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We introduce an operation on skew-symmetric matrices over $\mathbb{Z}/\ell\mathbb{Z}$ called switching, and also define a class of skew-symmetric matrices over $\mathbb{Z}/\ell\mathbb{Z}$ referred to as modular Eulerian matrices. We then show that these are closely related to the graded module categories over skew polynomial algebras at $\ell$-th roots of unity. As an application, we study the point simplicial complexes of skew polynomial algebras at cube roots of unity., Comment: 16 pages

We study the effect of diffuse surface scattering on the edge modes in two-dimensional chiral superconductors with time-reversal symmetry-breaking Cooper pairs, each carrying angular momentum $m \hbar$ ($m = 1,2,3, \cdots$). To elucidate the diffuse scattering effect, we formulate the inverse lifetime $\Gamma = \hbar/\tau$ corresponding to the broadening of the surface density of states (SDOS) for the edge mode. This derivation uses random S-matrix theory, which allows us to describe the surface effect in a unified way from the specular to the diffuse limit within the quasiclassical theory framework of superconductivity. We find that $\Gamma$ in the chiral states with $m \geq 2$ is larger than that in the chiral $p$-wave state ($m = 1$) because of the multiple edge mode branches in former superconducting states (the number of which equals $m$). Diffuse scattering between the different branches causes a significant broadening of SDOS. In contrast, the edge mode in the chiral $p$-wave state is robust to diffuse scattering because only a single edge mode branch exists. We also discuss SDOS at the diffuse limit, where the description in terms of $\Gamma$ is not useful. The diffuse scattering effect on SDOS can be understood qualitatively in terms of destructive interference analogous to that caused by impurity scattering in unconventional superconductors., Comment: 10 pages, 7 figures

The aim of this paper is to provide a new characterization of isomorphism classes of generalized Alexander quandles in terms of the underlying groups and their automorphisms. This extends the previous result [4, Theorem 1.4]. Additionally, we compute the number of generalized Alexander quandles up to quandle isomorphism arising from groups up to order 127 and their group automorphisms., Comment: 12 pages

Characteristic quasi-polynomials are the enumerative functions counting the number of elements in the complement of hyperplane arrangements modulo positive integers.In this paper, we compute the characteristic quasi-polynomials of the restriction of the Shi arrangement of type B by one given hyperplane. As a corollary, we determine whether period collapse occurs in the characteristic quasi-polynomial of the deletion of the Shi arrangement of type B. This implies the complete solution for the conjecture posed by Higashtiani, Tran and Yoshinaga., Comment: 29 pages

The goal of this paper is to study the possible monoids appearing as the associated monoids of the initial algebra of a finitely generated homogeneous $\Bbbk$-subalgebra of a polynomial ring $\Bbbk[x_1,\ldots,x_n]$. Clearly, any affine monoid can be realized since the initial algebra of the affine monoid $\Bbbk$-algebra is itself. On the other hand, the initial algebra of a finitely generated homogeneous $\Bbbk$-algebra is not necessarily finitely generated. In this paper, we provide a new family of non-finitely generated monoids which can be realized as the initial algebras of finitely generated homogeneous $\Bbbk$-algebras. Moreover, we also provide an example of a non-finitely generated monoid which cannot be realized as the initial algebra of any finitely generated homogeneous $\Bbbk$-algebra., Comment: 14 pages, 6 figures

The symmetric edge polytope (SEP) of a (finite, undirected) graph is a centrally symmetric lattice polytope whose vertices are defined by the edges of the graph. SEPs have been studied extensively in the past twenty years. Recently, T\'othm\'er\'esz and, independently, D'Al\'i, Juhnke-Kubitzke, and Koch generalized the definition of an SEP to regular matroids, as these are the matroids that can be characterized by totally unimodular matrices. Generalized SEPs are known to have symmetric Ehrhart $h^*$-polynomials, and Ohsugi and Tsuchiya conjectured that (ordinary) SEPs have nonnegative $\gamma$-vectors. In this article, we use combinatorial and Gr\"obner basis techniques to extend additional known properties of SEPs to generalized SEPs. Along the way, we show that generalized SEPs are not necessarily $\gamma$-nonnegative by providing explicit examples. We prove these polytopes to be "nearly" $\gamma$-nonnegative in the sense that, by deleting exactly two elements from the matroid, one obtains SEPs for graphs that are $\gamma$-nonnegative. This provides further evidence that Ohsugi and Tsuchiya's conjecture holds in the ordinary case., Comment: 20 pages. Proof of formula for gamma-vector added

