### Synopses & Reviews

In the 1970's, James developped a ``characterictic-free approach to the representation theory of the symmetric group on n letters, where Specht modules and certain bilinear forms on them play a crucial role. In this framework, we obtain a natural parametrization of the irreducible representations, but it is a major open problem to find explicit formulae for their dimensions when the ground field has positive characteristic. In a wider context, this problem is a special case of the problem of determining the irreducible representations of Iwahori--Hecke algebras at roots of unity. These algebras arise naturally in the representation theory of finite groups of Lie type, but they can be defined abstractly, as certain deformations of group algebras of finite Coxeter groups where the deformation depends on one or several parameters. One of the main aims of this book is to classify the irreducible representations of these Iwahori-Hecke algebras algebras at roots of unity. For this purpose, we develop an analogue of James' ``characterictic-free approach to the representation theory of Iwahori-Hecke algebras in general. The framework is provided by the Kazhdan-Lusztig theory of cells and the Graham-Lehrer theory of cellular algebras. When working over a ground field of characteristic zero, we also determine the dimensions of the irreducible representations, either by purely combinatorial algorithms (for algebras of classical type) or by explicit computations and tables (for algebras of exceptional type). The methods rely in an essential way on the ideas and results originating with the Lascoux-Leclerc-Thibon conjecture, which links Iwahori-Hecke algebras at roots of unity with the theory of canonical and crystal bases for the Fock space representations of certain affine Lie algebras. Thus, the main results of this book are obtained by an interaction of several branches of mathematics: Fock spaces and affine Lie algebras, the combinatorics of crystal bases, the theory of Kazhdan-Lusztig bases and cells, and computational methods.

#### Review

From the reviews: "This book unifies and summaries some of the work, mostly done during the last ten years, on representations of Iwahori-Hecke algebras of finite Coxeter groups. ... The book is very nicely written, striking the ideal balance between providing a uniform treatment of the finite Coxeter groups on the one hand, and presenting type-specific material on the other. ... In summary, this book is excellent. It will serve primarily as a reference for experts, but would also work well for self-study for a graduate student." (Matthew Fayers, Zentralblatt MATH, Vol. 1232, 2012)

#### Review

From the reviews:

"This book unifies and summaries some of the work, mostly done during the last ten years, on representations of Iwahori-Hecke algebras of finite Coxeter groups. ... The book is very nicely written, striking the ideal balance between providing a uniform treatment of the finite Coxeter groups on the one hand, and presenting type-specific material on the other. ... In summary, this book is excellent. It will serve primarily as a reference for experts, but would also work well for self-study for a graduate student." (Matthew Fayers, Zentralblatt MATH, Vol. 1232, 2012)

#### Synopsis

Fulfilling a valuable need in the field, this volume classifies the irreducible representations of Iwahori-Hecke algebras at roots of unity. The text develops an analogue of James' (1970) "characteristic-free approach to the representation theory of Iwahori-Hecke algebras in general.

#### Synopsis

The modular representation theory of Iwahori-Hecke algebras and this theory's connection to groups of Lie type is an area of rapidly expanding interest; it is one that has also seen a number of breakthroughs in recent years. In classifying the irreducible representations of Iwahori-Hecke algebras at roots of unity, this book is a particularly valuable addition to current research in this field. Using the framework provided by the Kazhdan-Lusztig theory of cells, the authors develop an analogue of James' (1970) "characteristic-free approach to the representation theory of Iwahori-Hecke algebras in general. Presenting a systematic and unified treatment of representations of Hecke algebras at roots of unity, this book is unique in its approach and includes new results that have not yet been published in book form. It also serves as background reading to further active areas of current research such as the theory of affine Hecke algebras and Cherednik algebras. The main results of this book are obtained by an interaction of several branches of mathematics, namely the theory of Fock spaces for quantum affine Lie algebras and Ariki's theorem, the combinatorics of crystal bases, the theory of Kazhdan-Lusztig bases and cells, and computational methods. This book will be of use to researchers and graduate students in representation theory as well as any researchers outside of the field with an interest in Hecke algebras.

### Table of Contents

Generic Iwahori-Hecke algebras.- Kazhdan-Lusztig cells and cellular bases.- Specialisations and decomposition maps.- Hecke algebras and finite groups of Lie type.- Representation theory of Ariki-Koike algebras.- Canonical bases in affine type A and Ariki's theorem.- Decomposition numbers for exceptional types.