On bijections that preserve complementarity of subspaces

TL;DR

This paper characterizes bijections between sets of subspaces that preserve complementarity or adjacency relations, showing they are induced by semilinear transformations, even in infinite-dimensional cases.

Contribution

It proves that bijections preserving one relation also preserve the other and are induced by semilinear maps, extending results to infinite-dimensional vector spaces.

Findings

Bijections preserving one relation also preserve the other.

Such bijections are induced by semilinear bijections.

Results include infinite-dimensional vector spaces.

Abstract

The set $G$ of all $m$-dimensional subspaces of a $2m$-dimensional vector space $V$ is endowed with two relations, complementarity and adjacency. We consider bijections from $G$ onto $G'$, where $G'$ arises from a $2m'$-dimensional vector space $V'$. If such a bijection $\phi$ and its inverse leave one of the relations from above invariant, then also the other. In case $m\geq 2$ this yields that $\phi$ is induced by a semilinear bijection from $V$ or from the dual space of $V$ onto $V'$. As far as possible, we include also the infinite-dimensional case into our considerations.