On geometry of linear involutions

2005/04/20 by Mark Pankov, Pankov, Mark
Engineering · Mathematics · #Advanced Topics in Algebra #Finite Group Theory Research #graph theory and CDMA systems #math.GR #msc:14M15 #msc:51M35

paper · pdf · doi:10.48550/arxiv.math/0504409

arxiv created 2005/04/20 · arxiv updated 2009/12/01

Abstract

Let V be an n-dimensional left vector space over a division ring R and n≥ 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)∈ \mathcal G_k× \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≠ 2k; for n=2k this fails. This result can be formulated as the following: if n≠ 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).

Related

• Transformations of Grassmannians and automorphisms of linear groups
• Apartments preserving transformations of Grassmannians of infinite-dimensional vector spaces
• Characterization of isometric embeddings of Grassmann graphs
• Geometrical characterization of semilinear isomorphisms of vector spaces and semilinear homeomorphisms of normed spaces
• Linear maps preserving product of involutions