Parking Structures: Fuss Analogs

TL;DR

This paper generalizes the concept of parking functions to Fuss analogs for all irreducible real reflection groups, proposing modules called $k$-$W$-parking spaces and exploring their properties and conjectured relationships.

Contribution

It introduces Fuss analogs of $W$-parking spaces for all finite reflection groups and proves partial results, extending previous type A results to broader classes.

Findings

Established a formula for the character of $k$-$W$-parking spaces.

Recovered a cyclic sieving phenomenon related to $k$-$W$-noncrossing partitions.

Connected $k$-$W$-parking spaces to actions on finite tori in crystallographic cases.

Abstract

For any irreducible real reflection group $W$ with Coxeter number $h$, Armstrong, Reiner, and the author introduced a pair of $W \times \ZZ_h$-modules which deserve to be called {\sf $W$-parking spaces} which generalize the type A notion of parking functions and conjectured a relationship between them. In this paper we give a Fuss analog of their constructions. For a Fuss parameter $k \geq 1$, we define a pair of $W \times \ZZ_{kh}$-modules which deserve to be called {\sf $k$-$W$-parking spaces} and conjecture a relationship between them. We prove the weakest version of our conjectures for each of the infinite families ABCDI of finite reflection groups, together with proofs of stronger versions in special cases. Whenever our weakest conjecture holds for $W$, we have the following corollaries. First, there is a simple formula for the character of either $k$-$W$-parking space. Second, we recover a cyclic sieving result due to Krattenthaler and M\"uller which gives the cycle structure of a generalized rotation action on $k$-$W$-noncrossing partitions. Finally, when $W$ is crystallographic, the restriction of either $k$-$W$-parking space to $W$ isomorphic to the action of $W$ on the finite torus $Q / (kh+1)Q$, where $Q$ is the root lattice.