4.
COHEN-MACAULAY MODULES AND HOLONOMIC MODULES OVER FILTERED RINGS - Miyahara, Hiroaki; Nishida, Kenji
We study Gorenstein dimension and grade of a module M over a ltered
ring whose assosiated graded ring is a commutative Noetherian ring. An equality or
an inequality between these invariants of a ltered module and its associated graded
module is the most valuable property for an investigation of ltered rings. We prove
an inequality G-dimM G-dim grM and an equality gradeM = grade grM, whenever
Gorenstein dimension of grM is nite (Theorems 2.3 and 2.8). We would say that the
use of G-dimension adds a new viewpoint for studying ltered rings and modules. We
apply these results to a ltered ring with a Cohen-Macaulay or Gorenstein...
8.
Green polynomials at roots of unity and its application - Morita, Hideaki
We consider Green polynomials at roots of unity.
We obtain a recursive formula for Green polynomials
at appropriate roots of unity,
which is described in a combinatorial manner.
The coefficients of the recursive formula are
realized by the number of permutations satisfying a certain condition,
which leads to interpretation of
a combinatorial property of certain graded modules of the symmetric group
in terms of representation theory.
9.
A formula of Lascoux-Leclerc-Thibon and representations of symmetric groups - Morita, Hideaki; Nakajima, Tatsuhiro
We consider Green polynomials at roots of unity,
corresponding to partitions which we call $l$-partitions.
We obtain a combinatorial formula for Green polynomials
corresponding to $l$-partitions
at primitive $l$-th roots of unity.
The formula is rephrased in terms of representation theory
of the symmetric group.
10.
Zero dimensional Gorenstein algebras with the action of the symmetric group $S_k$ - Morita, Hideaki; Watanabe, Junzo
We consider irreducible decompositions of certain Artinian algebras with
the action of the symmetric group.
The equi-degree monomial complete intersection can be thought of as a k-fold
tensor of an n dimensional vector space.
Otherwise put the tensor space can be given a commutative ring structure.
From this view point we show that, in the case n=2 or k=2,
the strong Lefschetz property can be
used efficiently to decompose the algebra into irreducible components.
We apply the result to determin a minimal generating set of certain
Gorenstein ideal.
Also we show that the Hilbert function of certain ring of invariants is a
q-anolog of the binomial coefficent.
11.
Decomposition of Green polynomials of type $A$ and DeConcini-Procesi-Tanisaki algebras of certain types - Morita, Hideaki
A class of graded representations of the symmetric group,
concerning with the cohomology ring of the corresponding flag variety,
are considered.
We point out a certain combinatorial property of the Poincar\'e
polynomial of these graded representations,
and interpret it in the language of representation theory
of the symmetric group.
1.
