

2.4. Bruhat Order

Contrary to the Coxeter groups (see p. 11), there is no nice subword order on G(m, 1, r). Since we often refer to the element represented by a given expression, we write ω = w if ω is an expression for w.

2 2 2 Example 2.4.1. Let w0 = ((ξ2, ξ 1, 3)), w1 = ((ξ 1, ξ2, 3)) and w2 = ((ξ 3, ξ2, 1)) be elements in G(4, 1, 3). The following table shows all the reduced expressions for these elements.

w Reduced expression(s) for w

2 w0 ω0 = s0s1s0 2 0 2 00 w1 ω1 = s0s1s0s1, ω1 = s1s0s1s0, ω1 = s0s1s0s1s0 2 0 2 w2 ω2 = s0s1s0s2s1, ω2 = s0s1s2s0s1

0 Firstly, ω0 is a subword of ω1, but it is not a subword of ω1. By the deﬁning relation of G(4, 1, 3), any non-reduced expression of w0 has at least `(w0) + 2 = 6 factors. Hence, no expression for w0 can be a subword of ω1. 0 On the other hand, ω0 is neither a subword of ω2 nor ω2. The only six factor 0 subwords of reduced expressions for w2 are ω2 and ω2. Thus, there is no expression for w0 which is a subword of any reduced expression for w2. Even if we deﬁne a weaker subword relation by:

w0 w if ω0 is a subword of ω for some reduced expressions ω, ω0

such that ω = w, and ω0 = w0, 46 2. COMPLEX REFLECTION GROUPS G(m, 1, r) the above shows that w0 w1 and w1 w2 but w0 6 w2. This subword relation is not transitive.

If we borrow Deﬁnition 1.2.13 directly for the deﬁnition of a Bruhat order on W = G(m, 1, r), we should take T as the set of conjugates of S in W . The following example shows that even when w < tw with respect to such Bruhat order for some w ∈ W and t ∈ T, no expression for w is a subword of any reduced expression for tw.

Example 2.4.2. In G(3, 1, 3), take w = ((ξ23, 2, 1))which has only two reduced ex- 2 2 t 2 −1 T pressions s0s1s2s1 and s0s2s1s2. Let = ((ξ3, 2, ξ 1))= s0s2s1s2s0 ∈ . The element tw = ((ξ1, 2, ξ3)) has four reduced expressions, namely s0s2s1s0s1s2, s2s0s1s0s1s2, s2s1s0s1s0s2 and s2s1s0s1s2s0. We can see `(w) < `(tw) but none of the reduced expressions of w is a subword of any reduced expression of tw. Moreover, any other expression of w will have more than ﬁve factors and it cannot be a subword of an expression of tw.

Now, we deﬁne the Bruhat order on W with a new choice of T which consists of all the conjugates of the generators by elements in Sr.

Definition 2.4.3 (Bruhat order on G(m, 1, r)). Let T = σ−1Sσ. For w, w0 ∈ σ∈Sr 0 0 W , deﬁne w < w if there exist elements w = w0, w1, . . . , wk =Sw ∈ W such that, for all 1 6 i 6 k,

(a) `(wi−1) < `(wi), and −1 T (b) wiwi−1 ∈ .

We write w0 6 w if w0 < w or w0 = w. Obviously, 6 is a partial order on W . From deﬁnition, w0 < w implies `(w0) < `(w). Moreover, (b) can be replaced by

0 (b ) wi = twi−1, for some t ∈ T.

By deﬁnition, t ∈ T if and only if t = ti for some 1 6 i 6 r or t is a transposition in Sr. Because of this restriction, the Bruhat order has a one-sided nature in contrast to that on Coxeter groups. 2.4. BRUHAT ORDER 47

Example 2.4.4. Let w = ((ξ21, ξ2)) and w0 = ((ξ2, ξ21)) be elements in G(3, 1, 2). 0 0 0 We have `(w ) + 1 = `(w) and w = s1w and hence w < w. The possible elements 0 0 0 t ∈ T = {t1, t2, s1} such that `(w t) > `(w ) are t2 and s1. However, both w t2 and ws1 have the same length as w, but they are not w. Hence, we cannot ﬁnd a sequence 0 of elements w = w0, w1, . . . , wk = w such that `(wi−1) < `(wi) and wi = wi−1ti for some ti ∈ T.

Our choice of T does allow us to relate the Bruhat order with subwords.

Proposition 2.4.5. Let w0 ∈ W and t ∈ T. If w0 < w = tw0, then there exist reduced expressions ω0 and ω such that ω0 = w0, ω = w and ω0 is a subword of ω.

Proof. We have two cases for t ∈ T.

0 ε1 εr 0 0 Case 1 — t = ti, let w = ((ξ a1, . . . , ξ ar)). By `(w ) < `(tiw ), we have 0 6 0 εi 6 m − 2. In one-line notation, w and w are identical except the ith terms. The distinguished reduced expressions for w and w0 can be written as

0 0 ω = d1 ··· di−1didi+1 ··· dr and ω = d1 ··· di−1didi+1 ··· dr.

εi+1 0 εi If εi =6 0, then di = si−1 ··· s0 u and di = si−1 ··· s0 u where u is an expression in D = {1, s1, . . . , s1 ··· si−1}. If εi = 0, then di = si−1 ··· s0u for some u ∈ D and 0 0 di ∈ {1, si−1, . . . , si−1 ··· s1}. Hence di is a subword of di and the proposition follows in this case. Case 2 — t is a transposition i 7→ j 7→ i where 1 6 i < j 6 r. We write

0 w = ((x1, . . . , xi−1, xi, xi+1, . . . , xj−1, xj, xj+1, . . . , xr)),

w = ((x1, . . . , xi−1, xj, xi+1, . . . , xj−1, xi, xj+1, . . . , xr)).

0 By length reduction sort, we can arrange the terms xi+1, . . . , xj−1 of w in ascending order with respect to the order ≺ . Then we move all the terms preceding xi and xj in front of xi, and move all the terms succeeding xi and xj after xj. Denote the resulting element by u0. The same sorting sequence is also a length reduction sort when it is applied to w. Denote the result by u. We now have a reduced expression ω1 such 48 2. COMPLEX REFLECTION GROUPS G(m, 1, r)

0 0 0 0 that w = ω1u , w = ω1u and `(w ) = `(ω1) + `(u ), `(w) = `(ω1) + `(u). Moreover, 0 u and u are identical except the terms xi and xj. Let xk1 ··· xkp be terms between 0 xi and xj. Since `(w ) < `(w), we can see that xi ≺ xj. Thus xi, xk1 , ··· , xkp , xj 0 are p + 2 consecutive terms of u , while xj, xk1 , ··· , xkp , xi are consecutive terms of u. Since xi ≺ xk1 ≺ · · · ≺ xkp ≺ xj, we have a length reduction sort which turns u 0 to u by swapping xi to the front of xj and then xj to the back of xkp . If κ is the 0 corresponding reduced expression for these sorting steps which turn u to u and ω2 is 0 0 a reduced expression for u , then ω1ω2 is a reduced expression for w and ω1κω2 is a reduced expression for w. The proposition then follows in this case.

We also have a weaker version of the strong exchange condition 1.2.12 for elements in W .

Proposition 2.4.6. Let w ∈ W and t ∈ T. If `(tw) < `(w), then there exist reduced expressions ω0 and ω for tw and w, respectively, such that ω0 is a subword of ω.

Proof. In the case that t = ti where 1 6 i 6 r, because of `(tw) < `(w), we can write

m−1 w = ((x1, . . . , xi−1, ξ ai, xi+1, . . . , xr)), and

tw = ((x1, . . . , xi−1, ai, xi+1, . . . , xr)).

The distinguished reduced expression of tw is obviously a subword of that of w. 0 When t is a conjugate of s1 by an element in Sr, we put w = tw. Thus we have w = tw0 and `(w0) < `(w), then we can apply Proposition 2.4.5.

The following lemma is useful in the sequel.

ε1 εr Lemma 2.4.7. Let w = ((ξ a1, ··· , ξ ar)) ∈ W .

(a) If iw (i + 1)w, then siw < w.

−εi (b) If εi =6 0, then ti w < w. 2.4. BRUHAT ORDER 49

Proof. Since iw (i + 1)w, by 2.2.2(c), we have `(siw) < `(w). Let w0 = siw, w1 = w. We have w1 = siw and si ∈ T, and the ﬁrst part follows.

−εi ε1 εr For the second part, let w0 = ti w = ((ξ a1, ··· , ai, ··· , ξ ar)). We have elements

2 εi w0, w1 = tiw0, w2 = ti w0, ··· , wεi = ti w0 = w. For each 1 6 j 6 εi, by 2.2.5 we get

`(wj−1) < `(wj). Since ti ∈ T, the second part follows. CHAPTER 3

Minimal Length Elements in Conjugacy Classes

As mentioned in Section 1.6, based on their reducibility theorem, Geck and Pfeiﬀer have proved that in an Iwahori-Hecke algebra, the character value is constant on the set of elements corresponding to minimal length elements in a conjugacy class in the Coxeter group. The character values of the other elements in the Iwahori-Hecke algebra can be calculated from the character values on the elements corresponding to a complete set of minimal length representative in conjugacy classes. We study the minimal length elements in conjugacy classes and prove an analogue of Geck-Pfeiﬀer reducibility theorem for W in this Chapter.

3.1. Conjugacy Classes of G(m, 1, r)

It is well-known that conjugacy classes of the wreath product, GoSr, of a group G by the symmetric group can be described by a matrix of non-negative integers (see for example [JK, 4.2]). We shall brieﬂy discuss the special case G(m, 1, r) = (Z/mZ)oSr. Based on the two-line notation, we can generalize the arguments used in symmetric groups. For any element u = ((1u, 2u, . . . , ru)) ∈ W , we can write

1u 2u ··· ru u−1 = .  1 2 ··· r    As a result, we have

1u 2u ··· ru (3.1.1) u−1wu = . 1wu 2wu ··· rwu   For a ﬁxed w ∈ W , start from an integer a0 ∈ r. For any positive integer k deﬁne

εk ak ∈ r and εk ∈ {0, 1, ··· , m − 1} by (ak−1)w = ξ ak. Since ak ∈ r for all k, there exists a smallest positive integer l such that a0, . . . , al−1 are all distinct but al = a0. 50 3.1. CONJUGACY CLASSES OF G(m, 1, r) 51

l ε l Thus, we have a0w = ξ a0 and ε ≡ i=1 εi (mod m). Under w, we have P ε1 ε1+ε2 ε1+···+εl−1 ε a0 7→ ξ a1 7→ ξ a2 7→ · · · 7→ ξ al−1 7→ ξ a0.

k k+1 This deﬁnes an element u in W which maps a0w to a0w for any positive integer k, while ju = j when j is not in {a0, . . . , al−1}. If we start from another element ak in the set {a0, . . . , al−1}, we have

εk+1 εk+1+···+εl εk+1+···+εl+ε1 ε ak 7→ ξ ak+1 7→ · · · 7→ ξ a0 7→ ξ a1 7→ · · · 7→ ξ ak, which deﬁnes the same element. We then call u a (ε, l)-cycle, or a cycle with label ε and length l.

If there is an integer b0 ∈ r \{a0, . . . , al−1}, we can repeat the process and get another cycle factor of w, say v which is given by

ν1 ν1+ν2 ν1+···+νn ν b0 7→ ξ b1 7→ ξ b2 7→ · · · 7→ ξ bn = ξ b0.

The two cycles u and v are said to be disjoint because

{a0, a1, . . . al−1} ∩ {b0, b1, . . . bn−1} = ∅.

Iteratively, we decompose w uniquely as a product of mutually commutative disjoint cycle factors and this product is called the disjoint cycle decomposition of w. In the disjoint cycle decomposition of w, suppose that the number of (ε, l)-cycles 0 0 is aεl for each pair (ε, l). We deﬁne an m × r matrix (aεl) such that aεl = aεl for ε =6 0 0 and aml = a0l. The matrix is called the cycle type of the element w. Furthermore, each row ai∗ corresponds uniquely to a partition with aij parts of value j. Thus we can label this cycle type by an m-tuple of partitions, or an m-partition.

Definition 3.1.2. An m-partition of r is an m-tuple λ = (λ(1), . . . , λ(m)) such (i) that each λ is a partition of some positive integer αi or an empty sequence with m (i) 1 αi = r. Here αi is taken to be zero when λ is empty. P 52 3. MINIMAL LENGTH ELEMENTS IN CONJUGACY CLASSES

Now, suppose w has a cycle factor i 7→ iw 7→ iw2 7→ · · · 7→ iwl = ξεi. Then by 3.1.1, u−1wu will have a cycle factor deﬁned by

iu 7→ (iw)u 7→ (iw2)u 7→ · · · 7→ (iwl)u = ξε(iu) which is also of length l and label ε. Thus, the cycle type of u−1wu is the same as that of w. Conversely, if w, w0 have the same cycle type, there is a one-one correspondence between the cycles in their disjoint cycle decompositions. In the correspondence, each (ε, l)-cycle of w

ε1 ε1+ε2 ε1+···+εl ε a0 7→ ξ a1 7→ ξ a2 7→ · · · 7→ ξ al = ξ a0, is associated with a cycle of w0

ν1 ν1+ν2 ν1+···+νl ε b0 7→ ξ b1 7→ ξ b2 7→ · · · 7→ ξ bl = ξ b0.

We then deﬁne an element u such that, for each pair of corresponding cycles and 0 6 i 6 l

ε1+···+εi ν1+···+νi (ξ ai)u = ξ bi.

Since ε1 + ··· + εl = ν1 + ··· + νl, the element u is well deﬁned. Moreover,

−1 −(ν1+···+νi)+(ε1+···+εi) (bi)u wu = ξ (aiw)u

−(ν1+···+νi)+(ε1+···+εi+εi+1) νi+1 = ξ (ai+1)u = ξ bi+1, and so w0 = u−1wu. As a result we have the following special case of [JK, 4.2.8].

Proposition 3.1.3. Two elements in W are conjugate to each other if and only if they have the same cycle type.

Example 3.1.4. Let w = ((ξ23, ξ5, ξ4, ξ1, ξ2)) and w0 = ((2, ξ21, 4, ξ5, 3)) be elements in G(3, 1, 5). Thus w has two cycle factors 1 7→ ξ23 7→ 4 7→ ξ1 and 2 7→ ξ5 7→ 3.1. CONJUGACY CLASSES OF G(m, 1, r) 53

ξ22. The other element w0 has cycle factors 1 7→ 2 7→ ξ21 and 3 7→ 4 7→ ξ5 7→ ξ3. Both of them have a (2, 2)-cycle and a (1, 3)-cycle and they have the cycle type

0 0 1 0 0 0 1 0 0 0 .   0 0 0 0 0     The corresponding 3-partition is ((3), (2), −). Deﬁne u such that

(1)u = 3, (ξ23)u = 4, (4)u = ξ5, (2)u = 1, (ξ5)u = 2, and that is 1u = 3, 2u = 1, 3u = ξ4, 4u = ξ5, 5u = ξ22.

It is straightforward to check that w0 = u−1wu.

Following [GP2, 3.4.2], the (ε, l)-cycle of the form

k 7→ k + 1 7→ · · · 7→ k + l − 1 7→ ξεk, for some positive integer k, is called a block of label ε and length l. We use a slightly (ε) different way to index this block and denote it by bk,l . Each block has a unique (0) (ε) ε reduced expression. We define tk to be the identity. When ε > 1, we write t1 = t1 (ε) (ε) ε (ε) and tk+1 = sktk sk. Note that as expressions, tk and tk are different when k =6 1. (ε) (ε) The reduced expression for bk,1 is simply tk . When l > 1, the reduced expression (ε) (ε) (ε) for bk,l is sk+l−2 ··· sk+1sktk . Without ambiguity, we identify bk,l with its unique reduced expression.

Definition 3.1.5. An element w ∈ W is said to standard if all cycles in the disjoint cycle decomposition of w are blocks.

Since disjoint cycles are commutative, a standard element can be written as

b(ε1) b(εp) k1,l1 ··· kp,lp . where k1 = 1, kj+1 = kj + lj, and (l1, . . . , lp) is a composition of r. It is determined by the two integer vectors ~ε = (ε1, . . . , εp) and ~l = (l1, . . . , lp), hence can be denoted by b~ε. ~l 54 3. MINIMAL LENGTH ELEMENTS IN CONJUGACY CLASSES

Remark 3.1.6. The standard elements deﬁned above are basically the same as those deﬁned in [HR, 2.10] by Halverson and Ram.

Example 3.1.7. In G(3, 1, 9), the following is a standard element

1 2 3 4 5 6 7 8 9 , 2 1 4 5 6 7 ξ23 8 ξ9   (0,2,0,1) (0) (2) (0) (1) because it is b(2,5,1,1) = b1,2b3,5b8,1b9,1.

Definition 3.1.8. Let b~ε be a standard element, where ~ε = (ε , . . . , ε ) and ~l 1 p ~ l = (l1, . . . , lp). The standard element is said to be specially standard if either εi = 0 for all 1 6 i 6 p, or there exists an integer 1 6 n 6 p such that

(a) l1 6 l2 6 ··· 6 ln, εi = 0 for all i > n, 1 6 ε1, . . . , εn 6 m − 1, and

(b) for any 1 6 i < j 6 n, we have li = lj implies εi 6 εj.

Remark 3.1.9. Start from a standard element b~ε in a conjugacy class C. If ~l

π ∈ Sp then we have a natural right action of π on ~ε and ~l. For instance, we have b(εi) 6 6 ~επ = (ε1π, . . . , εpπ). Since ki,li is a (εi, li)-cycle for each 1 i p, the standard 0 element b~επ is also in C. Conversely, since two standard elements b~ε and b~ε in the ~lπ ~l ~l 0 same conjugacy class have the same cycle type, there exists an element π ∈ Sp that 0 b~ε = b~επ. In particular, two specially standard elements in the same conjugacy class ~l 0 ~lπ are identical except the blocks with label 0. More precisely, the lengths of all label 0 blocks in one is a permutation of those in the other.

