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

TL;DR

This paper extends the concept of q,t-Fuss-Catalan numbers from type A to all finite complex reflection groups, exploring their algebraic and combinatorial properties and establishing proofs for specific groups.

Contribution

It generalizes the definition of q,t-Fuss-Catalan numbers to finite complex reflection groups and proves conjectures for dihedral and cyclic groups.

Findings

Conjectured properties of these polynomials are supported by computer experiments.

Proved algebraic properties for dihedral and cyclic groups.

Proposed connections to graded Hilbert series of modules in rational Cherednik algebras.

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. We generalize this construction to (finite) complex reflection groups and, based on computer experiments, we exhibit several conjectured algebraic and combinatorial properties of these polynomials with non-negative integer coefficients. We prove the conjectures for the dihedral groups and for the cyclic groups. Finally, we present several ideas how the q,t-Fuss-Catalan numbers could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras and thereby generalize known connections.