Embeddings of Grassmann graphs

TL;DR

This paper characterizes all isometric and $l$-rigid isometric embeddings of Grassmann graphs, providing a comprehensive description of their structure and properties.

Contribution

It offers a complete classification of isometric and $l$-rigid embeddings between Grassmann graphs of different dimensions.

Findings

All isometric embeddings are characterized explicitly.

Conditions for $l$-rigid isometric embeddings are established.

The structure of embeddings depends on dimensions and parameters.

Abstract

Let $V$ and $V'$ be vector spaces of dimension $n$ and $n'$, respectively. Let $k\in\{2,...,n-2\}$ and $k'\in\{2,...,n'-2\}$. We describe all isometric and $l$-rigid isometric embeddings of the Grassmann graph $\Gamma_{k}(V)$ in the Grassmann graph $\Gamma_{k'}(V')$.