Remark 3.1.10. Similar to symmetric groups, the groups G(m, 1, r) with r > 2 can also be presented as one generated by two elements, [Co, 5.4]. More precisely, a G(m, 1, r) is generated by {s, t} subject to

tmr = s2 = (ts)m(r−1) = t−rstrs = (t−1sts)3 = (t−jstjs)2 = e,

1 where j = 2,..., [ 2 r]. Here [x] is the largest integer less than or equal to x for a real number x. 3.2. MINIMAL ELEMENTS IN CONJUGACY CLASSES 55

We can identify s = ((2, 1, 3, . . . , r)) and t = ((2, 3, . . . , r, ξ1)). Note that s = (1 7→ 2 7→ 1) and t = (1 7→ 2 7→ 3 7→ · · · 7→ r 7→ ξ1) are elements with the smallest and largest order among the non-trivial cycles. Furthermore, the generators of W can be written in terms of s and t as

−(i−1) i−1 si = t st for 1 6 i 6 r − 1,

−1 r−1 r s0 = s1s2 ··· sr−1t = (st ) t , which can be veriﬁed easily.

3.2. Minimal Elements in Conjugacy Classes

We are going to generalize the Geck-Pfeiﬀer reducibility theorem 1.3.7 to the group W . Before that, we have to generalize the relation → of 1.3.5 on a Coxeter group to a relation on W .

Definition 3.2.1. The preorder relation → on W is deﬁned by w → w0 for 0 0 w, w ∈ W if w = w0, . . . , wn = w is a sequence of elements such that for all i, s s ki ki 0 0 −1 we have wi−1 −→ wi for some ski ∈ S. Here, w −→ w if w = ski wski with −1 0 `(ski w) < `(w) or `(wski ) < `(w). Also, we say w → w via the expression sk1 ··· skn .

Lemma 3.2.2. For w, w0 ∈ W , if w → w0 in W then `(w0) 6 `(w).

Proof. It suﬃces to prove the lemma for the case w → w0 via s ∈ S. If `(s−1w) < `(w), then w has a reduced expression starting with s by 2.3.2(a). Hence,

`(w0) = `(s−1ws) 6 `(s−1w) + 1 = `(w).

For the case `(ws) < `(w), if s =6 s0, then we have

`(w0) = `(sws) 6 `(ws) + 1 = `(w).

−1 Otherwise, by 2.3.2(d), w has a reduced expression ending with s0 . Hence `(ws0) + m − 1 = `(w), and then

0 −1 `(w ) = `(s0 ws0) 6 `(ws0) + m − 1 = `(w). 56 3. MINIMAL LENGTH ELEMENTS IN CONJUGACY CLASSES

The lemma then follows.

Remark 3.2.3. Although it is not obvious, when W is a Coxeter group the relation → defined above is in fact the one defined by Geck and Pfeiffer. For clarity, the relation defined in 1.3.5 is still denoted by →, while the one defined in 3.2.1 when W is a Coxeter group is temporarily denote by . In a Coxeter system (W, S), let w ∈ W, s ∈ S and let w sws. If `(sw) < `(w), then `(sw) = `(w) − 1. Together with the fact that `(sws) = `(sw) 1, we conclude that `(sws) 6 `(w). Similarly, we also have `(sws) 6 `(w) if `(ws) < `(w). Hence, w w0 implies w → w0. For the converse part, we only need to show that w → sws implies w sws, for all w ∈ W, s ∈ S. Assume the contrary that w → sws, but `(sw) > `(w) and `(ws) > `(w). Since w → sws, we have `(sws) 6 `(w). The conditions `(sw) > `(w) and `(ws) > `(w) imply `(sws) > `(w) and `(sw) = `(ws). In Coxeter groups, `(sw) = `(ws) and `(w) = `(sws) imply w = sws. Thus the two relations → and are the same in Coxeter groups.

The following example shows that, when W = G(m, 1, r) with m > 2 and r > 1, the relation w −→s w0 in W is stronger than that w0 is conjugate to w by the generator s and `(w0) 6 `(w). We need this stronger condition in the next chapter.

2 0 2 Example 3.2.4. In G(3, 1, 2), let w = ((ξ2, ξ 1)) and w = s1ws1 = ((ξ 2, ξ1)). We 2 2 0 then have s1w = ((ξ 1, ξ2)) and ws1 = ((ξ1, ξ 2)). In this case, we have `(w) = `(w ), 0 but `(s1w) > `(w) and `(ws1) > `(w). Hence w 6→ w via s1. On the other hand s0 −1 s1 w −→ ((2, 1)), but ((2, 1)) 6→ s0 ((2, 1))s0 and ((2, 1)) −→ ((2, 1)). This implies that w 6→ w0 via any expression.

One can generalize ∼ of 1.3.6 in an obvious way, but we only need ∼ for Sr in the generalization of the Geck-Pfeiﬀer reducibility theorem 1.3.7 to the following.

Theorem 3.2.5. Let C be a conjugacy class of W . (a) For each w ∈ C there exists a specially standard element w0 ∈ C such that w → w0. Moreover, all specially standard elements in C are of minimal length. 3.2. MINIMAL ELEMENTS IN CONJUGACY CLASSES 57

(b) Let w and w0 be both specially standard elements in C. If σ and σ0 are products of all label 0 blocks in the disjoint cycle decompositions of w and w0, respectively, then σ ∼ σ0.

We shall prove the ﬁrst part of the theorem by explicitly constructing an element w0 of minimal length and an expression via which w → w0. Before the construction, 0 we have to be aware that w → w via an expression, not via an element. If ω1 and 0 ω2 are two expressions for the same element, the fact w → w via ω1 does not imply 0 w → w via ω2.

Example 3.2.6. In G(3, 1, 7), suppose that

0 w = w0 = ((2, 3, 4, ξ1, 6, 7, ξ5)), w = ((2, 3, ξ1, 5, 6, 7, ξ4)),

ω1 = s4s3s2s1s5s4s3s6s5s4s2s3, and ω2 = s4s3s2s1s5s4s3s2s6s5s4s3.

First, ω1 and ω2 are expressions for the same element ((4, 5, 6, 7, 1, 2, 3)). Using Lemma 2.1.4, we have

s4 w0 −→ w1 = ((2, 3, 5, 6, ξ1, 7, ξ4)) [`(w0s4) < `(w0)]

s3 −→ w2 = ((2, 4, 6, 5, ξ1, 7, ξ3)) [`(w1s3) < `(w1)]

s2 −→ w3 = ((3, 6, 4, 5, ξ1, 7, ξ2)) [`(w2s2) < `(w2)]

s1 −→ w4 = ((6, 3, 4, 5, ξ2, 7, ξ1)) [`(w3s1) < `(w3)]

s5 −→ w5 = ((5, 3, 4, 6, 7, ξ2, ξ1)) [`(w4s5) < `(w4)]

s4 −→ w6 = ((4, 3, 5, 7, 6, ξ2, ξ1)) [`(w5s4) < `(w5)]

s3 −→ w7 = ((3, 4, 7, 5, 6, ξ2, ξ1)) [`(w6s3) < `(w6)]

s6 −→ w8 = ((3, 4, 6, 5, 7, ξ1, ξ2)) [`(w7s6) < `(w7)]

s5 −→ w9 = ((3, 4, 5, 6, ξ1, 7, ξ2)) [`(w8s5) < `(w8)]

s4 −→ w10 = ((3, 5, 4, ξ1, 6, 7, ξ2)) [`(s4w9) < `(w9)] 58 3. MINIMAL LENGTH ELEMENTS IN CONJUGACY CLASSES

s2 −→ w11 = ((2, 4, 5, ξ1, 6, 7, ξ3)) [`(s2w10) < `(w10)]

s3 −→ w12 = ((2, 3, ξ1, 5, 6, 7, ξ4)) [`(s3w11) < `(w11)]

0 and w → w via ω1.

Second, w → w7 via s4s3s2s1s5s4s3. Since `(s2w7) = `(w7) + 1 and `(w7s2) = 0 `(w7) + 1, so w → w via ω2 is not true.

The following lemma is a direct consequence of the weak exchange conditions 2.3.2.

Lemma 3.2.7. Let w ∈ W and s ∈ S = {s0, s1, . . . , sr−1}. If w has a reduced expression starting with s or ending with s−1, we have w −→s s−1ws.

Let ω be a reduced expression for y ∈ W where `(y) > 1. It is not true in general that if w = yu and `(w) = `(y) + `(u) then w → y−1wy via ω.

Example 3.2.8. Let w = ((5, 3, 1, 2, 4)) ∈ G(m, 1, 5). This element w has a reduced

s2 expression s2s3s1s2s3s4, and we have w −→ ((5, 1, 2, 3, 4)). The length of ((5, 1, 2, 3, 4)) 0 0 is 4, while the length of s3((5, 1, 2, 3, 4))s3 = ((5, 1, 4, 2, 3)) = w is 6. Hence w → w via s2s3 is not true.

ε1 εr Algorithm 3.2.9 (Standard element). Let w = ((ξ a1, . . . , ξ ar)) ∈ W . This algorithm determines a standard element w0 in the conjugacy class of w in W and outputs an expression ω such that w → w0 via ω.

(a) [Initialization] Set k := 1,L := r, M := r, ω := e, w0 := w. 0 (b) [Loop] At this stage, the ﬁrst L terms of w are in Cm{1,...,L}. If the ﬁrst L terms of w0 are replaced by 1,...,L in order, the result is a standard element. (i) If k =6 L, perform (1) search for the term of the form ξ∗M in w0; assign to k the term number, and to ε the exponent of ξ of this term; (2) if k =6 L, then perform

(A) if k = M − 1 then u := e else u := sk ··· sM−2, (ε) (B) if ε =6 0 then u := tk u, 3.2. MINIMAL ELEMENTS IN CONJUGACY CLASSES 59

(C) ω := ωu, w0 := u−1w0u, (D) M := M − 1, go to (1). (ii) If M > 2 then set L := M −1, M := M −1, k := 1 and repeat step (b). (c) [Output] Now w0 is a standard element and w → w0 via ω. Output w0 and ω.

We ﬁrst illustrate the algorithm by the following example.

Example 3.2.10. Let w = ((ξ35, ξ46, ξ23, 2, ξ21, ξ34)) be an element in G(5, 1, 6). Step (a) sets k = 1, L = M = 6, ω = e (the empty string) and w0 = w. Since k =6 L, step (b)(i) is performed. As ξ46 is in the second place, step (1) sets k = 2 and ε = 4. Now, we want to move ξ46 to the ﬁrst place then get rid of the power ξ4, and move the resulting term, 6, to the ﬁfth place. All these have to be done by conjugation. As 4 0 0 0 0 0 ξ 6 = 2w ≺ 1w , there is a reduced expression for w starting with s1, so w −→ s1w s1 by 3.2.7. After the removal of ξ4, the term, 6, is the greatest with respect to ≺ . Hence, we have

w0 −→s1 ((ξ46, ξ35, ξ23, 1, ξ22, ξ34))

s4 −→0 ((6, ξ35, ξ23, ξ41, ξ22, ξ34))

−→s1 ((ξ35, 6, ξ23, ξ42, ξ21, ξ34))

−→s2 ((ξ35, ξ22, 6, ξ43, ξ21, ξ34))

−→s3 ((ξ35, ξ22, ξ44, 6, ξ21, ξ33))

−→s4 ((ξ34, ξ22, ξ45, ξ21, 6, ξ33)).

0 The algorithm does not change w as the above. Instead, step (A) assign s2s3s4 to u. (4) (4) Step (B) turns u to t2 s2s3s4. Step (C) sets ω = t2 s2s3s4 then calculate and rename u−1w0u = ((ξ34, ξ22, ξ45, ξ21, 6, ξ33)) as w0. Step (D) sets M = 5. (4) When we get back to step (1), we have k = 3 and ε = 4. Step (2) sets u = t3 s3, (4) (4) 0 3 2 2 2 4 ω = t2 s2s3s4t3 s3, w = ((ξ 3, ξ 2, ξ 1, 5, 6, ξ 4)), and M = 4. As ξ 5 is the smallest and 5 is the greatest among the ﬁrst four terms, we also have

w −→ ((ξ33, ξ22, ξ21, 5, 6, ξ24)). 60 3. MINIMAL LENGTH ELEMENTS IN CONJUGACY CLASSES

When repeating step (1), we have k = L = 6. Thus, step (2) is skipped and 0 the last three terms of w form a block while the ﬁrst three terms are in C5{1, 2, 3}. Step (ii) sets L = M = 3, k = 1 and then step (b) is repeated.

Proposition 3.2.11. For any element w ∈ W , there exists a standard element w0 in the conjugacy class containing w such that w → w0.

Proof. We only need to show the validity of the above algorithm. Before enter- 0 ing the loop, w is of the form wLb, where wL is in the parabolic subgroup generated by {t1, s1, . . . , sL−1} and b is a standard element in the subgroup generated by ε {tL+1, sL+1, . . . , sr−1}. If the term ξ L is at the Lth place, then we have a block of length 1 and label ε. The algorithm then skips step (b)(i) and re-enter the loop with 0 0 0 (ε) (ε) w = wL−1b , where b = tL b = bL,1b and wL−1 is in the parabolic subgroup gener- ε ated by {t1, s1, . . . , sL−2}. Otherwise, ξ L is in the kth place for some 1 6 k 6 L − 1. If ε =6 0 then ξεL is the smallest term, with respect to ≺ , among the first L−1 terms of w0. If ξεL is not in the first place, then k =6 1 and w0 has a reduced expression 0 0 starting with sk−1 and w −→ sk−1w sk−1. The left multiplication by sk−1 moves this term one place to the left while the right multiplication by sk−1 does not move L to elsewhere. Among the first L − 1 terms, the smallest term ξεL is now in the (k − 1)th place. Similarly, we move ξεL to the first place and remove the power ξε by conjugation. The first term L is then the biggest among the first L − 1 terms, and we move it to the (L − 1)th place by conjugation. If we denote the resulting element 0 (ε) by w1, then we have w → w1 via u = tk sk ··· sL−2. If ε = 0, we can simply move 0 the term L from the kth place to the (L − 1)th place. In this case, we have w → w1 ν via u = sk ··· sL−2. If the term ξ (L − 1) is at the Lth place, we then have a block. Otherwise, similar to the above, we remove the power of ξ if necessary and than move it to the (L − 2)th place and so on, until we get a block. We then re-enter the loop when necessary. When the algorithm terminates, we get a standard element in the conjugacy class containing w. 3.2. MINIMAL ELEMENTS IN CONJUGACY CLASSES 61

Lemma 3.2.12. Let i, k, l, ε be integers such that 1 6 i 6 r − 1, 2 6 l 6 r, 0 6 ε 6 m and k ∈ r. Then

(ε) (ε) (a) sibk,l = bk,l si+1 if k 6 i < k + l − 2, (ε) (ε) (b) bk,l sk+l−2 = sk+l−3sk+l−2bk,l−1 if l > 2. Moreover, if each block is identiﬁed with its unique reduced expressions, all the expressions in the above equations are reduced.

Proof. In two-line notation,

(ε) ··· k k + 1 ··· i i + 1 ··· k + l − 2 k + l − 1 ··· bk,l = . ··· k + 1 k + 2 ··· i + 1 i + 2 ··· k + l − 1 ξεk ···   By Lemma 2.1.4, ﬁrst equality follows. Moreover, swapping i + 1 and i + 2 in the (ε) second row gives us a longer element. Hence, if we treat bk,l as the reduced expression (ε) sk+l−2 ··· sktk , the expressions on both sides of (a) are reduced. (ε) For the second one, bk,l sk+l−2 can be written as

(ε) (ε) (ε) sk+l−2sk+l−3bk,l−2sk+l−2 = sk+l−2sk+l−3sk+l−2bk,l−2 = sk+l−3sk+l−2bk,l−1, and these expressions are obviously reduced.

b(ν) b(ε) Let k,l1 and k+l1,l2 be two blocks in a standard element w. If these two blocks b(ε) b(ν) 0 are replaced by k,l2 and k+l2,l1 , the resulting element w is obviously conjugate to w and is still standard. This operation is called block exchange. The above lemma is important in proving the following three block exchange lemmas. The ﬁrst two exchange lemmas are complex reﬂection group analogues of [GP1, 2.4(b)] and [GP2, 3.4.5], and the third one is basically [GP1, 2.4(a)].

Lemma 3.2.13 (First block exchange lemma). Let 1 6 ε 6 m − 1 and l1, l2 be positive integers. Then b(0) b(ε) b(ε) b(0) k,l1 k+l1,l2 → k,l2 k+l2,l1 .

Proof. We prove the lemma for the case k = 1 and the general case follows by changing indices. First, we consider the case l1 = 1. 62 3. MINIMAL LENGTH ELEMENTS IN CONJUGACY CLASSES

b(0) b(ε) b(ε) When l1 = 1, we have 1,1 = e and 2,l2 = 1,l2+1s1. If l2 = 1 as well, then (ε) (ε) (ε) (ε) (0) b2,1 = s1b1,1s1 → b1,1 = b1,1b2,1. If l2 > 1, by Lemmas 3.2.7 and 3.2.12,

s − s b(0)b(ε) s1 b(ε) s2 l2 1 b(ε) b(ε) l2 b(ε) b(0) 1,1 2,l2 −→ 1,l2+1s2 −→ · · · −→ 1,l2+1sl2 = sl2 1,l2 sl2 −→ 1,l2 l2+1,1.

For the case l1 > 1, let σi = si+l1−1 ··· si where 1 6 i 6 l2. We start with b(0) b(ε) b(0) b(ε) −1 w0 = 1,l1 l1+1,l2 = 1,l1 1,l1+l2 σ1 . By Lemma 3.2.7, if l2 > 1 we have

s l1 b(0) b(ε) b(0) b(ε) w0 −→ sl1 1,l1 1,l1+l2 s1 ··· sl1−1 = 1,l1+1 1,l1+l2 s1 ··· sl1−1.

Since l1 > 1, the above can be conjugated by sl1−1 and, by Lemma 3.2.12, it

s − l1 1 b(0) b(ε) b(0) b(ε) −→ sl1−1 1,l1+1 1,l1+l2 s1 ··· sl1−2 = 1,l1+1 1,l1+l2 sl1+1 · s1 ··· sl1−2.

