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arXiv · 2402.03052

Cluster parking functions

Theo Douvropoulos·Matthieu Josuat-Vergès
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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.

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Theo Douvropoulos, Matthieu Josuat-Vergès. 2025-01-13. Cluster parking functions. https://arxiv.org/abs/2402.03052

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