The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces

TL;DR

This paper refines the enumeration of faces in the generalized cluster complex by linking it to hyperplane arrangements, Coxeter groups, and parking spaces, providing explicit formulas and combinatorial reciprocity results.

Contribution

It introduces a new enumeration method for faces of the generalized cluster complex using characteristic polynomials and connects it to parking spaces and noncrossing partitions.

Findings

Explicit formulas for face enumeration in terms of characteristic polynomials

Connection established between face counts and parking space enumeration

Generalization of Narayana and Kirkman number reciprocity

Abstract

The generalized cluster complex was introduced by Fomin and Reading, as a natural extension of the Fomin-Zelevinsky cluster complex coming from finite type cluster algebras. In this work, to each face of this complex we associate a parabolic conjugacy class of the underlying finite Coxeter group. We show that the refined enumeration of faces (respectively, positive faces) according to this data gives an explicit formula in terms of the corresponding characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents). This characteristic polynomial originally comes from the theory of hyperplane arrangements, but it is conveniently defined via the parabolic Burnside ring. This makes a connection with the theory of parking spaces: our results eventually rely on some enumeration of chains of noncrossing partitions that were obtained in this context. The precise relations between the formulas counting faces and the one counting chains of noncrossing partitions are combinatorial reciprocities, generalizing the one between Narayana and Kirkman numbers.