The number of trivial composition factors of the Steinberg module
type: Item
instance of: scholarly article
title: The number of trivial composition factors of the Steinberg module
zbMATH Open document ID: 0669.20036
DOI: 10.1007/BF01188519
author: Gerhard Hiss
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: Let G be a finite group with a split BN-pair. Let \(H=B\cap N\) and \(W=N/H\) the Weyl group of G. Let k be any field of characteristic r. The Steinberg module is the right ideal of kG spanned by the element \[ \sum_{b\in B}b\sum_{w\in W}(-1)^{l(w)}\dot w, \] where l(w) denotes the length of w, and ẇ is some arbitrary, but fixed, element of N mapping to w under the natural homomorphism. We determine the number of trivial composition factors of the Steinberg module. It turns out that this number is at most one. We apply our result to the finite groups of Lie type arising from simple algebraic groups. In particular, if \(G=G(q)\) is a (nontwisted) Chevalley group defined over the field with q elements, then the trivial module is a composition factor of the Steinberg module, if and only if r divides \(q+1\).
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 4095770
zbMATH Keywords: finite group
zbMATH Keywords: split BN-pair
zbMATH Keywords: Weyl group
zbMATH Keywords: Steinberg module
zbMATH Keywords: number of trivial composition factors
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: simple algebraic groups
zbMATH Keywords: Chevalley group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4095770