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James’ conjecture for Hecke algebras of exceptional type. I. (English) Zbl 1213.20005

Summary: 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:
\(\bullet\) the fact, recently proved by the first author, that all Hecke algebras of finite type are cellular in the sense of Graham-Lehrer, and
\(\bullet\) 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 MeatAxe and its variations, can be applied. In this part, we describe the theoretical foundations for this procedure.

MSC:

20C08 Hecke algebras and their representations
20C40 Computational methods (representations of groups) (MSC2010)

Software:

CHEVIE; MeatAxe
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Full Text: DOI arXiv

References:

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