Explicit presentation of relative Steinberg groups

TL;DR

This paper provides explicit presentations of relative Steinberg groups for various rings and root systems, facilitating a better understanding of their algebraic structure through generators and relations.

Contribution

It offers the first explicit generator-and-relation descriptions of relative Steinberg groups for general rings and specific root systems, advancing algebraic group theory.

Findings

Explicit presentations for relative linear Steinberg groups $ ext{St}(n, R, I)$ for $n \\geq 4$.

Explicit presentations for relative simply laced Steinberg groups $ ext{St}(\

\\Phi; R, I)$ for certain root systems.

Abstract

We find an explicit presentation of relative linear Steinberg groups $\mathrm{St}(n, R, I)$ for any ring $R$ and $n \geq 4$ by generators and relations as abstract groups. We also prove a similar result for relative simply laced Steinberg groups $\mathrm{St}(\Phi; R, I)$ for commutative $R$ and $\Phi \in \{\mathsf A_\ell, \mathsf D_\ell, \mathsf E_\ell \mid \ell \geq 3\}$.