The representation theory of Iwahori-Hecke algebras with unequal parameters.

Abstract

The Iwahori-Hecke algebras of finite Coxeter groups play an important role in many areas of mathematics. In this thesis we study the representation theory of the Iwahori-Hecke algebras of the Coxeter groups of type Bn and F4, in the unequal parameter case. We denote these algebras HQ and KQ respectively. This follows on from work done by Dipper, James, Murphy and Norton. We are interested in the Iwahori-Hecke algebras of type Bn and F4 since these are the only cases, apart from the dihedral groups, where the Coxeter generators lie in different conjugacy classes, and therefore the Iwahori-Hecke algebra can have unequal parameters. There are two parameters associated with these algebras, Q and q. Norton dealt with the case Q = q = 0, whilst Dipper, James and Murphy addressed the case q 6= 0 in type Bn. In this thesis we look at the case Q 6= 0; q = 0. We begin by constructing the simple modules for HQ, then compute the Ext quiver and find the blocks of HQ. We continue by observing that there is a natural embedding of the algebra of type n ð€€€ 1 in the algebra of type n, and this gives rise to the notion of an induced module. We look at the structure of the induced module associated with a given simple HQ-module. Here we are able to construct a composition series for the induced module and show that in a particular case the induced modules are self-dual. Finally, we look at KQ and find that the representation theory is related to representation theory of the Iwahori-Hecke algebra of type B3. Using this relationship we are able to construct the simple modules for KQ and begin the analysis of the Ext quiver.