Akihito Wachi

The strong Lefschetz property of the coinvariant ring of the Coxeter group of type H4

Abstract We prove that the coinvariant ring of the irreducible Coxeter group of type H4 has the strong Lefschetz property.

Generic Initial Ideals, Graded Betti Numbers, and k-Lefschetz Properties

We introduce the k-strong Lefschetz property and the k-weak Lefschetz property for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results are: 1. Let I be an ideal of R = K[x 1, x 2,…, x n ] whose quotient ring R/I has the n-SLP. Suppose that all kth differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I. 2. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.

Zero-dimensional Gorenstein Algebras with the Action of the Symmetric Group

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.

Strong Lefschetz Elements of the Coinvariant Rings of Finite Coxeter Groups

For the coinvariant rings of finite Coxeter groups of types other than H4, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.

Codimension one connectedness of the graph of associated varieties

Let

The Strong Lefschetz Property and the Schur–Weyl Duality

\pi

Complete Intersections with the SLP

be an irreducible Harish-Chandra

Cohomology Rings and the Strong Lefschetz Property

(\mathfrak{g}, K)

A Generalization of Lefschetz Elements

-module, and denote its associated variety by

Invariant Theory and Lefschetz Properties

AV(\pi)