Symmetry Classes of Polynomials

Let G be a subgroup of S_m and suppose χ is an irreducible complex character of G. Let H_d(G, χ) be the symmetry class of polynomials of degree d with respect to G and χ. Let V be an (d + 1)-dimensional inner product space over ℂ and V_χ(G) be the symmetry class of tensors associated with G and χ. A monomorphism H_d(G, χ) → V_χ(G) is given and it is used to obtain necessary and sufficient conditions for nonvanishing H_d(G, χ). The nonexistence of o-basis of H_d(S_m, χ^π) 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.