## University of Sydney Algebra Seminar

# Andrew Mathas (Sydney)

## Friday 6 May, 12:05-12:55pm, Carslaw 175

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Decomposition numbers for Hecke algebras of type *G(r,p,n)*

In this talk I will explain how the representation theory of the Hecke
algebras of the complex reflection groups of type *G(r,p,n)* is completely
determined by the representation theory of Ariki-Koike algebras, or the
Hecke algebras of type *G(r,1,n)*, when the parameters for the Hecke algebras
are ** (ε,q)-separated**. More precisely, when the parameters are

*(ε,q)*-separated there is an explicit algorithm for computing the decomposition numbers of the algebras of type

*G(r,p,n)*from the decomposition matrices of the algebras of type

*G(r,1,n)*.

The proof of this result relies on two Morita equivalences: the first
reduces the calculation of all decomposition numbers to the case of the
** l-splittable decomposition numbers** and the second Morita
equivalence allows us to compute these decomposition numbers using an
analogue of the cyclotomic Schur algebras for the Hecke algebras of
type

*. We then give a closed formula for the*

*G(r,p,n)**l*-splittable decomposition numbers using detailed calculations with some natural trace functions.

In proving these results, we develop a Specht module theory for these
algebras, explicitly construct their simple modules and introduce and study
analogues of the cyclotomic Schur algebras of type * G(r,p,n)* when the
parameters are

*(ε,q)*-separated.

This is joint work with Jun Hu.