An equivalent formulation of chromatic quasi-polynomials

Abstract

Given a central integral arrangement, the reduction of the arrangement modulo a positive integer $q$ gives rise to a subgroup arrangement in ${\mathbb{Z}}_q^{\ell }$. Kamiya et al. (2008) introduced the notion of characteristic quasi-polynomial, which enumerates the cardinality of the complement of this subgroup arrangement. Chen and Wang (2012) found a similar but more general setting that replacing the integral arrangement by its restriction to a subspace of ${\mathbb{R}}^{\ell }$, and evaluating the cardinality of the $q$-reduced complement will also lead to a quasi-polynomial in $q$. On an independent study, Brändén and Moci (2014) defined the so-called chromatic quasi-polynomial, and initiated the study of $q$-colorings on a finite list of elements in a finitely generated abelian group. The main purpose of this paper is to verify that the Chen–Wang quasi-polynomial and the Brändén–Moci chromatic quasi-polynomial are equivalent in the sense that the quasi-polynomials enumerate the cardinalities of isomorphic sets. Some applications including periodicity of the intersection posets of ${\mathbb{Z}}_q$-arrangements, an answer to a problem of Chen–Wang, and computation on the characteristic polynomials of $\mathbb{R}$-arrangements will also be discussed.

Introduction

In the simplest setting, when a finite list (multiset) $\mathcal{A}$ of vectors in ${\mathbb{Z}}^{\ell }$ is given, we may naturally associate to it an integral hyperplane arrangement $\mathcal{A}\left(\mathbb{R}\right)$ in ${\mathbb{R}}^{\ell }$. The study of a hyperplane arrangement typically goes along with the study of its characteristic polynomial as the polynomial carries combinatorial and topological information of the arrangement (e.g., [18]). In this paper, we are interested in an arithmetic-combinatorial method for studying the integral arrangements, generally known as the “finite field method” and first appeared in [8]. The method was developed into a systematic tool by Athanasiadis [3], after closely related techniques have been used by Björner–Ekedahl [4], and Blass–Sagan [5] to solve problems related to subspace arrangements. Roughly speaking, we can take coefficients modulo a positive integer $q$ and get the $q$-reduced arrangement $\mathcal{A}\left({\mathbb{Z}}_q\right)$ of subgroups in ${\mathbb{Z}}_q^{\ell }$ (e.g., [14]). The central result in the theory states that when $q$ is a sufficiently large prime, the arrangement $\mathcal{A}\left({\mathbb{Z}}_q\right)$ now is defined over the finite field ${\mathbb{F}}_q$, and the cardinality of its complement coincides with ${\chi }_{\mathcal{A}\left(\mathbb{R}\right)}\left(q\right)$, the evaluation of the characteristic polynomial ${\chi }_{\mathcal{A}\left(\mathbb{R}\right)}\left(t\right)$ of $\mathcal{A}\left(\mathbb{R}\right)$ at $q$ (e.g., [3, Theorem 2.2], [14, Theorem 2.5]).

The fundamental result mentioned above is efficiently applicable to compute the characteristic polynomials of several arrangements arising from root systems (e.g., [3]). Kamiya–Takemura–Terao [14] showed that the cardinality of the complement of $\mathcal{A}\left({\mathbb{Z}}_q\right)$ is actually a quasi-polynomial in $q$, and called this the characteristic quasi-polynomial of $\mathcal{A}$ (as its $1$-constituent agrees with ${\chi }_{\mathcal{A}\left(\mathbb{R}\right)}\left(t\right)$ by the fundamental theorem). Chen–Wang [7] considered the restriction of $\mathcal{A}\left(\mathbb{R}\right)$ to a subspace of ${\mathbb{R}}^{\ell }$, and proved a stronger result that after taking reduction modulo $q$ of the restricted arrangement, the cardinality of the complement is also a quasi-polynomial in $q$. More recently, a number of works (e.g., [2], [20], [24], [25]) extended the analysis on the deformations of root system arrangements and enhanced the calculation of the characteristic quasi-polynomials via the connection with Ehrhart theory.

In yet another consideration, given a finite list $\mathcal{A}$ in ${\mathbb{Z}}^{\ell }$, we can associate to it a toric arrangement $\mathcal{A}\left({\mathbb{S}}^1\right)$ in the torus ${\left({\mathbb{S}}^1\right)}^{\ell }$ (e.g., [10], [17]). Although we are concerned with the subtori (hypersurfaces) instead of the hyperplanes, it is still possible to formulate analogues of the “finite field method” (e.g., [1], [12], [15]). To derive a wider understanding of the characteristic polynomial of toric arrangements, an arithmetic generalization of the ordinary Tutte polynomial (e.g., [23]), the arithmetic Tutte polynomial was introduced [17]. These polynomials are currently receiving increasing attention (e.g., [9], [13], [21]). Among the others, Brändén–Moci [6] defined the Tutte quasi-polynomial associated to a finite list of elements in a finitely generated abelian group. This quasi-polynomial not only produces an interpolation between the Tutte polynomial and the arithmetic Tutte polynomial but also gives rise to the chromatic quasi-polynomial and the flow quasi-polynomial. These quasi-polynomials are group-theoretic counterparts of the graphic chromatic and flow polynomials, which proved to have an application to colorings and flows on CW complexes [11].

The Chen–Wang quasi-polynomial and the Brändén–Moci chromatic quasi-polynomial are natural generalizations of the characteristic quasi-polynomial, and arise independently in different contexts. In this paper, we prove that these two quasi-polynomials are indeed “equivalent” in the sense that any Chen–Wang quasi-polynomial is a chromatic quasi-polynomial and vice versa. The main idea of our proof is: we find a common concept that generalizes both quasi-polynomials, and the equivalence between them can be explained easily using a simple property of the concept.

The concept we will be discussing is $G$-arrangement with the associated $G$-Tutte polynomial, which was recently introduced by Liu, Yoshinaga and the author. This concept forms a common generalization of several arrangements and their “Tutte-like” polynomials in the literature, including all of the arrangements and (quasi-)polynomials mentioned above.

The quasi-polynomial equivalence establishes a connection between the two studies, from which many properties of one quasi-polynomial can be translated to those of the other. We will give some applications such as the periodicity of the intersection posets of ${\mathbb{Z}}_q$-arrangements, an answer to a problem of Chen–Wang, and computation on the characteristic polynomials of $\mathbb{R}$-arrangements (Section 5).¹

The remainder of the paper is organized as follows. In Section 2, we recall the definitions of the characteristic quasi-polynomial, the Chen–Wang quasi-polynomial, and the chromatic quasi-polynomial. In Section 3, after recalling the definitions and basic facts of $G$-arrangements and $G$-Tutte polynomials, we prove the equivalence of the chromatic quasi-polynomial and the Chen–Wang quasi-polynomial (Theorem 3.5). Then we prove the periodicity of the intersection posets of ${\mathbb{Z}}_q$-arrangements (Theorem 3.6). In Section 4, by using the language of chromatic quasi-polynomials, we give an answer to a problem asked by Chen–Wang (Problem 4.6). In Section 5, we generalize the fundamental theorem in the ordinary “finite field method” to $\mathbb{R}$-arrangements (Theorem 5.1).