Representations of the general linear group over symmetry classes of polynomials

summary:Let $V$ be the complex vector space of homogeneous linear polynomials in the variables $x_{1}, \ldots , x_{m}$. 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 $. \endgraf For any linear operator $T$ acting on $V$, there is a (unique) induced operator $K_{\chi } (T)\in {\rm End}(H_{d}(G,\chi ))$ acting on symmetrized decomposable polynomials by $$ K_{\chi }(T)(f_1\ast f_2\ast \ldots \ast f_d)=Tf_1\ast Tf_2\ast \ldots \ast Tf_d. $$ 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) $T\mapsto B_{\chi }^{G}(T)$