Overgroups of exterior powers of an elementary group. Normalizers

Abstract

We establish two characterizations of an algebraic group scheme ${\bigwedge }^m{\mathrm{GL}}_n$ over $\mathbb{Z}$. Geometrically, the scheme ${\bigwedge }^m{\mathrm{GL}}_n$ is a stabilizer of an explicitly given invariant form or, generally, an invariant ideal of forms. Algebraically, ${\bigwedge }^m{\mathrm{GL}}_n$ is isomorphic (as a scheme over $\mathbb{Z}$) to a normalizer of the elementary subgroup functor ${\bigwedge }^m{\mathrm{E}}_n$ and a normalizer of the subscheme ${\bigwedge }^m{\mathrm{SL}}_n$.
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.