[Paper Review] Catalan numbers for complex reflection groups

Iain J. Gordon, Stephen Griffeth|ArXiv.org|Dec 8, 2009

Advanced Algebra and Geometry4 citations

TL;DR

This paper constructs $(q,t)$-Catalan polynomials and $q$-Fuss-Catalan polynomials for all irreducible complex reflection groups using rational Cherednik algebras and the Knizhnik-Zamolodchikov connection. The key result is that these polynomials are in $\mathbb{N}[q]$ and generalize classical Catalan numbers, with explicit formulas derived from representation-theoretic data such as fake degrees and the permutation $\Psi$ on irreducible representations.

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.

Motivation & Objective

• To generalize classical Catalan numbers to all irreducible complex reflection groups, extending the notion to $q$-Fuss-Catalan and $(q,t)$-Catalan polynomials.
• To establish that these polynomials are in $\mathbb{N}[q]$, ensuring their integrality and combinatorial significance.
• To unify the construction across all complex reflection groups using representation theory of rational Cherednik algebras and the permutation $\Psi$ arising from the KZ connection.
• To confirm conjectures on cyclic sieving and Galois twists for these polynomials, particularly in the well-generated case.
• To provide explicit formulas for $q$-Fuss-Catalan numbers in terms of degrees, exponents, and the action of $\Psi$ on irreducible representations.

Proposed method

• Leverages Rouquier's formulation of shift functors for rational Cherednik algebras to define the $(q,t)$-Catalan polynomials via graded characters of irreducible representations.
• Uses Opdam's analysis of the permutation $\Psi$ on $\textsf{Irrep}(W)$, arising from the KZ connection, to define the generalized Catalan numbers via fake degrees and duality.
• Constructs the $q$-Fuss-Catalan polynomial as $C_W^{(m)}(q) = \prod_{i=1}^n \frac{[mh+1+e_i(\Psi^m(V^*)^*)]_q}{[d_i]_q}$, where $d_i$ are degrees of basic invariants and $e_i$ are exponents.
• Establishes that $C_W^{(m)}(q)$ is the Hilbert series of $(P/\Theta)^W$ for a homogeneous system of parameters $\Theta$ of degree $mh+1$ carrying $\Psi^m(V^*)$.
• Shows that $C_W^{(1)}(q)$ arises as the graded character of the irreducible representation $eL_{1+1/h}(\text{triv})$ in the spherical Cherednik algebra.
• Applies Galois twists to extend results to parameters $p/h$ with $p$ coprime to $h$, using automorphisms of $\mathbb{C}$ and the permutation $\Psi$.

Experimental results

Research questions

• RQ1 Can $q$-Fuss-Catalan numbers be defined for all irreducible complex reflection groups, not just well-generated ones?
• RQ2 Is the $q$-Fuss-Catalan polynomial $C_W^{(m)}(q)$ always in $\mathbb{N}[q]$ for any irreducible complex reflection group $W$?
• RQ3 How does the permutation $\Psi$ on irreducible representations relate to the duality and palindromic properties of fake degrees?
• RQ4 Can the $(q,t)$-Catalan polynomial be constructed uniformly for all $W$ using the representation theory of rational Cherednik algebras?
• RQ5 Do the generalized Catalan numbers exhibit cyclic sieving phenomena, and under what conditions?

Key findings

• The $q$-Fuss-Catalan polynomial $C_W^{(m)}(q)$ is shown to be in $\mathbb{N}[q]$ for all irreducible complex reflection groups, under Hypothesis 2.4.
• For well-generated groups, the formula simplifies to $C_W^{(m)}(q) = \prod_{i=1}^n \frac{[mh + d_i]_q}{[d_i]_q}$, matching the standard definition in the literature.
• The $q$-Fuss-Catalan number $C_W^{(m)}(q)$ is the Hilbert series of $(P/\Theta)^W$ where $\Theta$ is a system of parameters of degree $mh+1$ carrying $\Psi^m(V^*)$.
• The $(q,t)$-Catalan polynomial is realized as the graded character of the irreducible representation $eL_{1+1/h}(\text{triv})$ in the spherical Cherednik algebra.
• The construction confirms [2, Conjecture 4.3(i)] for well-generated groups and extends to Galois twists at parameters $p/h$ with $p$ coprime to $h$.
• Cyclic sieving is confirmed: $C_W^{(m)}(\zeta^t)$ is a positive integer for all $m$ and $t$, where $\zeta$ is a root of unity of order $d$, a regular number for $W$.