On geometry of linear involutions

Abstract

Let $V$ be an $n$-dimensional left vector space over a division ring $R$ and $n\ge 3$. Denote by ${\mathcal{G}}_k$ the Grassmann space of $k$-dimensional subspaces of $V$ and put ${\mathfrak{G}}_k$ for the set of all pairs $(S,U)\in {\mathcal{G}}_k\times {\mathcal{G}}_{n-k}$ such that $S+U=V$. We study bijective transformations of ${\mathfrak{G}}_k$ preserving the class of base subsets and show that these mappings are induced by semilinear isomorphisms of $V$ to itself or to the dual space $V^{\ast }$ if $n\ne 2k$; for $n=2k$ this fails. This result can be formulated as the following: if $n\ne 2k$ and the characteristic of $R$ is not equal to 2 then any commutativity preserving transformation of the set of $(k,n-k)$-involutions is extended to an automorphism of the group {\rm GL}(V).