Minimal equivariant embeddings of the Grassmannian and flag manifold

TL;DR

This paper constructs the smallest possible $ ext{SO}_n( ext{R})$-equivariant embedding of flag manifolds, including Grassmannians, into Euclidean space, significantly improving previous bounds.

Contribution

It establishes the minimal dimension for equivariant embeddings of flag manifolds and shows their uniqueness up to equivariant equivalence.

Findings

Minimal embedding dimension is $(n-1)(n+2)/2$

Embedding is unique up to equivariant equivalence

Significantly improves previous bounds by two orders of magnitude

Abstract

We show that the flag manifold $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$, with Grassmannian the special case $p=1$, has an $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding in an Euclidean space of dimension $(n-1)(n+2)/2$, two orders of magnitude below the current best known result. We will show that the value $(n-1)(n+2)/2$ is the smallest possible and that any $\operatorname{SO}_n(\mathbb{R})$-equivariant embedding of $\operatorname{Flag}(k_1,\dots, k_p, \mathbb{R}^n)$ in an ambient space of minimal dimension is equivariantly equivalent to the aforementioned one.