Stirling numbers for complex reflection groups

TL;DR

This paper extends the definition of q-analogues of Stirling numbers from Coxeter groups to all irreducible complex reflection groups, providing combinatorial interpretations and connections to algebraic structures.

Contribution

It generalizes the concept of q-Stirling numbers to all irreducible complex reflection groups G, linking them to Whitney numbers and algebraic Hilbert series.

Findings

Defined q-analogues of Stirling numbers for all irreducible complex reflection groups.

Provided combinatorial interpretations for groups G(m,p,n).

Connected q-Stirling numbers to conjectured Hilbert series of super coinvariant algebras.

Abstract

In an earlier paper, we defined and studied q-analogues of the Stirling numbers of both types for the Coxeter group of type B. In the present work, we show how this approach can be extended to all irreducible complex reflection groups G. The Stirling numbers of the first and second kind are defined via the Whitney numbers of the first and second kind, respectively, of the intersection lattice of G. For the groups G(m,p,n), these numbers and polynomials can be given combinatorial interpretations in terms of various statistics. The ordered version of ths q-Stirling numbers of the second kind also show up in conjectured Hilbert series for certain super coinvariant algebras.