Complete intersection Jordan types in height two

TL;DR

This paper classifies all possible Jordan type partitions for multiplication by a linear form in height two complete intersection Artinian algebras, linking them to vanishing higher Hessians and combinatorial structures.

Contribution

It provides a complete characterization of Jordan types in height two CI algebras, connecting algebraic, geometric, and combinatorial perspectives.

Findings

Characterization of Jordan type partitions via vanishing higher Hessians

Explicit combinatorial methods to construct partitions from branch labels and hook codes

Complete classification of Jordan types for these algebras

Abstract

We determine every Jordan type partition that occurs as the Jordan block decomposition for the multiplication map by a linear form in a height two homogeneous complete intersection (CI) Artinian algebra $A$ over an algebraically closed field $\sf k$ of characteristic zero or large enough. We show that these CI Jordan type partitions are those satisfying specific numerical conditions; also, given the Hilbert function $H(A)$, they are completely determined by which higher Hessians of $A$ vanish at the point corresponding to the linear form. We also show new combinatorial results about such partitions, and in particular we give ways to construct them from a branch label or hook code, showing how branches are attached to a fundamental triangle to form the Ferrers graph.