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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.

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