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Let $R$ be the Ehrhart ring of the stable set polytope of a cycle graph which is not Gorenstein. We describe the non-Gorenstein locus of $\mathrm{Spec} R$. Further, we show that $R$ is almost Gorenstein. Moreover, we show that the conjecture of Hibi and Tsuchiya is true., Comment: arXiv admin note: text overlap with arXiv:2201.02957, arXiv:2005.03259

In this paper, we give a criterion of the nearly Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ with connected components $G^{(1)}, \ldots, G^{(\ell)}$ is nearly Gorenstein if and only if (1) for each $i$, the Ehrhart ring of the stable set polytope of $G^{(i)}$ is Gorenstein and (2) $|\omega(G^{(i)})-\omega(G^{(j)})|\leq 1$ for any $i$ and $j$, where $\omega(G^{(i)})$ is the clique number of $G^{(i)}$. We also show that the Segre product of Cohen-Macaulay graded rings with linear non-zerodivisor which are Gorenstein on the punctured spectrum is also Gorenstein on the punctured spectrum if all but one rings are standard graded., Comment: Correction of typos. arXiv admin note: text overlap with arXiv:2005.03259

Let $P$ be a finite poset, $K$ a field, and $O(P)$ (resp. $C(P)$) the order (resp. chain) polytope of $P$. We study the non-Gorenstein locus of $E_K[O(P)]$ (resp. $E_K[C(P)]$), the Ehrhart ring of $O(P)$ (resp. $C(P)$) over $K$, which are each normal toric rings associated $P$. In particular, we show that the dimension of non-Gorenstein loci of $E_K[O(P)]$ and $E_K[C(P)]$ are the same. Further, we show that $E_K[C(P)]$ is nearly Gorenstein if and only if $P$ is the disjoint union of pure posets $P_1, \ldots, P_s$ with $|\mathrm{rank} P_i-\mathrm{rank} P_j|\leq 1$ for any $i$ and $j$.

In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $n\geq 3$ and there is no odd cycle without chord and length at least 5., Comment: Added an example, which shows that the symbolic powers of the canonical ideal of the Ehrhart ring of HSTAB(G) of a graph G is not equal to the ordinary power in general

Let P be a poset, O(P) the order polytope of P and C(P) the chain polytope of P. In this paper, we study the canonical ideal of the Ehrhart ring K[C(P)] of C(P) over a field K and characterize the level (resp. anticanonical level) property of K[C(P)] by a combinatorial structure of P. In particular, we show that if K[C(P)] is level (resp. anticanonical level), then so is K[O(P)]. We exhibit examples which show the converse does not hold. Moreover, we show that the symbolic powers of the canonical ideal of K[C(P)] are identical with ordinary ones and degrees of the generators of the canonical and anticanonical ideals are consecutive integers.

Let ${\cal R}_{\mathbb{K}}[H]$ be the Hibi ring over a field $\mathbb{K}$ on a finite distributive lattice $H$, $P$ the set of join-irreducible elements of $H$ and $\omega$ the canonical ideal of ${\cal R}_{\mathbb{K}}[H]$. We show the powers $\omega^{(n)}$ of $\omega$ in the group of divisors $\mathrm{Div}({\cal R}_{\mathbb{K}}[H])$ is identical with the ordinal powers of $\omega$, describe the $\mathbb{K}$-vector space basis of $\omega^{(n)}$ for $n\in\mathbb{Z}$. Further, we show that the fiber cones $\bigoplus_{n\geq 0}\omega^n/\mathfrak{m}\omega^n$ and $\bigoplus_{n\geq0}(\omega^{(-1)})^n/\mathfrak{m}(\omega^{(-1)})^n$ of $\omega$ and $\omega^{(-1)}$ are sum of the Ehrhart rings, defined by sequences of elements of $P$ with a certain condition, which are polytopal complex version of Stanley-Reisner rings. Moreover, we show that the analytic spread of $\omega$ and $\omega^{(-1)}$ are maximum of the dimensions of these Ehrhart rings. Using these facts, we show that the question of Page about Frobenius complexity is affirmative: $\lim_{p\to\infty}\mathrm{cx}_F({\cal R}_{\mathbb{K}}[H])= \dim(\bigoplus_{n\geq0}\omega^{(-n)}/\mathfrak{m}\omega^{(-n)})-1$, where $p$ is the characteristic of the field $\mathbb{K}$., Comment: Minor changes, including change of the title, correction of errors and adding some examples

Let $N_1, \ldots, N_d$ be positive integers with $N_1\leq\cdots\leq N_d$. Set $N=N_1\cdots N_{d-1}$. We show in this paper that an integer $r$ is a typical nonnegative rank of nonnegative tensors of format $N_1\times\cdots\times N_d$ if and only if $r\leq N$ and $r$ is greater than or equals to the generic rank of tensors over $\mathbb{C}$ of format $N_1\times\cdots\times N_d$. We also show that the maximal nonnegative rank of nonnegative tensors of format $N_1\times\cdots\times N_d$ is $N$.

