On bijections that preserve complementarity of subspaces

AbstractThe 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

Transformations of polar Grassmannians preserving certain intersecting relations

Let $\Pi$ be a polar space of rank $n\ge 3$. Denote by ${\mathcal G}_{k}(\Pi)$ the polar Grassmannian formed by singular subspaces of $\Pi$ whose projective dimension is equal to $k$. Suppose that $k$ is an integer not greater than $n-2$ and consider the relation ${\mathfrak R}_{i,j}$, $0\le i\le j\le k+1$ formed by all pairs $(X,Y)\in {\mathcal G}_{k}(\Pi)\times {\mathcal G}_{k}(\Pi)$ such that $\dim_{p}(X^{\perp}\cap Y)=k-i$ and $\dim_{p} (X\cap Y)=k-j$ ($X^{\perp}$ consists of all points of $\Pi$ collinear to every point of $X$). We show that every bijective transformation of ${\mathcal G}_{k}(\Pi)$ preserving ${\mathfrak R}_{1,1}$ is induced by an automorphism of $\Pi$ and the same holds for the relation ${\mathfrak R}_{0,t}$ if $n\ge 2t\ge 4$ and $k=n-t-1$. In the case when $\Pi$ is a finite classical polar space, we establish that the valencies of ${\mathfrak R}_{i,j}$ and ${\mathfrak R}_{i',j'}$ are distinct if $(i,j)\ne (i',j')$.Comment: 13 page

Embeddings of Grassmann graphs

Let $V$ and $V'$ be vector spaces of dimension $n$ and $n'$, respectively. Let $k\in\{2,...,n-2\}$ and $k'\in\{2,...,n'-2\}$. We describe all isometric and $l$-rigid isometric embeddings of the Grassmann graph $\Gamma_{k}(V)$ in the Grassmann graph $\Gamma_{k'}(V')$