Overgroups of elementary groups in polyvector representations

We initiate the study of subgroups $H$ of the general linear group $\mathrm{GL}_{\binom{n}{m}}(R)$ over a commutative ring $R$ that contain the $m$-th exterior power of an elementary group $\bigwedge^m\mathrm{E}_n(R)$. 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 $\bigwedge^2\mathrm{E}_n(R)$ all intermediate subgroups $H$ are parametrized by a single ideal of the ring $R$. Moreover, we characterize $\bigwedge^m\mathrm{GL}_n(R)$ 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.