Iarrobino's symmetric decomposition for self-dual modules

TL;DR

This paper extends Iarrobino's symmetric decomposition to self-dual modules over local algebras, analyzing their Hilbert functions and providing classifications for small degrees.

Contribution

It introduces a generalized symmetric decomposition for self-dual modules and offers new criteria for self-duality using Macaulay's inverse systems.

Findings

Classified local Hilbert functions for small degree modules

Generalized symmetric decomposition for self-dual modules

Extended criteria for self-duality in algebraic structures

Abstract

We generalize Iarrobino's symmetric decomposition for the associated graded algebra of an Artinian Gorenstein algebra to a symmetric decomposition of finite-length self-dual modules over a local algebra, and we deduce consequences for the Hilbert functions of such self-dual modules. We classify the local Hilbert functions for small degree modules. We generalize Kunte's criterion for self-duality in terms of Macaulay's inverse systems.