Periodic simple modules for \(SU_ 3(q\) 2) in the describing characteristic p\(\neq 2\)
type: Item
instance of: scholarly article
title: Periodic simple modules for \(SU_ 3(q\) 2) in the describing characteristic p\(\neq 2\)
zbMATH Open document ID: 0628.20033
DOI: 10.1007/BF01162873
author: Peter Fleischmann
published in: Mathematische Zeitschrift
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173943
review text: Considering the special-unitary resp. linear groups \(G=SU_ 3(q^ 2)\) resp. \(H=SL_ 3(p^ n)\) it is easy to see that they have isomorphic Sylow p-subgroups \((q=p^ n)\). Due to Steinberg it is well known, that G and H have exactly one simple projective KG, resp. KH-module, which is called ST, over the algebraic-closed field K with characteristic p. A KG- (KH-)-module M is called periodic, if there is a natural number \(r>1\) with \(\Omega ^ r(M)=\Omega (M)\), where \(\Omega ^ i(M)\) denotes the i- th Heller-module, that is the kernel of the i-th projective cover for a minimal projective resolution of M. I. Janiszczak showed, that for H there are no simple periodic KG-modules except ST, which is projective. It is therefore amazing that, in spite of the similar p-local structure of G and H, there are always periodic non- projective simple modules for G if \(p\neq 2\). The proof requires the theory of module-varieties as developed by Carlson et. al. Also a detailed investigation of the action of a p-Sylow group on irreducible Weyl-modules of G is given. This goes back to the polynomial representations of \(Gl_ m(k)\) as developed by Green and Carter-Lusztig. With a concrete formula for Garnir-relations one can determine the complete matrix-representation of a p-Sylow group of G on Weyl-modules. Finally, the module-varieties of some appropriate simple Weyl-modules can be calculated and, using Steinberg's tensorproduct theorem, simple periodic non-projective KG-modules can be constructed.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 4021445
zbMATH Keywords: special-unitary
zbMATH Keywords: linear groups
zbMATH Keywords: Sylow p-subgroups
zbMATH Keywords: Heller-module
zbMATH Keywords: projective cover
zbMATH Keywords: minimal projective resolution
zbMATH Keywords: simple periodic KG- modules
zbMATH Keywords: periodic non-projective simple modules
zbMATH Keywords: module-varieties
zbMATH Keywords: irreducible Weyl-modules
zbMATH Keywords: polynomial representations
zbMATH Keywords: Steinberg's tensor product theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4021445