Since the last expression is also reduced, we can apply a similar argument until

σ1 b(0) b(ε) b(0) b(ε) b(0) b(ε) −1 w0 −→ 1,l1+1 1,l1+l2 s3 ··· sl1+1 = 2,l1 s1 1,l1+l2 s3 ··· sl1+1 = 2,l1 1,l1+l2 σ2 .

b(0) b(ε) −1 6 6 If necessary, we further let wi = i+1,l1 1,l1+l2 σi+1 for 1 i l2 − 1. Similarly, we have wi−1 → wi via σi for 1 6 i 6 l2 − 1. On the other hand, including the case l2 = 1, we have

(ε) −1 wl2−1 = sl1+l2−2 ··· sl2 · sl1+l2−1 ··· sl2 sl2−1 ··· s1t1 · σl2

(ε) −1 = sl1+l2−1 ··· sl2 · sl1+l2−1 ··· sl2+1 · sl2−1 ··· s1t1 · σl2 b(0) b(ε) −1 b(ε) b(0) −1 = σl2 l2+1,l1 1,l2 σl2 = σl2 1,l2 l2+1,l1 σl2 .

b(ε) b(0) Hence, wl2−1 → 1,l2 l2+1,l1 via σl2 and the length is decreased by 2l1. The lemma for the case k = 1 is proved.

Lemma 3.2.14 (Second block exchange lemma). Let 1 6 ε, ν 6 m − 1 and l1 > l2 be positive integers. Then

b(ν) b(ε) b(ε) b(ν) k,l1 k+l1,l2 → k,l2 k+l2,l1 .

Proof. Once again, we only prove the lemma for the case k = 1. The general case follows from a modiﬁcation of indices. 3.2. MINIMAL ELEMENTS IN CONJUGACY CLASSES 63

For i = 1, . . . , l2, let σi = sl1+i−1 ··· s2i and γi = s2l2−2i+1 ··· s2l2−i. The expressions are clearly well-deﬁned except σl2 for l1 = l2. In this case, we deﬁne σl2 to be the identity. b(ν) b(ε) We start with w0 = 1,l1 l1+1,l2 whose one-line form is

ν ε ((2, 3, . . . , l1, ξ 1, l1 + 2, . . . , l1 + l2, ξ (l1 + 1), l1 + l2 + 1, . . . , r)).

b(ε) b(ν) As an expression, 1,l1+l2 1,l1+1s2 ··· sl1 corresponds to a length reduction sort which turns w0 to the identity. Thus, it is a reduced expression for w0. We ﬁrst consider the case l2 > 1. Since l1 > l2 > 2, applying Lemmas 3.2.7 and 3.2.12, we have

s l1 b(ε) b(ν) b(ε) b(ν) w0 −→ sl1 1,l1+l2 1,l1+1s2 ··· sl1−1 = 1,l1+l2 sl1+1 1,l1+1s2 ··· sl1−1

s − b(ε) b(ν) l1 1 b(ε) b(ν) = 1,l1+l2 1,l1+2s2 ··· sl1−1 −→ 1,l1+l2 1,l1+2sl1+1s2 ··· sl1−2.

The last expression is also reduced, so iteration can be applied if necessary and

σ1 b(ε) b(ν) b(ε) b(ν) −1 w0 −→ 1,l1+l2 1,l1+2s4 ··· sl1−1 = 1,l1+l2 1,l1+2σ2 .

b(ε) b(ν) −1 For i = 1, . . . , l2 − 1, let wi = 1,l1+l2 1,l1+i+1σi+1. Similar to the above, we have wi−1 → wi via σi if 1 6 i 6 l2 − 1. However, we can write

b(ε) b(ν) −1 wl2−1 = σl2 1,2l2 1,l1+l2 σl2 .

b(ε) b(ν) Hence, wl2−1 → 1,2l2 1,l1+l2 via σl2 and the length of wl2−1 is then reduced by 2(l1−l2) by the conjugation. b(ε) b(ν) Denote the element 1,2l2 1,l1+l2 by u1. Using Lemmas 3.2.7 and 3.2.12 again,

s − b(ε) b(ν) 2l2 1 b(ε) b(ν) u1 = s2l2−1 1,2l2−1 1,l1+l2 −→ 1,2l2−1 1,l1+l2 s2l2−1 b(ε) b(ν) b(ε) b(ν) b(ε) b(ν) = 1,2l2−1s2l2−2 1,l1+l2 = s2l2−3s2l2−2 1,2l2−1 1,l1+l2 = γ2 1,2l2−1 1,l1+l2 .

b(ε) b(ν) 6 6 6 6 If we write ui = γi 1,2l2−i 1,l1+l2 for 1 i l2, then ui → ui+1 via γi for 1 i l2 −1. Finally, γ l2 b(ε) b(ν) b(ε) b(ν) ul2 −→ 1,l2 1,l1+l2 γl2 = 1,l2 l2+1,l1+l2 .

σ1 If l2 = 1, we have w0 −→ u1, and we can apply the last step. The lemma then follows. 64 3. MINIMAL LENGTH ELEMENTS IN CONJUGACY CLASSES

The third block exchange lemma is for elements in Sr. Recall that the relation ∼ in 1.3.6 is deﬁned for Sr.

Lemma 3.2.15 (Third block exchange lemma). [GP1, 2.4(a)] Let 1 6 ε 6 m − 1 and l1, l2 be positive integers. Then

b(0) b(0) b(0) b(0) k,l1 k+l1,l2 ∼ k,l2 k+l2,l1 .

Proof. Let y = (sk+l1−1 ··· sk)(sk+l1 ··· sk+1) ··· (sk+l1+l2−2 ··· sk+l1−1). If we omit all the columns i 7→ i, we have y, in two-line notation,

k k + 1 ··· k + l1 − 1 k + l1 k + l1 + 1 ··· k + l1 + l2 − 1 . k + l1 k + l1 + 1 ··· k + l1 + l2 − 1 k k + 1 ··· k + l1 − 1    b(0) b(0) 0 b(0) b(0) Let w = k,l1 k+l1,l2 and w = k,l2 k+l2,l1 . As (sk+l1−2 ··· sk)(sk+l1+l2−2 ··· sk+l1 ) is a reduced expression for w, we can treat wy as an element formed by multiplying on the left to y ﬁrst by sk+l1 , then by sk+l1+1 and so on until the leftmost factor sk+l1−2 of w. By 2.2.2(c), we have `(wy) = `(w)+`(y). Furthermore w0 = y−1wy and `(w0) = `(w), so we have w ∼ w0.

Proof of Theorem 3.2.5. By the Proposition 3.2.11, we have w → u for some standard element u in the same conjugacy class of w. Then by the ﬁrst two block exchange lemmas 3.2.13 and 3.2.14, we have u → w0, where w0 is a specially standard element deﬁned in 3.1.8. In particular, if w is a minimal element in a conjugacy class, the specially standard element w0 will be of minimal length because w → w0 implies `(w) > `(w0). On the other hand, two specially standard elements in the same conjugacy class are identical except the blocks with label 0 (see Remark 3.1.9). The lengths of the label 0 blocks in one specially standard element is a permutation of those in the other. Thus all specially standard elements in the same conjugacy class are of the same length, and hence are minimal. For the second part, let w, w0 be two specially standard elements in a conjugacy class. Suppose that σ and σ0 are the products of all label 0 blocks in w and w0, 3.2. MINIMAL ELEMENTS IN CONJUGACY CLASSES 65 respectively. We can get σ0 from σ by exchanging label 0 blocks in w. By the third block exchange lemma 3.2.15, we have σ ∼ σ0.

Example 3.2.16. Let us continue Example 3.2.10 to ﬁnd a specially standard element w0 in the conjugacy class in G(5, 1, 6) containing

w = ((ξ35, ξ46, ξ23, 2, ξ21, ξ34)) and w → w0

(a) [Loop] Start from w0 = w.

−1 i u wi−1 wi = u wi−1u (4) (4) 3 2 2 3 3 2 4 2 3 1 t2 s2s3s4 t2 s2s3s4((ξ 5, ξ 3, 2, ξ 1, 6, ξ 4)) ((ξ 4, ξ 2, ξ 5, ξ 1, 6, ξ 3)) (4) (4) 3 2 2 3 3 2 2 2 2 t3 s3 t3 s3((ξ 4, ξ 2, ξ 1, 5, 6, ξ 3)) ((ξ 3, ξ 2, ξ 1, 5, 6, ξ 4)) (3) (3) 2 2 2 2 2 3 t1 s1 t1 s1((ξ 2, 3, ξ 1, 5, 6, ξ 4)) ((ξ 1, 3, 2, 5, 6, ξ 4))

(b) [Output] From the above, we get

3 4 2 2 3 2 2 (2) (0) (2) ((ξ 5, ξ 6, ξ 3, 2, ξ 1, ξ 4)) → ((ξ 1, 3, 2, 5, 6, ξ 4))= b1,1b2,2b4,3

4 4 3 via ω = s1t1s1s2s3s4s2s1t1s1s2s3t1s1. By exchanging blocks, we have

2 2 2 2 (2) (2) (0) 0 w → ((ξ 1, 3, 2, 5, 6, ξ 4)) → ((ξ 1, 3, 4, ξ 2, 6, 5))= b1,1b2,3b5,2 = w

4 4 3 via s1t1s1s2s3s4s2s1t1s1s2s3t1s1s3s2s4s3s5s4. The expression via which the blocks exchanged is given in the proofs of the block exchange lemmas 3.2.13, 3.2.14 and 3.2.15. Moreover, it is easy to see that w can be decomposed into the following disjoint cycles —

1 7→ ξ35 7→ 1, 2 7→ ξ46 7→ ξ24 7→ ξ22, and 3 7→ ξ23.

In this example, there is only one specially standard element in the conjugacy class, and we can write down w0 without using the algorithm. When there is more than one block with zero label, we do need the algorithm to get the desired w0.

By Remark 3.1.9, in each conjugacy class, there is only one specially standard element b~ε with the lengths l , . . . , l of all the label-0 blocks descending. ~l n+1 p 66 3. MINIMAL LENGTH ELEMENTS IN CONJUGACY CLASSES

Lemma 3.2.17. In each conjugacy class C, there is a unique specially standard element b(ε1) b(εn) b(0) b(0) k1,l1 ··· kn,ln kn+1,ln+1 ··· kp,lp , such that 1 6 ε1, . . . , εn 6 m − 1 and ln+1 > ··· > lp. This element is denoted by wC .

Corollary 3.2.18. The set {wC|C is a conjugacy class in W } is a complete set of representatives of minimal length, one in each conjugacy class of W . CHAPTER 4

Characters of Ariki-Koike Algebras

4.1. Ariki-Koike Algebras

The Hecke algebra of each complex reﬂection group G(m, 1, r) is called an Ariki- Koike algebra. These algebras are deﬁned and studied in [AK]. Suppose R is a commutative ring with 1. Let q, Q1,...,Qm be elements of R with q invertible.

Definition 4.1.1. The Ariki-Koike algebra H = H(W ) of W is an associative

R-algebra with 1 generated by the generators T0,T1,...,Tr−1 subject to the following relations

T0T1T0T1 = T1T0T1T0

Ti+1TiTi+1 = TiTi+1Ti for 1 6 i 6 r − 2

TiTj = TjTi for 0 6 i < j − 1 6 r − 2

(T0 − Q1) ··· (T0 − Qm) = 0

(Ti + 1)(Ti − q) = 0 for 1 6 i 6 r − 1.

i−1 The algebra H becomes the group algebra RW when q = 1 and Qi = ξ . The ﬁrst three relations are braid relations. The last two relations are deformations of order relations.

A certain set of commuting elements in the group algebra of Sr, denoted by {Li} is used by Murphy in [Mu1] to establish a new construction of Young’s seminormal representation of Sr. These elements are generalized in [DJ2] to the Hecke algebra of Sr. A detail survey of these elements can be found in [Mu2, p.492]. Of course, there are analogues of these elements in H. In [AK, p.226], they use Ti−1 ··· T1T0T1 ··· Ti−1. We 1−i follow [DJM2, p.386] and denote the element q Ti−1 ··· T1T0T1 ··· Ti−1 by Li. Such an element Li is a q-analogue of ti, and the following is an analogue of Lemma 2.1.13. 67 68 4. CHARACTERS OF ARIKI-KOIKE ALGEBRAS

Lemma 4.1.2. Let i, j be integers in r.

(a) If i =6 j − 1, j, then TiLj = LjTi.

(b) For any i, j, LiLj = LjLi. (c) If 1 6 i 6 r − 1 and ε > 1,

ε ε ε ε−k k TiLi+1 = Li Ti + (q − 1) Li Li+1 . ! Xk=1 (d) If 1 6 i 6 r − 1 and ε > 1,

ε ε ε ε−k k TiLi = Li+1Ti − (q − 1) Li Li+1 . ! Xk=1 (e) If 1 6 i 6 r − 1, then Ti commutes with LiLi+1 and Li + Li+1.

(f) If Q ∈ R and k =6 j then Tk commutes with (L1 − Q)(L2 − Q) ··· (Lj − Q).

Parts (a), (b), (e) and (f) can be found in [DJM2, 2.1], while parts (c) and (d) are straightforward modiﬁcations of [AK, 3.3(3), (4)].

Theorem 4.1.3. [AK, 3.10] The algebra H is a free R-module with basis

ε1 εr S B = {L1 ··· Lr Tσ|σ ∈ r and 0 6 εj 6 m − 1 for j ∈ r}.

Recall that, in an Iwahori-Hecke algebra, if w is an element in the corresponding

Coxeter group, we can deﬁne Tw to be the element Ti1 ··· Til where si1 ··· sil is any reduced expression for w. However, we do not have an analogue of Corollary 1.5.4 for 2 2 W . For instance, s1s0s1s0 and s0s1s0s1 are reduced expressions of the same element in G(m, 1, r), where m > 2. On the contrary, we have

2 2 2 T0T1T0 T1 = qL1L2 − (q − 1)L1L2T1,

2 2 2 2 T1T0 T1T0 = (q − q + 1)L1L2 − (q − 1)L1L2T1.

It is not surprising that they are not equal, because we need an order relation to 2 2 transform s1s0s1s0 to s0s1s0s1 in W —

2 2 s1s0s1s0 = s1s0s1s1s0s1s0 = s1s0s1s0s1s0s1 = s0s1s0s1s1s0s1 = s0s1s0s1. 4.1. ARIKI-KOIKE ALGEBRAS 69

In H, we write Tω = Ti1 ··· Tik when ω is the expression si1 ··· sik which is not necessarily reduced. ε ν ν ε Obviously, s1s0s1s0 and s0s1s0s1 are two reduced expressions for the element ((ξν1, ξε2, 3, . . . , r)). Hence, the relation

ε ν ν ε (4.1.4) s1s0s1s0 = s0s1s0s1 holds. Following [BM, p. 455], two expressions are said to be weakly braid-equivalent if one can be transformed to the other by the braid relations and 4.1.4. We state a weaker condition relating the reduced expressions for the same element, which is a direct consequence of [BM, 1.5].

Lemma 4.1.5. Two reduced expressions for the same element in W are weakly braid-equivalent.

The q-analogue of 4.1.4 is recorded in [BM, 2.2]. We quote it with our choice of parameters. ν ε ν ν ε ε+ν−i i i ε+ν−i (4.1.6) T1T0 T1T0 = T0 T1T0 T1 + (q − 1) T0 T1T0 − T0T1T0 i=1 X for 1 6 ε 6 m − 1, 0 6 ν 6 m − 1. The relation between two elements in H corresponding to two reduced expressions for the same element in W are given in [BM, 2.4]. We are going to state and prove a weaker version as follows. Recall that, if ω is an expression for w ∈ W , we write ω = w.

Proposition 4.1.7. Let ω, ω0 be two reduced expressions of the same elements in W . Then

Tω − Tω0 = rκTκ, where the summation is over some reduced expressionsX κ such that `(κ) < `(ω).

Proof. Since ω and ω0 are weakly braid-equivalent, there is a sequence of reduced 0 expressions of the same length ω = ω0, ω1, . . . , ωk = ω such that for each 0 6 i 6 k−1,

ωi = uiyivi and ωi+1 = uiyi+1vi, 70 4. CHARACTERS OF ARIKI-KOIKE ALGEBRAS for some expressions ui, vi, yi, yi+1. Moreover, yi and yi+1 are expressions on the two sides of a braid relation or 4.1.4. When they are the two sides of a braid relation, we have Tωi = Tωi+1 . Otherwise, by 4.1.6, Tωi − Tωi+1 is a linear combination of terms Tκ with κ reduced and `(κ) < `(ω). The Proposition follows easily.

Let Ω be a set of arbitrarily chosen reduced expressions, one for each element in W . As mentioned in [BM, p.459], the set {Tω|ω ∈ Ω} forms a basis for H. In particular, the set of all distinguished reduced expressions deﬁned in 2.2.7 forms a basis of H.

Corollary 4.1.8. Let Ω be a set of arbitrarily chosen reduced expressions, one for each element in W . The set {Tω|ω ∈ Ω} forms a basis for H. In particular,

B = {Tω|ω is the distinguished reduced expression for some w ∈ W } is a basis for the algebra H.

4.2. Characters of Ariki-Koike Algebras

In this section we assume that R is an arbitrary ﬁeld. We are going to prove that any character on H(G(m, 1, 2)) does have a constant value on

{Tω|ω is a reduced expression for some w ∈ Cmin},

where C is a conjugacy class of G(m, 1, 2). As in Coxeter groups, Cmin denotes the set of minimal length elements in C.

n1 n2 Instead of using a matrix or an m-partition, we use (ε1, l1) (ε2, l2) to denote the type of a conjugacy class containing elements of ni cycles with label εi and length li, where i = 1, 2. In the following table, we show the types of all conjugacy classes. In each conjugacy class C, the elements in Cmin and their possible reduced expressions are listed. 4.2. CHARACTERS OF ARIKI-KOIKE ALGEBRAS 71

Type of C w ∈ Cmin Reduced expression(s) for w (0, 1)2 ((1, 2)) e

(0, 2) ((2, 1)) s1 ε ε (ε, 1)(0, 1) ((ξ 1, 2)) s0 ε ν i ν ε−i (ε, 1)(ν, 1) ((ξ 1, ξ 2)) s0s1s0s1s0 , 0 6 i 6 ε ν ε i ε ν−i ((ξ 1, ξ 2)) s0s1s0s1s0 , 0 6 i 6 ν i ε−i i ε−i (ε, 2) ((ξ 2, ξ 1)), 0 6 i 6 ε s0s1s0

Here, 1 6 ε, ν 6 m − 1. The following is a direct consequence of the information shown in the above table.

