Minimal equivariant embeddings of the Grassmannian and flag manifold

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