Equality of Linear and Symplectic Orbits

TL;DR

The paper proves that symplectic and linear transvection groups have identical orbit structures on unimodular elements, leading to improved stability estimates for symplectic K-theory over certain rings.

Contribution

It establishes the equality of orbits under symplectic and linear transvection groups, enhancing understanding of their actions on unimodular elements and improving K_1 stability bounds.

Findings

Orbits of symplectic and linear transvection groups coincide.

Results lead to better injective stability estimates for symplectic K_1.

Applicable to symplectic modules over non-singular affine algebras.

Abstract

It is shown that the set of orbits of the action of the elementary symplectic transvection group on all unimodular elements of a symplectic module over a commutative ring of characteristic not 2 is identical with the set of orbits of the action of the corresponding elementary transvection group. This result is used to get improved injective stability estimates for $K_1$ of the symplectic transvection group over a non-singular affine algebras.