Rationalité des représentations de groupes finis
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)].
absolutely indecomposable representation
scientific article; zbMATH DE number 3853332