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 $V^{*}$ 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).