Orthogonal Frobenius reciprocity
Let \(K\) be a field of characteristic \(0\), \(H\) a subgroup of the finite group \(G\), and \(W\) a \(KH\)-module. If \(F_W\colon W\times W\to K\) is a non-degenerate symmetric \(H\)-invariant bilinear form, then \((W,F_W)\) is called an orthogonal \(KH\)-module, and \(F_W\) induces a symmetric non-degenerate \(G\)-invariant bilinear form \(F_W^G\) on \(W^G=KG\otimes_{KH}W\). Furthermore, if \((V,F_V)\) is an orthogonal \(KG\)-module, then the \(K\)-dual \(V^*\) of \(V\) becomes an orthogonal \(KG\)-module. The author proves that if \(F_V\) generates the space of all \(G\)-invariant symmetric bilinear forms on \(V\), then there is a canonical bijection between \(\text{Isom}_{KH}((W,F_W),(V,F_V))\) and \(\text{Isom}_{KG}((V^*,F^*_V),((W^G)^*,(F_W)^*))\). This result is applied to determine the rational isometry class of some irreducible orthogonal \(\mathbb{Q} S_n\)-modules.
scientific article; zbMATH DE number 1423612