On the decomposition rules of tensor products of the representations of the classical Weyl groups
The author derives decomposition rules for inner tensor products of ordinary irreducible representations of the series of Weyl groups, i.e. for the groups \(S_ n\), \(S_ 2\wr S_ n\) and \(S_ 2\wr S_ n\cap A_{2n}.\) The method used is an application of the fact that for ordinary characters \(\chi\) of a finite group G and \(\phi\) of a subgroup \(H\leq G\) we have \(\chi \otimes(\phi \uparrow G)=((\chi \downarrow H)\otimes \phi)\uparrow G.\) Therefore this method applies whenever the following conditions are satisfied: (i) An arbitrary ordinary character of G is a \({\mathbb{Z}}\)-linear combination of characters induced from some ''handy'' characters of ''nice'' subgroups H of G. (ii) The decompositions of the characters induced from handy characters of nice subgroups and the restriction of characters of G to nice subgroups can be calculated using partition methods. (iii) In the nice subgroups the inner tensor products of handy characters can be decomposed. Nice subgroups and handy characters are well known for \(S_ n:\) Young subgroups and their identity characters. For \(S_ 2\wr S_ n\) and \(S_ 2\wr S_ n\cap A_{2n}\) the author picks suitable ones from the known representation theory of these groups (although there exist \({\mathbb{Z}}\)- bases of transitive permutation characters, these are not the suitable ones since (ii) does not hold here). For related problems and results on more general wreath products, the interested reader may be referred to \textit{Ö. Celik}, \textit{H. Pahlings} and the reviewer [Mitt. Math. Semin. Gießen 119, 15-90 (1975; Zbl 0342.20004)], \textit{J. Tappe} and the reviewer [Discrete Math. 15, 151-161 (1976; Zbl 0335.20021)] and \textit{J. Tappe} [Mitt. Math. Semin. Gießen 121, 39-57 (1976; Zbl 0361.20017)].
scientific article; zbMATH DE number 3847645
scientific article