On Cohen-Macaulayness of S_n-invariant subspace arrangements

TL;DR

This paper investigates when certain symmetric group-invariant subspace arrangements are Cohen-Macaulay, providing new classifications and settling related conjectures using algebraic and computational methods.

Contribution

It classifies Cohen-Macaulayness of $X_$ arrangements based on the partition structure, extending previous results and resolving a conjecture by Sergeev and Veselov.

Findings

$X_$ is not Cohen-Macaulay with at least 4 distinct parts in $$

Classifies cases with 2 or 3 distinct parts for Cohen-Macaulayness

Settles a conjecture on Cohen-Macaulayness of algebras generated by deformed Newton sums

Abstract

Given a partition $\lambda$ of n, consider the subspace $E_\lambda$ of $C^n$ where the first $\lambda_1$ coordinates are equal, the next $\lambda_2$ coordinates are equal, etc. In this paper, we study subspace arrangements $X_\lambda$ consisting of the union of translates of $E_\lambda$ by the symmetric group. In particular, we focus on determining when $X_\lambda$ is Cohen-Macaulay. This is inspired by previous work of the third author coming from the study of rational Cherednik algebras and which answers the question positively when all parts of $\lambda$ are equal. We show that $X_\lambda$ is not Cohen-Macaulay when $\lambda$ has at least 4 distinct parts, and handle a large number of cases when $\lambda$ has 2 or 3 distinct parts. Along the way, we also settle a conjecture of Sergeev and Veselov about the Cohen-Macaulayness of algebras generated by deformed Newton sums. Our techniques combine classical techniques from commutative algebra and invariant theory, in many cases we can reduce an infinite family to a finite check which can sometimes be handled by computer algebra.