On the cyclic symmetry classes
Let \(\otimes^mV\) denote the \(m\)th tensor power of an \(n\)-dimensional complex vector space \(V\). For each permutation \(\pi\) in the symmetric group on \(m\) letters one can define a linear mapping \(P(\pi)\) on \({\otimes^m}V\) by defining \(P(\pi)(v_1\otimes\dots\otimes v_m)=v_{\pi^{-1}(1)}\otimes\dots\otimes v_{\pi^{-1}(m)}\) and extending it to the whole of \(\otimes^mV\) by linearity. Now for a subgroup \(G\) of \(S_m\) and an ordinary irreducible character \(\chi\) of \(G\) we can define the linear mapping \(T(G,\chi)\) on \({\otimes^m}V\) by setting \(T(G,\chi)={\chi(1)\over|G|}\sum_{\pi\in G}\chi(\pi)P(\pi)\). In this case the image of \(\otimes^mV\) under \(T(G,\chi)\) is called the symmetry class of tensors associated with \(G\) and \(\chi\) and is denoted by \(V_\chi(G)\). The purpose of the paper is to obtain a formula for the dimension of \(V_{\chi}(G)\) in the case when \(G\) is the direct product of cyclic subgroups. This result extends the result of \textit{L. J. Cummings} [in J. Algebra 40, 401-405 (1976; Zbl 0354.15012)] who considered only a cyclic subgroup \(G\) of \(S_m\) and where \(\chi\) is the trivial or a primitive linear character of \(G\).
scientific article; zbMATH DE number 922335