Subring subgroups in Chevalley groups with doubly laced root systems.
The author studies the lattice \(\mathcal L\) of subgroups of a Chevalley group \(G(\Phi,A)\) over a ring \(A\), containing its elementary subgroup \(E(\Phi,S)\) over a subring \(S\subseteq A\). The standard description asserts that for any \(H\in\mathcal L\) there exists a unique subring \(R\supseteq S\) of \(A\) such that \(H\) contains \(E(\Phi,R)\) as a normal subgroup. The standard description was obtained by Ya. Nuzhin for algebraic field extensions. On the other hand, it was expected that the standard description does not hold for transcendental extensions. This was recently proved by the author for simply laced root systems. Now, suppose that \(\Phi\) is doubly laced, i.e. \(\Phi=B_l\), \(C_l\) or \(F_4\), and that 2 is invertible in \(S\). In the article it is proven that in these settings the standard description of \(\mathcal L\) holds for an arbitrary pair of rings \(S\subseteq A\).
scientific article; zbMATH DE number 6138243
scientific article