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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 $m$-thexterior 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 crucialcase of the exterior squares, we state the subgroup lattice to be standard. Inother words, for $\bigwedge^2E_n(R)$ all intermediate subgroups $H$ areparametrized by a single ideal of the ring $R$. Moreover, we characterize$\bigwedge^mGL_n(R)$ as the stabilizer of a system of invariant forms. Thisresult is classically known for algebraically closed fields, here we prove thecorresponding group scheme to be smooth over $\mathbb{Z}$. So the last resultholds over arbitrary commutative rings.

Overgroups of elementary groups in polyvector representations