Proposition 4.2.1. Let C be a conjugacy class in G(m, 1, 2). The value of any character χ of the Hecke algebra H(G(m, 1, 2)) is constant on

{Tω|ω is a reduced expression for some w ∈ Cmin}.

Proof. In each conjugacy class C of the ﬁrst three types in the above table, there is only one element in Cmin and it has a unique reduced expression. There is nothing to prove. In the fourth type, for any 0 6 i 6 ε and 0 6 j 6 ν, we have

i ν ε−i ν ε j ε ν−j χ(T0 · T1T0 T1T0 ) = χ(T1T0 T1T0 ) = χ(T0 T1T0 · T1T0 ).

i ε−i ε For the last type, we have χ(T0T1T0 ) = χ(T1T0 ). The lemma then follows.

In general, in H(G(m, 1, r)) with r > 3, it is not true that the character values of 0 Tω,Tω0 are the same when ω and ω are minimal length elements in the same conjugacy class. Moreover, even when ω and ω0 are reduced expressions for the same element, the character values of Tω,Tω0 may be diﬀerent. 72 4. CHARACTERS OF ARIKI-KOIKE ALGEBRAS

Example 4.2.2. Assume Q1,Q2,Q3 are distinct elements in R. Deﬁne a matrix representation ρ : H(G(3, 1, 3)) → M6(R) by

Q1 0 0 0 0 0

 0 Q1 0 0 0 0   0 0 Q 0 0 0   2  ρ(T0) =   ,  0 0 0 Q 0 0   2     0 0 0 0 Q 0   3     0 0 0 0 0 Q3    

b21 0 a12 0 0 0

 0 b31 0 0 a13 0  a 0 b 0 0 0   21 12  ρ(T1) =   ,  0 0 0 b 0 a   32 23    0 a 0 0 b 0   31 13     0 0 0 a32 0 b23      b32 a23 0 0 0 0

a32 b23 0 0 0 0   0 0 b a 0 0   31 13  ρ(T2) =   .  0 0 a b 0 0   31 13     0 0 0 0 b a   21 12    0 0 0 0 a21 b12      qQi − Qj (q − 1)Qi where aij = and bij = . This representation is basically a special Qi − Qj Qi − Qj case of the matrix representation given in [AK, Section 3]. The elements ((ξ2, ξ1, ξ23)) and ((ξ22, 1, ξ23)) are minimal elements in the same 2 conjugacy class. Their distinguished reduced expressions are κ = s0s1s0s2s1s0s1s2 0 2 2 and κ = s0s1s2s1s0s1s2. It can be checked that

2 χρ(Tκ) = −q (q − 1)Q1Q2Q3(Q1 + Q2 + Q3), but

2 χρ(Tκ0 ) = q(q − 1)(q − 3q + 1)Q1Q2Q3(Q1 + Q2 + Q3). 4.2. CHARACTERS OF ARIKI-KOIKE ALGEBRAS 73

0 We have already noted that Tω and Tω0 may not be the same even when ω = ω and ω, ω0 are reduced. The following example shows that they may not have the 2 0 2 same character value. Let ω = s0s1s0s1s2s1 and ω = s1s0s1s0s2s1. They are reduced expressions for the same element ((ξ2, ξ23, 1)), but

2 χρ(Tω) = q(q − 1) Q1Q2Q3, but

2 2 χρ(Tω0 ) = −(q + 1)(q − 1) Q1Q2Q3.

Although the character values of Tω,Tω0 may be diﬀerent, the value of the character for Tω can be calculated from the values for Tω0 and Tκ with `(κ) < `(ω).

Lemma 4.2.3. If ω and ω0 are reduced expressions for the same element w ∈ W and χ is a character of H, then χ(Tω) = χ(Tω0 )+χ(h), where h is a linear combination of elements of the form Tκ with `(κ) < `(ω).

Proof. It follows directly from 4.1.7.

Furthermore, we can calculate the character value of any element h ∈ H from the character values of the elements in

{Tω|ω is the distinguished reduced expression of wC, for some C ∈ Cl(W )}, where Cl(W ) is the set of all conjugacy classes in W and wC is the element in 3.2.17 for C ∈ Cl(W ). To prove this, we need the following lemmas.

Lemma 4.2.4. Let ω, ω0 be reduced expressions for w, w0 ∈ W . Suppose that w → w0 and χ is a character of H. If `(w) = `(w0) then

χ(Tω) = χ(Tω0 ) + rκχ(Tκ);

Tκ∈B `(κX)<`(w) and if `(w) > `(w0) then

χ(Tω) = rκχ(Tκ).

Tκ∈B `(κX)<`(w)

Here, rκ ∈ R, and B is the basis deﬁned in 4.1.8. 74 4. CHARACTERS OF ARIKI-KOIKE ALGEBRAS

Proof. Thanks to Proposition 4.1.7, we only need to show that the summations are over κ with `(κ) < `(w) and κ reduced, but not necessarily satisfying Tκ ∈ B. Moreover, It suﬃces to consider the case that w → w0 via a generator s ∈ S. Thus we have w0 = s−1ws and either `(s−1w) < `(w) or `(ws) < `(w). If `(s−1w) < `(w), 0 then w has a reduced expression sω1 and hence w has an expression ω1s which is not necessarily reduced. If `(w) = `(w0), then the result follows from 4.1.7 and the fact 0 that χ(TsTω1 ) = χ(Tω1 Ts). If `(w) > `(w ), then ωs is not reduced and Tω1 Ts can be written as a linear combination of terms in Tκ with `(κ) < `(w) and κ reduced. By 4.1.7 again, the lemma follows.

Lemma 4.2.5. Let w, w0 be two specially standard elements in the same conjugacy class and χ be a character on H. If ω, ω0 are distinguished reduced expressions for 0 w, w , then χ(Tω) = χ(Tω0 ).

Proof. Since w and w0 are specially standard elements, they are identical except for the label 0 blocks. Let σ, σ0 be the products of all the blocks with zero label in 0 0 0 0 w, w respectively. We can write w = hσ and w = hσ . Moreover, if ω1, ω2 and ω2 0 0 0 are distinguished reduced expressions for h, σ and σ , then ω = ω1ω2 and ω = ω1ω2. By Theorem 3.2.5, we have σ ∼ σ0. It reduces quickly to the case that σ and 0 σ are elementarily strongly conjugate in Sr. Thus there exists τ ∈ Sr such that στ = τσ0 and `(στ) = `(σ) + `(τ), or τσ = σ0τ and `(τσ) = `(τ) + `(σ). In the ﬁrst 0 case, since σ, σ , τ ∈ Sr, we have

TσTτ = Tστ = Tτσ0 = Tτ Tσ0 .

On the other hand, h ∈ G(m, 1, k) for some k, while σ, σ0 and τ are in the parabolic subgroup generated by {sk+1, . . . , sr−1}. Hence

0 −1 0 −1 Tω = Tω1 Tω2 = Tω1 Tσ = Tω1 Tτ Tσ Tτ = Tτ Tω Tτ .

Hence χ(Tω) = χ(Tω0 ) in this case. Similarly, the lemma is also true in the other case. 4.2. CHARACTERS OF ARIKI-KOIKE ALGEBRAS 75

Theorem 4.2.6. For any conjugacy class C of W , denote the distinguished reduced expression for the element wC of 3.2.17 by ωC . If χ is a character of the Ariki-Koike algebra H(W ) and h ∈ H, then

χ(h) = rC χ(TωC ), C∈XCl(W ) where rC ∈ R and rC depends on h.

Proof. We only need to prove that the theorem is true for any basis element h = Tω ∈ B. Suppose ` = `(ω). We are going to prove this by induction on `. When ` = 0, then ω is the identity which is the unique elements in its conjugacy class.

When ` = 1, then ω = s for some generator s. If s = s0 or s = s1, then ω = ωC for some conjugacy class C. Otherwise, s and s1 are specially standard elements in the same conjugacy class and they have the same character values by Lemma 4.2.5.

Suppose the theorem is true for all Tκ, when κ is a distinguished reduced expression and `(κ) < `. If ω is not specially standard, there exists a specially standard elements w0 such that ω → w0. Let ω0 be the distinguished reduced expression for w0. By Lemma 4.2.4, we have

0 0 χ(Tω) = χ(Tω ) + rκχ(Tκ). Tκ∈B `(Xκ)<`

0 Denote the conjugacy class containing ω by C. Then we have w and wC are both

0 specially standard elements in C. By Lemma 4.2.5, we have χ(Tω ) = χ(TωC ). If

ω is specially standard, we have χ(Tω) = χ(TωC ). The theorem then follows from induction.

Remark 4.2.7. For Iwahori-Hecke algebras of type A with parameter q, Jones has constructed a Q[q, q−1]-basis for the centre in [J]. This basis is a q-analogue of the conjugacy class sum basis for the centre of the group algebra QSr. Francis has constructed integral bases for centres of Iwahori-Hecke algebras and centralizers of parabolic subalgebras of Iwahori-Hecke algebras of types A and B. Francis’ approach 76 4. CHARACTERS OF ARIKI-KOIKE ALGEBRAS is entirely combinatorial in nature and depends only on Geck-Pfeiﬀer reducibility theorem and his generalization of the theorem to J-conjugacy classes. Since we have an analogue of the reducibility theorem, a natural question is whether Francis’ approach can be generalized to G(m, 1, r). CHAPTER 5

Quasi-Parabolic Subgroups

The paper in the appendix has been published in the Journal of Algebra. It contains the deﬁnition of quasi-parabolic subgroups of W = G(m, 1, r) and the expositions of some properties of the cosets of double cosets of these subgroups. We are going to put in this chapter some results from the paper which are relevant to the next chapter.

5.1. Quasi-parabolic Subgroups

We have seen in Section 3.1 that conjugacy classes in W are indexed by m- ~ε partitions of r. Let bλ be a specially standard element as deﬁned in 3.1.8, where

~ε = (ε1, . . . , εn) and λ = (λ1, . . . , λn) is a composition of r. If p is the integer such that 1 6 εi 6 m − 1 for all 1 6 i 6 p and εp+1 = ··· = εn = 0. This specially standard element is in the subgroup is isomorphic to the direct product

G(m, 1, λ1) × G(m, 1, λ2) × · · · × G(m, 1, λp) × S(λp+1) × · · · × S(λn).

When p > 1, the subgroup is not conjugate to the ones which are generated by subsets of S, and hence they are not parabolic. For instance, the specially standard element (1,1) b(1,1) = t1t2 in G(3, 1, 2) is in the subgroups generated by {t1, t2}. Following [DS, 2.2], we call them quasi-parabolic. We study a slightly more general form of these subgroups in which the factors of symmetric groups are not grouped together at the end. They are labelled by the composition λ = (λ1, . . . , λn) and an index set which tells us which factors are complex reﬂection groups. Moreover, we allow some of the parts of λ to be zero. 77 78 5. QUASI-PARABOLIC SUBGROUPS

Denote the set of all compositions of r with n parts by Λ(n, r). Recall that λ Ri = {λ1 + ··· + λi−1 + 1, . . . , λ1 + ··· + λi}, and that n = {1, 2, . . . , n} for any positive integer n.

Definition 5.1.1. For any λ ∈ Λ(n, r), let nλ = n\{i|λi = 0}, and I ⊆ nλ. For λ each i ∈ nλ, suppose Wi is the subgroup of all elements w ∈ W which ﬁx the set λ λ λ Cm(r\Ri ) point-wise. Furthermore, Ri w = Ri when i ∈ nλ\I. Corresponding to this pair (λ, I), the quasi-parabolic subgroup W is deﬁned to be the product W λ, (λ,I) i∈nλ i λ that is, the subgroup generated by Wi , i ∈ nλ. Q

λ ∼ λ Sλ ∼ S From Deﬁnition 5.1.1, we have Wi = G(m, 1, λi) or Wi = i = (λi), when λ i ∈ I or i ∈ nλ\I, respectively. For simplicity, we take Wi = {e} when i ∈ n\nλ. The notion of the quasi-parabolic subgroup is a generalization of the parabolic subgroup, ∼ because if m = 1 then W(λ,∅) = Sλ. Moreover, when m = 2 and nλ = n, if I = p for some 1 6 p 6 r or I = ∅ then W(λ,I) is a quasi-parabolic subgroup of the type B Weyl group in [DS]. λ λSλ By 2.1.15, when i ∈ I, the group Wi can be written as a product Ci i , where λ λ λ Ci is the subgroup generated by {tp|p ∈ Ri }. The product i∈I Ci is denoted by

C(λ,I). Q

Lemma 5.1.2. If W(λ,I) is a quasi-parabolic subgroup, then W(λ,I) = C(λ,I)Sλ =

SλC(λ,I).

Proof. λ λ λ λ For any i, j ∈ I, we have Ci Cj = Cj Ci . Moreover, if k ∈ nλ then CλSλ = SλCλ. Hence W = W λ = Cλ Sλ = C S . Simi- i k k i (λ,I) i∈nλ i i∈I i i∈nλ i (λ,I) λ S larly, we also have W(λ,I) = λC(λ,IQ). Q Q

ι Sι S Applying the anti-involution ι in 2.1.16, we have C(λ,I) = C(λ,I), r = r, and hence the following corollary.

Corollary . ι 5.1.3 If W(λ,I) is quasi-parabolic, then W(λ,I) = W(λ,I). Hence, for ι ι any w ∈ W , (W(λ,I)w) = w W(λ,I). 5.2. COSETS OF QUASI-PARABOLIC SUBGROUPS 79

Similar to Sr, tableaux are useful for the study of W . For each w = ((1w, . . . , rw)), the corresponding λ-tableau t is formed by ﬁlling in a λ-diagram by iw in order along successive rows. The particular one corresponding to e is denoted by tλ. The set of λ-tableaux for all the elements in W is denoted by T(λ). Obviously δ : T(λ) → W is a bijection. This one-one correspondence gives rise to left and right actions of W on T(λ) such that for all w ∈ W and t ∈ T(λ), we deﬁne tw = δ−1(δ(t)w) and wt = δ−1(wδ(t)). In particular, tλw is the tableau for w.

In order to read the index set easily, we label the ith row by A when i ∈ nλ\I and by B when i ∈ I. There is no label for an empty row.

Example 5.1.4. If λ = (5, 4, 0, 1) and I = {2}, then the (λ, I)-tableaux tλ and t for w = ((ξ10, 7, 2, ξ25, 4, 9, 1, ξ26, ξ8, ξ23)) ∈ G(3, 1, 10) are given as follows.

A 1 2 3 4 5 A ξ10 7 2 ξ25 4 B 6 7 8 9 B 9 1 ξ26 ξ8 tλ = t = tλw =

A 10 A ξ23

5.2. Cosets of Quasi-parabolic Subgroups

For any t ∈ T(λ), the sequence of entries in the ith row of t is denoted by rowit. t tλ λ t t Hence if = w, then Ri w is the set of entries in rowi . Naturally, rowi is the ε ε sequence formed by replacing each term ξ a in rowit with ξ a = a.

Definition 5.2.1. Let W(λ,I) be a quasi-parabolic subgroup of W , where λ ∈ 0 Λ(n, r) and I ⊆ nλ. Two λ-tableaux t and t are said to be row-equivalent with respect to I if

0 (RE1) when i ∈ I, rowit is a permutation of rowit and 0 (RE2) when i ∈ nλ\I, rowit is a permutation of rowit.

Lemma 5.2.2. Let w, w0 be elements in W . Denote the corresponding λ-tableaux for w and w0 by t and t0 respectively. The following statements are equivalent. 80 5. QUASI-PARABOLIC SUBGROUPS

0 (a) The elements w and w are in the same coset in W(λ,I)\W . (b) The tableaux t and t0 are row-equivalent with respect to I.

Proof. Let w0 = uw for some u ∈ W , and u = u where u ∈ W λ. (λ,I) i∈nλ i i i t When i ∈ I, ui permutes and possibly changes exponents ofQξ of the entries in rowi . 0 When i ∈ nλ\I, ui only permutes the entries in the rowit. Therefore, t and t are row- equivalent with respect to I. Conversely, (RE1) and (RE2) guarantee the existence λ t0 t 0 of ui ∈ Wi such that rowi = rowi(ui ) for each i ∈ nλ. Hence w = uw, where u = u which is in W . i∈nλ i (λ,I) LemmaQ 5.2.2 gives us an easy way to check whether two elements in W are in the same right coset by simply examining the corresponding tableaux.

In Sr, the tableaux with increasing rows are said to be row-standard. With the order deﬁned in 2.1.6, we deﬁne row-standard λ-tableaux for W .

Definition 5.2.3. A λ-tableau t is said to be row-standard with respect to I if

(RS1) for all i ∈ I, entries in rowit are in r and

(RS2) for all i ∈ nλ, rowit is ascending with respect to the order 4 .

Definition 5.2.4. A λ-tableau t is said to be descending row-standard with respect to I if

m−1 (a) for all i ∈ I, entries in rowit are in ξ r and

(b) for all i ∈ nλ, rowit is descending with respect to the order 4 .

When the set I is clear from the quasi-parabolic subgroup under consideration, we shall drop the phase “with respect to I”.

Example 5.2.5. The following tableaux are row-standard and descending row- standard and they are row-equivalent to t in Example 5.1.4.

