On Cohen-Macaulayness of Sn-Invariant Subspace Arrangements

Aaron Brookner, David Corwin, Pavel Etingof, Steven V. Sam

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Abstract

Given a partition λ of n, consider the subspace Eλ of ℂn where the first λ1 coordinates are equal, the next λ2 coordinates are equal, etc. In this paper, we study subspace arrangements Xλ consisting of the union of translates of Eλ by the symmetric group. In particular, we focus on determining when Xλ 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 Xλ 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.

Original languageEnglish
Pages (from-to)2104-2126
Number of pages23
JournalInternational Mathematics Research Notices
Volume2016
Issue number7
DOIs
StatePublished - 1 Jan 2016

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