Transformations of polar Grassmannians preserving certain intersecting relations

TL;DR

This paper characterizes transformations of polar Grassmannians that preserve specific intersecting relations, showing they are induced by automorphisms of the underlying polar space, with applications to finite classical polar spaces.

Contribution

It proves that bijections preserving certain relations are induced by automorphisms, extending understanding of symmetry transformations in polar Grassmannians.

Findings

Transformations preserving ${rak R}_{1,1}$ are induced by automorphisms.

Preserving ${rak R}_{0,t}$ also implies automorphism induction under specified conditions.

Valencies of relations are distinct in finite classical polar spaces, aiding in classification.

Abstract

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 $(i,j)\ne (i',j')$.