# Subgroups of Chevalley groups of types $B_l$ and $C_l$ containing the group over a subring and corresponding carpets

## TL;DR

This paper extends the classification of subgroups in Chevalley groups of types B_l and C_l over rings, focusing on those containing elementary subgroups over subrings, and explores their Bruhat and Gauss decompositions.

## Contribution

It generalizes previous results to arbitrary weight lattices for types B_l and C_l, introducing carpet subgroups and analyzing their decompositions.

## Key findings

- Extended subgroup classification to types B_l and C_l
- Introduced and characterized carpet subgroups
- Analyzed Bruhat and Gauss decompositions for these subgroups

## Abstract

We continue study of subgroups of a Chevalley group $G_P(\Phi,R)$ over a ring $R$ with a root system $\Phi$ and a weight lattice $P$, containing the elementary subgroup $E_P(\Phi,K)$ over a subring $K$ of $R$. Recently A. Bak and A. Stepanov considered the symplectic case (i. e. the case of simply connected group of type $\Phi=C_l$) in characteristic 2. In this article we extend their result for groups with arbitrary weight lattice of types $B_l$ and $C_l$. Similarly to the work of Nuzhin that handles the case of an algebraic extension $R$ of a nonperfect field $K$ of bad characteristic, we use in the description a special kind of carpet subgroups. In the second half of the article we study Bruhat and Gauss decompositions for these carpet subgroups.