Projective Subgrassmannians of Polar Grassmannians

TL;DR

This paper classifies certain subspaces within polar Grassmannians, showing most are parabolic and that automorphisms are generally derived from the underlying polar space, extending previous classification results.

Contribution

It completes a classification of subgrassmannians in polar geometries, identifying when they are parabolic and describing exceptions in detail.

Findings

Most subspaces are parabolic residues of flags.

Automorphisms of Grassmannians are induced by polar space automorphisms.

Exceptions occur for Grassmannians of 2-spaces in projective m-spaces.

Abstract

In this short note, completing a sequence of studies by Cooperstein, Kasikova and Shult, we consider the k-Grassmannians of a number of polar geometries of finite rank n. We classify those subspaces that are isomorphic to the j-Grassmannian of a projective m-space. In almost all cases, these are parabolic, that is, they are the residues of a flag of the polar geometry. Exceptions only occur when the subspace is isomorphic to the Grassmannian of 2-spaces in a projective m-space and we describe these in some detail. This Witt-type result implies that automorphisms of the Grassmannian are almost always induced by automorphisms of the underlying polar space.