Free extensions and Jordan type

Abstract

Free extensions of graded Artinian algebras were introduced by T. Harima and J. Watanabe, and were shown to preserve the strong Lefschetz property. The Jordan type of a multiplication map m by a nilpotent element of an Artinian algebra is the partition determining the sizes of the blocks in a Jordan matrix for m. We show that a free extension C of the Artinian algebra A with fiber B is a deformation of the usual tensor product. This has consequences for the generic Jordan types of A,B and C: we show that the Jordan type of C is at least that of the usual tensor product in the dominance order (Theorem 2.5). In particular this gives a different proof of the T. Harima and J. Watanabe result concerning the strong Lefschetz property of a free extension. Examples illustrate that a non-strong-Lefschetz graded Gorenstein algebra A with non-unimodal Hilbert function may nevertheless have a non-homogeneous element with strong Lefschetz Jordan type, and may have an A-free extension that is strong Lefschetz. We apply these results to algebras of relative coinvariants of linear group actions on a polynomial ring.