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 fibre ring corresponds to a fibre bundle over a manifold. If the manifold is Kähler, then its cohomology ring satisfies the strong Lefschetz property, which means multiplication by a linear form has the largest possible Jordan type. In this paper, we study the behaviour of strong Lefschetz and Jordan type with respect to free extensions, using relative coinvariant rings of finite groups 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 is a non-parabolic reflection subgroup, then the relative coinvariant ring cannot have a linear element of strong Lefschetz Jordan type. However, we give examples where these rings , some with non-unimodal Hilbert functions, nevertheless have (non-homogeneous) elements of strong Lefschetz Jordan type. Some of these examples are related to open combinatorial questions proposed and partially solved by G. Almkvist.