Jordan type of an Artinian algebra, a survey

Abstract

We consider Artinian algebras A over a field 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 (l, A) where l 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 l on A, and it is a finer invariant than the pair (l, 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 k[l] 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.