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Symmetry classes of polynomials. (English) Zbl 1338.05272
Summary: Let \(G\) be a subgroup of \(S_m\) and suppose \(\chi\) is an irreducible complex character of \(G\). Let \(H_d(G,\chi)\) be the symmetry class of polynomials of degree \(d\) with respect to \(G\) and \(\chi\). Let \(V\) be an \((d+1)\)-dimensional inner product space over \(\mathbb{C}\) and \(V_\chi(G)\) be the symmetry class of tensors associated with \(G\) and \(\chi\). A monomorphism \(H_d(G,\chi)\to V_\chi(G)\) is given and it is used to obtain necessary and sufficient conditions for nonvanishing \(H_d(G,\chi)\). The nonexistence of o-basis of \(H_d(S_m,\chi^\pi)\) for a certain class of irreducible characters of \(S_m\) is concluded. The dimensions of symmetry classes of polynomials with respect to the irreducible characters of \(S_m\) and \(A_m\) are computed.

05E05 Symmetric functions and generalizations
15A69 Multilinear algebra, tensor calculus
20C30 Representations of finite symmetric groups
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