Representations of the general linear group over symmetry classes of polynomials

摘要

  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 $chi$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $K_{chi} (T)in{m End}(H_d(G,chi))$ acting on symmetrized decomposable polynomials by $K_{chi}(T)(f_1st f_2stldotsst f_d)=Tf_1st Tf_2stldotsst Tf_d.$ In this paper, we show that the representation $Tmapsto K_{chi} (T)$ of the general linear group $GL(V)$ is equivalent to the direct sum of $chi(1)$ copies of a representation (not necessarily irreducible) $Tmapsto B_{chi}^G(T)$.