Cluster parking functions
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.
Cite
@article{arxiv.2402.03052,
title = {Cluster parking functions},
author = {Theo Douvropoulos and Matthieu Josuat-Vergès},
journal= {arXiv preprint arXiv:2402.03052},
year = {2025}
}