2.
Garsia-Haiman modules for hook partitions and Green polynomials with two variables - Morita, Hideaki
We consider Garsia-Haiman modules for the symmetric groups,
a doubly graded generalization of Springer modules.
Our main interest lies in singly graded submodules
of a Garsia-Haiman module.
We show that these submodules satisfy a certain combinatorial property,
and verify that this property is implied by a behavior of Macdonald polynomials
at roots of unity.
3.
Faces of arrangements of hyperplanes and Arrow's impossibility theorem - Abe, Takuro
In \cite{T}, Terao introduced an admissible map of chambers of a
real central arrangement, and completely classified it.
An admissible map is a generalization of
a social welfare function and Terao's classification is
that of Arrow's impossibility theorem in economics.
In this article we consider an admissible map not of chambers but
faces, and show that an admissible map of faces is a projection to a component if an
arrangement is indecomposable and its cardinality is not less than three.
From the view point
of Arrow's theorem, our result corresponds to the impossibility theorem
of a welfare function which
permits the...
6.
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.
7.
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.
8.
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.
9.
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.
