### School of Mathematics and Statistics

### Monica Vazirani

University of San Diego

### 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.