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 ′ , where G ′ arises from a 2m ′ -dimensional vector space V ′ . If such a bijection ϕ and its inverse leave one of the relations from above invariant, then also the other. In case m ≥ 2 this yields that ϕ is induced by a semilinear bijection from V or from the dual space of V onto V ′ .As far as possible, we include also the infinite-dimensional case into our considerations.2000 Mathematics Subject Classification: 51A10, 51A45, 05C60.

The Distant Graph of the Projective Line Over a Finite Ring with Unity

“…These graphs have been thoroughly investigated by many authors, for example in [5,6,20]. In [19] finite Grassmann graphs are uniquely determined as distance-regular graphs, however, this special case of the Grassmann graph G (n, 2n, q) is not characterized.…”

“…G(M n (q), Δ) can be described using the notion of the Grassmann graph [6,Theorem 3.2]. These graphs have been thoroughly investigated by many authors, for example in [5,6,20].…”

Geometries on σ-Hermitian matrices

“…In the matrix-geometric setting two n-dimensional subspaces of K 2n are called adjacent (∼) if, and only if, their intersection has dimension n − 1. However, adjacency can be expressed in terms of being distant and vice versa [5,Theorem 3.2]. See also [12, 3.2.4], where complementary subspaces are called opposite.…”

Opposite Relation on Dual Polar Spaces and Half-Spin Grassmann Spaces

“…(1) =⇒ (2). Show that every point S = S 1 , S 2 on the line joining S 1 with S 2 is as required (this line consists of all elements of O δ (Π) containing S 1 ∩ S 2 ).…”

“…Two elements of G k (V ) are called opposite if the distance between them is equal to the diameter. It follows from Blunck-Havlicek's results [2] (see also [5]) that the adjacency relation can be characterized in terms of the relations to be opposite: distinct S 1 , S 2 ∈ G k (V ) are adjacent if and only if there exists S ∈ G k (V ) \ {S 1 , S 2 } such that every element of G k (V ) opposite to S is opposite to S 1 or S 2 . In particular, this implies that every bijective transformation of G k (V ) preserving the relation to be opposite in both directions is an automorphism of the Grassmann graph.…”