Representation groups for the Schur index
Let G be a finite group, \(\chi\) an irreducible complex character of G and K an algebraic number field containing the values of \(\chi\). Then \(\chi\) determines a central simple component A in the group algebra KG and thus an element [A] in the Brauer group Br(K). By class field theory the order of A in Br(K) coincides with the Schur index \(m_ K(\chi)\). As usual, Br(K) can be identified with \(H^ 2(\Gamma,\bar K^{\times})\) where \(\bar K\) denotes the algebraic closure of K and \(\Gamma\) the Galois group of \(\bar K\) over K. The author constructs, for any prime p, a group extension \(1\to M\to P\to P/M\to 1\) inside G, an epimorphism \(\Gamma\) \(\to P/M\) and a compatible homomorphism \(M\to \bar K^{\times}\) such that the corresponding change of groups \(H^ 2(P/M,M)\to H^ 2(\Gamma,\bar K^{\times})\) maps the cohomology class of \(1\to M\to P\to P/M\to 1\) onto the p-component \([A]_ p\) of [A]. This method gives a unifying approach to several results on the Schur index.
scientific article; zbMATH DE number 3910594
scientific article