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September 2020 Unitriangular shape of decomposition matrices of unipotent blocks
Olivier Brunat, Olivier Dudas, Jay Taylor
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Ann. of Math. (2) 192(2): 583-663 (September 2020). DOI: 10.4007/annals.2020.192.2.7

Abstract

We show that the decomposition matrix of unipotent $\ell$-blocks of a finite reductive group $\mathbf{G}(\mathbb{F}_q)$ has a unitriangular shape, assuming $q$ is a power of a good prime and $\ell$ is very good for $\mathbf{G}$. This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand--Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.

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Olivier Brunat. Olivier Dudas. Jay Taylor. "Unitriangular shape of decomposition matrices of unipotent blocks." Ann. of Math. (2) 192 (2) 583 - 663, September 2020. https://doi.org/10.4007/annals.2020.192.2.7

Information

Published: September 2020
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2020.192.2.7

Subjects:
Primary: 20C20 , 20C33

Keywords: decomposition numbers , finite groups of Lie type , Generalised Gelfand--Graev representations

Rights: Copyright © 2020 Department of Mathematics, Princeton University

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Vol.192 • No. 2 • September 2020
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