In this paper, we state criteria of a Hibi ring to be level, non-Gorenstein and almost Gorenstein and to be non-level and almost Gorenstein in terms of the structure of the partially ordered set defining the Hibi ring. We also state a criterion of a ladder determinantal ring defined by 2-minors to be non-Gorenstein and almost Gorenstein in terms of the shape of the ladder.

Let $m$, $n$ and $p$ be integers with $3\leq m\leq n$ and $(m-1)(n-1)+1\leq p\leq (m-1)m$. We showed in previous papers that if $p\geq (m-1)(n-1)+2$, then typical ranks of $p\times n\times m$-tensors over the real number field are $p$ and $p+1$ if and only if there exists a nonsingular bilinear map $\mathbb{R}^m\times \mathbb{R}^n\to\mathbb{R}^{mn-p}$. We also showed that the "if" part also valid in the case where $p=(m-1)(n-1)+1$. In this paper, we consider the case where $p=(m-1)(n-1)+1$ and show that the typical ranks of $p\times n\times m$-tensors over the real number field are $p$ and $p+1$ in several cases including the case where there is no nonsingular bilinear map $\mathbb{R}^m\times \mathbb{R}^n\to\mathbb{R}^{mn-p}$. In particular, we show that the "only if" part of the above mentioned fact does not valid for the case $p=(m-1)(n-1)+1$.

In this paper, we study the minimal generating system of the canonical module of a Hibi ring. Using the results, we state a characterization of a Hibi ring to be level. We also give a characterization of a Hibi ring to be of type 2. Further, we show that the degrees of the elements of the minimal generating system of the canonical module of a Hibi ring form a set of consecutive integers., Comment: Changed the subtitle. Shortened the last two sections

Let $m,n\geq 3$, $(m-1)(n-1)+2\leq p\leq mn$, and $u=mn-p$. The set $\mathbb{R}^{u\times n\times m}$ of all real tensors with size $u\times n\times m$ is one to one corresponding to the set of bilinear maps $\mathbb{R}^m\times \mathbb{R}^n\to \mathbb{R}^u$. We show that $\mathbb{R}^{m\times n\times p}$ has plural typical ranks $p$ and $p+1$ if and only if there exists a nonsingular bilinear map $\mathbb{R}^m\times\mathbb{R}^n\to\mathbb{R}^{u}$. We show that there is a dense open subset $\mathscr{O}$ of $\mathbb{R}^{u\times n\times m}$ such that for any $Y\in\mathscr{O}$, the ideal of maximal minors of a matrix defined by $Y$ in a certain way is a prime ideal and the real radical of that is the irrelevant maximal ideal if that is not a real prime ideal. Further, we show that there is a dense open subset $\mathscr{T}$ of $\mathbb{R}^{ n\times p \times m}$ and continuous surjective open maps $\nu\colon\mathscr{O}\to\mathbb{R}^{u\times p}$ and $\sigma\colon\mathscr{T}\to\mathbb{R}^{u\times p}$, where $\mathbb{R}^{u \times p}$ is the set of $u\times p$ matrices with entries in $\mathbb{R}$, such that if $\nu(Y)=\sigma(T)$, then $\mathrm{rank} T=p$ if and only if the ideal of maximal minors of the matrix defined by $Y$ is a real prime ideal.

