Co-quasi-invariant spaces for finite complex reflection groups

TL;DR

This paper provides a uniform study of quotient spaces formed by diagonal quasi-invariant polynomials under complex reflection groups, introducing a universal symmetric function that encodes their graded Hilbert series.

Contribution

It introduces an explicit universal symmetric function that describes the Hilbert series for quotient spaces associated with complex reflection groups, highlighting their combinatorial structure.

Findings

Universal symmetric function for Hilbert series

Explicit formulas for G(r,n) groups

Positive coefficient expansion in symmetric functions

Abstract

We study, in a global uniform manner, the quotient of the ring of polynomials in l sets of n variables, by the ideal generated by diagonal quasi-invariant polynomials for general permutation groups W=G(r,n). We show that, for each such group W, there is an explicit universal symmetric function that gives the N^l-graded Hilbert series for these spaces. This function is universal in that its dependance on l only involves the number of variables it is calculated with. We also discuss the combinatorial implications of the observed fact that it affords an expansion as a positive coefficient polynomial in the complete homogeneous symmetric functions.