  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.
