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

الأصل · EN

Let Φ be an irreducible crystallographic root system with Weyl group W, coroot lattice Q and Coxeter number h. Recently the second named author defined a uniform W-isomorphism ζ between the finite torus Q/(mh+1)Q and the set of non-nesting parking fuctions Park⁽ᵐ⁾(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.