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Masahiko Yoshinaga

Publications and source records attributed to Masahiko Yoshinaga.

At least 37 records · Page 2Linked to original sources
arXiv2023

Finite record sets of chip-firing games

A finite graph with an assignment of non-negative integers to vertices gives chip-firing games. Chip-firing games determine languages (sets of words) called the record sets of legal games. Björner, Lovász and Shor found several properties that are satisfied by record sets. In this paper, we will find two more properties of record sets. Under the assumption that the record set is finite and the game fires only two vertices, these properties characterize the record sets of graphs.

math.CO↗
arXiv2022

Free reflection multiarrangements and quasi-invariants

To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close relation between these modules. Berest-Chalykh freeness results for the module of quasi-invariants lead to new free complex reflection multiarrangements. K. Saito's primitive derivative gives a linear map between certain spaces of quasi-invariants. We also establish a close relation between non-homogeneous quasi-invariants for root systems and logarithmic vector fields for the extended Catalan arrangements. As an application, we prove the freeness of Catalan arrangements corresponding to the non-reduced root system $BC_N$.

math.QA↗
arXiv2022

Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes

Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the CW-complexes in connection with the diagonality of magnitude homology groups. We prove that the Asao-Izumihara complex is homotopy equivalent to a wedge of spheres for pawful graphs introduced by Y. Gu. The result can be considered as a homotopy type version of Gu's result. We also formulate a slight generalization of the notion of pawful graphs and find new non-pawful diagonal graphs of diameter $2$.

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arXiv2021

$G$-Tutte polynomials and abelian Lie group arrangements

We introduce and study the notion of the $G$-Tutte polynomial for a list $\mathcal{A}$ of elements in a finitely generated abelian group $Γ$ and an abelian group $G$, which is defined by counting the number of homomorphisms from associated finite abelian groups to $G$. The $G$-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Brändén-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy polynomial for a finite CW-complex. As in the classical case, $G$-Tutte polynomials carry topological and enumerative information (e.g., the Euler characteristic, point counting and the Poincaré polynomial) of abelian Lie group arrangements. We also discuss differences between the arithmetic Tutte and the $G$-Tutte polynomials related to the axioms for arithmetic matroids and the (non-)positivity of coefficients.

math.CO↗
arXiv2021

Magnitude homology of metric spaces and order complexes

Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order complexes of posets. In a metric space, an interval (the set of points between two chosen points) has a natural poset structure, which is called the interval poset. Under additional assumptions on sizes of $4$-cuts, we show that the magnitude chain complex can be constructed using tensor products, direct sums and degree shifts from order complexes of interval posets. We give several applications. First, we show the vanishing of higher magnitude homology groups for convex subsets of the Euclidean space. Second, magnitude homology groups carry the information about the diameter of a hole. Third, we construct a finite graph whose $3$rd magnitude homology group has torsion.

math.AT↗
arXiv2021

Feynman graphs and Hyperplane arrangements defined over $\mathbb{F}_1$

Motivated by some computations of Feynman integrals and certain conjectures on mixed Tate motives, Bejleri and Marcolli posed questions about the $\mathbb{F}_1$-structure (in the sense of torification) on the complement of a hyperplane arrangement, especially for an arrangement defined in the space of cycles of a graph. In this paper, we prove that an arrangement has an $\mathbb{F}_1$-structure if and only if it is Boolean. We also prove that the arrangement in the cycle space of a graph is Boolean if and only if the cycle space has a basis consisting of cycles such that any two of them do not share edges.

math.CO↗
arXiv2021

Divides with cusps and Kirby diagrams for line arrangements

The complement of a complexified real line arrangement is an affine surface. It is classically known that such a space has a handle decomposition up to $2$-handles. We will describe the handle decomposition induced from Lefschetz hyperplane section theorem for such a space. To describe the Kirby diagram, we introduce the notion of the divide with cusps which is a generalization of the divide introduced by A'Campo.

math.GT↗
arXiv2021

Period collapse in characteristic quasi-polynomials of hyperplane arrangements

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 quasi-polynomial 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 $\ge 1$, occurs for any value of the lcm period $\ge 2$, 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.

