Overgroups of exterior powers of an elementary group. I. Levels and normalizers

TL;DR

This paper characterizes groups between exterior powers of elementary groups and general linear groups, establishing a unique ideal level for such groups and identifying their normalizers and related structures.

Contribution

It provides the first comprehensive description of overgroups of exterior powers of elementary groups, including explicit equations and the coincidence of various normalizers.

Findings

Unique ideal levels correspond to overgroups for n ≥ 3m.

Normalizers of exterior power groups coincide with certain algebraic group schemes.

Explicit equations for exterior powers of algebraic groups are derived.

Abstract

In the present paper, we prove the first part in the standard description of groups $H$ lying between $m$-th exterior power of elementary group $E(n,R)$ and the general linear group $GL_{\binom{n}{m}}(R)$. We study structure of the exterior power of elementary group and its relative analog $E\left(\binom{n}{m},R,A\right)$. In the considering case $n \geq 3m$, the description is explained by the classical notion of level: for every such $H$ we find unique ideal $A$ of the ring $R$. Motivated by the problem, we prove the coincidence of the following groups: normalizer of the exterior power of elementary group, normalizer of the exterior power of special linear group, transporter of the exterior power of elementary group into the exterior power of special linear group, and an exterior power of general linear group. This result mainly follows from the found explicit equations for the exterior power of algebraic group scheme $GL_n(\_)$.