Cluster parking functions

TL;DR

This paper explores the topological and algebraic properties of cluster parking functions associated with finite real reflection groups, revealing their homotopy type and connections to diagonal coinvariants.

Contribution

It establishes that the simplicial complex of cluster parking functions has the homotopy type of a wedge of spheres and links their homology to Gordon's quotient of diagonal coinvariants.

Findings

Homotopy type of cluster parking functions is a wedge of spheres

Homology corresponds to a sign-twisted parking representation

Properties of the parking function poset are characterized

Abstract

The cluster complex on one hand, parking functions on the other hand, are two combinatorial (po)sets that can be associated to a finite real reflection group. Cluster parking functions are obtained by taking an appropriate fiber product (over noncrossing partitions). There is a natural structure of simplicial complex on these objects, and our main goal is to show that it has the homotopy type of a (pure) wedge of spheres. The unique nonzero homology group (as a representation of the underlying reflection group) is a sign-twisted parking representation, which is the same as Gordon's quotient of diagonal coinvariants. Along the way, we prove some properties of the poset of parking functions. We also provide a long list of remaining open problems.