Incidence matrices, permutation characters, and the minimal genus of a permutation group
type: Item
instance of: scholarly article
title: Incidence matrices, permutation characters, and the minimal genus of a permutation group
zbMATH Open document ID: 1008.20002
author: Kay Magaard
author: Daniel Frohardt
author: Robert Guralnick
published in: Journal of Combinatorial Theory. Series A
publication date: 2003-01-13T00:00:00Z
review text: Let \(V\) be a non-degenerate symplectic, orthogonal, or unitary space of dimension \(n\) over \(\text{GF}(q)\). The authors consider the incidence matrix \(H\) of totally singular 1-spaces of \(V\) against \(k\)-spaces of fixed isometry type (totally singular or non-degenerate). The main theorem is that if \(2\leq k<n\) then \(M:=HH^t\) is non-singular except in a particular case when \(M=0\), or in certain cases when \(k\) is large with respect to \(n\). As a corollary, the following is shown for an almost simple classical group \(G\) with natural module \(V\) of dimension \(n\) (with \(G\) not having a graph automorphism in the linear case): if \(2\leq k<n-1\), \(P\) is the stabiliser of a singular 1-space of \(V\), and \(K\) is the stabiliser of a totally singular or non-degenerate \(k\)-space of \(V\), then the permutation module \(1^G_P\) is a submodule of \(1^G_K\) except in 2 known cases. The excluded cases are when \(K\) is the stabiliser of a maximal totally singular subspace of \(V\) or of a 2-space containing no singular points, or the orthogonal complement of such. There is a further corollary, stating that if \(S\) is a simple classical group, then the minimal genus among all almost simple groups with socle \(S\) is realised by a non-\(k\)-space action or by one of 4 possible actions on subspaces of the natural module.
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 05B20
zbMATH DE Number: 1760669
zbMATH Keywords: incidence matrices
zbMATH Keywords: permutation characters
zbMATH Keywords: genus
zbMATH Keywords: almost simple classical groups
zbMATH Keywords: permutation modules
reviewed by: H. D. Macpherson
MaRDI profile type: Publication
OpenAlex ID: W1977482091
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DOI: 10.1006/JCTA.2001.3229
description: scientific article; zbMATH DE number 1760669