On embeddings of Grassmann graphs in polar Grassmann graphs

TL;DR

This paper investigates how Grassmann graphs can be embedded into polar Grassmann graphs, showing they can be reduced to embeddings into simpler structures, and explores distance-preserving embeddings of dual polar graphs.

Contribution

It establishes reduction methods for embeddings of Grassmann and dual polar graphs into polar Grassmann graphs, clarifying their structural relationships.

Findings

Embeddings of Grassmann graphs can be reduced to embeddings into Grassmann graphs or polar space collinearity graphs.

3-embeddings of dual polar graphs with diameter ≥ 3 can be reduced to Grassmann graph embeddings.

The results unify understanding of embeddings between complex geometric graph structures.

Abstract

We establish that every embedding of a Grassmann graph in a polar Grassmann graph can be reduced to an embedding in a Grassmann graph or to an embedding in the collinearity graph of a polar space. Also, we consider $3$-embeddings, i.e. embeddings preserving all distances not greater than $3$, of dual polar graphs whose diameter is not less than $3$ in polar Grassmann graphs formed by non-maximal singular subspaces. Using the same arguments we show that every such an embedding can be reduced to an embedding in a Grassmann graph.