A ξ10 ξ25 2 4 7 A 7 4 2 ξ25 ξ10 B 1 6 8 9 B ξ21 ξ26 ξ28 ξ29

A ξ23 A ξ23 5.3. SHORTEST AND LONGEST COSET REPRESENTATIVES 81

5.3. Shortest and Longest Coset Representatives

If Sλ is a parabolic subgroup of Sr and σ ∈ Sr, there are a unique shortest element and a unique longest element in the coset Sλσ. This is also true for cosets of quasi-parabolic subgroups W(λ,I) in W .

Proposition 5.3.1. Let W(λ,I) be a quasi-parabolic subgroup. In each right coset of W(λ,I) in W , there exists a unique minimal element with respect to the Bruhat order. Furthermore, the corresponding tableau is row-standard.

Proof. Let T be the set of λ-tableaux corresponding to elements in W(λ,I)w for a ﬁxed w ∈ W . From this coset, take an element u = ((x1, . . . , xr)). For each i ∈ I, λ λ change rowi(t u) to rowi(t u). The resulting tableau is denoted by t1. The tableau t1 is in T and it satisﬁes (RS1). The element u1 = δ(t1) is in the coset. By 2.4.7(b), u1 < u (the Bruhat order in W ) unless u1 = u. Then for all i ∈ nλ, we sort rowit1 to ascending order by 2.1.7. As a result we get a row-standard tableau t ∈ T, and the element d = δ(t) is in the coset. By 2.4.7(a), d < u1 unless d = u1. Hence d < u unless u = d. Since there is only one row-standard tableau in T, so d is independent of the choice of u. Thus, d is the unique element of shortest length in the coset.

By the fact that w < w0 implies `(w) < `(w0), we have the following corollary.

Corollary 5.3.2. The unique minimal element with respect to the Bruhat order in a right coset of a quasi-parabolic subgroup is the unique shortest length element in the coset.

Proposition 5.3.3. Let W(λ,I) be a quasi-parabolic subgroup. In each right coset of W(λ,I) in W , there exists a unique maximal element with respect to the Bruhat order. This element is also the unique longest length element and its corresponding tableau is descending row-standard.

Proof. We use exactly the same argument in the proof of Proposition 5.3.1 except the following. For each i ∈ I, all the entries are changed from ξεa to ξm−1a. Then all the rows are sorted to descending order with respect to ≺ . 82 5. QUASI-PARABOLIC SUBGROUPS

Following [DS, p.4334], we call the shortest length representative in a right coset

W(λ,I)w a distinguished right coset representative. The set of all distinguished right coset representatives for the cosets in W(λ,I)\W is denoted by D(λ,I), while the set of + all longest right coset representatives is denoted by D(λ,I).

Corollary 5.3.4. Let λ ∈ Λ(n, r) and I ⊆ nλ. The set of λ-tableaux which are row-standard (respectively descending row-standard) with respect to I is in one-one + correspondence with D(λ,I) (respectively D(λ,I)).

The following corollary follows from 5.1.3, 5.3.1 and 5.3.4.

Corollary 5.3.5. Let W(µ,J) be a quasi-parabolic subgroup. For each w ∈ W , there are a unique shortest and a unique longest element in the left coset wW(µ,J). ι + ι Moreover, D(µ,J) and (D(µ,J)) are the sets of all shortest and respectively longest left coset representatives.

Note that there may be more than one minimal element in the left coset with respect to the Bruhat order because of its one-side nature. For instance, the set 2 2 2 2 S {s0s1s0 = ((ξ2, ξ 1)), s0s1s0s1 = ((ξ1, ξ 2))} is a left 2-coset of G(3, 1, 2). Using arguments similar to those in Example 2.4.4, we can see that both of them are minimal with respect to the Bruhat order. Naturally, if we deﬁne the other side Bruhat order, we shall have analogous results.

Remark . µ µ 5.3.6 Using the notation in 5.1.1, let W(µ,J) = W1 ··· Wn . Since the µ µ −1 −1 factors are pair-wise commutative and Wj = (Wj ) for all j, we have W(µ,J) = −1 −1 ι −1 W(µ,J). So (W(µ,J)w) = w W(µ,J). When m equals 1 or 2, we have w = w ι −1 + ι + −1 for all w ∈ W . As a result, D(µ,J) = D(µ,J) and (D(µ,J)) = (D(µ,J)) . When m > 2, although wι =6 w−1 in general (see Example 2.1.18), yet we have the following proposition.

Proposition . −1 ι 5.3.7 If W(µ,J) is quasi-parabolic, then D(µ,J) = D(µ,J).

ε1 εr Proof. Let w = πc, where π = ((a1, . . . , ar)) ∈ Sr and c = ((ξ 1, . . . , ξ r)) ∈

−1 m−ε1 m−εr ι ε1 εr C. We then have w = ((ξ b1, . . . , ξ br)) and w = ((ξ b1, . . . , ξ br)), where 5.4. CHARACTERIZING A DOUBLE COSET BY AN n × n ARRAY 83

−1 apπ = bp for all p ∈ r. Note that m − εp ≡ 0 (mod m) if and only if εp ≡ 0 µ −1 µ ι µ −1 (mod m). Hence, rowi(t w ) ⊆ r if and only if rowi(t w ) ⊆ r, and rowi(t w ) µ ι −1 is ascending if and only if rowi(t w ) is ascending. Thus w ∈ D(µ,J) if and only if ι w ∈ D(µ,J), and the result follows.

+ ι + −1 Unfortunately, (D(µ,J)) =6 (D(µ,J)) in general. For instance, in G(3, 1, 1), we + + ι 2 + −1 have D(1,{1}) = (D(1,{1})) = {((ξ 1))}, while (D(1,{1})) = {((ξ1))}.

For a right coset W(λ,I)w, we can easily construct the shortest element and the longest element from the row-standard and the descending row-standard tableaux λ which are row-equivalent to t w. For a left coset wW(µ,J), we ﬁrst construct the short- ι est element and the longest element in W(µ,J)w , then apply the anti-automorphism ι to the resulting elements.

5.4. Characterizing a Double Coset by an n × n Array

Let λ, µ be compositions of r. We may assume both λ and µ are in Λ(n, r) because zero parts can be added to the one with less parts. We generalize [JK, 1.3.8] to the following lemma which gives a condition for elements being in the same double coset. In order to avoid complicated speciﬁcation of indices, we assume i and j are in the set n throughout this section, unless otherwise stated.

0 Lemma 5.4.1. Elements w, w are in the same double coset in W(λ,I) \W/W(µ,J) if and only if

λ µ λ 0 µ (a) card(Ri w ∩ Rj ) = card(Ri w ∩ Rj ), when i ∈ I or j ∈ J and λ ε µ λ 0 ε µ (b) for each ε, card(Ri w ∩ ξ Rj ) = card(Ri w ∩ ξ Rj ), when i∈ / I and j∈ / J.

0 0 Proof. If w, w are in the same double coset, then w = uwv for some u ∈ W(λ,I) λ 0 λ λ and v ∈ W(µ,J). For each i, we have Ri w = (Ri u)wv = Ri wv. Since v is in S λ 0 µ λ µ λ µ µ, then Ri w ∩ Rj = Ri wv ∩ Rj v = (Ri w ∩ Rj )v for every j. In particular, (a) λ λ µ µ holds. When i∈ / I and j∈ / J, we have Ri u = Ri and Rj v = Rj . We then have λ 0 ε µ λ ε µ λ ε µ Ri w ∩ ξ Rj = Ri wv ∩ ξ Rj v = (Ri w ∩ ξ Rj )v for every ε, and (b) follows. 84 5. QUASI-PARABOLIC SUBGROUPS

Conversely, let w, w0 be elements satisfying the conditions. Let us first fix a λ µ λ 0 µ j ∈ J. From (a) and (b), we have card(Ri w ∩ CmRj ) = card(Ri w ∩ CmRj ) for µ ∼ λ µ all i. Therefore, there exists vj ∈ Wj = G(m, 1, µj) such that (Ri w ∩ CmRj )vj = λ 0 µ Sµ Ri w ∩ CmRj for all i. Now, for a fixed j∈ / J, there exists vj ∈ j such that λ µ λ 0 µ λ ε µ λ 0 ε µ (Ri w ∩ Rj )vj = Ri w ∩ Rj for all i ∈ I, and (Ri w ∩ ξ Rj )vj = Ri w ∩ ξ Rj for all i∈ / I. The product v = v1 ··· vn, which is in W(µ,J), satisfies the conditions that: λ λ 0 λ λ 0 Ri wv = Ri w for i ∈ I, and Ri wv = Ri w for i∈ / I. Hence, there exists u ∈ W(λ,I) such that w0 = uwv.

Theorem 5.4.2. Let λ = (λ1, . . . , λn) and µ = (µ1, . . . , µn) be compositions in

Λ(n, r) and I ⊆ nλ,J ⊆ nµ. There is a one-one correspondence between the set of double cosets W(λ,I)\W/W(µ,J) and the set of n × n arrays (zij) whose entries are sequences of non-negative integers such that

(a) zij = (zij) is a sequence of one term, when i ∈ I or j ∈ J;

(b) zij = (zij0, . . . , zij(m−1)) is a sequence of m terms, when i∈ / I and j∈ / J; n n (c) zij = λi and zij = µj, where zij is the sum of the terms in zij. j=1 i=1 P P Proof. By Lemma 5.4.1, the double coset W(λ,I)wW(µ,J) is characterized by the following system of non-negative integers.

λ µ (i) zij = card(Ri w ∩ Rj ), when i ∈ I or j ∈ J; λ ε µ (ii) zijε = card(Ri w ∩ ξ Rj ), when i∈ / I and j∈ / J.

These integers form an array (zij) which satisﬁes the conditions in the theorem. Thus, we have an injection from the set of double cosets W(λ,I) \W/W(µ,J) to the set of square arrays mentioned in the theorem.

Conversely, let (zij) be an n × n array satisfying the conditions. We divide a

λ-diagram into n parts such that there are zij boxes in the ith row of the jth part. µ We ﬁrst ﬁll in the jth part by the elements in Rj . Then, for each pair i∈ / I and 2 j∈ / J, to the entries of the jth part in the ith row, we multiply ξ to zij1 of them, ξ to zij2 of the remaining entries, and so on. If the resulting tableau is denoted by t, then the double coset W(λ,I)δ(t)W(µ,J) is obviously characterized by the array. 5.4. CHARACTERIZING A DOUBLE COSET BY AN n × n ARRAY 85

Example 5.4.3. Corresponding to λ = (5, 2, 2) and µ = (3, 6, 0), I = {2} and

J = {1}, we have quasi-parabolic subgroups W(λ,I) and W(µ,J) of G(3, 1, 9). Let w = ((5, ξ28, 3, ξ9, ξ2, ξ21, ξ6, ξ7, 4)). The tableau corresponding to w and the 3 × 3 array corresponding to the double coset W(λ,I)wW(µ,J) are displayed as follows.

A 5 ξ28 3 ξ9 ξ2 (2) (1, 1, 1) (0, 0, 0) 2 t = B ξ 1 ξ6 (zij) = (1) (1) (0)    A ξ7 4 (0) (1, 1, 0) (0, 0, 0)    

Corollary 5.4.4. The number of double cosets in W(λ,I) \W/W(µ,J) is equal to

λ µ λ1 λn µ1 µn the coeﬃcient of x y = x1 ··· xn y1 ··· yn in the formal power series

−1 1−m (1 − xiyj) (1 − xiyj) . i,j i/∈I Y andYj∈ /J Proof. The number of n × n arrays satisfying the conditions in Theorem 5.4.2 is equal to the coeﬃcient of xλyµ in m

2 2 2 2 (1 + xiyj + xi yj + ··· )  (1 + xiyj + xi yj + ··· ) , i∈I i/∈I orYj∈J  andYj∈ /J    which can be rewritten as  

−1 −m −1 1−m (1 − xiyj) (1 − xiyj) = (1 − xiyj) (1 − xiyj) . i∈I i/∈I i,j i/∈I orYj∈J andYj∈ /J Y andYj∈ /J Hence the corollary follows. CHAPTER 6

q-Permutation Modules

The deﬁnition of multipartitions has been quoted in 3.1.2. There is a one-one correspondence between the set of m-partitions of r and the set of conjugacy classes of W . Thus, there is a one-one correspondence between the set of partitions and the set of non-isomorphic irreducible ordinary representations of W . The construction of these irreducible representations can be found in [Can]. The irreducible representations of a semi-simple Ariki-Koike algebra is also in one-one correspondence with the set of m-partitions [A1, 3.1]. Ariki has deﬁned Specht modules for this case in [A3, 2.3]. Following the same line of arguments in [Mu3] for the construction of Specht modules and irreducible modules of Iwahori-Hecke algebras of type A, Dipper, James and Mathas have constructed a version of Specht modules for W [DJM2, 3.28]. The construction is based on a Murphy type basis of the Ariki-Koike algebra H and the Specht modules are submodules of certain quotient modules of H. Following the way of constructing Specht modules in [DJ1, section 4], Du and Rui have constructed another version of Specht modules which are submodules of H in [DR, 2.1].

When m = 1, the cyclic modules mλH in [DJM2] are q-permutation modules deﬁned in [DJ1, section 3]. Further if q = 1, each mλ is the sum of elements in a parabolic subgroup of the symmetric group. When m = 2, the cyclic modules are naturally associated with quasi-parabolic subgroups. However, when m > 2, these mλ are not naturally associated with subgroups of W . In this chapter, we attempt to construct q-permutation modules, each of which is a cyclic module generated by an element xλ. Under specialization, xλ is a product of a power of ξ and the sum of elements in a group which is isomorphic to a subgroup of W .

86 6.1. MULTICOMPOSITIONS AND MULTITABLEAUX 87

6.1. Multicompositions and Multitableaux

Multicompositions and multitableaux are widely used in the literature for Weyl groups of type B and more generally for W . We ﬁrst recall the deﬁnitions and clarify the notation.

Definition 6.1.1. An m-composition is an m-tuple λ = λ(1), . . . , λ(m) . The ith (i) (i) (i) component λ is either an empty sequence or a composition λ1 , . . . , λni of the (i) (i) (i) (i) positive integer αi = λ = λ1 + ··· + λni . When λ is a composition, we assume (i) (i) that λni =6 0. If λ is an empty sequence, we denote it by − and set αi = 0.

Moreover, λ is an m-composition of r = α1 + ··· + αm.

When all λ(i) are partitions or empty, the composition λ is an m-partition deﬁned in 3.1.2. Related to an m-composition λ, we have two important 1-compositions.

(6.1.2) λ˜ = (α1, . . . , αm), and

(6.1.3) λ¯ = λ(1) · λ(2) ····· λ(m).

Here the dot · means concatenation of sequences. Furthermore, we write

(6.1.4) λˆ = µ, −,..., −, λ(m) , where µ = λ(1) · λ(2) ····· λ(m−1).

Definition 6.1.5. [DJM2, 3.11(i)] Let λ and µ be m-compositions of r. We say that λ dominates µ, and write λ D µ, if

k−1 j k−1 j (i) (k) (i) (k) λ + λi > µ + µi i=1 i=1 i=1 i=1 X X X X for all k and j with 1 6 k 6 m and j > 0. If λ D µ and λ =6 µ then we write λ . µ.

Definition 6.1.6. Let λ be an m-composition of r. (a) An m-diagram of the m-composition λ, or a λ-diagram, is an m-tuple of Young diagrams (λ(1)-diagram, λ(2)-diagram, . . . , λ(m)-diagram). Moreover, if λ(i) is empty, we denote this component by −. 88 6. q-PERMUTATION MODULES

(b) An m-tableau of shape λ, or a λ-tableau for an element w ∈ W , is an m-tuple of Young tableaux t = (t(1), t(2),..., t(m)) formed by ﬁlling the r boxes of the λ-diagram by 1w, . . . , rw along successive rows in the ﬁrst component then the second component and so on. The particular one corresponding to the identity is denoted by tλ. The set of all λ-tableaux is denoted by T(λ).

For a fixed λ, the set T(λ) is in one-one correspondence with W . If we write the element in W corresponding to an m-tableau t as δ(t), the action of W on T(λ) can be naturally defined similar to that in Chapter 5 (page 79). For our purpose, we confine to m-tableaux corresponding to elements in Sr. We index the rows of a λ-diagram the same way as the rows of the λ¯-diagram.

For instance, if there are nk rows in the kth component of the λ-diagram for all k, then the jth row of the ith component is called the (n1 + ··· + ni−1 + j)th row. Then λ we deﬁne ni = ∅ when the ith component is empty, and

λ ni = {n1 + ··· + ni−1 + 1, . . . , n1 + ··· + ni} otherwise.

Example 6.1.7. Let µ be the 4-composition ((3, 2), −, (1, 0, 2), (3, 1)). Then we have

µ˜ = (5, 0, 3, 4),

µ¯ = (3, 2, 1, 0, 2, 3, 1), and

µˆ = ((3, 2, 1, 0, 2), −, −, (3, 1)).

6 1 2 3 9 10 11 tµ =  , − , ,  , 4 5 12    7 8      6.1. MULTICOMPOSITIONS AND MULTITABLEAUX 89

1 2 3  4 5  9 10 11 tµˆ =  6 , − , − ,  .    12         7 8      The following shows the tableau tµ¯ and how its rows are indexed.

1st row 1 2 3 2nd row 4 5 3rd row 6 4th row 5th row 7 8 6th row 9 10 11 7th row 12

Hence we have the following.

µ µ µ µ n1 = {1, 2}, n2 = ∅, n3 = {3, 4, 5}, n4 = {6, 7}.

Definition 6.1.8. Let λ be an m-partition. A λ-tableau is standard if all its entries are in r and the rows and columns of each component are increasing. The set of all standard λ-tableaux is denoted by Ts(λ).

The λ-tableaux of type µ and the corresponding converting function in the next two deﬁnitions are essentially the same as [DJM2, 4.1, 4.2] except the way the rows of a µ-tableau are indexed.

