On bijections that preserve complementarity of subspaces

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 $G^{\prime }$, where $G^{\prime }$ arises from a $2m^{\prime }$-dimensional vector space $V^{\prime }$. If such a bijection $\varphi$ and its inverse leave one of the relations from above invariant, then also the other. In case $m\ge 2$ this yields that $\varphi$ is induced by a semilinear bijection from $V$ or from the dual space of $V$ onto $V^{\prime }$. As far as possible, we include also the infinite-dimensional case into our considerations.

References in corpus (2)

• The Connected Components of the Projective Line over a Ring
• Projective Representations II. Generalized chain geometries

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• Graphs, defined by Weyl distance or incidence, that determine a vector space
• Automorphisms of infinite Johnson graph
• The distant graph of the projective line over a finite ring with unity
• Order preserving transformations of the Hilbert grassmannian
• Automorphisms of infinite-dimensional hypercube graph
• Opposite relation on dual polar spaces and half-spin Grassmann spaces