On Cohen-Macaulayness of S_n-Invariant Subspace Arrangements

Given a partition λ of n, consider the subspace E_λ of ℂⁿ where the first λ₁ coordinates are equal, the next λ₂ 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 (CM). 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 CM when λ has at least four distinct parts, and handle a large number of cases when λ has two or three 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.