### School of Mathematics and Statistics

### Andrew Mathas

University of Sydney

### Decomposition matrices of Iwahori-Hecke algebras.

**Friday 19th March, 12-1pm, Carslaw 375.**
Iwahori-Hecke algebras are an important class of algebras
which arise naturally in the study of Coxeter groups and the
groups of Lie type. It has been known for some time that the
decomposition matrices of the Iwahori-Hecke algebras are
unitriangular when the associated Coxeter group is a
symmetric group or a Coxeter group of type B.

Recently Meinolf Geck ("Kazhdan-Lusztig cells and
decomposition numbers", Rep. Theory. 2 (1998), 264-277) has
shown that the decomposition matrices of the Iwahori-Hecke
algebras of Weyl groups are always unitriangular. The proof
uses Lusztig's a-function and the theory of cells and in
this way relies upon some very deep positivity properties of
the Kazhdan-Lusztig polynomials.

This talk will be a survey of some of the very beautiful
ideas underpinning these results.