James' Conjecture for Hecke algebras of exceptional type, I

TL;DR

This paper advances the verification of James' Conjecture for Hecke algebras of exceptional type by leveraging cellular algebra structures and explicit W-graph data to compute decomposition numbers.

Contribution

It completes the program of determining decomposition numbers and verifying James' Conjecture for exceptional type Hecke algebras using new theoretical tools.

Findings

Hecke algebras of finite type are cellular.

Explicit W-graphs for E7 and E8 are determined.

Decomposition numbers can be computed with standard techniques.

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 $E_7$ 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.