Overgroups of elementary groups in polyvector representations

TL;DR

This paper studies subgroups of general linear groups containing exterior powers of elementary groups, characterizing their structure via ideals and invariant forms, extending classical results to arbitrary commutative rings.

Contribution

It introduces a framework for understanding overgroups of exterior powers of elementary groups over any commutative ring, including a classification of intermediate subgroups and a characterization of stabilizers.

Findings

Subgroups containing exterior powers are parametrized by ideals.

The subgroup lattice for exterior squares is standard and fully described.

The group of exterior automorphisms is characterized as stabilizers of invariant forms.

Abstract

We initiate the study of subgroups $H$ of the general linear group $GL_{\binom{n}{m}}(R)$ over a commutative ring $R$ that contain the $m$-th exterior power of an elementary group $\bigwedge^mE_n(R)$. Each such group $H$ corresponds to a uniquely defined level $(A_0,\dots,A_{m-1})$, where $A_0,\dots,A_{m-1}$ are ideals of $R$ with certain relations. In the crucial case of the exterior squares, we state the subgroup lattice to be standard. In other words, for $\bigwedge^2E_n(R)$ all intermediate subgroups $H$ are parametrized by a single ideal of the ring $R$. Moreover, we characterize $\bigwedge^mGL_n(R)$ as the stabilizer of a system of invariant forms. This result is classically known for algebraically closed fields, here we prove the corresponding group scheme to be smooth over $\mathbb{Z}$. So the last result holds over arbitrary commutative rings.