Catalan numbers for complex reflection groups

TL;DR

This paper extends the theory of Catalan numbers to complex reflection groups by constructing new (q,t)-Catalan and q-Fuss-Catalan polynomials using advanced algebraic tools like Cherednik algebras and KZ connections.

Contribution

It introduces a novel construction of (q,t)-Catalan and q-Fuss-Catalan polynomials for all irreducible complex reflection groups, expanding their algebraic and combinatorial understanding.

Findings

Construction of (q,t)-Catalan polynomials for complex reflection groups

Development of q-Fuss-Catalan polynomials for these groups

Integration of Cherednik algebra shift functors and KZ analysis

Abstract

We construct (q,t)-Catalan polynomials and q-Fuss-Catalan polynomials for any irreducible complex reflection group W. The two main ingredients in this construction are Rouquier's formulation of shift functors for the rational Cherednik algebras of W, and Opdam's analysis of permutations of the irreducible representations of W arising from the Knizhnik-Zamolodchikov connection.