Overgroups of elementary groups in polyvector representations

R. A. Lubkov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 40, Tome 531 (2024), pp. 101-116

Résumé

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.