The characteristic quasi-polynomials of the arrangements of root systems and mid-hyperplane arrangements

TL;DR

This paper explores the properties of characteristic quasi-polynomials associated with arrangements of root systems and mid-hyperplanes, providing formulas and results for specific cases and highlighting their mathematical significance.

Contribution

It introduces general properties of characteristic quasi-polynomials and derives explicit formulas for root system arrangements and low-dimensional mid-hyperplane arrangements.

Findings

A formula for the generating function of root system arrangements.

Explicit determination of generating functions for mid-hyperplane arrangements in dimensions less than six.

Insights into the structure and computation of characteristic quasi-polynomials.

Abstract

Let $q$ be a positive integer. In our recent paper, we proved that the cardinality of the complement of an integral arrangement, after the modulo $q$ reduction, is a quasi-polynomial of $q$, which we call the characteristic quasi-polynomial. In this paper, we study general properties of the characteristic quasi-polynomial as well as discuss two important examples: the arrangements of reflecting hyperplanes arising from irreducible root systems and the mid-hyperplane arrangements. In the root system case, we present a beautiful formula for the generating function of the characteristic quasi-polynomial which has been essentially obtained by Ch. Athanasiadis and by A. Blass and B. Sagan. On the other hand, it is hard to find the generating function of the characteristic quasi-polynomial in the mid-hyperplane arrangement case. We determine them when the dimension is less than six.