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arXiv · 2008.01206

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

Christakis A. Pallikaros·Harold N. Ward
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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.

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Christakis A. Pallikaros, Harold N. Ward. 2020-08-03. Linear degenerations of algebras and certain representations of the general linear group. https://arxiv.org/abs/2008.01206

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