An equivalent formulation of chromatic quasi-polynomials

Abstract

Given a central integral arrangement, the reduction of the arrangement modulo positive integers $q$ gives rise to a subgroup arrangement in $(\mathbb{Z}/q\mathbb{Z}{)}^{\ell }$. Kamiya-Takemura-Terao (2008) introduced the notion of characteristic quasi-polynomials, which uses to evaluate the cardinality of the complement of the subgroup arrangement. Chen-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 $q$-reduction complement will also lead to a quasi-polynomial in $q$. On an independent study, Brändén-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's quasi-polynomial and the Brändén-Moci's chromatic quasi-polynomial are equivalent in the sense that the quasi-polynomials enumerate the cardinalities of isomorphic sets.

arXiv comment12 pages

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• Affine and toric hyperplane arrangements
• Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials
• $G$-Tutte polynomials and abelian Lie group arrangements
• Characteristic and Ehrhart polynomials
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• Eulerian polynomials for subarrangements of Weyl arrangements
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Cited by in corpus (1)

• Period collapse in characteristic quasi-polynomials of hyperplane arrangements