q,t-Fuss-Catalan numbers for complex reflection groups

TL;DR

This paper generalizes q,t-Fuss-Catalan numbers from type A symmetric groups to complex reflection groups, exploring their algebraic and combinatorial properties and potential connections to rational Cherednik algebras.

Contribution

It extends the definition of q,t-Fuss-Catalan numbers to complex reflection groups and discusses their conjectured properties and possible links to rational Cherednik algebra modules.

Findings

Proposed a generalized construction of q,t-Fuss-Catalan numbers for complex reflection groups.

Conjectured algebraic and combinatorial properties of these polynomials.

Suggested potential relations to graded Hilbert series in rational Cherednik algebra context.

Abstract

In type A, the q,t-Fuss -Catalan numbers can be defined as a bigraded Hilbert series of a module associated to the symmetric group S_n. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in q and t. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress.