Embeddings of flag manifolds

Question

Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a sub-vector space of dimension $a_i$.

Then $\mathbb{F}(a_1,\dots,a_k)$ embeds in the product of Grassmannians $G(a_1,V)\times\dots\times G(a_k,V)$ which in turn embeds in $\mathbb{P}^{N_1}\times\dots\times\mathbb{P}^{N_k}$ via the product of the Plücker embeddings. Now we can embed $\mathbb{P}^{N_1}\times\dots\times\mathbb{P}^{N_k}$ in a projective space $\mathbb{P}^N$ via the Segre embedding.

Finally, we get an embedding $\mathbb{F}(a_1,\dots,a_k)\hookrightarrow\mathbb{P}^N$. Is this embedding the minimal rational homogeneous embedding of $\mathbb{F}(a_1,\dots,a_k)$?

Answer 1

In general there is a more efficient way: $a_1,\ldots,a_k$ determines a Young diagram, and you can realize the flag variety as the stabilizer of a point in the unique closed orbit of ${\mathbb P}(U)$, where $U$ is the representation of $GL(V)$ corresponding to this diagram. Its dimension is given by the "hook formula".

Answer 2

Victor Petrov essentially answered your question showing that this projective embedding is, in general, not minimal. I'll just try to explain why this other embedding is, in fact, minimal by dimension. (I'm assuming everything is complex.)

First, the embedding. Let $n$ and $a_1,\ldots,a_k$ be as in your question, $F=\mathbb F(a_1,\ldots,a_k)$ and $G=SL_{n+1}$. Consider the dominant $G$-weight $\lambda=\omega_{a_1}+\ldots+\omega_{a_k}$ where the $\omega_i$ are the fundamental weights. Let $L_\lambda$ be the corresponding irreducible representation with highest weight vector $v_\lambda$. (Your $F$ is $G/P$ where $P$ is the parabolic subgroup preserving the line $\mathbb Cv_\lambda$.) Consider the projectivization $\mathbb P(L_\lambda)$ and the point $\mathrm v_\lambda$ therein corresponding to the line $\mathbb Cv_\lambda$. Then, $F$ can be realized as the (closed) orbit $G\mathrm v_\lambda\subset \mathbb P(L_\lambda)$.

Now, the minimality. Suppose we have a minimal projective embedding $\iota:F\hookrightarrow\mathbb P(U)$. Consider the pullback $\mathcal L=\iota^*(\mathcal O_{\mathbb P(U)}(1))$. The minimality implies that $\Gamma(F,\mathcal L)=U^*$. However, every line bundle on $F$ is $G$-equivariant (see Theorem 1 in http://www.math.harvard.edu/~lurie/papers/bwb.pdf) and every equivariant line bundle with global sections on $F$ is $\mathcal L_\mu$ for some $\mu$ which is a $\mathbb Z_{>0}$-linear combination of the $\omega_{a_i}$ (by Borel-Weil-Bott, again, see Lurie's text). So $\mathcal L=\mathcal L_\mu$ for some $\mu$ but $\Gamma(F,\mathcal L_\mu)=L_\mu^*$ and $\dim L_\mu\ge\dim L_\lambda$, i.e. $\dim U\ge \dim L_\lambda$.