We study an upper bound of ranks of $n$-tensors with size $2\times\cdots\times2$ over the complex and real number field. We characterize a $2\times 2\times 2$ tensor with rank 3 by using the Cayley's hyperdeterminant and some function. Then we see another proof of Brylinski's result that the maximal rank of $2\times2\times2\times2$ complex tensors is 4. We state supporting evidence of the claim that 5 is a typical rank of $2\times2\times2\times2$ real tensors. Recall that Kong and Jiang show that the maximal rank of $2\times2\times2\times2$ real tensors is less than or equal to 5. The maximal rank of $2\times2\times2\times2$ complex (resp. real) tensors gives an upper bound of the maximal rank of $2\times\cdots\times 2$ complex (resp. real) tensors., Comment: 13 pages, no fugiure

Let $3\leq m\leq n$. We study typical ranks of $m\times n\times ((m-1)n-1)$ tensors over the real number field. The number $(m-1)n-1$ is a minimal typical rank of $m\times n\times ((m-1)n-1)$ tensors over the real number field. We show that a typical rank of $m\times n\times ((m-1)n-1)$ tensors over the real number field is less than or equal to $(m-1)n$ and in particular, $m\times n\times ((m-1)n-1)$ tensors over the real number field has two typical ranks $(m-1)n-1, (m-1)n$ if $m\leq \rho(n)$, where $\rho$ is the Hurwitz-Radon function defined as $\rho(n)=2^b+8c$ for nonnegative integers $a,b,c$ such that $n=(2a+1)2^{b+4c}$ and $0\leq b<4$.

In this paper, we study the ring of invariants under the action of SL(m,K)\times SL(n,K) and SL(m,K)\times SL(n,K)\times SL(2,K) on the 3-dimensional array of indeterminates of form m\times n\times 2, where K is an infinite field. And we show that if m=n\geq 2, then the ring of SL(n,K)\times SL(n,K)-invariants is generated by n+1 algebraically independent elements over K and the action of SL(2,K) on that ring is identical with the one defined in the classical invariant theory of binary forms. We also reveal the ring of SL(m,K)\times SL(n,K)-invariants and SL(m,K)\times SL(n,K)\times SL(2,K)-invariants completely in the case where m\neq n., Comment: Changed the expresseion of Proposition 5.6 since it was misleading

Tensor type data are used recently in various application fields, and then a typical rank is important. Let $3\leq m\leq n$. We study typical ranks of $m\times n\times (m-1)n$ tensors over the real number field. Let $\rho$ be the Hurwitz-Radon function defined as $\rho(n)=2^b+8c$ for nonnegative integers $a,b,c$ such that $n=(2a+1)2^{b+4c}$ and $0\leq b<4$. If $m \leq \rho(n)$, then the set of $m\times n\times (m-1)n$ tensors has two typical ranks $(m-1)n,(m-1)n+1$. In this paper, we show that the converse is also true: if $m > \rho(n)$, then the set of $m\times n\times (m-1)n$ tensors has only one typical rank $(m-1)n$., Comment: 20 pages

In this paper, we study typical ranks of 3-tensors and show that there are plural typical ranks for m\times n\times p tensors over R in the following cases: (1) 3\leq m\leq \rho(n) and (m-1)(n-1)+1\leq p\leq (m-1)n, where \rho\ is the Hurwitz-Radon function, (2) m=3, n\equiv 3\pmod 4 and p=2n-1, (3) m=4, n\equiv 2\pmod 4, n\geq 6 and p=3n-2, (4) m=6, n\equiv 4\pmod 8, n\geq 12 and p=5n-4. (5) m=10, n\equiv 24\pmod{32} and p=9n-8.

4x4x3 absolutely nonsingular tensors are characterized by their determinant polynomial. Non-quivalence among absolutely nonsingular tensors with respect to a class of linear transformations, which do not chage the tensor rank,is studied. It is shown theoretically that affine geometric invariants of the constant surface of a determinant polynomial is useful to discriminate non-equivalence among absolutely nonsingular tensors. Also numerical caluculations are presented and these invariants are shown to be useful indeed. For the caluculation of invarinats by 20-spherical design is also commented. We showed that an algebraic problem in tensor data analysis can be attacked by an affine geometric method., Comment: 24 pages, 3 figures, 5 tables

In various application fields, tensor type data are used recently and then a typical rank is important. Although there may be more than one typical ranks over the real number field, a generic rank over the complex number field is the minimum number of them. The set of $n$-tensors of type $p_1\times p_2\times\cdots\times p_n$ is called perfect, if it has a typical rank $\max(p_1,\ldots,p_n)$. In this paper, we determine perfect types of $n$-tensor., Comment: 11 pages, no figures

Quantum communication is concerned with the complexity of entanglement of a state and statistical data analysis is concerned with the complexity of a model. A common key word for both is "rank". In this paper we will show that both community is tracing the same target and that the methods used are slightly different. Two different methods, the range criterion method from quantum communication and the determinant polynomial method, are shown as an examples., Comment: 6 pages, presented at "The 21st Quantum Information Technology Symposium (QIT21)", Chofugaoka Chofu-shi, Tokyo, Japan, November 4th, 2009

