Opposite relation on dual polar spaces and half-spin Grassmann spaces

TL;DR

This paper characterizes the collinearity relation in half-spin Grassmann spaces using the opposite relation in collinearity graphs, and demonstrates that this characterization fails for dual polar spaces.

Contribution

It provides a new characterization of collinearity in half-spin Grassmann spaces and clarifies its limitations in dual polar spaces.

Findings

Collinearity in half-spin Grassmann spaces can be characterized by opposite relations.

The same characterization does not apply to dual polar spaces.

The paper clarifies the structural differences between these spaces.

Abstract

We characterize the collinearity (adjacency) relation of half-spin Grassmann spaces in terms of the relation to be opposite in the corresponding collinearity graphs. Also we show that this characterization does not hold for dual polar spaces.