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

Abstract

We study Jordan types of linear forms for graded Artinian Gorenstein algebras with arbitrary codimension. We introduce rank matrices of linear forms for such algebras which represent the ranks of multiplication maps in various degrees. Rank matrices correspond to Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices which occur for linear forms with vanishing third power. As a consequence for such algebras we show that Jordan types with parts of length at most four are uniquely determined by at most three parameters.