2016/09/11 by Sulzgruber, Robin, Thiel, Marko
#05E99 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1609.03128
Let Φ be an irreducible crystallographic root system with Weyl group W, coroot lattice \checkQ and Coxeter number h. Recently the second named author defined a uniform W-isomorphism ζ between the finite torus \checkQ/(mh+1)\checkQ 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.