math.CO↗
arXiv2021

The Primitive Derivation and Discrete Integrals

The modules of logarithmic derivations for the (extended) Catalan and Shi arrangements associated with root systems are known to be free. However, except for a few cases, explicit bases for such modules are not known. In this paper, we construct explicit bases for type $A$ root systems. Our construction is based on Bandlow-Musiker's integral formula for a basis of the space of quasiinvariants. The integral formula can be considered as an expression for the inverse of the primitive derivation introduced by K. Saito. We prove that the discrete analogues of the integral formulas provide bases for Catalan and Shi arrangements.

math.CO↗
arXiv2021

What is $-Q$ for a poset $Q$?

In the context of combinatorial reciprocity, it is a natural question to ask what "$-Q$" is for a poset $Q$. In a previous work, the definition "$-Q:=Q\times\mathbb{R}$ with lexicographic order" was proposed based on the notion of Euler characteristic of semialgebraic sets. In fact, by using this definition, Stanley's reciprocity for order polynomials was generalized to an equality for the Euler characteristics of certain spaces of increasing maps between posets. The purpose of this paper is to refine this result, that is, to show that these spaces are homeomorphic if the topology of $Q$ is metrizable.

math.CO↗
arXiv2020

$G$-index, topological dynamics and marker property

Given an action of a finite group $G$, we can define its index. The $G$-index roughly measures a size of the given $G$-space. We explore connections between the $G$-index theory and topological dynamics. For a fixed-point free dynamical system, we study the $\mathbb{Z}_p$-index of the set of $p$-periodic points. We find that its growth is at most linear in $p$. As an application, we construct a free dynamical system which does not have the marker property. This solves a problem which has been open for several years.

math.DS↗
arXiv2020

Eulerian polynomials for subarrangements of Weyl arrangements

Let $\mathcal{A}$ be a Weyl arrangement. We introduce and study the notion of $\mathcal{A}$-Eulerian polynomial producing an Eulerian-like polynomial for any subarrangement of $\mathcal{A}$. This polynomial together with shift operator describe how the characteristic quasi-polynomial of a new class of arrangements containing ideal subarrangements of $\mathcal{A}$ can be expressed in terms of the Ehrhart quasi-polynomial of the fundamental alcove. The method can also be extended to define two types of deformed Weyl subarrangements containing the families of the extended Shi, Catalan, Linial arrangements and to compute their characteristic quasi-polynomials. We obtain several known results in the literature as specializations, including the formula of the characteristic polynomial of $\mathcal{A}$ via Ehrhart theory due to Athanasiadis (1996), Blass-Sagan (1998), Suter (1998) and Kamiya-Takemura-Terao (2010); and the formula relating the number of coweight lattice points in the fundamental parallelepiped with the Lam-Postnikov Eulerian polynomial due to the third author.

math.CO↗
arXiv2019

Double coverings of arrangement complements and $2$-torsion in Milnor fiber homology

We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.

math.AG↗
arXiv2017

A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements

A general vanishing result for the first cohomology group of affine smooth complex varieties with values in rank one local systems is established. This is applied to the determination of the monodromy action on the first cohomology group of the Milnor fiber of some line arrangements, including the monomial arrangement and the exceptional reflection arrangement of type $G_{31}$.

math.AG↗
arXiv2017

Euler characteristic reciprocity for chromatic, flow and order polynomials

The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define "negative sets" to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using "negative posets", we establish Stanley's reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials.

math.CO↗
arXiv2016

Characteristic polynomials of Linial arrangements for exceptional root systems

The (extended) Linial arrangement $\mathcal{L}_Φ^m$ is a certain finite truncation of the affine Weyl arrangement of a root system $Φ$ with a parameter $m$. Postnikov and Stanley conjectured that all roots of the characteristic polynomial of $\mathcal{L}_Φ^m$ have the same real part, and this has been proved for the root systems of classical types. In this paper we prove that the conjecture is true for exceptional root systems when the parameter $m$ is sufficiently large. The proof is based on representations of the characteristic quasi-polynomials in terms of Eulerian polynomials.

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