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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.

Representations of Hecke Algebras at Roots of Unity