Yasuhide Numata

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.

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.

Signed-eliminable graphs and free multiplicities on the braid arrangement

We define specific multiplicities on the braid arrangement by using signed graphs. To consider their freeness, we introduce the notion of signed-eliminable graphs as a generalization of Stanley�s classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of certain deformations of the braid arrangement in terms of directed graphs.

Exponents of 2-multiarrangements and multiplicity lattices

We introduce a concept of multiplicity lattices of 2-multiarrangements, determine the combinatorics and geometry of that lattice, and give a criterion and method to construct a basis for derivation modules effectively.

Graph presentations for moments of noncentral Wishart distributions and their applications

We provide formulas for the moments of the real and complex noncentral Wishart distributions of general degrees. The obtained formulas for the real and complex cases are described in terms of the undirected and directed graphs, respectively. By considering degenerate cases, we give explicit formulas for the moments of bivariate chi-square distributions and 2 × 2 Wishart distributions by enumerating the graphs. Noting that the Laguerre polynomials can be considered to be moments of a noncentral chi-square distributions formally, we demonstrate a combinatorial interpretation of the coefficients of the Laguerre polynomials.

Separation of integer points by a hyperplane under some weak notions of discrete convexity

We give some sufficient conditions of separation of two sets of integer points by a hyperplane. Our conditions are related to the notion of convexity of sets of integer points and are weaker than existing notions.

On Computation of the First Baues–Wirsching Cohomology of a Freely-Generated Small Category

The Baues–Wirsching cohomology is one of the cohomologies of a small category. Our aim is to describe the first Baues–Wirsching cohomology of the small category generated by a finite quiver freely. We consider the case where the coefficient is a natural system obtained by the composition of a functor and the target functor. We give an algorithm to obtain generators of the vector space of inner derivations. It is known that there exists a surjection from the vector space of derivations of the small category to the first Baues–Wirsching cohomology whose kernel is the vector space of inner derivations.

The Strong Lefschetz Property and the Schur–Weyl Duality

The purpose of this chapter is to illustrate a role played by the SLP in connection with the theory of Artinian rings and the Schur–Weyl duality. We assume that the reader is familiar with commutative algebra but perhaps without knowledge of representation theory, but we are hopeful that the expert in representation theory may also find the following sections of interest.

Complete Intersections with the SLP

The main result of this chapter is Theorem 4.10. This may be regarded as a generalization of Theorem 3.34 which states that the SLP is preserved by tensor products. Using the main theorem, we give some examples of complete intersections with the strong Lefschetz property.

Cohomology Rings and the Strong Lefschetz Property

The Lefschetz property originates in the Hard Lefschetz Theorem for compact Kahler manifolds, so it is natural that some results discussed in the former chapters have geometric backgrounds. For example, Corollary 4.17 on the flat extension can be understood from the cohomology ring of projective space bundles in a geometric setting.