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 functions $\operatorname{Park}^{(m)}(\Phi)$. 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.
@article{10_37236_6714,
author = {Robin Sulzgruber and Marko Thiel},
title = {On parking functions and the zeta map in types {\(B\),} {\(C\)} and {\(D\)}},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {1},
doi = {10.37236/6714},
zbl = {1386.05195},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6714/}
}
TY - JOUR
AU - Robin Sulzgruber
AU - Marko Thiel
TI - On parking functions and the zeta map in types \(B\), \(C\) and \(D\)
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/6714/
DO - 10.37236/6714
ID - 10_37236_6714
ER -
%0 Journal Article
%A Robin Sulzgruber
%A Marko Thiel
%T On parking functions and the zeta map in types \(B\), \(C\) and \(D\)
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/6714/
%R 10.37236/6714
%F 10_37236_6714
Robin Sulzgruber; Marko Thiel. On parking functions and the zeta map in types \(B\), \(C\) and \(D\). The electronic journal of combinatorics, Tome 25 (2018) no. 1. doi: 10.37236/6714
