Linear degenerations of algebras and certain representations of the general linear group

TL;DR

This paper studies the structure of algebraic degenerations and representations of the general linear group on the space of algebra structure vectors, revealing composition factors and $GL(V)$-module structure.

Contribution

It introduces the notion of linear degeneration to analyze algebra degenerations over fields with more than two elements, extending known results.

Findings

Determined the composition factors of $G$-submodules of the structure vector space.

Established the $GL(V)$-structure of the space of algebra structures.

Extended degeneration results to fields with $||>2$.

Abstract

Let $\boldsymbol{\Lambda}\,(=\mathbb{F}^{n^{3}})$, where $\mathbb{F}$ is a field with $|\mathbb{F}|>2$, be the space of structure vectors of algebras having the $n$-dimensional $\mathbb{F}$-space $V$ as the underlying vector space. Also let $G=GL(V)$. Regarding $\boldsymbol{\Lambda}$ as a $G$-module via the `change of basis' action of~$G$ on~$V$, we determine the composition factors of various $G$-submodules of~$\boldsymbol{\Lambda}$ which correspond to certain important families of algebras. This is achieved by introducing the notion of linear degeneration which allows us to obtain analogues over $\mathbb{F}$ of certain known results on degenerations of algebras. As a result, the $GL(V)$-structure of~$\boldsymbol{\Lambda}$ is determined.