Gap conditions for representations of symmetric groups
type: Item
instance of: scholarly article
title: Gap conditions for representations of symmetric groups
zbMATH Open document ID: 0881.57041
DOI: 10.1016/S0022-4049(96)00019-9
author: Karl Heinz Dovermann
author: Marcel Herzog
published in: Journal of Pure and Applied Algebra
publication date: 1998-02-05T00:00:00Z
review text: In the process of constructing smooth group actions with exotic fixed point behaviour, one usually needs linear representations satisfying gap conditions between dimensions of fixed point sets for certain pairs of subgroups of the group. Standard methods need so-called admissible representations, while a much more complicated method due to \textit{A. Bak} and \textit{M. Morimoto} [Forum Math. 8, No. 3, 267-302 (1996; Zbl 0860.57028)] can do with almost admissible representations. In a 1993 preprint, Laitinen and Morimoto studied a specific representation giving rise to almost admissible representations. For every Oliver group, Bak-Morimoto surgery then shows existence of one fixed point actions on spheres, in particular for \(G=S(n), n\geq 5\), the symmetric group of order \(n\). Using intricate group theoretic arguments and calculations with the software package GAP, the authors modify this approach to yield admissible representations for many groups. This makes standard surgery applicable to produce one fixed point actions, in particular for \(G=S(n), n\geq 6\). (It is known that admissible representations do not exist for \(S(5)\).) Moreover, for Oliver groups with these admissible representations, the authors produce -- modifying ideas of Pawałowski -- smooth actions on spheres whose fixed point set is any given closed smooth stably parallelizable manifold all of whose components have the same dimension.
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 57S25
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20B30
zbMATH DE Number: 1038608
zbMATH Keywords: equivariant surgery
zbMATH Keywords: admissible representation
zbMATH Keywords: gap condition
zbMATH Keywords: symmetric group
reviewed by: Martin Raussen
describes a project that uses: GAP
MaRDI profile type: Publication
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full work available at URL: s0022-4049(96)00019-9
OpenAlex ID: W2070389530
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description: scientific article; zbMATH DE number 1038608