Definition . µ 6.1.9 Suppose λ and µ are m-compositions of r. Let n = |n1 | + µ t ··· + |nm|.A λ-tableau is of type µ if (a) each entry in t is an integer in n, and (b) for each i ∈ n, the number of entries with value i in t is the same as the number of entries in the i-th row of tµ. 90 6. q-PERMUTATION MODULES

Definition 6.1.10. Suppose λ and µ are m-compositions of r. Let s be a λ- tableau with entries in r. Deﬁne µ(s) to be the λ-tableau obtained from s by replacing each entry k by i if k is in the ith row of tµ.

Now, we are in the position to deﬁne some special semistandard tableaux which are important to the sequel.

Definition 6.1.11. Suppose λ is an m-partition of r and µ is an m-composition of r. Let s = (s(1),..., s(m)) be a λ-tableau of type µ. Then s is strongly semistandard if (SS1) the entries in each row of each component of s are non-decreasing, (SS2) the entries in each column of each component are strictly increasing; and µ µ (SSS3) if i is an entry in the kth component of s, then i ∈ nk or i ∈ nm. Let Tsss(λ, µ) denote the set of all strongly semistandard λ-tableaux of type µ.

Note that if (SSS3) is replaced by

µ (SS3) if i ∈ nl is an entry in the kth component of s, then l > k, then we have the semistandard tableau deﬁned in [DJM2, 4.4]. Obviously, all strongly semistandard tableaux are semistandard. The set of semistandard λ-tableaux of type µ is denoted by Tss(λ, µ).

Example 6.1.12. As in Example 6.1.7, let µ = ((3, 2), −, (1, 0, 2), (3, 1)). Fur- thermore, let λ be ((3, 2, 1), (2), (2, 2), −). For the following λ-tableaux

1 2 3 7 9 s =  4 5 , 8 10 , , −  , 11 12    6      1 2 3 6 8 t =  4 5 , 9 11 , , −  , 7 10    12      6.2. q-PERMUTATION MODULES 91 their corresponding λ-tableaux of type µ are

1 1 1 5 6 µ(s) =  2 2 , 5 6 , , −  , 6 7    3      1 1 1 3 5 µ(t) =  2 2 , 6 6 , , −  . 5 6    7     

By deﬁnition, µ(s) is semistandard and µ(t) is strongly semistandard.

6.2. q-Permutation Modules

The q-permutation modules for Iwahori-Hecke algebras of type A are defined in [DJ1, Section 3]. The q-permutation modules for Iwahori-Hecke algebras of type B are defined independently in [DS, 3.2] and [DJM1, 3.1]. In both types, when the Hecke algebras are specialized to Weyl groups, the q-permutation modules become permutation modules. In type A, permutation modules are isomorphic to coset spaces of parabolic subgroups. In type B, they are isomorphic to coset spaces of quasi- parabolic subgroups. Dipper, James and Mathas have generalized the notion to some cyclic modules based on which they have defined Specht modules and have constructed all irreducible modules for the Ariki-Koike algebras in [DJM2, Section 3]. Their cyclic modules under specialization are not naturally associated with subgroups. We are going to define q-permutation modules which are also cyclic. Under specialization, they are “almost isomorphic” to coset spaces of some subgroups. ι We need the anti-automorphism ι on H, defined by Ti = Ti for all 0 6 i 6 r [DJM2, 2.3], in the subsequent definitions. 92 6. q-PERMUTATION MODULES

Definition 6.2.1. Let λ = (λ(1), λ(2), . . . , λ(m)) be an m-composition. Suppose (i) a1 = 0 and ai+1 = ai + |λ | for 1 6 i 6 m − 1. For 2 6 i 6 m − 1, deﬁne

|λ(i)| u0 = hι (L − Q ) ··· (L − Q ) h λ,i i  j 1 j i−1  i j=1  Y    (i) where hi = (T|λ(i)| ··· Tai+1−1) ··· (T2 ··· Tai+1)(T1 ··· Tai ), when |λ | > 0. When (i) 0 |λ | = 0, the element uλ,i is deﬁned to be the identity of H. Furthermore, deﬁne

0 0 0 uλ = uλ,m−1 ··· uλ,2.

Definition . 0 6.2.2 The element vλ is deﬁned to be the product uλuλ, where uλ = uλ,2 ··· uλ,m, and

uλ,i = (L1 − Qi)(L2 − Qi) ··· (Lai − Qi) for 2 6 i 6 m.

˜ 0 0 From the above definitions, if λ = µ˜ then we have uλ,i = uµ,i, uλ,i = uµ,i, uλ = uµ, 0 0 ˜ uλ = uµ and vλ = vµ. Labelling these elements by λ instead of λ, we can associate these elements more conveniently with any given composition. The elements uλ,i and uλ have been defined by Dipper, James and Mathas in [DJM2, 3.1]. They denote uλ,i by ua,i where a is uniquely determined by λ˜. They have also defined the element mλ = uaxλ¯ and the cyclic module mλH, where xλ¯ has the meaning as defined in 1.6.1. We are going to define a new cyclic module.

Definition 6.2.3 (q-permutation modules). If λ is an m-composition, the element xλ is deﬁned to be vλxλ¯ and the cyclic module xλH is called a q-permutation module.

Note that our q-permutation modules xλH is related to Dipper, James and Mathas’ 0 cyclic module mλH by xλH = uλmλH. The q-permutation modules are naturally related with subgroups. 6.2. q-PERMUTATION MODULES 93

i−1 When q = 1 and Qi = ξ , we can check that

i−1 i−1 i−1 uλ,i = (t1 − ξ )(t2 − ξ ) ··· (tai − ξ ), and

|λ(i)| 0 i−2 uλ,i = sai+j−1 ··· sj(tj − 1) ··· (tj − ξ )sj ··· sai+j−1 j=1 Y |λ(i)| i−2 = (tai+j − 1) ··· (tai+j − ξ ). j=1 Y i (i) If (tj + ξ ) is denoted by τj , then vλ becomes the product of all the following factors, 0 0 0 uλ,2 uλ,3 uλ,m−1 (0) (0) (0) (0) (0) (0) τa2+1 ··· τa3 τa3+1 ··· τa4 τam−1+1 ··· τam uλ,2 (1) (1) (1) (1) ... (1) (1) τ1 ··· τa2 τa3+1 ··· τa4 τam−1+1 ··· τam uλ,3 (2) (2) (2) (2) (2) (2) τ ··· τa2 · τ ··· τa3 1 a2+1 τam−1+1 ··· τam uλ,4 (3) (3) (3) (3) (3) (3) . τ1 ··· τa2 · τa2+1 ··· τa3 · τa3+1 ··· τa4 . . (m−3) (m−3) . τam−1+1 ··· τam . . (m−1) (m−1) (m−1) (m−1) (m−1) (m−1) (m−1) (m−1) uλ,m τ1 ··· τa2 · τa2+1 ··· τa3 · τa3+1 ··· τa4 ··· τam−1+1 ··· τam

On the other hand, all the factors of uλ are mutually commutative. Under specialization, uλ can be rewritten as

am kj kj+1 m−1 (tj − ξ )(tj − ξ ) ··· (tj − ξ ), j=1 Y λ where kj = i if j is in the ith component of t . Hence, under specialization,

m−1 ai+1 \i−1 m−1 vλ = (tj − 1)(tj − ξ) ··· (tj − ξ ) ··· (tj − ξ ). i=1 j=a +1 Y Yi The sign denotes the omission of the factor underneath. Obviously,

m−1 2 m−1 b (tj − ξ) ··· (tj − ξ ) = 1 + tj + tj + ··· + tj , 94 6. q-PERMUTATION MODULES which is the sum of elements of the subgroup generated by tj. Now, denote

\i−1 m−1 (tj − 1)(tj − ξ) ··· (tj − ξ ) ··· (tj − ξ )

(m−1)(i−1) 1−i 1−i 1−i 2−i 1−\i 1−i m−i = ξ (ξ tj − ξ )(ξ tj − ξ ) ··· (ξ tj − 1) ··· (ξ tj − ξ )

1−i 1−i 1−i m−1 = ξ (ξ tj − ξ) ···(ξ tj − ξ )

1−i 1−i 1−i 2 1−i m−1 = ξ 1 + ξ tj + (ξ tj) + ··· + (ξ tj) , by v. This element v is the product of of ξ1−i and the sum of elements of the 1−i i−1 subgroup generated by ξ tj in RW . Furthermore, we have vtj = ξ v. Let Cλ be the subgroup generated by

1−i λ (i) ξ tj j is an entry in t , 1 6 i 6 m − 1 . n o Under specialization, xλ is a product of a power of ξ and the sum of the elements in Wλ = CλSλ¯ . The module xλH is naturally associated with the subgroup Wλ.

This module is not a permutation module under specialization because Wλ is not a subgroup of W if there exists a non-empty ith component of tλ for some 2 6 i 6 m−1.

For any m-composition λ, the subgroup Wλˆ is quasi-parabolic, and the module xλˆ H will specialize to a permutation module.

Lemma . S 6.2.4 Suppose λ is an m-composition. If σ ∈ λ˜ , then vλTσ = Tσvλ.

Proof. By Lemma 4.1.2(f), the element Tσ commutes with each uλ,i and hence 0 with uλ. To show Tσ commutes with uλ it suﬃces to show that for each 2 6 i 6 m−1, 0 S Tσ commutes with uλ,i. By deﬁnition, hi = Tw for some w ∈ r and w in two-line notation is

1 2 ··· αi αi + 1 αi + 2 ··· ai+1 ai+1 + 1 ··· r ,   ai + 1 ai + 2 ··· ai+1 1 2 ··· ai ai+1 + 1 ··· r   (i) (1) (i−1) where αi = |λ | and ai = |λ | + ··· + |λ |. Let α(i) be the composition

α(i) (αi, α1, . . . , αi−1, αi+1, . . . , αm). Immediately, the α(i)-tableau t w is row-standard. S Thus w is a distinguished right α(i) -coset representative. Similarly, we can write 6.2. q-PERMUTATION MODULES 95 w−1 as

1 2 ··· ai ai + 1 ai + 2 ··· ai+1 ai+1 + 1 ··· r .   αi + 1 αi + 2 ··· ai+1 1 2 ··· αi ai+1 + 1 ··· r   −1 Hence w is a distinguished right Sα-coset representative, where α is the composition S S (α1, . . . , αm). Thus w is a distinguished left α-coset representative. Since σ ∈ λ˜ = −1 Sα, we have hiTσ = TwTσ = Twσ. Let π = wσw and so wσ = πw. Obviously S π ∈ α(i) , so we also have Tπw = Tπhi. Hence, we have hiTσ = Tπhi.

For each j,(L1 − Qj ) ··· (Lαi − Qj ) commutes with all Tk except k = αi. Thus Tπ i−1 commutes with j=1(L1 − Qj) ··· (Lαi − Qj).

− − ι ι Finally, sinceQTw 1 Tπ = TσTw 1 , then hiTπ = Tσhi. The result then follows.

Corollary 6.2.5. The elements vλ and xλ¯ commute.

(1) (2) (m) Lemma 6.2.6. Suppose λ = (λ , λ , . . . , λ ) is an m-composition. Let a1 = 0, (i) ai+1 = ai + |λ |. If ai < k 6 ai+1 and 1 6 i 6 m − 1, then

e, when k = ai + 1 −(k−ai−1) ι vλLk = q QivλTσTσ where σ =  Tai+1 ··· Tk−1, otherwise.

Proof. 0 0 For simplicity, we write ui = uλ,i, ui = uλ,i. It suﬃces to show that

0 −(k−ai−1) ι uλuλ(Lk − q QiTσTσ) = 0.

S ι 0 S Since σ ∈ λ˜ , so Tσ and Tσ commute with all uj and uj. Each hj ∈ H( aj+1 ), so 0 uj and ui are commutative if j < i. Therefore,

0 −(k−ai−1) ι 0 −(k−ai−1) ι uλuλ(Lk − q QiTσTσ) = uλuλq Tσ(Lai+1 − Qi)Tσ

−(k−ai−1) ι 0 0 0 0 = q Tσum−1 ··· uiui(Lai+1 − Qi)ui+1 ··· umui−1 ··· u2u2 ··· ui−1Tσ.

Because of (Lai+1 − Qi), the factor (T1 ··· Tai ) of hi commutes with ui(Lai+1 −

Qi)ui+1 ··· um. The remaining factor of hi commutes with (L1 − Q1) ··· (L1 − Qi−1). 96 6. q-PERMUTATION MODULES

0 Thus, the factor uiui(Lai+1 − Qi)ui+1 ··· um of the above expression can be written as

|λ(i)| hι (L − Q ) ··· (L − Q ) h u (L − Q )u ··· u i  j 1 j i−1  i i ai+1 i i+1 m j=1  Y  |λ(i)|  ι  = h (L − Q ) ··· (L − Q ) (T (i) ··· T ) ··· (T ··· T ) i  j 1 j i−1  |λ | ai+1−1 2 ai+1 j=2  Y 

(L1 − Q1) ··· (L1 − Qi−1)ui(Lai+1− Qi)ui+1 ··· um(T1 ··· Tai )

= ··· (L1 − Q1)(L1 − Q2) ··· (L1 − Qm) ··· = 0

Hence, the result follows.

Definition . b εr ε1 S b 6.2.7 If = Lr ··· L1 Tσ ∈ B with σ ∈ r, deﬁne n0( ) to be

ε1 + ··· + εr. Here, B is the basis in 4.1.3.

Lemma . m 6.2.8 For any integer k ∈ r, the element Lk is a linear combination of terms lTσ ∈ B such that n0(l) 6 m, l is in L1,...,Lk and σ ∈ Sk.

Proof. We are going to prove the lemma by induction on k. When k = 1, the m lemma follows from (T0 − Q1) ··· (T0 − Qm) = 0. Assume the lemma is true for Lk . By 4.1.2(d),

m −1 m−1 Lk+1 = q Lk+1 TkLkTk

m−1 −1 m−1 −1 m−1−j j = q TkLk LkTk + q (q − 1) Lk Lk+1 LkTk. j=1 ! X m The lemma for Lk+1 then follows from the inductive assumption and 4.1.2(d).

For proving the independence of a certain subset of H, we need a linear order on the basis B.

0 ε0 Definition . εr ε1 0 εr 1 6.2.9 Let l = Lr ··· L1 and l = Lr ··· L1 be elements in L =

εr εr−1 ε1 0 {Lr Lr−1 ··· L1 }. We write l < l , if either

0 (a) εr > εr, or 0 0 0 (b) εr = εr, . . . , εk+1 = εk+1 and εk > εk, for some 1 6 k < r. 6.2. q-PERMUTATION MODULES 97

Furthermore, if l is a linear order on Sr such that the identity is the minimal 0 0 element, then we deﬁne a linear order on the basis B as follows. Let b = lTσ, b = l Tσ0 0 0 0 such that l, l ∈ L and σ, σ ∈ Sr. We write b < b when (a) l < l0, or (b) l = l0 and σ l σ0.

The following subsets of B are frequently referred to in the proof of the subsequent lemmas.

Definition . εa ε1 6.2.10 If L = La ··· L1 with εa =6 0, deﬁne

P (L) = R-span of {lTσ ∈ B|l ∈ L, σ ∈ Sa, n0(l) 6 n0(L), and l > L}, and

0 P (L) = R-span of {lTσ ∈ B|l ∈ L, σ ∈ Sa, n0(l) < n0(L), and l > L}.

In the proofs of the following lemmas, we use Lemma 4.1.2(c) and (d) frequently. Note that the terms on the right hand sides in the 4.1.2(c) and (d) are terms b ∈ B with n0(b) = ε if 0 6 ε 6 m − 1. When ε > m, the two parts of the lemma are still valid, but the n0 values of the terms on the right hand sides may be less than ε.

Lemma 6.2.11. Let ε, j, k be integers such that 1 6 ε 6 m − 1 and 1 6 j 6 k 6 r − 1. Then ε k−j+1 ε Tk ··· TjLjTj ··· Tk = q Lk+1 + h,

ε where h ∈ P (Lk+1).

Proof. When ε = 1, the result follows directly from the deﬁnition of Lk+1. In the case that ε > 2, we prove it by induction on k − j. If k − j = 0, by Lemma 4.1.2(d),

ε ε ε−1 ε−1 ε TjLj Tj = Lj+1TjTj − (q − 1) Lj Lj+1 + ··· + LjLj+1 + Lj+1 Tj ε ε−1 ε−1 (6.2.12) = qLj+1 − (q − 1) Lj Lj+1 + ··· + LjLj+1 Tj.

Assume the result holds for positive integers j, k such that j 6 k. Then by the induction hypothesis,

ε k−j+1 ε Tk+1(Tk ··· TjLjTj ··· Tk)Tk+1 = q Tk+1Lk+1Tk+1 + Tk+1hTk+1, 98 6. q-PERMUTATION MODULES

ε k−j+2 ε 0 0 where h ∈ P (Lk+1). By 6.2.12, the ﬁrst term becomes q Lk+2 + h , where h ∈ ε P (Lk+2).

εk+1 ε1 The element h can be written in terms of the form Lk+1 ··· L1 Tσ, where εk+1 < ε,

ε1 + ··· + εk+1 6 ε, and σ ∈ Sk+1. Moreover,

εk+1 ε1 εk ε1 εk+1 −1 Tk+1Lk+1 ··· L1 TσTk+1 = Lk ··· L1 Tk+1Lk+1 Tk+1Tk+1TσTk+1.

ε By 6.2.12 again, the above is in P (Lk+2).

Lemma 6.2.13. Let λ be an m-composition. Suppose 2 6 i 6 m − 1, |λ(i)| > 0, (1) (i−1) and ai = |λ | + ··· + |λ |. Then

(i) hιLi−1 ··· Li−1h = q|λ |ai Li−1 ··· Li−1 + h, i |λ(i)| 1 i ai+1 ai+1

i−1 i−1 where h ∈ P (Lai+1 ··· Lai+1).

