Rationalité des représentations de groupes finis
type: Item
instance of: scholarly article
title: Rationalité des représentations de groupes finis
zbMATH Open document ID: 0537.20003
DOI: 10.1016/0021-8693(83)90195-3
author: Charles Kratzer
published in: Journal of Algebra
publication date: 1983-01-01T00:00:00Z
review text: Let \(\pi\) be a finite group, K a field and \(R(K\pi)\) the ring of representations of \(\pi\) over K. For a field extension \(i:F\subset K\) let \(i_*:R(F\pi)\to R(K\pi)\) denote the map induced by \(V\mapsto K\otimes_ FV.\) The article deals with the following question: Supposing there is a finite group G of automorphisms of K with fixed field F and a representation \([V]\in R(K\pi)\) invariant under G, i.e. such that \(\sigma_*[V]=[V]\) for all \(\sigma\in G\). Under which conditions is [V] realizable over F, i.e. does any representation [W]\(\in R(F\pi)\) with \(i_*[W]=[V]\) exist? The first part of the paper is grouped around a general theorem: An absolutely indecomposable representation [V] invariant under G is realizable over F if and only if the class of the algebra \(End_{F\pi}(V)/Rad End_{F\pi}(V)\) is trivial in the Brauer group of the center of this algebra. The second, more specific part, is restricted to fields of characteristic 0. Necessary and sufficient conditions in terms of the Adams operators \(\psi^ k:R(K\pi)\to R(K\pi)\) for the realizability over F of an absolutely irreducible representation [V] are proved in case \(\pi\) is nilpotent, K arbitrary resp. \(\pi\) is metacyclic, K a cyclotomic field, and G a certain group of automorphisms of K. These results continue works by \textit{P. Roquette} [Arch. Math. 9, 241-250 (1958; Zbl 0083.250)], and by \textit{D. M. Goldschmidt} and \textit{I. M. Isaacs} [J. Algebra 33, 191-199 (1975; Zbl 0297.20018)], and generalize the classical criterion by \textit{G. Frobenius} and \textit{I. Schur} [Sitzungsber. Preuss. Akad. Wiss., Berlin 186-208 (1906; JFM 37.0161.01)].
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 3853332
zbMATH Keywords: realizability of representations
zbMATH Keywords: ring of representations
zbMATH Keywords: absolutely indecomposable representation
zbMATH Keywords: Brauer group
zbMATH Keywords: Adams operators
zbMATH Keywords: absolutely irreducible representation
MaRDI profile type: Publication
full work available at URL: 0021-8693(83)90195-3
OpenAlex ID: W2014611349
cites work: Q3277493
cites work: Q5340151
cites work: Q5645586
cites work: Schur indices in finite groups
cites work: Opérations d'Adams en théorie des représentations linéaires des groupes finis
cites work: Realisierung von Darstellungen endlicher nilpotenter Gruppen. (Realization of representations of finite-dimensional nilpotent groups)
cites work: The Schur subgroup of the Brauer group
description: scientific article; zbMATH DE number 3853332