On Cohen-Macaulayness of S_n-invariant subspace arrangements

Abstract

Given a partition $λ$ of n, consider the subspace $E_λ$ of $C^n$ where the first $λ_1$ coordinates are equal, the next $λ_2$ coordinates are equal, etc. In this paper, we study subspace arrangements $X_λ$ consisting of the union of translates of $E_λ$ by the symmetric group. In particular, we focus on determining when $X_λ$ 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 $λ$ are equal. We show that $X_λ$ is not Cohen-Macaulay when $λ$ has at least 4 distinct parts, and handle a large number of cases when $λ$ 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.

16 pages, supporting computations included as ancillary files, v2: Conjecture 5.13 changed to Proposition, Proposition 7.5 added