147 research outputs found
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 , , , ..., , , ..., be integers with and . And let be the universal
matrix with the property that -minors of first rows and
first columns are all zero, for , ..., ( and
). For an integer with , we denote by the
matrix consisting of the first rows of . 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 be a polynomial ring over a field with
variables , ..., , \mmmm the irrelevant maximal ideal of , a
monomial ideal in and the polarization of in the polynomial ring
with 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 .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 generates a
Cohen-Macaulay ASL, then is pure and, if is moreover Buchsbaum, then
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
is a Cohen-Macaulay poset with unique minimal element and is a poset
ideal of , then is also Cohen-Macaulay
A sufficient condition for a Hibi ring to be level and levelness of Schubert cycles
Let be a field, a finite distributive lattice and the set of all
join-irreducible elements of . We show that if is
pure for any , 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
Typical rank of tensors with over the real number field
Tensor type data are used recently in various application fields, and then a
typical rank is important. Let . We study typical ranks of
tensors over the real number field. Let be the
Hurwitz-Radon function defined as for nonnegative integers
such that and . If , then
the set of tensors has two typical ranks
. In this paper, we show that the converse is also true: if , then the set of tensors has only one
typical rank .Comment: 20 page
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