# $F_4$ flag variety

As flag variety or a homogeneous variety is a quotient $\Sigma=G/P$ of a reductive Lie group $G$ by one of its parabolic subgroups $P$. The subgroup $P$ fixes a flag of subspaces of standard representation $V$ of $G$. There is an embedding of projective varieties $\Sigma\subset \mathbb P V_{\lambda}$, where $V_{\lambda}$ is some highest weight representation of $G$.

For the exceptional Lie group of type $G_2$, if we consider its highest weight representation for highest weight $\omega_2$ then we have an embedding of a homogeneous variety $\Sigma\subset \mathbb P V_{\omega_2}$. Since its a subvariety of Gr(2,7), which can easily be seen to be a "flag variety", so we can some how realise this $G_2$ variety as a flag variety.

If we consider an exceptional Lie group $F_4$ and take its highest weight representation with highest weight $\omega_1$ then $V_{\omega_1}$ is 26 dimensional and we have an embedding of $F_4$ homogeneous variety $\Sigma ^{15}\subset \mathbb P V_{\omega_1}$, which is a codimension 10 embedding. My question is that how can we realise this variety as a "flag variety" or is it also a subvariety of some other standard flag variety?

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Why is this community wiki? –  José Figueroa-O'Farrill Oct 27 '11 at 18:13
But this 15-dimensional variety already is a flag variety of $F_4$; it's a quotient of $F_4$ by one of its maximal parabolic subgroups. Are you asking whether there is some classical Lie group $G$ that contains $F_4$ and that acts on $\Sigma$ in such a way that $\Sigma = G/P$? (I think this question was essentially asked before on MO, but I can't find it now.) In that case, the answer is 'no'. Also, is your $F_4$ complex or are you dealing with one of the real forms? –  Robert Bryant Oct 27 '11 at 22:41
I am working over the complex numbers. Actually I am more interested in tautological vector bundles on these varieties, which are $P$ representations. For $G_2$ we can restrict the bundles on Gr(2,7) to $G_2$ variety. What "flag" of which finite dimensional "vector space" is fixed by $P$ in the case I mentioned? –  Asad Jamal Oct 27 '11 at 23:14
Is the question "how to realize this homogeneous space as parametrizing certain subspaces"? If so, I think it's a very interesting one in general, and only partially known. For that specific $F_4$-variety, though, I think it's a hypersurface in the octonionic projective plane. To get a concrete picture, you'll want to look at Albert algebras. Unraveling the definition of OP^2, this does parametrize certain (1-dimensional) subspaces. The more interesting parts will be for other fundamental representations... (I'm assuming you're dealing with complex forms, btw.) –  Dave Anderson Oct 27 '11 at 23:25
Some insight as to why F_4 is harder than G_2: G_2 is the subgroup fixed by a group of automorphisms of D_4, which is a classical group and hence is easily defined by matrices. On the other hand, F_4 is the subgroup fixed by a group of automorphisms of E_6, which is not a classical group. (The connection to E_6 is why OP^2 shows up.) –  Alexander Woo Oct 28 '11 at 4:11

Section 9.1 of Carr-Garibaldi contains a nice explanation of this: http://arxiv.org/abs/math/0503201

In general, this paper explains how to realize G/P as flags of special kinds of subspaces for any type (except $E_8$)

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Edited only to correct the author's name: "Skip Garibaldi" is a full name, not two co-authors or a hyphenated surname. –  Noam D. Elkies Oct 29 '11 at 0:30
whoops, I meant to write Carr-Garibaldi –  Steven Sam Oct 29 '11 at 2:07
Ah, that explains it: normally one wouldn't give the author's full name in any case. (Evidently I didn't recognize the book...) –  Noam D. Elkies Oct 29 '11 at 4:42

Boris Rosenfelds book "The Geometry of Lie Groups" seems to address that question in Thm. 7.35 (page 358). Be warned that Rosenfelds book is both fascinating (for the wealth of its knowldege) and very frustrating (for reasons that you can find out for yourself).

If I understand these matters correctly (a non-negligible "if"), there are notions of "points", "lines", "planes" and "symplecta" as submanifolds of the Cayley plane $\mathbb{O}P^2$. These are submanifolds of $\mathbb{O} P^2$ that are isometric to $\mathbb{C}P^1$, $\mathbb{C}P^2$, $\mathbb{C}Sp^5$, resp. the "absolute hermitian conic of $\mathbb{O}P^2$" (whatever that may be). The parabolic quotients $F_4/P$ then represent flags of these structures, with incidence requirements. (I think this approach goes back to Freudenthal, so you might find a clearer description in his writings.)

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A related question is this: mathoverflow.net/questions/16474/matrix-representation-for-f-4/… (I would link to Marty's comment on his own answer if I knew how to do this.) –  Christian Nassau Oct 28 '11 at 8:10
This does seem like it answers the question (at least the way I've interpreted it). I googled for some terms that I learned from Rosenfeld's book, and I found the following paper arxiv.org/abs/math/9810140 of Landsberg and Manivel which seems like it might be a more accessible reference. In particular, their section 6.2 seems to be very relevant, but I can't claim that I fully understand what's going on there –  Faisal Oct 28 '11 at 8:50

As far as I remember $F_4/P_{\omega_1}$ is a hyperplane section of $E_6/P_{\omega_1}$.

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