# Algebra Seminar

### A strong multiplicity one result of Hecke algebra modules

Friday 30th July, 12-1pm, Carslaw 375.

Let S_n denote the symmetric group on n letters and let F denote a field. Given an irreducible S_n-module M the Branching Rule, shows how Res_{S_{n-1}}^{S_n} M decomposes into irreducibles over F, is well known and a wonderful source of combinatorics, when char F = 0.

One can hope to find the Branching Rule for the finite Hecke algebra H_n^{\fin}, which is a deformation of the symmetric group, or for the cyclotomic Hecke algebra H_n^\lambda, which is a deformation of the wreath product S_n \wr \Z/r\Z. However, these algebras are rarely semisimple, for instance when char F is prime, when lambda is large, or when q is a root of unity. The restriction of irreducible modules is thus not a direct sum of irreducibles and can be quite complicated.

Kleshchev has given a partial Branching Rule for the symmetric group in prime characteristic---he describes the socle of Res_{S_{n-1}}^{S_n} M, and in particular shows that it is multiplicity-free and isomorphic to the cosocle of restriction. In [GV], we prove that the socle of restriction of any irreducible module of the cyclotomic Hecke algebra is multiplicity free and isomorphic to the cosocle.

We do this by proving the result for a large class of modules of the affine Hecke algebra that includes all cyclotomic Hecke algebra irreducibles.