From Anderson to Zeta

TL;DR

This paper constructs a uniform bijection between affine Weyl group elements and a finite torus for all irreducible crystallographic root systems, generalizing known maps and connecting to parking functions.

Contribution

It introduces a new uniform bijection for all irreducible crystallographic root systems, extending previous type-specific maps and linking affine Weyl groups to parking functions.

Findings

Established a uniform bijection for all root systems.

Connected the bijection to existing maps in special cases.

Provided a new perspective on affine Weyl groups and parking functions.

Abstract

For an irreducible crystallographic root system $\Phi$ and a positive integer $p$ relatively prime to the Coxeter number $h$ of $\Phi$, we give a natural bijection $\mathcal{A}$ from the set $\widetilde{W}^p$ of affine Weyl group elements with no inversions of height $p$ to the finite torus $\check{Q}/p\check{Q}$. Here $\check{Q}$ is the coroot lattice of $\Phi$. This bijection is defined uniformly for all irreducible crystallographic root systems $\Phi$ and is equivalent to the Anderson map $\mathcal{A}_{GMV}$ defined by Gorsky, Mazin and Vazirani when $\Phi$ is of type $A_{n-1}$. Specialising to $p=mh+1$, we use $\mathcal{A}$ to define a uniform $W$-set isomorphism $\zeta$ from the finite torus $\check{Q}/(mh+1)\check{Q}$ to the set of $m$-nonnesting parking functions $\mathsf{Park}_{\Phi}^{(m)}$ of $\Phi$. The map $\zeta$ is equivalent to the zeta map $\zeta_{HL}$ of Haglund and Loehr when $m=1$ and $\Phi$ is of type $A_{n-1}$.