arXiv subjects

Akihiro Higashitani

Publications and source records attributed to Akihiro Higashitani.

At least 73 records · Page 4Linked to original sources

arXiv2020

Homogeneous quandles arising from automorphisms of symmetric groups

Quandle is an algebraic system with one binary operation, but it is quite different from a group. Quandle has its origin in the knot theory and good relationships with the theory of symmetric spaces, so it is well-studied from points of view of both areas. In the present paper, we investigate a special kind of quandles, called generalized Alexander quandles $Q(G,ψ)$, which is defined by a group $G$ together with its group automorphism $ψ$. We develop the quandle invariants for generalized Alexander quandles. As a result, we prove that there is a one-to-one correspondence between generalized Alexander quandles arising from symmetric groups $\Sf_n$ and the conjugacy classes of $\Sf_n$ for $3 \leq n \leq 30$ with $n \neq 6,15$, and the case $n=6$ is also discussed.

math.GR↗

arXiv2020

Feedback game on $3$-chromatic Eulerian triangulations of surfaces

In this paper, we study the feedback game on $3$-chromatic Eulerian triangulations of surfaces. We prove that the winner of the game on every $3$-chromatic Eulerian triangulation of a surface all of whose vertices have degree $0$ modulo $4$ is always fixed. Moreover, we also study the case of $3$-chromatic Eulerian triangulations of surfaces which have at least two vertices whose degrees are $2$ modulo $4$, and in particular, we determine the winner of the game on a concrete class of such graphs, called an octahedral path.

cs.DM↗

arXiv2019

Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions

In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.

math.RT↗

arXiv2019

Toric ideals of Minkowski sums of unit simplices

In this paper, we discuss the toric ideals of Minkowski sums of unit simplices. More precisely, we prove that the toric ideal of Minkowski sum of unit simplices has a squarefree initial ideal and is generated by quadratic binomials. Moreover, we also prove that Minkowski sums of unit simplices have the integer decomposition property. Those results are a partial contribution to Oda conjecture and Bøgvad conjecture.

math.CO↗

arXiv2019

An algebraic reduction of Hedetniemi's conjecture

For a graph $G$, let $χ(G)$ denote the chromatic number. In graph theory, the following famous conjecture posed by Hedetniemi has been studied: For two graphs $G$ and $H$, $χ(G\times H)=\min\{χ(G),χ(H)\}$, where $G \times H$ is the tensor product of $G$ and $H$. In this paper, we give a reduction of Hedetniemi's conjecture to an inclusion relation problem on ideals of polynomial rings, and we demonstrate computational experiments for partial solutions of Hedetniemi's conjecture along such a strategy using Gröbner basis.

math.CO↗

arXiv2019

Permutations with small maximal $k$-consecutive sums

Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $(π_1,\ldots,π_n)$ of integers $1,\ldots,n$, we consider $k$-consecutive sums of $π$, i.e., $s_i:=\sum_{j=0}^{k-1}π_{i+j}$ for $i=1,\ldots,n$, where we let $π_{n+j}=π_j$. What we want to do in this paper is to know the exact value of $$\mathrm{msum}(n,k):=\min\left\{\max\{s_i : i=1,\ldots,n\} -\frac{k(n+1)}{2}: π\in S_n\right\},$$ where $S_n$ denotes the set of all permutations of $1,\ldots,n$. In this paper, we determine the exact values of $\mathrm{msum}(n,k)$ for some particular cases of $n$ and $k$. As a corollary of the results, we obtain $\mathrm{msum}(n,3)$, $\mathrm{msum}(n,4)$ and $\mathrm{msum}(n,6)$ for any $n$.

math.CO↗

arXiv2019

Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon

By generalising the notion of a unimodular sequence, we create an expression for the winding number of certain ordered sets of lattice points. Since the winding number of the vertices of a Fano polygon is necessarily one, we use this expression as a restriction to classify all Fano polygons without T-singularities and whose basket of residual singularities is of the form $\left\{ \frac{1}{r}(1,s_{1}), \frac{1}{r}(1,s_{2}), \ldots, \frac{1}{r}(1,s_{k}) \right\}$ for $k,r \in \mathbb{Z}_{>0}$, and $1 \leq s_{i} < r$ is coprime to $r$.

math.CO↗

arXiv2018

Universal conditions on $h^*$-vectors of lattice simplices

In this paper, we will prove that given a lattice simplex with its $h^*$-polynomial $\sum_{i \geq 0}h_i^*t^i$, if $h_{k+1}^*=\cdots=h_{2k}^*=0$ holds, then there exists a lattice simplex of degree $k$ whose $h^*$-polynomial coincides with $\sum_{i=0}^k h_i^*t^i$. Moreover, we will present the examples showing that the condition $h_{k+1}^*=h_{k+2}^*=\cdots=h_{2k-1}^*=0$ is necessary.

