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 H 4 has the strong Lefschetz property.

Sperner property and finite-dimensional Gorenstein algebras associated to matroids

“…Stanley [15] has shown the Sperner property of the strong Bruhat ordering on W from the Lefschetz property of R W (except for type H 4 ). The Lefschetz property of R W of type H 4 has been confirmed in [12]. Since R W is Gorenstein, it has a presentation as in Proposition 1.5.…”

The Jordan type of graded Artinian Gorenstein algebras

“…The Lefschetz properties for Artinian K-algebras are algebraic abstractions inspired by the Hard Lefschetz theorem on the cohomology of smooth projective varieties (see [La] or [Ru,Chapter 7]). Nowadays there are lots of contexts where the Lefschetz properties have been introduced, for instance Kahler manifolds, convex polytopes, Coxeter groups and tropical varieties (see [Ka,Be,HL,KN,NW,St,St2,BN,GZ]). These structures share some algebraic properties, in particular the existence of a certain cohomology ring.…”

Lefschetz elements of Artinian Gorenstein algebras and Hessians of homogeneous polynomials

“…In fact, the coinvariant algebra of the Weyl group is isomorphic to the cohomology ring of the corresponding flag variety. In [11] and [12], it has been shown that the coinvariant algebra of any finite Coxter group has the Lefschetz property and that the set of the Lefschetz elements is the complement of the union of the reflection hyperplanes except for type H 4 case. The determination of the set of the Lefschetz elements is still open for H 4 because of the computational complexity.…”