Parking Structures: Fuss Analogs

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.