math.CO↗

arXiv2018

Levelness of toric rings arising from order and chain polytopes

Let $K[\mathcal{O}(P)]$ denote the toric ring of the order polytope $\mathcal{O}(P)$ of a finite partially ordered set $P$ and $K[\mathcal{C}(P)]$ that of the chain polytope $\mathcal{C}(P)$. It will be shown that $β_{p, p+j}(K[\mathcal{O}(P)]) = β_{p, p+j}(K[\mathcal{C}(P)])$ for all $j \geq 0$, where $p$ is the projective dimension of $K[\mathcal{O}(P)]$ (and that of $K[\mathcal{C}(P)]$). In particular, $K[\mathcal{O}(P)]$ is level if and only if $K[\mathcal{C}(P)]$ is level.

math.CO↗

arXiv2018

Arithmetic aspects of symmetric edge polytopes

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^\ast$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^\ast$-polynomial is $γ$-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (2011).

math.CO↗

arXiv2018

Smooth centrally symmetric polytopes in dimension 3 are IDP

In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric $3$-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.

math.CO↗

arXiv2017

Existence of regular unimodular triangulations of dilated empty simplices

Given integers $k$ and $m$ with $k \geq 2$ and $m \geq 2$, let $P$ be an empty simplex of dimension $(2k-1)$ whose $δ$-polynomial is of the form $1+(m-1)t^k$. In the present paper, the necessary and sufficient condition for the $k$-th dilation $kP$ of $P$ to have a regular unimodular triangulation will be presented.

math.CO↗

arXiv2017

Distribution of accumulation points of roots for type $(n-1,1)$ Coxeter groups

In this paper, we investigate the set of accumulation points of normalized roots of infinite Coxeter groups for certain class of their action. Concretely, we prove the conjecture proposed in [6, Section 3.2] in the case where the equipped Coxeter matrices are of type $(n-1,1)$, where $n$ is the rank. Moreover, we obtain that the set of such accumulation points coincides with the closure of the orbit of one point of normalized limit roots. In addition, in order to prove our main results, we also investigate some properties on fixed points of the action.

math.GR↗

arXiv2017

Lattice simplices of maximal dimension with a given degree

It was proved by Nill that for any lattice simplex of dimension $d$ with degree $s$ which is not a lattice pyramid, the inequality $d+1 \leq 4s-1$ holds. In this paper, we give a complete characterization of lattice simplices satisfying the equality, i.e., the lattice simplices of dimension $(4s-2)$ with degree $s$ which are not lattice pyramids. It turns out that such simplices arise from binary simplex codes. As an application of this characterization, we show that such simplices are counterexamples for the conjecture known as "Cayley conjecture". Moreover, by modifying Nill's inequaitly slightly, we also see the sharper bound $d+1 \leq f(2s)$, where $f(M)=\sum_{n=0}^{\lfloor \log_2 M \rfloor} \lfloor M/2^n \rfloor$ for $M \in \mathbb{Z}_{\geq 0}$. We also observe that any lattice simplex attaining this sharper bound always comes from a binary code.

math.CO↗

arXiv2017

Universal inequalities in Ehrhart Theory

In this paper, we show the existence of universal inequalities for the $h^*$-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the $h^*$-polynomial which are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients $h^*_1$ and $h^*_2$ of the $h^*$-vector $(h^*_0,h^*_1,\ldots,h^*_d)$ of a lattice polytope of any degree satisfy Scott's inequality if $h^*_3=0$.

math.CO↗

arXiv2016

Interlacing Ehrhart Polynomials of Reflexive Polytopes

It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann ζ function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method is to prove a stronger property called interlacing.

math.CO↗

arXiv2016

Non-level semi-standard graded Cohen-Macaulay domain with $h$-vector $(h_0,h_1,h_2)$

Let $k$ be an algebraically closed field of characteristic 0, and $A$ a Cohen-Macaulay graded domain with $A_0=k$. If $A$ is semi-standard graded (i.e., $A$ is finitely generated as a $k[A_1]$-module), it has the $h$-vector $(h_0, h_1, ..., h_s)$, which encodes the Hilbert function of $A$. From now on, assume that $s=2$. It is known that if $A$ is standard graded (i.e., $A=k[A_1]$), then $A$ is level. We will show that, in the semi-standard case, if $A$ is not level, then $h_1+1$ divides $h_2$. Conversely, for any positive integers $h$ and $n$, there is a non-level $A$ with the $h$-vector $(1, h, (h+1)n)$. Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings).

math.AC↗

arXiv2016

Lattice multi-polygons

We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a unimodular sequence in $\mathbb{Z}^2$. This formula implies the generalized twelve-point theorem in [12]. We then introduce the notion of lattice multi-polygons which is a generalization of lattice polygons, state the generalized Pick's formula and discuss the classification of Ehrhart polynomials of lattice multi-polygons and also of several natural subfamilies of lattice multi-polygons.

math.CO↗