Jordan degree type for cxdimension three Gorenstein algebras of small Sperner number

Journal article, 2026

The Jordan type PA,ℓ of a linear form ℓ acting on a graded Artinian algebra A over a field k is the partition describing the Jordan block decomposition of the multiplication map mℓ, which is nilpotent. The Jordan degree type SA,ℓ is a finer invariant, describing also the initial degrees of the simple submodules of A in a decomposition of A as a direct sum of k[ℓ]-modules. The set of Jordan types of A or Jordan degree types (JDT) of A as ℓ varies, is an invariant of the algebra. This invariant has been studied for codimension two graded algebras. We here extend the previous results to certain codimension three graded Artinian Gorenstein (AG) algebras - those of small Sperner number. Given a Gorenstein sequence T - one possible for the Hilbert function of a codimension three graded AG algebra - the irreducible variety Gor(T) parametrizes all Gorenstein algebras of Hilbert function T. We here completely determine the JDT possible for all pairs (A,ℓ),A∈Gor(T), for Gorenstein sequences T of the form T=(1,3,s^k,3,1) for Sperner number s=3,4,5 and arbitrary multiplicity k. For s=6 we delimit the prospective JDT, without verifying that each occurs.