The coinvariant algebra of the symmetric group as a direct sum of induced modules

Abstract

Let $R_n$ be the coinvariant algebra of the symmetric group $S_n$. The algebra has a natural gradation. For a fixed $l$ ($1\leq l \leq n$), let $R_n(k;l)$ ($0\leq k\leq l-1$) be the direct sum of all the homogeneous components of $R_n$ whose degrees are congruent to $k$ modulo $l$. In this article, we will show that for each $l$ there exists a subgroup $H_{l}$ of $S_n$ and a representation $\Psi(k;l)$ of $H_{l}$ such that each $R_n(k;l)$ is induced by $\Psi(k;l)$.