Overgroups of exterior powers of an elementary group. Normalizers

TL;DR

This paper characterizes the algebraic group scheme $igwedge^m GL_n$ over $Z$ as a stabilizer of invariant forms and as a normalizer of elementary subgroups, aiding classification of overgroups and solving a linear preserver problem.

Contribution

It provides two explicit characterizations of $igwedge^m GL_n$ over $Z$, linking geometric invariants with algebraic normalizers, advancing the classification of overgroups.

Findings

$igwedge^m GL_n$ is a stabilizer of an invariant form or ideal.

$igwedge^m GL_n$ is isomorphic to a normalizer of elementary and special linear subgroups.

Results facilitate the classification of overgroups and linear preserver problems.

Abstract

We establish two characterizations of an algebraic group scheme $\bigwedge^m GL_n$ over $\mathbb{Z}$. Geometrically, the scheme $\bigwedge^m GL_n$ is a stabilizer of an explicitly given invariant form or, generally, an invariant ideal of forms. Algebraically, $\bigwedge^m GL_n$ is isomorphic (as a scheme over $\mathbb{Z}$) to a normalizer of the elementary subgroup functor $\bigwedge^m E_n$ and a normalizer of the subscheme $\bigwedge^m SL_n$. Our immediate goal is to apply both descriptions in the "sandwich classification" of overgroups of the elementary subgroup. Additionally, the results can be seen as a solution of the linear preserver problem for algebraic group schemes over $\mathbb{Z}$, providing a more functorial description that goes beyond geometry of the classical case over fields.