# Representations of the general linear group over symmetry classes of polynomials

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## Abstract

Let *V* be the complex vector space of homogeneous linear polynomials in the variables *x*_{1},..., *x*_{ m }. Suppose *G* is a subgroup of *S*_{ m }, and *χ* is an irreducible character of *G*. Let *H*_{ d }(*G*, *χ*) be the symmetry class of polynomials of degree *d* with respect to *G* and *χ*.

For any linear operator
In this paper, we show that the representation

*T*acting on*V*, there is a (unique) induced operator*K*_{ χ }(*T*) ∈ End(*H*_{ d }(*G*,*χ*)) acting on symmetrized decomposable polynomials by$${K_\chi }\left( T \right)\left( {{f_1} * {f_2} * \cdots * {f_d}} \right) = T{f_1} * T{f_2} * \cdots * T{f_d}.$$

*T*↦*K*_{ χ }(*T*) of the general linear group*GL*(*V*) is equivalent to the direct sum of*χ*(1) copies of a representation (not necessarily irreducible)*T*↦*B*_{ χ }^{ G }(*T*).## Keywords

symmetry class of polynomials general linear group representation irreducible character induced operator## MSC 2010

20C15 15A69 05E05## Preview

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© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017