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

Abstract

Let $\mathbf{\Lambda }(={\mathbb{F}}^{n^3})$, 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 $\mathbf{\Lambda }$ as a $G$-module via the `change of basis' action of~$G$ on~$V$, we determine the composition factors of various $G$-submodules of~$\mathbf{\Lambda }$ 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~$\mathbf{\Lambda }$ is determined.

26 pages