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Representations of the general linear group over symmetry classes of polynomials. (English) Zbl 06861580
Summary: Let $$V$$ be the complex vector space of homogeneous linear polynomials in the variables $$x_1,\dots ,x_m$$. Suppose $$G$$ is a subgroup of $$S_m$$, and $$\chi$$ is an irreducible character of $$G$$. Let $$H_d(G,\chi)$$ be the symmetry class of polynomials of degree $$d$$ with respect to $$G$$ and $$\chi$$.
For any linear operator $$T$$ acting on $$V$$, there is a (unique) induced operator $$K_{\chi}(T)\in\text{End}(H_d(G,\chi))$$ acting on symmetrized decomposable polynomials by $K_{\chi}(T)(f_1\ast f_2\ast\dots\ast f_d)=Tf_1\ast Tf_2\ast\dots\ast Tf_d.$ In this paper, we show that the representation $$T\mapsto K_{\chi}(T)$$ of the general linear group $$GL(V)$$ is equivalent to the direct sum of $$\chi(1)$$ copies of a representation (not necessarily irreducible) $$T\mapsto B_{\chi}^G(T)$$.
##### MSC:
 20C15 Ordinary representations and characters 15A69 Multilinear algebra, tensor calculus 05E05 Symmetric functions and generalizations
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