Embeddings of Grassmann graphs

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 $Γ_{k}(V)$ in the Grassmann graph $Γ_{k'}(V')$.

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