Overgroups of exterior powers of an elementary group. Levels

TL;DR

This paper characterizes groups lying between an exterior power of an elementary group and a general linear group over a commutative ring, using the concept of a level ideal to describe their structure.

Contribution

It provides a partial classification of such groups by associating a unique ideal of the ring to each group, advancing the understanding of their algebraic structure.

Findings

Established a correspondence between groups and ideals in the ring.

Proved the existence and uniqueness of the level ideal for each group.

Extended the standard description to a broader class of groups.

Abstract

We prove a first part of the standard description of groups $H$ lying between an exterior power of an elementary group $\bigwedge^m E_n(R)$ and a general linear group $GL_{n \choose m}(R)$ for a commutative ring $R$, $2\in R^*$ and $n\geqslant 3m$. The description uses the classical notion of a level: for every group $H$ we find a unique ideal $A$ of the ground ring $R$ which describes $H$.