Structure Theorems for the Symmetric Groups Acting on its Natural Module

Abstract

This paper gives an explicit structure theorem for the symmetric group acting on the symmetric algebra of its natural module. Let $G$ be the symmetric group on $x_1,..., x_n$ and let $d_i$ be the $i^{\text{th}}$ elementary symmetric polynomial in the $x_i$'s. We show that if we take monomial representations discussed in \cite[Section 3]{Kemper} to be the modules $V_I$, then we have an isomorphism of $kG$-modules $k[x_1,..., x_n] \cong \Oplus_{\{n\} \subseteq I \subseteq [n]} k[d_I] \otimes_k V_I$.