Free Extensions and Jordan type

TL;DR

This paper studies free extensions of Artinian algebras, showing they are deformations of tensor products, and explores implications for Jordan types and applications to group action invariants.

Contribution

It establishes that free extensions are deformations of tensor products and analyzes their impact on Jordan types and algebraic invariants.

Findings

Free extensions are deformations of tensor products.

Jordan type of a free extension dominates that of the tensor product.

Applications to invariants of linear group actions on polynomials.

Abstract

Free extensions of commutative Artinian algebras were introduced by T. Harima and J. Watanabe. The Jordan type of a multiplication map $m$ by a nilpotent element of an Artinian algebra is the partition determining the sizes of the blocks in a Jordan matrix for $m$. We show that a free extension of the Artinian algebra $A$ with fibre $B$ is a deformation of the usual tensor product. This has consequences for the generic Jordan types of $A,B$ and $C$, showing that the Jordan type of $C$ is at least that of the usual tensor product in the dominance order. We give applications to algebras of relative coinvariants of linear group actions on a polynomial ring.