# When is a Specht ideal Cohen-Macaulay?

## TL;DR

This paper characterizes when Specht ideals lead to Cohen-Macaulay quotients, showing specific shape conditions on partitions and establishing characteristic-dependent properties.

## Contribution

It provides a complete characterization of Cohen-Macaulay Specht ideals based on partition shape and field characteristic, including new results on radicalness and Cohen-Macaulayness.

## Key findings

- Cohen-Macaulay Specht ideals correspond to specific partition shapes.
- The converse holds in characteristic zero.
- The ideal for partition (n-3,3) is not Cohen-Macaulay in characteristic 2.

## 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 ${\rm char}(K)=2$.