Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials

Tran, Tan Nhat; Yoshinaga, Masahiko

doi:10.1016/j.jcta.2019.02.003

Cited by 14 publications

(12 citation statements)

References 23 publications

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“…However, to date, no proof has been known. Recently, a complete description of the constituents of the characteristic quasi-polynomial was found via the theory of toric arrangements due to the second author and the third author [TY19]. Using this approach, we are able to affirmatively prove that this is always the case.…”

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confidence: 64%

“…Both methods are powerful and interesting. The former gives explicitly the coefficients of the quasipolynomials which in turn produces the lcm period the best known candidate for period so far, and it also opens an unexpected connection with the theory of toric arrangements [TY19,LTY21]. The latter, generally, applies only to the central case (see §4) but enables one to apply many results in the Ehrhart theory to further discover interesting properties of the arrangement.…”

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confidence: 91%

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Period collapse in characteristic quasi-polynomials of hyperplane arrangements

Higashitani¹,

Tran²,

Yoshinaga³

2021

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Given an integral hyperplane arrangement, Kamiya-Takemura-Terao (2008& 2011 introduced the notion of characteristic quasi-polynomial, which enumerates the cardinality of the complement of the arrangement modulo a positive integer. The most popular candidate for period of the characteristic quasi-polynomials is the lcm period. In this paper, we initiate a study of period collapse in characteristic quasi-polynomials stemming from the concept of period collapse in the theory of Ehrhart quasi-polynomials. We say that period collapse occurs in a characteristic quasipolynomial when the minimum period is strictly less than the lcm period. Our first main result is that in the non-central case, with regard to period collapse anything is possible: period collapse occurs in any dimension ≥ 1, occurs for any value of the lcm period ≥ 2, and the minimum period when it is not the lcm period can be any proper divisor of the lcm period. Our second main result states that in the central case, however, no period collapse is possible in any dimension, that is, the lcm period is always the minimum period.

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confidence: 64%

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confidence: 91%

Period collapse in characteristic quasi-polynomials of hyperplane arrangements

Higashitani¹,

Tran²,

Yoshinaga³

2021

Preprint

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“…One of the most important properties is that the first constituent of the characteristic quasi-polynomial is equal to the characteristic polynomial of a hyperplane arrangement [12]. The characteristic quasi-polynomial is also important in the context of toric arrangements [4,9,16]. We also note that Suter's computations (Example 6.7) is deeply related to characteristic quasi-polynomials.…”

Section: Rational Polytopes With Gcd-property and Hyperplane Arrangem...mentioning

confidence: 99%

Ehrhart quasi-polynomials of almost integral polytopes

de Vries,

Yoshinaga

2021

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A lattice polytope translated by a rational vector is called an almost integral polytope. In this paper we investigate Ehrhart quasi-polynomials of almost integral polytopes. We study the relationship between the shape of the polytopes and algebraic properties of the Ehrhart quasi-polynomials. In particular, we prove that lattice zonotopes and centrally symmetric lattice polytopes are characterized by Ehrhart quasi-polynomials of their rational translations.

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“…These facts will be made clear in Theorem 4.4 and Propositions 4.3, 4.5. More generally, we will give other interpretations for every chromatic quasi-polynomial and their constituents through subspace and toric viewpoints in our forthcoming paper [TY18].…”

Section: Application To Hyperplane Arrangementsmentioning

confidence: 99%

An equivalent formulation of chromatic quasi-polynomials

Tran

2018

Preprint

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Given a central integral arrangement, the reduction of the arrangement modulo positive integers q gives rise to a subgroup arrangement in (Z/qZ) ℓ . 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 R ℓ , and evaluating the cardinality of the q-reduction complement will also lead to a quasipolynomial 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.

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Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials

Cited by 14 publications

References 23 publications

Period collapse in characteristic quasi-polynomials of hyperplane arrangements

Period collapse in characteristic quasi-polynomials of hyperplane arrangements

Ehrhart quasi-polynomials of almost integral polytopes

An equivalent formulation of chromatic quasi-polynomials

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