Regularity of Cohen-Macaulay Specht ideals

Abstract

For a partition $\lambda$ of $n\in \mathbb{N}$, let $I_{\lambda }^{\mathrm{Sp}}$ be the ideal of $R=K[x_1,\dots ,x_n]$ generated by all Specht polynomials of shape $\lambda$. In the previous paper, the second author showed that if $R/I_{\lambda }^{\mathrm{Sp}}$ is Cohen-Macaulay, then $\lambda$ is either $(n-d,1,\dots ,1),(n-d,d)$, or $(d,d,1)$, and the converse is true if $\mathrm{char}(K)=0$. In this paper, we compute the Hilbert series of $R/I_{\lambda }^{\mathrm{Sp}}$ for $\lambda =(n-d,d)$ or $(d,d,1)$. Hence, we get the Castelnuovo-Mumford regularity of $R/I_{\lambda }^{\mathrm{Sp}}$, when it is Cohen-Macaulay. In particular, $I_{(d,d,1)}^{\mathrm{Sp}}$ has a $(d+2)$-linear resolution in the Cohen-Macaulay case.

12 pages. To appear in J.Algebra