Characteristic polynomials of Linial arrangements for exceptional root systems

TL;DR

This paper proves that for exceptional root systems, the roots of the characteristic polynomial of the Linial arrangement all share the same real part when the parameter is large, extending known results from classical types.

Contribution

It establishes the conjecture for exceptional root systems using Eulerian polynomial representations, for sufficiently large parameters.

Findings

Roots of characteristic polynomials have the same real part for large m in exceptional root systems.

The proof uses representations of characteristic quasi-polynomials via Eulerian polynomials.

Confirmed conjecture previously known only for classical root systems.

Abstract

The (extended) Linial arrangement $\mathcal{L}_{\Phi}^m$ is a certain finite truncation of the affine Weyl arrangement of a root system $\Phi$ with a parameter $m$. Postnikov and Stanley conjectured that all roots of the characteristic polynomial of $\mathcal{L}_{\Phi}^m$ have the same real part, and this has been proved for the root systems of classical types. In this paper we prove that the conjecture is true for exceptional root systems when the parameter $m$ is sufficiently large. The proof is based on representations of the characteristic quasi-polynomials in terms of Eulerian polynomials.