Quillen-Suslin theory for a structure theorem for the Elementary Symplectic Group

TL;DR

This paper introduces a new symmetrical set of elementary symplectic generators, proving their sufficiency for the elementary symplectic group and using them to analyze group actions and subgroup normality.

Contribution

It presents a novel set of generators for the elementary symplectic group and applies Quillen's Local Global Principle to establish subgroup normality with these generators.

Findings

New symmetrical generators for ESp_{2n}(R)

Alternative proof of normality of ESp_{2n}(R) in Sp_{2n}(R)

Enhanced understanding of group actions on unimodular rows

Abstract

A new set of elementary symplectic elements is described, It is shown that these also generate the elementary symplectic group {\rm ESp}$_{2n}(R)$. These generators are more symmetrical than the usual ones, and are useful to study the action of the elementary symplectic group on unimodular rows. Also, an alternate proof of, {\rm ESp}$_{2n}(R)$ is a normal subgroup of {\rm Sp}$_{2n}(R)$, is shown using the Local Global Principle of D. Quillen for the new set of generators.