On parking functions and the zeta map in types B,C and D

TL;DR

This paper explores the zeta map for root systems of types B, C, and D, establishing combinatorial models and bijections with lattice paths, extending previous work on type A and connecting to parking functions.

Contribution

It generalizes the zeta map to types B, C, and D, providing new combinatorial models and bijections with lattice paths for these root systems.

Findings

Defined models for finite torus and parking functions using labelled lattice paths

Established the zeta map as a bijection between these combinatorial objects

Created new bijections between square and ballot lattice paths

Abstract

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking fuctions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.