On parking functions and the zeta map in types \(B\), \(C\) and \(D\) (Q1691102)

Summary: 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.