On the doubly transitive permutation representations of \(\text{Sp}(2n,\mathbb{F}_2)\).
type: Item
instance of: scholarly article
title: On the doubly transitive permutation representations of \(\text{Sp}(2n,\mathbb{F}_2)\).
zbMATH Open document ID: 1050.20032
DOI: 10.1016/S0021-8693(02)00551-3
author: N. S. Narasimha Sastry
author: Peter Sin
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: Let \(G\) denote the finite symplectic group \(\text{Sp}(2n,2)\) of degree \(2n\) over the field \(\mathbb{F}_2\) of order 2. It is well known that \(G\) has two doubly transitive permutation actions on the cosets of the two non-isomorphic orthogonal groups of degree \(2n\). These actions have degree \(2^{n-1}(2^n-1)\) and \(2^{n-1}(2^n+1)\), respectively. The purpose of the paper under review is to investigate the submodule structure of the reduction modulo 2 of the corresponding permutation lattices. Let \(\theta_1\) and \(\theta_2\) be the corresponding permutation characters. As the authors remark, when restricted to elements of odd order, \(\theta_1+\theta_2\) agrees with the permutation character of \(G\) acting on the points of the natural vector space \(V\) of dimension over \(2n\) over \(\mathbb{F}_2\). In turn, one knows that the reduction modulo 2 of the permutation lattice of \(G\) acting on the points of \(V\) has the same Brauer character as that of the exterior algebra \(\bigwedge V\) of \(V\), considered as a module for \(G\). Thus, the composition factors of the reduction modulo 2 of each doubly transitive permutation lattice occur among the composition factors of the exterior algebra. Moreover, on elements of odd order, the generalized character \(\theta_2-\theta_1\) agrees with the Brauer character of the spin module for \(G\). This spin module is irreducible of degree \(2^n\) and it occurs with multiplicity 1 in \(\bigwedge V\), as a composition factor of \(\bigwedge^nV\). With this extra knowledge, we now know that the composition factors of the reduction modulo 2 of the two doubly transitive permutation lattices are almost identical, except that the one of larger dimension contains the spin module, while the other does not. Working over the algebraic closure \(k\) of \(\mathbb{F}_2\), the authors construct filtrations for the modular reductions of the permutation lattices which have subquotients that are modules for the symplectic group \(\text…
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1868074
zbMATH Keywords: symplectic groups
zbMATH Keywords: doubly transitive actions
zbMATH Keywords: modular reductions
zbMATH Keywords: exterior algebras
zbMATH Keywords: Weyl modules
zbMATH Keywords: filtrations
zbMATH Keywords: fundamental modules
zbMATH Keywords: permutation lattices
zbMATH Keywords: composition factors
MaRDI profile type: Publication
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full work available at URL: s0021-8693(02)00551-3
OpenAlex ID: W1995385037
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description: scientific article; zbMATH DE number 1868074