Preface

One of the major open problems in the representation theory of finite groups is the determination of the irreducible representations of the symmetric group Sn over a field of characteristic p > 0. Thanks to the work of James [179] in the 1970s, we do have a natural parametrisation of the irreducible representations in the framework of the theory of Specht modules, but explicit combinatorial formulae for their dimensions are not known in general! Note that the analogous problem in characteristic 0 has been solved for a long time, by the work of Frobenius around 1900. In a wider context, this problem is a special case of the problem of determining the irreducible representations of Iwahori–Hecke algebras. These algebras arise naturally in the representation theory of finite groups of Lie type, but they can also be defined abstractly as certain deformations of group algebras of finite Coxeter groups, where the deformation depends on one or several parameters. For the purposes of this introduction, let us assume that all the parameters are integral powers of a fixed element in the base field. If this base parameter has infinite order and the base field has characteristic 0, then we are in the “generic case” where the algebras are semisimple; this case is quite well understood [132], [231]. Also note that, both for historical reasons and as far as applications are concerned, the case where all parameters are equal is particularly important. The main focus in this text will be on the “modular case” where the algebras are non-semisimple. This situation typically occurs over fields of positive characteristic (a familiar phenomenon from the representation theory of finite groups), but it also occurs over fields of characteristic 0 when the base parameter is a root of unity. While leading to a highly interesting and rich theory in its own right, it turns out that the study of the characteristic 0 situation also provides a crucial step for understanding the positive characteristic case, which is most important for applications to finite groups of Lie type. Over the last two decades, there has been considerable progress on the characteristic 0 situation. One of the most spectacular advances is the “LLT conjecture” [208] (where “LLT” stands for Lascoux, Leclerc, Thibon) and its proof by Ariki [7], [10]. This brings deep geometric methods and the combinatorics of crystal/canonical bases of quantum groups into the picture, opening the way for a variety