On embedded products of Grassmannians

TL;DR

This paper investigates the embeddings of products of Grassmannians, providing conditions under which a given embedding can be expressed via automorphisms and collineations related to standard Plücker and Segre embeddings.

Contribution

It establishes a sufficient condition for when an arbitrary embedding of Grassmannian products aligns with the standard embedding through automorphisms and collineations.

Findings

Characterization of embeddings via automorphisms and collineations

Conditions for the equivalence of embeddings to standard forms

Extension of classical embedding results to product Grassmannians

Abstract

Let $\Gamma'$ and $\Gamma$ be two Grassmannians. The standard embedding $\phi:\Gamma'\times\Gamma\to \bar{P}$ is obtained by combining the Pl\"ucker and Segre embeddings. Given a further embedding $\eta: \Gamma'\times\Gamma \to P'$, we find a sufficient condition for the existence of $\alpha\in Aut(\Gamma)$ and of a collineation $\psi: \bar{P} \to P'$ such that $\eta=({\rm id}_{\Gamma'}\times\alpha)\phi\psi$.