Almost Gorenstein Hibi rings

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

On the analytic spread and the reduction number of the ideal of maximal minors

Let $m$, $n$, $a_1$, ..., $a_r$, $b_1$, ..., $b_r$ be integers with $1\leq a_1<...<a_r\leq m$ and $1\leq b_1<...<b_r\leq n$. And let $x$ be the universal $m\times n$ matrix with the property that $i$-minors of first $a_i-1$ rows and first $b_i-1$ columns are all zero, for $i=1$, ..., $r+1$ ($a_{r+1}=m+1$ and $b_{r+1}=n+1$). For an integer $u$ with $1\leq u\leq m$, we denote by $U$ the $u\times n$ matrix consisting of the first $u$ rows of $x$. In this paper, we consider the analytic spread and the reduction number of the ideal of maximal minors of $U

Behavior of local cohomology modules under polarization

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.pd

On the canonical ideal of the Ehrhart ring of the chain polytope of a poset

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

On the generators of the canonical module of a Hibi ring: a criterion of level property and the degrees of generators

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 section

Action of special linear groups to the tensor of indeterminates, classical invariants of binary forms and hyperdeterminant

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 misleadin

On the discrete counterparts of Cohen-Macaulay algebras with straightening laws

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

A sufficient condition for a Hibi ring to be level and levelness of Schubert cycles

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

On the generating poset of Schubert cycles and the characterization of Gorenstein property

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 Educatio

Perfect type of n-tensors

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 figure