On Cohen-Macaulayness of algebras generated by generalized power sums

TL;DR

This paper investigates when subalgebras generated by generalized power sums are Cohen-Macaulay, revealing that such cases are rare and closely linked to quantum integrability and Cherednik algebra representation theory, with new results confirming conjectures.

Contribution

The paper proves Cohen-Macaulayness for certain q,t-deformed power sum algebras and generalizations, confirming a prior conjecture using representation-theoretic and deformation techniques.

Findings

Cohen-Macaulayness is rare among these algebras.

Established Cohen-Macaulayness for q,t-deformed power sums.

Computed Hilbert series for quasi-invariants in specific cases.

Abstract

Generalized power sums are linear combinations of i-th powers of coordinates. We consider subalgebras of the polynomial algebra generated by generalized power sums, and study when such algebras are Cohen-Macaulay. It turns out that the Cohen-Macaulay property of such algebras is rare, and tends to be related to quantum integrability and representation theory of Cherednik algebras. Using representation-theoretic results and deformation theory, we establish Cohen-Macaulayness of the algebra of $q,t$-deformed power sums defined by Sergeev and Veselov, and of some generalizations of this algebra, proving a conjecture from arXiv:1410.5096. We also apply representation-theoretic techniques to studying m-quasi-invariants of deformed Calogero-Moser systems. In an appendix to this paper, M. Feigin uses representation theory of Cherednik algebras to compute Hilbert series for such quasi-invariants, and show that in the case of one light particle, the ring of quasi-invariants is Gorenstein.