Bijective PC-maps of the unipotent radical of the Borel subgroup of the classical symplectic group

TL;DR

This paper characterizes all bijections of the unipotent radical of the Borel subgroup in classical symplectic groups that preserve commutators, showing they are composed of standard automorphisms and central maps.

Contribution

It provides a complete description of bijective commutator-preserving maps for the unipotent radical of the Borel subgroup in classical symplectic groups over fields with 6F=F.

Findings

Any such bijection is a composition of a standard automorphism and a central map.

Central maps act as right multiplication by a central element.

The result applies to groups of rank at least 4 over fields satisfying 6F=F.

Abstract

Every commutator preserving bijection of the unipotent radical $Up(2n, R)$ of the Borel subgroup of the classical symplectic group of rank at least 4 over a field $F$ such that $6F=F$ is shown to be the composition of a standard automorphism of $Up(2n,R)$ and a central map. The latter is a bijection which acts as the right multiplication by a matrix from the center of $Up(2n,F)$.