Let \(a_{ij}\in \mathbb{C}\), \(i\geq 1\), \(1\leq j\leq N\), be nonzero complex numbers. Polynomials \(Q_i=Q_i(x_1,\ldots,x_N) =\sum^N_{j=1}a_{ij}x^i_j\) are called generalized power sums. The main question studied in this paper is when the algebra generated by \(Q_i\), \(i\geq 1\) inside \(\mathbb{C}[x_1,\ldots,x_N]\) is Cohen-Macaulay (shortly, \(CM\)).
Specifically, following [A. Brookner et al., “On Cohen-Macaulayness of \(S^n\)-invariant subspace arrangements”, Preprint, arXiv:1410.5096], for various collections of positive integers \((r_1,\ldots,r_k)\) with \(\sum r_i = N\), the authors study the \(CM\) property of algebras of generalized power sums with symmetry type \((r_1,\ldots,r_k)\) (i.e.symmetric in the first \(r_1\) variables, the next \(r_2\) variables, etc.). Using representation theoretic results and deformation theory, the authors establish Cohen-Macaulayness of the algebra of \(q,t\)-deformed power sums defined by A. N. Sergeev and A. P. Veselov [Commun. Math. Phys. 245, No. 2, 249–278 (2004; Zbl 1062.81097)] and of some generalizations of this algebra, proving a conjecture of A.Brookner, D.Corwin, P.Etingof, S.Sam.
Representation-theoretic techniques are also applied to studying \(m\)-quasi-invariants of deformed Calogero-Moser systems. In an appendix to this paper, M.Feigin uses representation theory of I. Cherednik algebras [Prog. Math. 243, 79–95 (2006; Zbl 1097.20007)] to compute Hilbert series for such quasi-invariants, and shows that in the case of one light particle, the ring of quasi-invariants is Gorenstein.