On Geometry of Linear Involutions

Let V be an n-dimensional left vector space over a division ring R and n ≥ 3. Denote by Gk the Grassmann space of k-dimensional subspaces of V and put Gk for the set of all pairs (S, U) ∈ Gk ×Gn−k such that S+U = V . We study bijective transformations of Gk 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 6= 2k; for n = 2k this fails. This result can be formulated as the following: if n 6= 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 GL(V).