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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.03052 |
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Table of Contents:
- 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.