On Parking Functions and the Zeta Map in Types B, C and D

Let Φ be an irreducible crystallographic root system with Weyl group W , coroot lattice Q̌ and Coxeter number h. Recently the second named author defined a uniform W -isomorphism ζ between the finite torus Q̌/(mh + 1)Q̌ and the set of non-nesting parking functions Park(Φ). If Φ is of type An−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 (Bn, Cn and Dn). 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 ζ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.