James' conjecture for Hecke algebras of exceptional type, I

Abstract

In this paper, and a second part to follow, we complete the programme (initiated more than 15 years ago) of determining the decomposition numbers and verifying James' conjecture for Iwahori-Hecke algebras of exceptional type. The new ingredients which allow us to achieve this aim are:•the fact, recently proved by the first author, that all Hecke algebras of finite type are cellular in the sense of Graham-Lehrer, and•the explicit determination of W-graphs for the irreducible (generic) representations of Hecke algebras of type E7 and E8 by Howlett and Yin. Thus, we can reduce the problem of computing decomposition numbers to a manageable size where standard techniques, e.g., Parker's MeatAxe and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure. © 2008 Elsevier Inc. All rights reserved.