Let $\mathbb{K}[G]$ be the edge ring of a finite simple graph $G$. Investigating properties of the $h$-vector of $\mathbb{K}[G]$ is of great interest in combinatorial commutative algebra. However, there are few families of graphs for which the $h$-vector has been explicitly determined. In this paper, we compute the $h$-vectors of a certain family of graphs that satisfy the odd-cycle condition, generalizing a result of the second and third named authors. As a corollary, we obtain a characterization of the graphs in this family whose edge rings are almost Gorenstein., Comment: 16 pages, 6 figures, comments welcome

Let $Q$ be an affine monoid, $\Bbbk[Q]$ the associated monoid $\Bbbk$-algebra, and $\Bbbk[\overline{Q}]$ its normalization, where we let $\Bbbk$ be a field. In this paper, in the case where $\Bbbk[Q]$ is homogeneous (i.e., standard graded), a difference of the Hilbert series of $\Bbbk[Q]$ and $\Bbbk[\overline{Q}]$ is discussed. More precisely, we prove that if $\Bbbk[Q]$ satisfies Serre's condition $(S_2)$, then the degree of the $h$-polynomial of $\Bbbk[Q]$ is always greater than or equal to that of $\Bbbk[\overline{Q}]$. Moreover, we also show counterexamples of this statement if we drop the assumption $(S_2)$., Comment: 11 pages, 2 figures

A flag variety is a smooth projective homogeneous variety. In this paper, we study Newton-Okounkov polytopes of the flag variety $Fl(\mathbb{C}^4)$ arising from its cluster structure. More precisely, we present defining inequalities of such Newton-Okounkov polytopes of $Fl(\mathbb{C}^4)$. Moreover, we classify these polytopes, establishing their equivalence under unimodular transformations.

This article provides a comprehensive exposition about inequalities that the coefficients of Ehrhart polynomials and $h^*$-polynomials satisfy under various assumptions. We pay particular attention to the properties of Ehrhart positivity as well as unimodality, log-concavity and real-rootedness for $h^*$-polynomials. We survey inequalities that arise when the polytope has different normality properties. We include statements previously unknown in the Ehrhart theory setting, as well as some original contributions in this topic. We address numerous variations of the conjecture asserting that IDP polytopes have a unimodal $h^*$-polynomial, and construct concrete examples that show that these variations of the conjecture are false. Explicit emphasis is put on polytopes arising within algebraic combinatorics. Furthermore, we describe and construct polytopes having pathological properties on their Ehrhart coefficients and roots, and we indicate for the first time a connection between the notions of Ehrhart positivity and $h^*$-real-rootedness. We investigate the log-concavity of the sequence of evaluations of an Ehrhart polynomial at the non-negative integers. We conjecture that IDP polytopes have a log-concave Ehrhart series. Many additional problems and challenges are proposed., Comment: 44 pages. To appear in EMS Surveys in Mathematical Sciences

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 paper we study the following problem: Given a rational $d$-polytope $P \subset \mathbb R^d$, is there a nice way to know Ehrhart quasi-polynomials of translated polytopes $P+ \mathbf v$ for all $\mathbf v \in \mathbb Q^d$? We provide a way to compute such Ehrhart quasi-polynomials using a certain toric arrangement and lattice point counting functions of translated cones of $P$. This method allows us to visualize how constituent polynomials of $\mathrm{ehr}_{P+\mathbf v}$ change in the torus $\mathbb R^d/\mathbb Z^d$. We also prove that information of $\mathrm{ehr}_{P+\mathbf v}$ for all $\mathbf v \in \mathbb Q^d$ determines the rational $d$-polytope $P \subset \mathbb R^d$ up to translations by integer vectors, and characterize all rational $d$-polytopes $P \subset \mathbb R^d$ such that $\mathrm{ehr}_{P+\mathbf v}$ is symmetric for all $\mathbf v \in \mathbb Q^d$., Comment: 26 pages

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field $\mathbb{F}_p$. We apply symmetric (row and column) elementary transformations to the adjacency matrix over $\mathbb{F}_p$ in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over $\mathbb{F}_p$ for any odd prime $p$. Our key tool is quadratic residues of $\mathbb{F}_p$., Comment: 17 pages, 2 figures, 3 tables. Comments are welcome

Edge ideals of finite simple graphs $G$ on $n$ vertices are the ideals $I(G)$ of the polynomial ring $S$ in $n$ variables generated by the quadratic monomials associated with the edges of $G$. In this paper, we consider the possible pairs of dimensions and depths of $S/I(G)$ for connected graphs with a fixed number of vertices. We discuss such pairs in the case where dimension is relatively large. As a corollary, we completely determine the pairs for connected graphs with small number of vertices. We also study the possible pairs for connected chordal graphs., Comment: 11 pages, comments are welcome

