Subring subgroups in symplectic groups in characteristic 2

TL;DR

This paper extends the understanding of subgroup structures within symplectic groups over rings, especially in characteristic 2, by describing the lattice of subgroups containing elementary subgroups through a new framework involving form rings.

Contribution

It generalizes previous subgroup lattice descriptions to cases where 2=0 in the ring, introducing form rings and expanding the classification of subgroups in symplectic groups.

Findings

Lattice of subgroups splits into sandwiches parametrized by intermediate rings.

Introduces form rings to describe subgroup structures in characteristic 2.

Generalizes Nuzhin's theorem for algebraic extensions of rings.

Abstract

In 2012 the second author obtained a description of the lattice of subgroupsof a Chevalley group $G(\Phi,A)$, containing the elementary subgroup $E(\Phi,K)$ over a subring $K\subseteq A$ provided $\Phi=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$ is invertible in $K$. It turns out that this lattice splits into a disjoint union of "sandwiches", parametrized by intermediate subrings between $K$ and $A$. In the current article a similar result is proved for $\Phi=B_n$ or $C_n$, $n\ge3$, and $2=0$ in $K$. In this settings one has to introduce more sandwiches, namely, the sandwiches are parametrized by form rings $(R,\Lambda)$ such that $K\subseteq\Lambda\subseteq R\subseteq A$. In particular, this result, generalizes a part of Ya.\,N.\,Nuzhin's theorem of 2013 concerning root systems $\Phi=B_n,$ $C_n$, $n\ge3$, where the same description of the subgroup lattice is obtained under the condition that $A$ is an algebraic extension of~$K$.