Algebra Seminar

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.