Let $G$ be a flat finite-type group scheme over a scheme $S$, and $X$ a noetherian $S$-scheme on which $G$-acts. We define and study $G$-prime and $G$-primary $G$-ideals on $X$ and study their basic properties. In particular, we prove the existence of minimal $G$-primary decomposition and the well-definedness of $G$-associated $G$-primes. We also prove a generalization of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts type theorem on graded rings for $F$-regular and $F$-rational properties., Comment: 54pages, added Example 6.16 and the reference [8]. The final version

We define the concept of a doset Hibi ring and a generalized doset Hibi ring which are subrings of a Hibi ring and are normal affine semigrouprings. We apply the theory of (generalized) doset Hibi rings to analyze the rings of absolute orthogonal invariants and absolute special orthogonal invariants and show that these rings are normal and Cohen-Macaulay and has rational singularities if the characteristic of the base field is zero and is F-rational otherwise. We also state criteria of Gorenstein property of these rings., Comment: We added a down to earth proof of the result of DeConcini and Procesi about the rings of absolute orthogonal invariants and absolute special orthogonal invariants

The determination of the maximal ranks of a set of a given type of tensors is a basic problem both in theory and application. In statistical applications, the maximal rank is related to the number of necessary parameters to be built in a tensor model. Based on this classical theorem by Bosch we will show the tight bound for 2 x n x n tensors by simple row and column operations, symmetrization and mathematical induction, which has been given by several authors based on eigenvalue theories., Comment: 17 pages, no figures

Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A+T is diagonalizable for a given 3-tensor A with 2 slices over the complex and real number field., Comment: 12 pages, no figure

High dimensional array data, tensor data, is becoming important in recent days. Then maximal rank of tensors is important in theory and applications. In this paper we consider the maximal rank of 3 tensors. It can be attacked from various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and Atkinson-Lloyd(1980). They treated the problem in the complex field, and we will present various bounds over the real field by proving several lemmas and propositions, which is real counterparts of their results., Comment: 13 pages, no figure v2: correction and improvement

Let $S=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ with $n$ variables $x_1$, ..., $x_n$, $\mmmm$ the irrelevant maximal ideal of $S$, $I$ a monomial ideal in $S$ and $I'$ the polarization of $I$ in the polynomial ring $S'$ with $\rho$ variables. We show that each graded piece $H_\mmmm^i(S/I)_\aaa$, $\aaa\in\ZZZ^n$, of the local cohomology module $H_\mmmm^i(S/I)$ is isomorphic to a specific graded piece $H_{\mmmm'}^{i+\rho-n}(S'/I')_\alphaaa$, $\alphaaa\in\ZZZ^\rho$, of the local cohomology module $H_{\mmmm'}^{i+\rho-n}(S'/I')$, where $\mmmm'$ is the irrelevant maximal ideal of $S'$., Comment: Now available from http://lib1.kyokyo-u.ac.jp/kiyou/kiyoupdf/no109/bkue10902.pdf

Let $m$, $n$, $a_1$, ..., $a_r$, $b_1$, ..., $b_r$ be integers with $1\leq a_1<...

The homogeneous coordinate ring of a Schubert variety (a Schubert cycle for short) is an algebra with straightening law generated by a distributive lattice. This paper gives a simple method to study the set of all the join-irreducible elements of this distributive lattice, and gives a simple proof of the criterion of the Gorenstein property of Schubert cycles., Comment: To appear in Bulletin of Kyoto University of Education

Let $K$ be a field, $D$ a finite distributive lattice and $P$ the set of all join-irreducible elements of $D$. We show that if $\{y\in P\mid y\geq x\}$ is pure for any $x\in P$, then the Hibi ring $\RRRRR_K(D)$ is level. Using this result and the argument of sagbi basis theory, we show that the homogeneous coordinate rings of Schubert subvarieties of Grassmannians are level.

We study properties of a poset generating a Cohen-Macaulay algebra with straightening laws (ASL for short). We show that if a poset $P$ generates a Cohen-Macaulay ASL, then $P$ is pure and, if $P$ is moreover Buchsbaum, then $P$ is Cohen-Macaulay. Some results concerning a Rees algebra of an ASL defined by a straightening closed ideal are also established. And it is shown that if $P$ is a Cohen-Macaulay poset with unique minimal element and $Q$ is a poset ideal of $P$, then $P\uplus Q$ is also Cohen-Macaulay.