Composition series of $\gl(m)$ as a module for its classical subalgebras over an arbitrary field

TL;DR

This paper determines the composition series of the $ ext{gl}(m)$ module when restricted to its classical subalgebras over any field, providing detailed structure and identification of all composition factors.

Contribution

It explicitly constructs the composition series of $ ext{gl}(V)$ as an $L(f)$-module for classical subalgebras over arbitrary fields, extending known results to a more general setting.

Findings

Explicit composition series for $gl(V)$ as an $L(f)$-module.

Multiple identifications of all composition factors.

Generalization to arbitrary fields and forms.

Abstract

Let $F$ be an arbitrary field and let $f:V\times V\to F$ be a non-degenerate symmetric or alternating bilinear form defined on an $F$-vector space of finite dimension $m\geq 2$. Let $L(f)$ be the subalgebra of $gl(V)$ formed by all skew-adjoint endomorphisms with respect to $f$. We find a composition series for the $L(f)$-module $gl(V)$ and furnish multiple identifications for all its composition factors.