Improved $K_1$-stability for the embedding $D_5$ into $E_6$

TL;DR

This paper improves the understanding of $K_1$-stability for the embedding of $D_5$ into $E_6$, showing stability under weaker conditions and exploring implications for overgroups of subsystem subgroups.

Contribution

It advances previous results by weakening assumptions for $K_1$-stability and demonstrates applications to overgroups of subsystem subgroups.

Findings

Surjective stability holds under weaker ring assumptions.

Enhanced understanding of $K_1$-stability for the $D_5$ into $E_6$ embedding.

Applications to studying overgroups of subsystem subgroups.

Abstract

This paper is dedicated to the surjective stability of the $K_1$-functor for Chevalley groups for the embedding $D_5$ into $E_6$. This case was already studed by Plotkin. In the present paper, we improve his result by showing that surjective stability holds under a weaker assumption on a ring. Another result of the present paper shows how the $K_1$-stability can help to study overgroups of subsystem subgroups.