Characterization of apartments in polar Grassmannians

TL;DR

This paper characterizes apartments in all polar Grassmannians of non-maximal singular subspaces, extending previous work on dual polar spaces and half-spin Grassmannians, and relates to embeddings of polar Johnson graphs.

Contribution

It provides a new characterization of apartments in polar Grassmannians beyond the maximal case, generalizing earlier results.

Findings

Characterization of apartments in all polar Grassmannians of non-maximal singular subspaces.

Connection between embeddings of polar Johnson graphs and Grassmannian apartments.

Extension of previous characterizations in dual polar spaces and half-spin Grassmannians.

Abstract

Buildings of types $\textsf{C}_n$ and $\textsf{D}_n$ are defined by rank $n$ polar spaces. The associated building Grassmannians are polar and half-spin Grassmannians. Apartments in dual polar spaces and half-spin Grassmannians were characterized in \cite{CKS}. We characterize apartments in all polar Grassmannians consisting of non-maximal singular subspaces. This characterization is a partial case of more general results concerning embeddings of polar Johnson graphs in polar Grassmann graphs.