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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 fξ"aj0 6 " 6 m − 1; 1 6 a 6 rg; where " 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 = f1; : : : ; rg 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 cor- responding 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 irre- ducible 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 reducibil- ity 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 representa- tives. 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].
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