Eigenvalues of unipotent elements in cross-characteristic representations of finite classical groups.
type: Item
instance of: scholarly article
title: Eigenvalues of unipotent elements in cross-characteristic representations of finite classical groups.
zbMATH Open document ID: 1158.20004
published in: Journal of Algebra
publication date: 2008-05-21T00:00:00Z
review text: This paper is a significant contribution to a long-term program intended to study minimum polynomials of group elements in (modular) irreducible representations \(\Theta\) of degree \(>1\). An important part of this program deals with elements \(g\), of prime-power order \(s\), of finite classical groups \(G\) defined over fields of characteristic \(p\), where the characteristic of the representation \(\Theta\) in question is coprime to \(p\). Assume in addition that \(g\) is semisimple (that is, \(s\) and \(p\) are coprime). In a previous paper [J. Algebra 243, No. 1, 228-263 (2001); corrigendum 296, No. 1, 249-252 (2006; Zbl 1017.20036)] the authors treated the case where \(g\) belongs to a parabolic subgroup of \(G\). In general, an optimal lower bound for the degree of the minimum polynomial of \(\Theta(g)\) has recently been proved by the second author and the reviewer [in Proc. Lond. Math. Soc. (3) 97, No. 3, 623-668 (2008; Zbl 1173.20012)]. The paper under review handles the case where \(p\mid s\), i.e. \(g\) is unipotent (and \(G\) is a quasi-simple classical group). One can summarize the main results of the paper as follows: either the spectrum of \(\Theta(g)\) consists of all \(s\)-roots of unity, or \((G,g,\Theta)\) belongs to a short list of explicitly described exceptions. The major kind of exceptions is that \(g^{s/p}\) is a transvection, \(\Theta\) is a Weil representation, and either \(G=\text{Sp}(2m,p)\) with \(p>2\), or \(G=\text{SU}(m,2)\). Other exceptions can occur only for some specific representations of \(\text{Sp}(4,p)\), \(\text{SU}(3,p)\), \(\text{SL}(2,p)\), \(\text{SL}(2,p^2)\), \(\text{Sp}(8,3)\), \(\text{Sp}(6,3)\), \(\text{Sp}(6,2)\), \(\text{Sp}(4,9)\), \(\text{SU}(7,2)\), \(\text{Sp}(5,2)\), and \(\text{SU}(4,3)\). The authors also prove a lower bound for the multiplicity of each eigenvalue of \(\Theta(g)\) when the rank of \(G\) is large enough. One should consult the paper for precise formulations of all these interesting results.
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 5277608
zbMATH Keywords: finite classical groups
zbMATH Keywords: cross-characteristic representations
zbMATH Keywords: unipotent elements
zbMATH Keywords: eigenvalues
zbMATH Keywords: minimum polynomials
zbMATH Keywords: elements of prime power order
zbMATH Keywords: Hall-Higman type theorems
author: L. Di Martino
author: A. Zalesski
reviewed by: Pham Huu Tiep
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.12.024
OpenAlex ID: W2012407279
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DOI: 10.1016/J.JALGEBRA.2007.12.024
description: scientific article; zbMATH DE number 5277608