Proof. We are going to prove the lemma by induction on |λ(i)|. When |λ(i)| = 1, the result follows from Lemma 6.2.11. Denote Tk ··· Tai+k−1 by hi,k. Assume the lemma holds for |λ(i)| = p. When |λ(i)| = p + 1, by Lemma 6.2.11

ι i−1 i−1 ι ι i−1 i−1 ι i−1 hiLp+1 ··· L1 hi = hi,1 ··· hi,pLp ··· L1 (hi,p+1Lp+1hi,p+1)hi,p ··· hi,1

ai ι ι i−1 i−1 i−1 = q hi,1 ··· hi,pLp ··· L1 Lai+p+1hi,p ··· hi,1

ι ι i−1 i−1 0 + hi,1 ··· hi,pLp ··· L1 h hi,p ··· hi,1,

0 i−1 ι ι for some h ∈ P (Lai+p+1). Since Lai+p+1 commutes with hi,1 ··· hi,p, we can apply the induction hypothesis to the ﬁrst term and rewrite it as

(p+1)ai i−1 i−1 ai i−1 q Lai+p+1 ··· Lai+1 + q Lai+p+1h.

i−1 i−1 i−1 i−1 i−1 Since h ∈ P (Lai+p ··· Lai+1), we have Lai+p+1h ∈ P (Lai+p+1 ··· Lai+1). 0 ε We can write h as a linear combination of terms in the form Lai+p+1 l Tσ such S that ε < i − 1, n0(l) 6 i − 1 − ε, l is in L1,...,Lai+p only and σ ∈ ai+p+1. Hence, ι ι i−1 i−1 0 the second term in the above equation, hi,1 ··· hi,pLp ··· L1 h hi,p ··· hi,1, is a linear combination of the terms

ε ι ι i−1 i−1 Lai+p+1(hi,1 ··· hi,p lLp ··· L1 )Tσhi,p ··· hi,1. 6.2. q-PERMUTATION MODULES 99

We can see that the element in the brackets is in H(G(m, 1, ai + p)). Then, by i−1 i−1 Lemma 4.1.2(c), (d) and Lemma 6.2.8, the above terms are in P (Lai+1 ··· Lai+1) and the lemma then follows.

(1) (2) (m) Lemma 6.2.14. Suppose λ = (λ , λ , . . . , λ ) is an m-composition. Let αi = (i) |λi |, a1 = 0, ai+1 = ai + αi, and let a = am. Then

m−1 ε 0 m−1 m−1 vλ = q L + h, where h ∈ P (La ··· L1 ), and ε = αiai. i=2 X Proof. Suppose Tσ1 l1Tσ2 l2 ··· Tσk lk is an element in H(G(m, 1, a)) such that l1, . . . , lk ∈ L ∩ H(G(m, 1, a)) and σ1, . . . , σk ∈ Sa. Thanks to 4.1.2(c), (d) and 6.2.8 0 m−1 m−1 again, if n0(l1)+···+n0(lk) < (m−1)a, then the element will be in P (La ··· L1 ). 0 ι i−1 i−1 Let li = hiLi ··· Lαi hi. The element vλ is the sum of the ﬁrst term

0 0 m−1 m−1 m−2 m−2 f = lm−1 ··· l2(L1 ··· La2 )(La2+1 ··· La3 ) ··· (Lam−1+1 ··· Lam ),

0 m−1 m−1 and terms in P (La ··· L1 ). S Since hi ∈ H( ai+1 ), we have

0 0 m−2 m−2 m−1 m−1 f = (lm−1Lam−1+1 ··· Lam ) ··· (l2La2+1 ··· La3 )(L1 ··· La2 ).

Applying the anti-automorphism ι to the statement of Lemma 6.2.13, we have

0 αiai i−1 i−1 ι i−1 i−1 li = q Lai+1 ··· Lai+1 + h, where h ∈ P (Lai+1 ··· Lai+1). We can write f as sum of

m−1 ε m−1 m−1 q La ··· L1 , where ε = αiai, i=2 X and a linear combination of terms of the form

m−1 m−1 m−i m−i 0 h = La ··· Lai+1+1TσlLai+1 ··· Lai+1h ,

S 1 ai+1 0 where αi =6 0, σ ∈ ai+1 , l = L1 ··· Lai+1 , and h ∈ H(G(m, 1, ai)). Moreover n0(li) 6 αi(i − 1) and ai+1 < i − 1. Then

m−i m−i 0 ai+1 ai+1 m−i m−i 1 ai 0 TσlLai+1 ··· Lai+1h = TσLai+1 ··· Lai+1 Lai+1 ··· Lai+1L1 ··· Lai h ai+1 ai+1 m−i m−i 00 = TσLai+1 ··· Lai+1 Lai+1 ··· Lai+1h , 100 6. q-PERMUTATION MODULES

00 ai+1 for some h ∈ H(G(m, 1, ai)). Since k=ai+1 k < αi(i − 1), the term h is in 0 m−1 m−1 P (La ··· L1 ). The lemma hence follows.P

Denote the set of distinguished Sλ¯ -coset representatives in Sr by Dλ¯ .

Lemma 6.2.15. For any m-composition λ, the algebra H has a basis

{TσlTd|σ ∈ Sλ¯ , l ∈ L and d ∈ Dλ¯ }.

Proof. Since the number of elements in the above set is mrr!, we only need to show that it is a spanning set. Take any element lTπ ∈ B with l ∈ L and π ∈ Sr. We then write π = σd, where σ ∈ Sλ¯ and d ∈ Dλ¯ . By Lemma 4.1.2(c) and (d), we have

ε ε ε ε−k k Li Ti = TiLi+1 − (q − 1) Li Li+1 , ! Xk=1 ε ε ε ε−k k Li+1Ti = TiLi + (q − 1) Li Li+1 . ! Xk=1 Apply them to lTσ, we have

lTσTd = rwTwlwTd, X where rw ∈ R, lw ∈ L and the summation is over a set of some elements w 6 σ with respect to Bruhat order on Sr. Thus, each w is a subword of a reduced expression for σ, then it is also an element in Sλ¯ . The set spans H and hence it is basis.

Proposition 6.2.16. Let λ be an m-composition. The q-permutation module xλH is free with basis

εa+1 εr Bλ = {xλLa+1 ··· Lr Td|d ∈ Dλ¯ , 0 6 εi 6 m − 1 for all a + 1 6 i 6 r}, where a = r − |λ(m)|.

Proof. To show that Bλ spans xλH, we only need to prove that xλh is in the

ε1 εr S span of Bλ for any h = L1 ··· Lr Tσ, where σ ∈ r. By Lemma 6.2.6, for any 0 0 S S 1 6 i 6 a the product xλLi can be written as xλ¯ vλh where h ∈ H( λ˜ ∩ a). By 6.2. q-PERMUTATION MODULES 101

0 0 Lemma 6.2.4, the element h commutes with vλ. Thus, we have xλLi = xλ¯ h vλ and 00 S S we can repeatedly apply Lemma 6.2.6 to xλh until for some h ∈ H( λ˜ ∩ a)

00 εa+1 εr εa+1 εr 00 xλh = xλ¯ h vλLa+1 ··· Lr Tσ = vλxλ¯ La+1 ··· Lr h Tσ.

00 Then we write h Tσ as a linear combination of the elements in the form of TπTd, with

S εa+1 εr π ∈ λ¯ and d ∈ Dλ¯ . Now, we only need to show that vλxλ¯ La+1 ··· Lr TπTd is in the span. We now take a reduced expression si1 ··· si` for π. If si is a factor of this

εa+1 εr product, then Ti commutes with La+1 ··· Lr when i < a. Otherwise, we have i > a

εi εi+1 εi εi+1 εi εi+1 and we apply Lemma 4.1.2(e) to Li Li+1 Ti. If εi = εi+1, then Li Li+1 Ti = TiLi Li+1 .

If εi > εi+1, then we have

εi εi+1 εi−εi+1 εi+1 εi+1 Li Li+1 Ti = Li TiLi Li+1 .

εi εi+1 By part (c) of the lemma, this expression is the sum of TiLi Li+1 and some terms 0 0 εi εi+1 0 0 in form of Li Li+1 with both εi and εi+1 not exceeding εi. When εi < εi+1, the 0 0 εi εi+1 εi εi+1 0 expression is the sum of TiLi Li+1 and some terms in form of Li Li+1 with both εi

0 εa+1 εr and εi+1 not exceeding εi+1. Thus, vλxλ¯ La+1 ··· Lr TπTd is a linear combination of 0 0 εa ε 0 +1 r 0 S 0 the terms of the form vλxλ¯ Tπ La+1 ··· Lr Td, where π ∈ λ¯ . Since xλ¯ Tπ = q xλ¯ for some integer , the set Bλ spans xλH.

We now order the elements in the basis in Lemma 6.2.15 to b1,..., bmrr!, where b S k = Tσk lkTdk with σk ∈ λ¯ , lk ∈ L, and dk ∈ Dλ¯ , such that i < j if either

(a) σi l σj; or

(b) σi = σj and liTdi < ljTdj with respect to the order deﬁned in 6.2.9.

ε m−1 m−1 b 0 By Lemma 6.2.14, vλ is the sum of q L1 ··· La and terms = lTσ in B with 0 n0(l) < a(m − 1) and σ ∈ Sa. Then by 4.1.2(c) and (d), when vλ is written in terms of basis 6.2.15, the factor li of each non-zero term bi has n0 value less than

m−1 m−1 εa+1 εr a(m − 1) except the L1 ··· La term. When xλLa+1 ··· Lr Td is written as a linear combination of elements in the basis 6.2.15 ordered as the above, the ﬁrst term

ε m−1 m−1 εa+1 εr is q L1 ··· La La+1 ··· Lr Td. The independence of the set Bλ then follows. 102 6. q-PERMUTATION MODULES

6.3. Semistandard Basis of xλH

The Robinson-Schensted algorithm establishes a one-one correspondence between the set of all r × r permutation matrices and the set of standard pairs

{(s, t)|s, t are standard ν-tableaux for some partition ν of r}.

In [K2], Knuth generalized it to an algorithm (RSK) which establishes a one-one correspondence between the set of all n × n matrices (zij) of non-negative integers satisfying n n

zij = λi and zij = µj, j=1 i=1 X X and the set of semistandard pairs. More precisely, [K2, Theorem 2] can be restated as follows.

Theorem 6.3.1. Let λ = (λ1, . . . , λn) and µ = (µ1, . . . , µn) be compositions of r. The set of n × n matrices (zij) with row sums λi and column sums µj is in one- to-one correspondence with the set of semistandard pairs (s, t) such that s and t are ν-tableaux of type λ and of type µ, respectively, for some partition ν of r.

It follows that for any compositions λ and µ, there is a one-one correspondence between Sλ\Sr/Sµ and the set of semistandard pairs of tableaux of type λ and type µ. We also use the terms standard pairs, semistandard pairs, etc. for pairs of m- tableaux of the same shape of an m-partitions. In particular, we are interested in strongly semistandard-standard pairs in which the ﬁrst one is strongly semistandard and the second one is standard.

Lemma 6.3.2. There is a one-one correspondence between the set Bλ deﬁned in 6.2.16 and the set of strongly semistandard-standard pairs (s, t) such that s is of type λ.

Proof. λ λ λ λ Let n = |n1 |+···+|nm|, and let a = |n1 |+···+|nm−1|. Take µ as the m- r composition (−,..., −, (1 )). Thus, Wλˆ is quasi-parabolic and Wµ is the trivial group. 6.3. SEMISTANDARD BASIS OF xλH 103

The set of Wλˆ -cosets is the same as the set of (Wλˆ ,Wµ)-double cosets. Without loss of generality, we discard all the rows of zeros from row n + 1 in Theorem 5.4.2. Then, there is a one-one correspondence between the set of double cosets Wλˆ -cosets and the set of n×r arrays (zij) whose entries are sequences of non-negative integers such that

(a) zij = (zij) is a sequence of one term, when i 6 a;

(b) zij = (zij0, . . . , zij(m−1)) is a sequence of m terms, when i > a; r n ¯ (c) zij = λi and zij = 1, where zij is the sum of the terms in zij. j=1 i=1 P P For each 1 6 k 6 m, form an n × r matrix Zk from (zij) by replacing the rows zi∗ by λ rows of zeros when i∈ / nk and i 6 a. Also replace each of the m-term sequences zi∗ by its kth entry when i > a. By RSK-algorithm, Zk is associated with a semistandard (k) λ λ t(k) 1-tableau s with all the entries in nk ∪ nm and a standard tableau with entries corresponding to the non-zero columns of Zk. Then we get a strongly semistandard ν-tableau s = (s(1),..., s(m)) and a standard ν-tableau t = (t(1),..., t(m)) for some m-partition ν. It is clear that if we start with a strongly semistandard-standard pair, by reversing the above process, we can get a Wλˆ -coset.

Finally, the number of Wλˆ -cosets is the same as the number of elements in Bλ.

From the basis of the cyclic modules in [DJM2], we are going to construct another basis for our q-permutation module. Let us ﬁrst quote some deﬁnitions and results in the paper.

Definition 6.3.3. [DJM2, 3.5, 3.8, 3.14] Let λ be an m-partition and µ be an m-composition. Deﬁne

(a) the element mµ to be uµxµ¯ , µ (b) the module M to be the right ideal mµH, and ι s t (c) the element mst to be Tδ(s)mλTδ(t), where , are standard λ-tableaux. 104 6. q-PERMUTATION MODULES

Definition 6.3.4. [DJM2, 4.8] Suppose that λ is an m-partition and µ is an ss s m-composition. For any s ∈ T (λ, µ) and t ∈ T (λ), the element mst by

mst = mst. s∈Ts(λ) µX(s)=s

Theorem 6.3.5. [DJM2, 4.14] Let µ be an m-composition of r. Then M µ is free as an R-module with basis

ss s {mst | s ∈ T (λ, µ) and t ∈ T (λ) for some m-partition λ of r} .

0 By multiplying uµ to the basis elements in the above theorem and picking up all the non-zero ones, we get a basis for xµH. The new basis elements are associated with strongly semistandard tableaux.

Lemma 6.3.6. If s is a semistandard λ-tableau of type µ which is not strongly 0 semistandard, then uµmst = 0.

Proof. It suffices to show that for any standard λ-tableau s such that µ(s) = s, 0 the product uµmst is zero. Since s is semistandard but not strongly semistandard, there exists an integer j which is an entry in both s(k) and (tµ)(l), for some l =6 m and k < l. Without loss of generality, we assume that j is the smallest one with this property. Suppose that (1) (l−1) (1) (l−1) aλ = |λ | + ··· + |λ | and aµ = |µ | + ··· + |µ |. Obviously aλ > aµ, and we let p = aλ − aµ. Since s is semistandard, the integers 1, . . . , aµ are in the first (l − 1) components of s. Let b1, . . . , bp be the other entries in the first (l − 1) components of s such that aµ < b1 < ··· < bp. By the choice of j, we have b1 = j.

To avoid complicated subscripts, we write S(ν) instead of Sν for any composition

ν. Write δ(s) as σd, where σ ∈ S(aλ) and d is a distinguished right S(aλ)-coset representative. Hence, d corresponds to a row-standard (aλ, r − aλ)-tableau with the

ﬁrst row (1, . . . , aµ, j, b2, . . . , bp). We can use the length reduction sort to arrange the last r − (aµ − 1) terms in order. Let the corresponding elements be d1, and obviously 6.3. SEMISTANDARD BASIS OF xλH 105 d1 ∈ S(aµ + 1). Let

e if j = aµ + 1, d2 =  saµ+1 ··· sj−1 if j > aµ + 1.

 (l) Hence, we have d = d1d2. Since j 6 aµ + |aµ |, the element d2 is in S(µ˜) and hence

ι 0 S aµ+1 ι Td2 commutes with uµ. For the element d1, it is in (1 , r − aµ − 1), and so Td1 0 0 commutes with uµ,l−1 ··· uµ,2 and the right factor (L1 − Q1) ··· (L1 − Ql−1)T1 ··· Taµ 0 S ι of uµ,l. Furthermore, Tσ is in (aλ) and then Tσ commutes with uλ,l ··· uλ,m. Hence,

0 0 0 ι ι ι st t uµm = uµ,m−1 ··· uµ,2Td2 Td1 Tσuλ,2 ··· uλ,mxλ¯ Tδ( )

ι 0 0 = ··· Td1 (L1 − Q1) ··· (L1 − Ql−1)T1 ··· Taµ uµ,l−1 ··· uµ,2

ι uλ,l ··· uλ,mTσuλ,2 ··· uλ,l−1xλ¯ Tδ(t)

ι 0 0 = ··· Td1 (L1 − Q1) ··· (L1 − Ql−1)uλ,l ··· uλ,mT1 ··· Taµ uµ,l−1 ··· uµ,2

ι Tσuλ,2 ··· uλ,l−1xλ¯ Tδ(t)

= ··· (L1 − Q1)(L1 − Q2) ··· (L1 − Qm) ··· = 0.

Hence the lemma follows.

Now, we state and prove the following theorem for our q-permutation module.

Theorem 6.3.7. Let µ be an m-composition of r. Then xµH is free as an R- module with basis

0 Tsss t Ts uµmst | s ∈ (λ, µ) and ∈ (λ) for some m-partition λ of r . Proof. 0 µ By 6.3.5, the previous Lemma and the fact that xµH = uµM , the above set spans xµH. From Lemma 6.3.2, this set has the same number of elements as the basis in Proposition 6.2.16 and hence it is a basis. 106 6. q-PERMUTATION MODULES

6.4. Specht Series and Young’s Rule for xλH

In this section, we ﬁrst summarize the construction of Specht modules and Specht series for mλH in [DJM2]. Using their Specht modules, we are going to construct Specht series for the q-permutation modules.

Definition 6.4.1. [DJM2, 3.21(ii)] If λ is an m-composition, then N λ is deﬁned to be the R-module spanned by

s {mst|s, t ∈ T (λ) for some m-partition µ . λ} .

Proposition 6.4.2. [DJM2, 3.22] If λ is an m-composition, then N λ is a two- sided ideals of H.

Definition 6.4.3. [DJM2, 3.28] For an m-partition λ, let zλ be the element λ λ λ λ mλ + N in H/N . The Specht module S is the submodule of H/N generated by zλ as an H-module.

Theorem 6.4.4 (Young’s rule for M µ). [DJM2, 4.15] Suppose that µ is an m- composition. Then there is a ﬁltration of M µ,

µ M = M1 > M2 > ··· > Mk+1 = 0,

∼ λi such that for each i with 1 6 i 6 k there exists an m-partition λi with Mi/Mi+1 = S .

