Jordan degree type for codimension three Gorenstein algebras of small Sperner number (2406.06322v1)

Abstract: The Jordan type $P_{A,\ell}$ of a linear form $\ell$ acting on a graded Artinian algebra $A$ over a field $\sf k$ is the partition describing the Jordan block decomposition of the multiplication map $m_\ell$, which is nilpotent. The Jordan degree type $\mathcal S_{A,\ell}$ is a finer invariant, describing also the initial degrees of the simple submodules of $A$ in a decomposition of $A$ as ${\sf k}[\ell]$-modules. The set of Jordan types of $A$ or Jordan degree types (JDT) of $A$ as $\ell$ 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 AG algebra - the irreducible variety $\mathrm{Gor}(T)$ parametrizes all Gorenstein algebras of Hilbert function $T$. We here completely determine the JDT possible for all pairs $(A,\ell), A\in \mathrm{Gor}(T)$, 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.