Jordan types with small parts for Artinian Gorenstein algebras of codimension three

TL;DR

This paper investigates the Jordan types of linear forms in graded Artinian Gorenstein algebras of codimension three, introducing rank matrices to classify possible Jordan types with small parts, and providing a complete classification for certain cases.

Contribution

It introduces rank matrices for linear forms in Artinian Gorenstein algebras and classifies all such matrices for codimension three with vanishing third power, linking them to Jordan types.

Findings

Established a 1-1 correspondence between rank matrices and Jordan degree types.

Classified all rank matrices for codimension three algebras with vanishing third power.

Showed Jordan types with parts of length at most four are determined by at most three parameters.

Abstract

We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determined by at most three parameters.