Jordan type of an Artinian algebra, a survey

TL;DR

This survey reviews the Jordan type and Jordan degree type invariants of Artinian algebras, highlighting recent developments, their differences from Lefschetz properties, and open problems in the study of these algebraic structures.

Contribution

It provides a comprehensive overview of Jordan type and Jordan degree type invariants in Artinian algebras, emphasizing recent research and unresolved questions.

Findings

Jordan type is a finer invariant than Lefschetz properties.

Jordan degree type incorporates initial degrees of decomposition strings.

The survey identifies open problems in the classification of Artinian algebra invariants.

Abstract

We consider Artinian algebras $A$ over a field $\mathsf{k}$, both graded and local algebras. The Lefschetz properties of graded Artinian algebras have been long studied, but more recently the Jordan type invariant of a pair $(\ell,A)$ where $\ell$ is an element of the maximal ideal of $A$, has been introduced. The Jordan type gives the sizes of the Jordan blocks for multiplication by $\ell$ on $A$, and it is a finer invariant than the pair $(\ell,A)$ being strong or weak Lefschetz. The Jordan degree type for a graded Artinian algebra adds to the Jordan type the initial degree of ``strings'' in the decomposition of $A$ as a $\mathsf{k}[\ell]$ module. We here give a brief survey of Jordan type for Artinian algebras, Jordan degree type for graded Artinian algebras, and related invariants for local Artinian algebras, with a focus on recent work and open problems.