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Preprints, Working Papers, ... Year : 2019

Unitriangular Shape of Decomposition Matrices of Unipotent Blocks

Olivier Brunat
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  • PersonId : 831177
Olivier Dudas
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  • PersonId : 883084
Jay Taylor
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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. 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.

Dates and versions

hal-02334862 , version 1 (27-10-2019)

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Olivier Brunat, Olivier Dudas, Jay Taylor. Unitriangular Shape of Decomposition Matrices of Unipotent Blocks. 2019. ⟨hal-02334862⟩
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