Worpitzky partitions for root systems and characteristic quasi-polynomials

TL;DR

This paper introduces a new lattice partition related to root systems, generalizes classical formulas, and applies these to analyze characteristic quasi-polynomials of arrangements, proving polynomiality and exploring zero locations.

Contribution

It generalizes Worpitzky partitions to root systems and uses them to study characteristic quasi-polynomials of arrangements, including proving polynomiality for Shi arrangements.

Findings

Characteristic quasi-polynomial of Shi arrangement is a polynomial.

Verified the functional equation for Linial arrangement's characteristic polynomial.

Partial results supporting Riemann hypothesis for certain exceptional root systems.

Abstract

We introduce a partition of (coweight) lattice points inside the dilated fundamental parallelepiped into those of partially closed simplices. This partition can be considered as a generalization and a lattice points interpretation of the classical formula of Worpitzky. This partition, and the generalized Eulerian polynomial, recently introduced by Lam and Postnikov, can be used to describe the characteristic (quasi)polynomials of Shi and Linial arrangements. As an application, we prove that the characteristic quasi-polynomial of the Shi arrangement turns out to be a polynomial. We also present several results on the location of zeros of characteristic polynomials, related to a conjecture of Postnikov and Stanley. In particular, we verify the "functional equation" of the characteristic polynomial of the Linial arrangement for any root system, and give partial affirmative results on "Riemann hypothesis" for the root systems of type $E_6, E_7, E_8$, and $F_4$.