The $g$-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined by tilting theory. We give a classification of complete $g$-fans of rank 2. More explicitly, our first main result asserts that every complete sign-coherent fan of rank 2 is a $g$-fan of some finite dimensional algebra. Our proof is based on three fundamental results, Gluing Theorem, Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level of finite dimensional algebras. For each of 16 convex sign-coherent fans $\Sigma$ of rank 2, our second main result gives a characterization of algebras $A$ of rank 2 satisfying $\Sigma(A)=\Sigma$. As a by-product of our method, we prove that for each positive integer $N$, there exists a finite dimensional algebra $A$ of rank 2 such that the Hasse quiver of the poset of 2-term silting complexes of $A$ has precisely $N$ connected components., Comment: 37 pages, v2: Fixed typos, updated references and added section 6

We study restricted chain-order polytopes associated to Young diagrams using combinatorial mutations. These polytopes are obtained by intersecting chain-order polytopes with certain hyperplanes. The family of chain-order polytopes associated to a poset interpolate between the order and chain polytopes of the poset. Each such polytope retains properties of the order and chain polytope; for example its Ehrhart polynomial. For a fixed Young diagram, we show that all restricted chain-order polytopes are related by a sequence of combinatorial mutations. Since the property of giving rise to the period collapse phenomenon is invariant under combinatorial mutations, we provide a large class of rational polytopes that give rise to period collapse., Comment: 18 pages, 3 figures

The goal of this paper is to characterization generalized Alexander quandles of finite groups in the language of the underlying groups. Firstly, we prove that if finite groups $G$ are simple, then the quandle isomorphic classes of generalized Alexander quandles of $G$ one-to-one correspond to the conjugacy classes of the automorphism groups of $G$. This correspondence can be also claimed for the case of symmetric groups. Secondly, we give a characterization of generalized Alexander quandles of finite groups $G$ under some assumptions in terms of $G$. As corollaries of this characterization, we obtain several characterizations in some particular groups, e.g., abelian groups and dihedral groups. Finally, we perform a characterization of generalized Alexander quandles arising from groups with their order up to $15$., Comment: 33 pages; 11 tables; comments are welcome

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant $H^*$-polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycles and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope., Comment: 15 pages, 1 figure

The Gelfand-Tsetlin and the Feigin-Fourier-Littelmann-Vinberg polytopes for the Grassmannians are defined, from the perspective of representation theory, to parametrize certain bases for highest weight irreducible modules. These polytopes are Newton-Okounkov bodies for the Grassmannian and, in particular, the GT-polytope is an example of a string polytope. The polytopes admit a combinatorial description as the Stanley's order and chain polytopes of a certain poset, as shown by Ardila, Bliem and Salaza. We prove that these polytopes occur among matching field polytopes. Moreover, we show that they are related by a sequence of combinatorial mutations that passes only through matching field polytopes. As a result, we obtain a family of matching fields that give rise to toric degenerations for the Grassmannians. Moreover, all polytopes in the family are Newton-Okounkov bodies for the Grassmannians., Comment: 23 pages, 3 figures

Two vertex colorings of a graph are Kempe equivalent if they can be transformed into each other by a sequence of switchings of two colors of vertices. It is PSPACE-complete to determine whether two given vertex $k$-colorings of a graph are Kempe equivalent for any fixed $k\geq 3$, and it is easy to see that every two vertex colorings of any bipartite graph are Kempe equivalent. In this paper, we consider Kempe equivalence of {\it almost} bipartite graphs which can be obtained from a bipartite graph by adding several edges to connect two vertices in the same partite set. We give a conjecture of Kempe equivalence of such graphs, and we prove several partial solutions and best possibility of the conjecture, but it is more lately proved by Cranston and Feghali that this conjecture is false in general., Comment: 23 pages

The $h$-vectors of homogeneous rings are one of the most important invariants that often reflect ring-theoretic properties. On the other hand, there are few examples of edge rings of graphs whose $h$-vectors are explicitly computed. In this paper, we compute the $h$-vector of a special family of graphs, by using the technique of initial ideals and the associated simplicial complex.

