When is a Specht ideal Cohen–Macaulay?

Abstract

For a partition $\lambda$ of $n$, let $I_{\lambda }^{\text{Sp}}$ be the ideal of $R=K\left[x_1,\dots ,x_n\right]$ generated by all Specht polynomials of shape $\lambda$. We show that if $R∕I_{\lambda }^{\text{Sp}}$ is Cohen–Macaulay then $\lambda$ is of the form either $\left(a,1,\dots ,1\right)$, $\left(a,b\right)$, or $\left(a,a,1\right)$. We also prove that the converse is true in the $\text{char}\left(K\right)=0$ case. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R∕I_{\left(n-3,3\right)}^{\text{Sp}}$ is not Cohen–Macaulay if and only if $\text{char}\left(K\right)=2$.