arXiv subjects
Publications and source records attributed to Takayuki Hibi.
Motivated by pseudo-Gorenstein rings in commutative algebra, introduced by Herzog et al., we define pseudo-Gorenstein$^{*}$ graphs and classify them in several natural graph families using independence polynomials.
A finite poset (partially ordered set) $P$ with ${\hat 0}$ is called of distributive type if every interval $[{\hat 0}, a]$, $a \in P$, of $P$ is a distributive lattice. From a viewpoint of ASL's (algebras with straightening laws), the join-meet toric ring on a finite distributive lattice is generalized to an ASL on a finite poset of distributive type. Our target is the questions when a finite poset of distributive lattice is Cohen--Macaulay and when the ASL on it is Gorenstein. We focus on a natural class of finite posets of distributive type and study various aspects of the above questions.
It is well known that the Plücker ideal defining the Grassmannian is generated by quadratic Plücker relations. These relations form a reverse lexicographic Gröbner basis and endow the Plücker algebra with the structure of an algebra with straightening laws (ASL). In this paper, we study quadratically generated projections of the Grassmannian of lines $\mathrm{Gr}(2,n)$. We then combinatorially characterize the Gorenstein ASL subalgebras of the Plücker algebra of $\mathrm{Gr}(2,n)$.
In this paper, we provide a complete description of the minimal primes of ideals generated by adjacent $2$-minors, in terms of the so-called admissible sets and associated lattice ideals. We prove that for these ideals, the properties of being unmixed, Cohen-Macaulay, level, Gorenstein, and complete intersection are equivalent. Moreover, we give a combinatorial characterization of all convex collections of cells satisfying any of these equivalent properties. Finally, we study the radicality of these ideals and derive necessary combinatorial conditions based on minimal non-radical configurations.
A lattice polytope $\mathcal{P} \subset \mathbb{R}^n$ of dimension $n$ is called level* if (i) $\mathcal{P}$ is normal, (ii) $(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n \neq \emptyset$ and (iii) for each $N = 2,3, \ldots$ and for each $\textbf{a} \in N(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n$, there is $\textbf{a}_0 \in (\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n$ together with $\textbf{a}' \in (N-1)\mathcal{P} \cap \mathbb{Z}^n$ for which $\textbf{a} = \textbf{a}_0 + \textbf{a}'$, where $N\mathcal{P} = \{N\textbf{a} : \textbf{a} \in \mathcal{P}\}$. A normal polytope $\mathcal{P} \subset \mathbb{R}^n$ of dimension $n$ is called pseudo-Gorenstein* [4] if $ |(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n| = 1. $ A pseudo-Gorenstein* polytope $\mathcal{P}$ is level* if and only if $\mathcal{P}$ is reflexive up to translation. In the present paper, level* polytopes together with pseudo-Gorenstein* polytopes arising from discrete polymatroids of bounded powers of edge ideals are studied.
Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G) \subset S$ the edge ideal of a finite graph $G$ on $n$ vertices. Given a vector $\mathfrak{c}\in\mathbb{N}^n$ and an integer $q\geq 1$, we denote by $(I(G)^q)_{\mathfrak{c}}$ the ideal of $S$ generated by those monomials belonging to $I(G)^q$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Let $δ_{\mathfrak{c}}(I(G))$ denote the largest integer $q$ for which $(I(G)^q)_{\mathfrak{c}}\neq (0)$. Since $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal, it follows that its minimal set of monomial generators is the set of bases of a discrete polymatroid $\mathcal{D}(G,\mathfrak{c})$. In the present paper, a classification of Gorenstein polytopes of the form ${\rm conv}(\mathcal{D}(G,\mathfrak{c}))$ is studied.
For four elements of a Noetherian ring we construct complexes of free modules of length three (resp. five) by an explicit description of the homomorphisms of the free modules. We provide exactness criteria for them. As an application we use these results in order to describe explicit the minimal free resolution of the Hartshorne--Rao module of a monomial curve lying on a smooth quadric. Also it provides an example of linearly generated module with syzygies of arbitrary high degree.
In this paper, we introduce the concept of complementary edge ideals of graphs and study their algebraic properties and invariants.
It has been conjectured that the toric ideal of the base ring of a discrete polymatroid is generated by symmetric exchange binomials. In the present paper, we give several classes of discrete polymatroids which yield toric ideals generated by symmetric exchange binomials. Especially, we are interested in the discrete polymatroids arising from bounded powers of edge ideals of finite graphs.
Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $\mathfrak{c}\in\mathbb{Z}_{>0}^n$, the ideal $I_{\mathfrak{c}}$ is the ideal generated by those monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Let $δ_{\mathfrak{c}}(I)$ be the largest integer $q$ for which $(I^q)_{\mathfrak{c}}\neq 0$. Let $I(G) \subset S$ denote the edge ideal of a finite graph $G$ on the vertex set $V(G) = \{x_1, \ldots, x_s\}$. In our previous work, it is shown that $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. Let $\mathcal{W}(\mathfrak{c},G)$ denote the minimal system of monomial generators of $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$. It follows that $\mathcal{W}(\mathfrak{c},G)$ satisfies the symmetric exchange property. In the present paper, the question when $\mathcal{W}(\mathfrak{c},G)$ enjoys the strong exchange property, or equivalently, when $\mathcal{W}(\mathfrak{c},G)$ is of Veronese type is studied.
Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $\mathfrak{c}\in\mathbb{N}^n$, the ideal $I_{\mathfrak{c}}$ is the ideal generated by those monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Let $δ_{\mathfrak{c}}(I)$ be the largest integer $q$ for which $(I^q)_{\mathfrak{c}}\neq 0$. For a finite graph $G$, its edge ideal is denoted by $I(G)$. Let $\mathcal{B}(\mathfrak{c},G)$ be the toric ring which is generated by the monomials belonging to the minimal system of monomial generators of $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$. In a previous work, the authors proved that $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. It follows that $\mathcal{B}(\mathfrak{c},G)$ is a normal Cohen--Macaulay domain. In this paper, we study the Gorenstein property of $\mathcal{B}(\mathfrak{c},G)$.
Given integers $2 \leq p \leq c \leq q$, we construct a finite simple graph $G$ with $ν_1(G) = p$ and $ν(G) = q$ for which the squarefree power $I(G)^{[k]}$ of the edge ideal $I(G)$ of $G$ has linear quotients for each $c \leq k \leq q$ and is not linearly related for each $1 \leq k < c$, where $ν_1(G)$ is the induced matching number of $G$ and $ν(G)$ is the matching number of $G$.
In 2017, Ehrenborg, Govindaiah, Park, and Readdy defined the van der Waerden complex ${\tt vdW}(n,k)$ to be the simplicial complex whose facets correspond to all the arithmetic sequences on the set $\{1,\ldots,n\}$ of a fixed length $k$. To complement a classification of the Cohen--Macaulay van der Waerden complexes obtained by Hooper and Van Tuyl in 2019, a classification of van der Waerden complexes with linear resolution is presented. Furthermore, we show that the Stanley--Reisner ring of a Cohen--Macaulay van der Waerden complex is level.
The pure $O$-sequences of the form $(1,a,a,\ldots)$ are classified.
Let $G$ be a finite graph and $κ(G)$ the vertex connectivity of $G$. A chordal graph $G$ is called chordal$^*$ if no vertex of $G$ is adjacent to all other vertices of $G$. Using the syzygy theory in commutative algebra, it is proved that every chordal$^*$ graph $G$ on $n$ vertices satisfies $κ(G) \leq (n - 1) - \lceil2\sqrt{n}-2\,\rceil$. Furthermore, given an integer $0 \leq κ\leq (n - 1) - \lceil2\sqrt{n}-2\,\rceil$, a chordal$^*$ graph $G$ on $n$ vertices satisfying $κ(G) = κ$ is constructed.
Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and let $I \subset S$ be a monomial ideal. For a vector $\mathfrak{c}\in\mathbb{N}^n$, we set $I_{\mathfrak{c}}$ to be the ideal generated by monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Also, let $δ_{\mathfrak{c}}(I)$ be the largest integer $k$ such that $(I^k)_{\mathfrak{c}}\neq 0$. It is shown that for every graph $G$ with edge ideal $I(G)$, the ideal $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. Moreover, we show that for each integer $s=1, \ldots δ_{\mathfrak{c}}(I(G))$, the Castelnuovo--Mumford regularity of $(I(G)^s)_{\mathfrak{c}}$ is bounded above by $δ_{\mathfrak{c}}(I(G))+s$.
Let $I(G)$ be the edge ideal of a gapfree graph $G$. An open conjecture of Nevo and Peeva states that $I(G)^q$ has linear resolution for $q\gg 0$. We present a promising approach to this challenging conjecture by investigating the stronger property of linear quotients. Specifically, we make the conjecture that if $I(G)^q$ has linear quotients for some integer $q\geq 1$, then $I(G)^{s}$ has linear quotients for all $s\geq q$. We give a partial solution to this conjecture, and identify conditions under which only finitely many powers need to be checked. It is known that if $G$ does not contain a cricket, a diamond, or a $C_4$, then $I(G)^q$ has linear resolution for $q \geq 2$. We construct a family of gapfree graphs $G$ containing cricket, diamond, $C_4$ together with $C_5$ as induced subgraphs of $G$ for which $I(G)^q$ has linear quotients for $q \ge 2$.
Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$. It follows from the lower bound theorem of Ehrhart polynomials that, when $c > 0$, \[ {\rm vol}(\mathcal{P}) \geq (d \cdot c(\mathcal{P}) + (d-1) \cdot b(\mathcal{P}) - d^2 + 2)/d!, \] where ${\rm vol}(\mathcal{P})$ is the (Lebesgue) volume of $\mathcal{P}$. Pick's formula guarantees that, when $d = 2$, the above inequality is an equality. In the present paper several classes of lattice polytopes for which the equality here holds will be presented.