For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $\Delta(A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $\Delta(A)$. Using $g$-vectors of 2-term silting complexes, $\Delta(A)$ gives a nonsingular fan $\Sigma(A)$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-fan. We give several basic properties of $\Sigma(A)$ including sign-coherence, sign decomposition, idempotent reductions, Jasso reductions, pairwise positivity and a connection with Newton polytopes of $A$-modules. Moreover, $\Sigma(A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-polytope of $A$. We call $A$ $g$-convex if $P(A)$ is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of $A$. There are precisely 7 convex $g$-polyogons up to isomorphism. We give a classification of algebras whose $g$-polytopes are smooth Fano. We study $g$-fans and $g$-polytopes of two important classes of algebras. We show that the $g$-fan of a classical or generalized preprojective algebra is given by the Coxeter fan. It is $g$-convex if and only if it is of type $A$ or $B$, and in this case, its $g$-polytope is the dual polytope of the short root polytope. Moreover we classify Brauer graph algebras which are $g$-convex, and describe their $g$-polytopes as the root polytopes of type $A$ or $C$., Comment: 63 pages, v4: The old section on Jacobian algebras has been removed and will be included in a separate paper, as it requires a more general setting than other sections. v2: Fix typos. We remove the part of convex g-polygons, which will be discussed in our next paper

In this paper, we study the root distributions of Ehrhart polynomials of free sums of certain reflexive polytopes. We investigate cases where the roots of the Ehrhart polynomials of the free sums of $A_d^\vee$'s or $A_d$'s lie on the canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ on the complex plane $\mathbb{C}$, where $A_d$ denotes the root polytope of type A of dimension $d$ and $A_d^\vee$ denotes its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} \leq 1$ or $m+n \leq 7$, $A_2^\vee \oplus (A_1^\vee)^{\oplus n}$ and $A_3^\vee \oplus (A_1^\vee)^{\oplus n}$ for any $n$ satisfy this property. We also perform computational experiments for other types of free sums of $A_n^\vee$'s or $A_n$'s., Comment: 16 pages, 3 tables

The main objects of the present paper are (i) Hibi rings (toric rings arising from order polytopes of posets), (ii) stable set rings (toric rings arising from stable set polytopes of perfect graphs), and (iii) edge rings (toric rings arising from edge polytopes of graphs satisfying the odd cycle condition). The goal of the present paper is to analyze those three toric rings and to discuss their structures in the case where their class groups have small rank. We prove that the class groups of (i), (ii) and (iii) are torsionfree. More precisely, we give descriptions of their class groups. Moreover, we characterize the posets or graphs whose associated toric rings have rank $1$ or $2$. By using those characterizations, we discuss the differences of isomorphic classes of those toric rings with small class groups., Comment: 25 pages, 20 figures. Comments are welcome

The noncommutative projective scheme $\operatorname{\mathsf{Proj_{nc}}} S$ of a $(\pm 1)$-skew polynomial algebra $S$ in $n$ variables is considered to be a $(\pm 1)$-skew projective space of dimension $n-1$. In this paper, using combinatorial methods, we give a classification theorem for $(\pm 1)$-skew projective spaces. Specifically, among other equivalences, we prove that $(\pm 1)$-skew projective spaces $\operatorname{\mathsf{Proj_{nc}}} S$ and $\operatorname{\mathsf{Proj_{nc}}} S'$ are isomorphic if and only if certain graphs associated to $S$ and $S'$ are switching (or mutation) equivalent. We also discuss invariants of $(\pm 1)$-skew projective spaces from a combinatorial point of view., Comment: 14 pages, v2: minor modifications, v3: removed an example

It is known that the homogeneous coordinate ring of a Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is a family of plabic graphs giving a cluster structure of the same Grassmannian. Such plabic graphs are related by the operation called square move which can be considered as the mutation in cluster theory. By using a plabic graph, we also obtain the Newton--Okounkov polytope which gives a toric degeneration of the Grassmannian. The purposes of this article is to survey these phenomena and observe the behavior of Newton--Okounkov polytopes under the operation called the combinatorial mutation of polytopes. In particular, we reinterpret some operations defined for Newton--Okounkov polytopes using the combinatorial mutation., Comment: 38 pages, to appear in Advanced Studies in Pure Mathematics, v2: minor changes

In this paper, we study the Ehrhart polynomial of the dual of the root polytope of type C of dimension $d$, denoted by $C_d^*$. We prove that the roots of the Ehrhart polynomial of $C_d^*$ have the same real part $-1/2$, and we also prove that the Ehrhart polynomials of $C_d^*$ for $d=1,2,\cdots$ has the interlacing property., Comment: 10 pages

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., Comment: 20 pages, title changed, author list updated, Section 6 significantly improved: the lcm period proved to be the minimum period in the central case