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  • Print publication year: 2005
  • Online publication date: January 2010

Preface

Summary

The subject of this book is representation theory of symmetric groups. We explain a new approach to this theory based on the recent work of Lascoux, Leclerc, Thibon, Ariki, Grojnowski, Brundan, Kleshchev, and others. We are mainly interested in modular representation theory, although everything works in arbitrary characteristic, and in case of characteristic 0 our approach is somewhat similar to the theory of Okounkov and Vershik [OV], described in Chapter 2 of this book. The methods developed here are quite general and they apply to a number of related objects: finite and affine Iwahori–Hecke algebras of type A, cyclotomic Hecke algebras, spin-symmetric groups, Sergeev algebras, Hecke–Clifford superalgebras, affine and cyclotomic Hecke–Clifford superalgebras, …. We concentrate on symmetric and spin-symmetric groups though.

We now outline some of the ideas which lead to the new approach. Let us concentrate on the modular case, as this is where things get really interesting. So let F be a field of characteristic p > 0, and Sn be the symmetric group. Irreducible FSn-modules were classified by James. His approach is as follows (see [J] for details). Let Sλ be the Specht module corresponding to a partition λ of n (the Specht construction works over any field and even over ℤ). This module has a canonical Sn-invariant bilinear form. The form is non-zero if and only if λ is p-regular, that is no non-zero part of λ is repeated p or more times.