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

Abstract

The Jordan type $P_{A,\ell }$ 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 $m_{\ell }$, which is nilpotent. The Jordan degree type $S_{A,\ell }$ 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 $k\left[\ell \right]$-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 $\mathrm{Gor}\left(T\right)$ parametrizes all Gorenstein algebras of Hilbert function T. We here completely determine the JDT possible for all pairs $(A,\ell ),A\in \mathrm{Gor}\left(T\right)$, 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.