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

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 $GL_n(\_)$.