When is a Specht ideal Cohen-Macaulay?

Abstract: For a partition $\lambda$ of $n$, let $I^{\rm Sp}\lambda$ be the ideal of $R=K[x_1, \ldots, x_n]$ generated by all Specht polynomials of shape $\lambda$. We show that if $R/I^{\rm Sp}\lambda$ is Cohen--Macaulay then $\lambda$ is of the form either $(a, 1, \ldots, 1)$, $(a,b)$, or $(a,a,1)$. We also prove that the converse is true if ${\rm char}(K)=0$. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R/I^{\rm Sp}_{(n-3,3)}$ is NOT Cohen--Macaulay if and only if ${\rm char}(K)=2$.