Subgroups of the general linear group containing an elementary subgroup in a reducible representation

Let $R$ be a commutative ring, $G=\mathrm{GL}(mn,R)$ be the general linear group of degree $mn$ over $R$. We construct and study a wide class of overgroups of the elementary group $E^m(n,R)\cong E(n,R)$ in the representation which is the direct sum of $m$ copies of the vector representation. When $R=K$ is a field and $n$ is large enough with respect to $m$, this allows us to give a complete description of all subgroups intermediate between $E^m(n,K)$ and $G$. This is a very broad generalization of some results by Z. I. Borewicz, N. A. Vavilov and others.