On the dimensions of cyclic symmetry classes of tensors (Q1271025)

Let \(V\) be an \(n\)-dimensional complex vector space and suppose \(V^{\otimes m}\) is equipped with the standard action of the symmetric group \(S_m\) permuting the factors. If \(G\) is a subgroup of \(S_m\) one associates to \(G\) the symmetry class of tensors \(T(G,\chi)\), where \(\chi\) is a character. The authors study the case when \(G\) is a cyclic group. The main result of the paper under review relates the dimension of the symmetry class of tensors in a very nice way with a natural generalization of the Ramanujan sum \(C_m(h)=\sum\exp(2\pi iht/m)\), the summation running over all \(t\), \(1\leq t\leq m-1\), and \((t,m)=1\).