Grassmannian fixed point ratios
type: Item
instance of: scholarly article
title: Grassmannian fixed point ratios
zbMATH Open document ID: 0970.20003
DOI: 10.1023/A:1005278605358
author: Kay Magaard
author: Daniel Frohardt
published in: Geometriae Dedicata
publication date: 2001-07-08T00:00:00Z
review text: Let \(G\) be a group of permutations on the finite set \(\Omega\) of cardinality \(n\). If \(g\) is a nonidentity element of \(G\), let \(f(g)=|\Omega^g|/n\), where \(\Omega^g\) is the set of fixed points of \(g\). This has long been studied, particularly in the case of primitive permutation groups. If \(G\) is primitive, these numbers are typically very close to \(0\) unless we are considering natural actions. See [\textit{M. W. Liebeck} and \textit{A. Shalev}, J. Am. Math. Soc. 12, No.~2, 497-520 (1999; Zbl 0916.20003)] on asymptotic results for actions other than these natural ones. This paper provides a major contribution to the subject. These types of results have had many applications to many different problems in group theory -- in particular to probabilistic methods. In this paper, the authors study the case where \(G\) is a finite almost simple classical group (i.e., linear, unitary, symplectic or orthogonal) and \(\Omega\) is taken to be an orbit of nonsingular or totally singular subspaces for \(G\) in the natural module. They then give quite good estimates for \(f(g)\) in terms of the size of the field and dimensions of eigenspaces of \(g\). Let \(n\) be the dimension of the natural module and suppose the orbit consists of subspaces of dimension or codimension \(k\leq n/2\). They show that typically \(f(g)<(4000+150n)/q^{(n-6)/2}\) or \(g\) has an eigenspace of codimension \(d\) with \(dk<n/2\) and \(f(g)<q^{-dk}+2000/q^{(n-6)/2}\). For small values of \(n\) and \(q\), one can do a bit better, but asymptotically these results are quite sharp. One motivation for these results was a conjecture of the reviewer and \textit{J. G. Thompson} [J. Algebra 131, No.~1, 303-341 (1990; Zbl 0713.20011)] about monodromy groups of coverings of Riemann surfaces. The conjecture is that for a fixed genus \(g\), the nonabelian simple composition factors of the monodromy group of a cover \(\phi\colon X\to Y\) with \(X\) of genus \(g\) are either alternating groups or are in …
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 51A05
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 30F10
zbMATH DE Number: 1547909
zbMATH Keywords: Grassmannians
zbMATH Keywords: fixed point ratios
zbMATH Keywords: numbers of fixed points
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: almost simple classical groups
zbMATH Keywords: monodromy groups
zbMATH Keywords: coverings of Riemann surfaces
reviewed by: Robert Guralnick
MaRDI profile type: Publication
full work available at URL: a:1005278605358
OpenAlex ID: W81772674
description: scientific article; zbMATH DE number 1547909