Moreover, if λ is an m-partition, then the number of i such that λ = λi is equal to the number of semistandard λ-tableaux of type µ.

Corollary 6.4.5 (Young’s rule for xµH). Suppose that µ is an m-composition.

Then there is a ﬁltration of xµH,

xµH = M˜ 1 > M˜ 2 > ··· > M˜ p+1 = 0,

∼ λi such that for each i with 1 6 i 6 p there exists an m-partition λi with M˜ i/M˜ i+1 = S .

Moreover, for any m-partition λ, the number of i such that λ = λi is equal to the number of strongly semistandard λ-tableaux of type µ. 6.4. SPECHT SERIES AND YOUNG’S RULE FOR xλH 107

Proof. As in the proof of [DJM2, Theorem 4.15], the semistandard tableaux of type µ are arranged in order, s1 > s2 > ··· > sk, such that j > i if λi . λj where λn µ is the shape of sn for all n. The module Mi is deﬁned to be the R-submodule of M spanned by t Ts msj t|j > i, and ∈ (λj) . Then the modules M1,...,Mk+1 are right H-submodules and the quotient module

λi Mi/Mi+1 is isomorphic to the Specht module S .

Let sn1 > sn2 > ··· > snp be the subsequence of all strongly semistandard tableaux of type µ. By deﬁnition and Lemma 6.3.6, for a ﬁxed integer i, if ni > ni+1, then

0 0 0 uλMni > uλMni+1 = ··· = uλMni+1 .

0 ˜ Then denote the right H-module uµMni by Mi. Then we have

xµH = M˜ 1 > M˜ 2 > ··· > M˜ p+1 = 0.

λ ∼ ˜ ˜ For an arbitrary ﬁxed i, let λ = λni , s = sni . We claim that S = Mi/Mi+1. The theorem follows directly from the claim. ˜ ˜ ∼ We only need to show that Mi/Mi+1 = Mni /Mni+1. Since the two modules have 0 ˜ the same rank, we can deﬁne an R-map Φ such that Φ(uµmst + Mi+1) = mst + Mni+1. Let

msth = h1 + rvmsv, v Ts ∈X(λ) where h1 ∈ Mni+1. Thus

0 ˜ 0 ˜ 0 ˜ uµmst + Mi+1 h = uµmsth + Mi+1 = rvuλmsv + Mi+1, v Ts  ∈X(λ)   0 ˜ because uµh1 ∈ Mi+1. Hence Φ is an H-isomorphism and the claim is then proved.

Remark 6.4.6. Corresponding to each m-partition λ, Du and Rui have constructed another version of Specht module which is an H-submodule generated by an element of the form mλh for some special h ∈ H. Naturally, it is worth investigating the possibility of constructing Specht module as a submodule generated by an element 108 6. q-PERMUTATION MODULES of the form xλh. We are not attempting to answer this question here, but may be in the future. CHAPTER 7

Other Imprimitive Complex Reﬂection Groups

7.1. Presentation of G(m, p, r) m In this chapter, we assume that m, p, r are integers and p divides m. Let d = . p The presentations of all irreducible complex reflection groups are given in [BMR, Appendix 2]. As mentioned there, the presentations of the imprimitive ones were known before [BMR] was written. For instance, the presentations of G(m, 1, r) and G(m, m, r) by r generators are given in [She2, 4.12] which was published in 1953. m 1 m In [She2], these groups are referred to as γn and m γn , respectively. We start from the definition of imprimitive complex reflection groups as a presentation (see [BMR] and [A2, p.165]).

Definition . 0 7.1.1 The group G(m, p, r) is generated by {s0, s1, s1, ··· , sr−1} subjected to the following relations

0 0 0 0 s2s1s1s2s1s1 = s1s1s2s1s1s2;

0 0 0 s1s2s1 = s2s1s2;

0 0 s0s1s1 = s1s1s0; 0 0 0 0 0 s1s0s1s1s1s1 ··· = s0s1s1s1s1s1s1 ···; p + 1 factors p + 1 factors | {z } | {z } sisi+1si = si+1sisi+1, if 1 6 i 6 r − 2;

0 0 s1si = sis1 if 3 6 i 6 r − 1;

sisj = sjsi, if |i − j| > 2; and

d 0 2 2 2 s0 = (s1) = s1 = ··· = sr−1 = e, where e is the identity. 109 110 7. OTHER IMPRIMITIVE COMPLEX REFLECTION GROUPS

It is well-known that the group deﬁned above is a subgroup of W = G(m, 1, r). Based on the matrix representation in [A2, section 1], we let

p g0 = ((ξ 1, 2, . . . , r)),

0 −1 g1 = ((ξ 2, ξ1, 3, . . . , r)), and

gi = ((1, . . . , i − 1, i + 1, i, i + 2, . . . , r)), 1 6 i 6 r − 1.

0 be elements in G(m, 1, r). As in [A2, p.166], deﬁne t2 = g1g1 and ti+1 = gitigi for 2 6 i 6 r − 1. That is in one-line notation, we have

−1 ti = ((ξ 1, 2, . . . , i − 1, ξi, i + 1, . . . , r)).

Lemma 7.1.2. Let w be an element in the subgroup

ε1 εr Wp = {((ξ a1, . . . , ξ ar)) ∈ W |ε1 + ··· + εr ≡ 0 (mod p)}.

ε ε2 εr S In the notation introduced above, w can be written as g0t2 ··· tr σ, with σ ∈ r and 1 ε = (ε + ··· + ε ). In particular, W is generated by {g , g0 , g , . . . , g }. p 1 r p 0 1 1 r−1

ε1 εr ε1 εr Proof. For any w = ((ξ a1, . . . , ξ ar)), we write it as ((ξ 1, . . . , ξ r))σ where

σ = ((a1, . . . , ar)). By composition of functions we have

ε ε2 εr pε−ε2−···−εr ε2 εr g0t2 ··· tr = ((ξ 1, ξ 2, . . . , ξ r)).

ε ε2 εr By deﬁnition, we have pε − ε2 − · · · − εr = ε1 and so w = g0t2 ··· tr σ. 0 From the above, Wp is a subgroup of the group generated by {g0, g1, g1, . . . , gr−1}. 0 Since we also have {g0, g1, g1, . . . , gr−1} ⊆ Wp, the second part of the lemma follows.

Proposition 7.1.3. The group G(m, p, r) is isomorphic to Wp. 7.2. REDUCED EXPRESSIONS FOR ELEMENTS IN G(2, 2, r) 111

Proof. 0 By Lemma 7.1.2, Wp is the subgroup generated by {g0, g1, g1, . . . , gr−1}. 0 We ﬁrst verify that all the relations in the deﬁnition 7.1.1 hold when g1, gi are sub- 0 stituted into s1, si. By composition of functions, we have

0 −1 g2g1g1 = ((ξ 1, 3, ξ2, 4, . . . , r)),

0 −1 g1g1g2 = ((ξ 1, ξ3, 2, 4, . . . , r)), and hence

0 0 −2 0 0 g2g1g1g2g1g1 = ((ξ 1, ξ2, ξ3, 4, . . . , r))= g1g1g2g1g1g2;

0 0 −1 0 g1g2g1 = ((ξ 3, 2, ξ1, 4, . . . , r))= g2g1g2;

0 p−1 g0g1g1 = ((ξ 1, ξ2, 3, . . . , r)), and

0 0 p−2 p−1 0 p−2 p−1 g1g0g1(g1g1) = ((ξ1, ξ 2, 3, . . . , r))(g1g1) = ((ξ 1, ξ2, 3, . . . , r)).

The last two lines above imply that

0 0 0 0 0 g1g0g1g1g1g1 ··· = g0g1g1g1g1g1g1 ···. p + 1 factors p + 1 factors | {z } | {z } Each of the other relations in 7.1.1 is either a deﬁning relation of G(m, 1, r) or straightforward to verify. Thus the mapping, which maps si to gi for all 0 6 i 6 r − 1 and 0 0 s1 to g1, can be lifted to an epimorphism G(m, p, r) → Wp. Since G(m, p, r) and Wp both have dmr−1r! elements, the result follows.

Thus any imprimitive complex reflection group can be defined as a subgroup of G(m, 1, r), and its elements in one-line form satisfy an extra condition. Do we have an algorithm similar to the one in Chapter 2 such that we can get reduced expression easily and effectively? We have a positive answer for G(2, 2, r) which is exactly the real reflection group of type Dr.

7.2. Reduced expressions for elements in G(2, 2, r)

The entries in the one-line form of each element in G(2, 2, r) are integers. The order ≺ is the usual order < on integers. For 1 6 i 6 r − 1, when si is multiply on 0 the left of an element, it also swaps the ith and the (i + 1)th terms. However, s1 does 112 7. OTHER IMPRIMITIVE COMPLEX REFLECTION GROUPS not only swap the ﬁrst two terms, it changes the signs of both terms. That is

0 s1((x1, x2, x3, . . . , xr))= (( − x2, −x1, x3, . . . , xr)).

Thus, if we can sort an element to the identity by swapping adjacent terms or swapping and changing signs of the ﬁrst two term, we can get an expression for the element.

Algorithm 7.2.1 (Reduced expression for an element in G(2, 2, r)). Given an element w = ((x1, . . . , xr)) in G(2, 2, r). The algorithm outputs an expression u = 0 u1 ··· un for w with u1, . . . , un ∈ {s1, s1, . . . , sr−1} and the number n = n(w). (a) [Initialization] Set n := 0, j := 2, and u := e. (b) [Loop] At this stage, the first j − 1 terms of w are integers in ascending order and all terms but possibly the first one are positive. If j 6 r, then perform (i) sort the first j terms of w in ascending order;

(ii) suppose that xj occupies the pth position; set u := u · (sj−1 ··· sp) and n := n + j − p;

(iii) rename the entries of w in order as x1, . . . , xr;

(iv) if x2 > 0, then set j := j + 1 and repeat step (b), else 0 (1) replace the ﬁrst two terms by −x2, −x1 and u := u · s1 and n :=

n + 1, (Note: Since x1 < x2 < 0, we have 0 < −x2 < −x1 here.) (2) sort the ﬁrst j terms which are positive integers in ascending order,

(3) suppose that −x2, −x1 occupy the pth, kth positions respectively,

set u := u · (s2 ··· sk−1s1 ··· sp−1) and n := n + p + k − 3;

(4) rename the entries of w in order as x1, . . . , xr, set j := j + 1, and repeat step (b). (c) [Output] Output u and n.

Example 7.2.2. For the element w = ((2, −4, −1, 3)) ∈ G(2, 2, 4), by the above 0 algorithm, we can get an expression s1s2s1s2s3 and n(w) = 5.

Similar to the case of G(m, 1, r), the above algorithm terminates and the output u is an expression for the input element w. We are going to prove that the output expression is reduced. 7.2. REDUCED EXPRESSIONS FOR ELEMENTS IN G(2, 2, r) 113

Lemma 7.2.3. Let w = ((1w, ··· , rw)) ∈ G(2, 2, r). Then

n(w) = 2 · card{(i, j)|i < j, |iw| < |jw|, jw < 0} + card{(i, j)|i < j, |iw| > |jw|}, where n(w) is the number of factors n output by Algorithm 7.2.1.

Proof. We write n(w) as the sum n2 + ··· + nr, where

nj = 2 · card{i|i < j, |iw| < |jw|, jw < 0} + card{i|i < j, |iw| > |jw|}.

We are going to show that the number nj is the number of times that jw swaps with iw for i < j in the algorithm. These swaps are called swaps made by jw. At the (j − 1)th time when we enter the loop of Algorithm, the ﬁrst j − 1 terms are in ascending order with all terms positive except, possibly, the ﬁrst one. These terms in order are named as x1, . . . , xj−1. From the algorithm,

{|x1|, x2, . . . , xj−1} = {|1w|,..., |(j − 1)w|}.

Thus the number of swaps made by jw with x1, x2, . . . , xj−1 is the same as that made by jw with 1w, ··· , (j − 1)w.

Case 1 — x1 > 0. When jw > 0, the number of swaps made by jw is card{i|i < j, xi > jw} which is nj. When jw < 0, this term will make j − 1 swaps to the ﬁrst term. After it turns to positive by some latter terms, the number of swaps made by jw will be card{i|i < j, xj < |jw|}. We must not forget that the swap between |jw| and |kw| with j < k is counted as a swap by kw. It is not diﬃcult to see that the number of swaps made by jw is also nj.

Case 2 — x1 < 0. When jw > 0, the number of swaps made by jw is card{i|1 < i < j, xi > jw} which is nj. If |x1| < jw, there is no more swap made by jw. If

|x1| > jw, there is one more swap made by jw after x1 is turned into positive. In both situations, the number of swaps made by jw is nj.

When jw < 0, it makes (j − 2) swaps up to the second term. If 0 > x1 > jw, two 0 swaps corresponding to s1 and s1 are made and the ﬁrst two terms become |x1| and 114 7. OTHER IMPRIMITIVE COMPLEX REFLECTION GROUPS

|jw|. Thus the number of swaps made by jw is

j + card{i|1 < i < j, |xi| < |jw|} = j − 1 + card{i|i < j, |xi| < |jw|}.

0 If 0 > jw > x1, the swap corresponding to s1 changes the ﬁrst two terms to |jw| and

|x1|. The number of swaps made by jw is also

j − 1 + card{i|i < j, |xi| < |jw|}, which is nj and the lemma follows.

Lemma 7.2.4. Let

w = ((x1, . . . , xr)),

0 w = (( − x2, −x1, x3, . . . , xr)),

00 w = ((x1, . . . , xk−1, xk+1, xk, xk+2, . . . , xr)).

0 (a) If both x1, x2 are negative, then n(w) = 1 + n(w ); and (b) if kw > (k + 1)w, then n(w) = 1 + n(w00).

Proof. If x1 < x2 < 0, the algorithm swaps the ﬁrst two terms into −x2, −x1 0 0 and turns w to w , and hence n(w) = 1+n(w ). If x2 < x1 < 0, the algorithm turns w 0 0 to ((−x1, −x2, x3, . . . , xr))by two swaps corresponding to s1s1 and it also turns w into 0 0 the same element by one swap corresponding to s1. Again, we have n(w) = 1 + n(w ) in this case. For (b), we let

N1(w) = {(i, j)|i < j, |iw| < |jw|, jw < 0},

N2(w) = {(i, j)|i < j, |iw| > |jw|}.

00 00 00 The sets N1(w ) and N2(w ) are similarly deﬁned. Since w and w are identical except 00 the kth and (k + 1)th terms, we have N1(w) \{(k, k + 1)} = N1(w ) \{(k, k + 1)} and 00 00 00 N2(w)\{(k, k +1)} = N2(w )\{(k, k +1)}. As kw = (k +1)w > (k +1)w = kw , we have four cases — (1) kw > (k+1)w > 0; (2) kw > 0 > (k+1)w and |kw| > |(k+1)w|; (3) kw > 0 > (k + 1)w and |kw| < |(k + 1)w|; or (4) 0 > kw > (k + 1)w. In case 1, 7.2. REDUCED EXPRESSIONS FOR ELEMENTS IN G(2, 2, r) 115

00 00 (k, k + 1) is in N2(w) but not N1(w),N1(w ) nor N2(w ). In case 2, (k, k + 1) is again 00 in N2(w) only. In case 3, (k, k + 1) is in N1(w) and N2(w ) only. In case 4, (k, k + 1) 00 is also in N1(w) and N2(w ) only. By the formula in 7.2.3, we have n(w) = n(w0) + 1 in all four cases.

Corollary 7.2.5. If s ∈ S and w ∈ W , then n(sw) = n(w) 1.

Proof. For any 1 6 k 6 r − 1, we have either kw > (k + 1)w or kw < (k + 1)w, 0 and hence n(skw) = n(w)1 by Lemma 7.2.4. By the same lemma, we have n(s1w) = n(w) 1 if 1w and 2w have the same sign. Let 1w = −a1 and 2w = a2 with a1, a2 0 0 being positive integers. Thus 1(s1w) = −a2 and 2(s1w) = a1. By the formula 7.2.3, 0 0 if a1 < a2, we have n(s1w) = n(w) + 1. If a1 > a2, then n(s1w) = n(w) − 1. Thus 0 0 n(s1w) = n(w) 1. Similarly, when 1w > 0 > 2w, we can check that n(s1w) = n(w) 1.

Since Algorithm 7.2.1 and Lemma 7.2.4 for G(2, 2, r) are exact analogues of Al- gorithm 2.1.7 and Lemma 2.2.2 for G(m, 1, r), we can prove the following proposition by the same argument used to prove Proposition 2.2.4.

Proposition 7.2.6. For any w ∈ G(2, 2, r), the value of n(w) output in Algo- rithm 7.2.1 equals the length `(w) of w and the output u is a reduced expression for w.

Remark 7.2.7. In G(2, 2, r), we have an efficient way to get reduced expressions for elements in the one-line form. Elements in G(m, p, r) can also be written in the one-line form. It is difficult but still possible to find an algorithm for getting reduced expressions of elements in G(m, p, r). If such an algorithm exists, we shall be able to derive length formulae from it. We shall then understand the groups and hence their Hecke algebras better. Hopefully, we can generalize the reducibility theorem and get some results for the characters. Another topic for our future research is to find integral bases for centres of the Ariki-Koike algebras. We have the reducibility theorem for G(m, 1, r). Very likely, we 116 7. OTHER IMPRIMITIVE COMPLEX REFLECTION GROUPS can use Francis’ approach to construct the integral bases for the centres. Naturally, we should aim at the goal of constructing an integral basis for the centre of the Hecke algebra of an arbitrary complex reflection group. APPENDIX - Quasi-Parabolic Subgroups of G (m; 1; r)

This appendix contains third party copyright materials and due to the copyright restrictions, the entire appendix has been removed from a digital form of thesis.

However, a full text of appendix is accessible through the “Journal of Algebra”, Volume 246, pp. 471 - 490 in 2001, or hardcopy is kept at the University of New South Wales Library, Sydney, Australia.

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