Abstract
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.
Highlights
IntroductionThe combinatorial objects Vert(An−1) and Diag(An−1) may be viewed as corresponding to the type An−1 cases of more general algebraic objects associated to any irreducible crystallographic root system Φ
One of the most well-studied objects in algebraic combinatorics is the space of diagonal harmonics of the symmetric group Sn
We prove that the combinatorial zeta map coincides with the uniform zeta map of the given type, and is a bijection
Summary
The combinatorial objects Vert(An−1) and Diag(An−1) may be viewed as corresponding to the type An−1 cases of more general algebraic objects associated to any irreducible crystallographic root system Φ These are, respectively, the finite torus Q/(h + 1)Qand the set of non-nesting parking functions Park(Φ) of Φ. We define the combinatorial bijections ζB, ζC , and ζD which correspond to the uniform bijection ζ in those types Besides their meaning in the theory of reflection groups these maps can be appreciated from a purely combinatorial point of view. As such ζB and ζC are new bijections between lattice paths in an n × n-square and ballot paths with 2n steps, both of which are well-known to be counted by the central binomial coefficients. An extended abstract [ST15] of this paper containing mostly the results of Section 4 has appeared in the conference proceedings of FPSAC 2015 in Daejeon
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