Recursions and Proofs in Coxeter-Catalan combinatorics

Abstract: In a significant --yet absolutely understandable-- deviation from traditions of logic, secularism, and platonic dialectic, combinatorialists the world around have celebrated Catalan objects with a reverence better suited to mystical, preternatural endeavors. Various sects have been formed through the years by mathematicians who study particular aspects of the Catalan doctrine, including the Coxeter-Catalan sect of which the speaker might be a member.

One of the central objects in Coxeter-Catalan combinatorics is the noncrossing partition lattice $NC(W)$ associated to a finite Coxeter group $W$ and its sibling object, the cluster complex $Y(W)$. These two objects encode much of the geometric group theory, combinatorics, and representation theory of $W$, and they have fascinating stuctural and enumerative properties; in particular, the zeta polynomials of certain intervals in $NC(W)$ and the (almost) colored $f$-vectors of $Y(W)$ all have product formulas given in terms of invariants of $W$ (generalizing formulas of Kreweras and Loday for the symmetric group case). A central open problem in the area since at least the early 2000's has been to give case-free proofs of these product formulas, i.e. proofs that do not depend on the classification of finite Coxeter groups. In this talk, I will present the first such proof, in collaboration with Matthieu Josuat-Verges, solving the more general Fuss version of the problem; in our approach, we develop a collection of recursions that are shown to be satisfied by both the combinatorial objects and the product formulas.

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar