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arXiv · 1301.0947

Structure Theorems for the Symmetric Groups Acting on its Natural Module

Robert Mckemey
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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 $k[x_1,..., x_n] \cong \Oplus_{\{n\} \subseteq I \subseteq [n]} k[d_I] \otimes_k V_I$.

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Robert Mckemey. 2013-01-05. Structure Theorems for the Symmetric Groups Acting on its Natural Module. https://arxiv.org/abs/1301.0947

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