Free extensions and Lefschetz properties, with an application to rings of relative coinvariants

Abstract

Graded Artinian algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of compact topological manifolds. In this analogy, a free extension of a base ring with a fiber ring corresponds to a fiber bundle over a manifold. For rings, as with manifolds, it is a natural question to ask: to what extent do properties of the base and the fiber carry over to the extension? For example, if the base and fiber both satisfy a strong Lefschetz property, can we conclude the same for the extension? Or, more generally, can one determine the generic Jordan type for the extension given the generic Jordan types of the base and fiber? We address these questions using the relative coinvariant rings as prototypical models. We show that if $V$ is a vector space and if the subgroup $W$ of the general linear group Gl(V), is a non-modular finite reflection group and $K\subset W$ is a non parabolic reflection subgroup, then the relative coinvariant ring $R^K_W$ cannot have a linear element of strong Lefschetz Jordan type. However, we give examples where these rings $R^K_W$, some with non-unimodal Hilbert functions, nevertheless have (non-homogeneous) elements of strong Lefschetz Jordan type. Some of these examples give rise to open questions concerning Lefschetz properties of certain algebras $A(m,n)$, related to combinatorial questions proposed and partially answered by G. Almqvist.