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 Zqℓ. 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 Rℓ, 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 Zq-arrangements, an answer to a problem of Chen–Wang, and computation on the characteristic polynomials of R-arrangements will also be discussed.