Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parking Spaces for Complex Reflection Groups

Jason Stack

TL;DR

This work extends noncrossing and parking-function theory to irreducible well-generated complex reflection groups by defining both combinatorial $W$-noncrossing parking spaces and algebraic $W$-parking spaces, proving a $(W\times C)$-equivariant isomorphism between them, and thereby enumerating $W$-noncrossing parking functions. It generalizes to Fuss analogues, establishing $(W\times \mathbb{Z}_{kh})$-equivariant isomorphisms and cardinalities $(kh+1)^n$, and provides explicit case-by-case bijections for the families $G(d,1,n)$ and $G(d,d,n)$, including hsop constructions illustrated by examples like $G_{10}$. The results hold for all such complex groups except $G_{34}$, $E_7$, and $E_8$, with computational verification underpinning the exceptional cases. By bridging combinatorial models with invariant-theoretic algebra, the paper broadens Coxeter-Catalan phenomena into the realm of complex reflection groups and their Fuss generalizations.

Abstract

We answer an open problem of arXiv:1204.1760 and arXiv:1205.4293, extending their work to irreducible well--generated complex reflection groups $W$. We define a combinatorial $W$-noncrossing parking space and an algebraic $W$-parking space for such $W$, and exhibit a $(W \times C)$-equivariant isomorphism between the two. As a consequence of this isomorphism, we enumerate the $W$-noncrossing parking functions. Finally, we extend our results to the Fuss case. We prove the results for all such complex reflection groups except $G_{34}$, $E_7,$ and $E_8$.

Parking Spaces for Complex Reflection Groups

TL;DR

This work extends noncrossing and parking-function theory to irreducible well-generated complex reflection groups by defining both combinatorial -noncrossing parking spaces and algebraic -parking spaces, proving a -equivariant isomorphism between them, and thereby enumerating -noncrossing parking functions. It generalizes to Fuss analogues, establishing -equivariant isomorphisms and cardinalities , and provides explicit case-by-case bijections for the families and , including hsop constructions illustrated by examples like . The results hold for all such complex groups except , , and , with computational verification underpinning the exceptional cases. By bridging combinatorial models with invariant-theoretic algebra, the paper broadens Coxeter-Catalan phenomena into the realm of complex reflection groups and their Fuss generalizations.

Abstract

We answer an open problem of arXiv:1204.1760 and arXiv:1205.4293, extending their work to irreducible well--generated complex reflection groups . We define a combinatorial -noncrossing parking space and an algebraic -parking space for such , and exhibit a -equivariant isomorphism between the two. As a consequence of this isomorphism, we enumerate the -noncrossing parking functions. Finally, we extend our results to the Fuss case. We prove the results for all such complex reflection groups except , and .

Paper Structure

This paper contains 37 sections, 17 theorems, 92 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Noncrossing Partitions
  3. Parking Functions
  4. Nonnesting Parking Functions
  5. Noncrossing Parking Functions
  6. Fuss Analogues
  7. Algebraic Parking Space
  8. Outline
  9. Background
  10. Noncrossing Partitions in Real Reflection Groups
  11. Noncrossing Parking Functions in Real Reflection Groups
  12. Fuss Analogues
  13. Complex Reflection Groups
  14. Notation
  15. Classification
  16. ...and 22 more sections

Key Result

Theorem 1.3

For all such $W$, there is a $(W\times C)$-equivariant isomorphism between the $W$-noncrossing parking functions and the algebraic $W$-parking space.

Figures (12)

  • Figure 1: A noncrossing partition in type $A_{7}$ with $\pi=\{\{1,5,6\},\{2,3,4\},\{7,8\}\}.$
  • Figure 2: A noncrossing parking function in type $A_{7}$ with noncrossing partition $\pi$ as in \ref{['fig: Type A noncrossing partition']}.
  • Figure 3: Noncrossing partitions in real reflection groups for types $A$, $B$, and $D$.
  • Figure 4: Parking functions in real reflection groups for types $A$, $B$, and $D$.
  • Figure 5: A $G(3,1,6)$-noncrossing partition with $\pi$ as in \ref{['partition pi for d,1,n example']}.
  • ...and 7 more figures

Theorems & Definitions (71)

  • Example 1.1
  • Example 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2: bessis2003dualwatt2002Kpi
  • Example 2.3
  • Definition 2.4
  • ...and 61 more