Masahiko Yoshinaga
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View article: The quasi-polynomiality of mod q permutation representations for a linear finite group action on a lattice
The quasi-polynomiality of mod q permutation representations for a linear finite group action on a lattice Open
View article: Reconstrucion of oriented matroids from Varchenko-Gelfand algebras
Reconstrucion of oriented matroids from Varchenko-Gelfand algebras Open
The algebra of $R$-valued functions on the set of chambers of a real hyperplane arrangement is called the Varchenko-Gelfand (VG) algebra. This algebra carries a natural filtration by the degree with respect to Heaviside functions, giving r…
View article: The small‐scale limit of magnitude and the one‐point property
The small‐scale limit of magnitude and the one‐point property Open
The magnitude of a metric space is a real‐valued function whose parameter controls the scale of the metric. A metric space is said to have the one‐point property if its magnitude converges to 1 as the space is scaled down to a point. Not e…
View article: Ehrhart quasi-polynomials and parallel translations
Ehrhart quasi-polynomials and parallel translations Open
Given a rational polytope \(P \subset \mathbb R^d\), the numerical function counting lattice points in the integral dilations of \(P\) is known to become a quasi-polynomial, called the Ehrhart quasi-polynomial \(\operatorname{ehr}_P\) of \…
View article: Is magnitude 'generically continuous' for finite metric spaces?
Is magnitude 'generically continuous' for finite metric spaces? Open
Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the metric varies. Motivated by applications in topological data…
View article: $q$-deformation of chromatic polynomials and graphical arrangements
$q$-deformation of chromatic polynomials and graphical arrangements Open
We first observe a mysterious similarity between the braid arrangement and the arrangement of all hyperplanes in a vector space over the finite field $\mathbb{F}_q$. These two arrangements are defined by the determinants of the Vandermonde…
View article: Free Reflection Multiarrangements and Quasi-Invariants
Free Reflection Multiarrangements and Quasi-Invariants Open
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 re…
View article: Magnitude homology and homotopy type of metric fibrations
Magnitude homology and homotopy type of metric fibrations Open
In this article, we show that each two metric fibrations with a common base and a common fiber have isomorphic magnitude homology, and even more, the same magnitude homotopy type. That can be considered as a generalization of a fact proved…
View article: The quasi-polynomiality of mod q permutation representation for a linear finite group action on a lattice
The quasi-polynomiality of mod q permutation representation for a linear finite group action on a lattice Open
For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod q-anal…
View article: Topology of hyperplane arrangements via real structure
Topology of hyperplane arrangements via real structure Open
This note is a survey on the topology of hyperplane arrangements. We mainly focus on the relationship between topology and the real structure, such as adjacent relations of chambers and stratifications related to real structures.
View article: A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups
A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups Open
Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration…
View article: A construction of homotopically non-trivial embedded spheres for hyperplane arrangements
A construction of homotopically non-trivial embedded spheres for hyperplane arrangements Open
We introduce the notion of locally consistent system of half-spaces for a real hyperplane arrangement. We embed a sphere in the complexified complement by shifting the real unit sphere into the imaginary direction indicated by the half-spa…
View article: $q$-deformation of Aomoto complex
$q$-deformation of Aomoto complex Open
A degree one element of the Orlik-Solomon algebra of a hyperplane arrangement defines a cochain complex known as the Aomoto complex. The Aomoto complex can be considerd as the ``linear approximation'' of the twisted cochain complex with co…
View article: The small-scale limit of magnitude and the one-point property
The small-scale limit of magnitude and the one-point property Open
The magnitude of a metric space is a real-valued function whose parameter controls the scale of the metric. A metric space is said to have the one-point property if its magnitude converges to 1 as the space is scaled down to a point. Not e…
View article: Integral expressions for derivations of multiarrangements
Integral expressions for derivations of multiarrangements Open
The construction of an explicit basis for a free multiarrangement is not easy in general. Inspired by the integral expressions for quasi-invariants of quantum Calogero-Moser systems, we present integral expressions for specific bases of ce…
View article: Ehrhart quasi-polynomials and parallel translations
Ehrhart quasi-polynomials and parallel translations Open
Given a rational polytope $P \subset \mathbb R^d$, the numerical function counting lattice points in the integral dilations of $P$ is known to become a quasi-polynomial, called the Ehrhart quasi-polynomial $\mathrm{ehr}_P$ of $P$. In this …
View article: Causal order complex and magnitude homotopy type of metric spaces
Causal order complex and magnitude homotopy type of metric spaces Open
In this paper, we construct a pointed CW complex called the magnitude homotopy type for a given metric space $X$ and a real parameter $\ell \geq 0$. This space is roughly consisting of all paths of length $\ell$ and has the reduced homolog…
View article: Betti numbers and torsions in homology groups of double coverings
Betti numbers and torsions in homology groups of double coverings Open
Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases assoc…
View article: Ehrhart Quasi-Polynomials of almost integral polytopes
Ehrhart Quasi-Polynomials of almost integral polytopes Open
In this thesis we characterize centrally symmetric lattice polytopes and lattice zonotopes through properties of the Ehrhart quasi-polynomials of almost integral polytopes. To this end, we introduce the notion of GCD-property and symmetry …
View article: What is $$-Q$$ for a poset Q?
What is $$-Q$$ for a poset Q? Open
View article: Free reflection multiarrangements and quasi-invariants
Free reflection multiarrangements and quasi-invariants Open
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 re…
View article: Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes
Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes Open
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 diagon…
View article: Period collapse in characteristic quasi-polynomials of hyperplane arrangements
Period collapse in characteristic quasi-polynomials of hyperplane arrangements Open
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 mo…
View article: The Primitive Derivation and Discrete Integrals
The Primitive Derivation and Discrete Integrals Open
he 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 cons…
View article: Divides with cusps and Kirby diagrams for line arrangements
Divides with cusps and Kirby diagrams for line arrangements Open
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 hyperplan…
View article: $G$-index, topological dynamics and marker property
$G$-index, topological dynamics and marker property Open
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 sy…
View article: Finite record sets of chip-firing games
Finite record sets of chip-firing games Open
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 …
View article: Eulerian polynomials and polynomial congruences
Eulerian polynomials and polynomial congruences Open
We prove that the Eulerian polynomial satisfies certain polynomial congruences. Furthermore, these congruences characterize the Eulerian polynomial.
View article: Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials
Combinatorics of certain abelian Lie group arrangements and chromatic quasi-polynomials Open
View article: A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups
A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups Open
Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration…