James' Conjecture for Hecke algebras of exceptional type, I

Details James' Conjecture for Hecke algebras of exceptional type, I James' Conjecture for Hecke algebras of exceptional type, I

2007-12-11

arxiv.org/abs/0712.1620v2

Mathematics

Representation Theory

24 pages; corrected some misprints, added Remark 4.10

Scientific paper

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 $E_7$ and $E_8$ 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 {\sf MeatAxe} and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure.

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