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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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