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

2020/08/03 by Pallikaros, Christakis A., Ward, Harold N.
#14D06 (Primary) 14R20 #20C99 (Secondary) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2008.01206

Abstract

Let \boldsymbolΛ (=\mathbbF^n³), where \mathbbF is a field with |\mathbbF|>2, be the space of structure vectors of algebras having the n-dimensional \mathbbF-space V as the underlying vector space. Also let G=GL(V). Regarding \boldsymbolΛ as a G-module via the `change of basis' action of~G on~V, we determine the composition factors of various G-submodules of~\boldsymbolΛ 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 \mathbbF of certain known results on degenerations of algebras. As a result, the GL(V)-structure of~\boldsymbolΛ is determined.

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