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Cluster parking functions

Theo Douvropoulos, Matthieu Josuat-Vergès

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.

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.
Paper Structure (20 sections, 42 theorems, 134 equations, 3 figures)

This paper contains 20 sections, 42 theorems, 134 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Parking functions
  3. The cluster complex
  4. Cluster parking functions
  5. Definitions and background
  6. Poset topology
  7. Finite real reflection groups and noncrossing partitions
  8. Parking functions
  9. The cluster complex
  10. Combinatorial models in terms of dissections of polygons
  11. Cluster parking functions
  12. Definition and properties
  13. Combinatorial models and examples
  14. Topology on noncrossing partitions
  15. Topology of the parking function poset
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $W$ be a finite irreducible real reflection group of rank $n$, and let $h$ be its Coxeter number. The simplicial complexes $\mathop{\mathrm{CPF}}\nolimits^{(m)}(W)$ and $\mathop{\mathrm{CPF}}\nolimits^{(m),+}(W)$ (defined in Section sec:cpf) have the homotopy type of a wedge of $(n-1)$-dimension respectively ( i.e., these are parking characters twisted by the sign character).

Figures (3)

  • Figure 1: The graph $\mathop{\mathrm{CPF}}\nolimits_3$, drawn on a torus.
  • Figure 2: The complex $\Delta^+$ in type $A_2$ with a bipartite Coxeter element (left). Illustration of the poset $\Theta'(w)$ (right).
  • Figure 3: Regions with $\operatorname{mfl}(R)>0$ in $\operatorname{Cat}^{(1)}(A_2)$ and $\operatorname{Cat}^{(1)}(B_2)$

Theorems & Definitions (96)

  • Theorem 1.1: Theorem \ref{['theo:topcpf']} below
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 86 more