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

On Cohen-Macaulayness of algebras generated by generalized power sums