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Let $G=SO_7$ and let $P$ be the maximal parabolic corresponding to the fundamental weight $\varpi_3$. Since $\varpi_3$ is minuscule, $G/P$ may be fairly easy to study. Is the structure of $G/P$ known in literature ? What are the Plücker type relations (quadratic) in this case ? Are the relations same as they are in the Grassmannian $G(3,7)$ ? I have a feeling that the quotient is $\mathbb P^2$.

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If $G = SO(7)$ and $P$ corresponds to $\varpi_3$, then $$ G/P \cong Q^6 $$ is a 6-dimensional quadric. In particular, there is a unique Plücker relation in this case (the equation of the quadric).

The isomorphism can be sen as a combination of a general isomorphism $$ SO(2n-1)/P_{\varpi_{n-1}} \cong SO(2n)/P_{\varpi_{n}} $$ and the triality isomorphism $$ SO(8)/P_{\varpi_{4}} \cong SO(8)/P_{\varpi_{1}} \cong Q^6. $$

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  • $\begingroup$ Ah, good point about triality! $\endgroup$ – Sam Hopkins May 15 at 4:32
  • $\begingroup$ Is there a reference why $SO(8)/P_{\varpi_1} \cong Q^6$ ? $\endgroup$ – icmes imrf May 16 at 0:56
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Standard Monomial Theory (https://en.wikipedia.org/wiki/Standard_monomial_theory) provides a uniform description of the coordinate ring of any minuscule $G/P$ (and in fact with more work applies to other $G/P$ as well). In the minuscule case the main result, due to Seshadri, is that a basis of the coordinate ring is given by multichains in the corresponding minuscule poset. However, this result only says that in principle there is some way to express a product of standard monomials as a sum of standard monomials (in other words, we have an "algebra with a straightening law"). But we might not know the explicit coefficients in this expression.

Fortunately, for the case you care about (the maximal orthogonal Grassmannian) the specific relations actually have been worked out. Namely, the paper "Pfaffians and Shuffling Relations for the Spin Module" (https://arxiv.org/abs/1203.2943) by Chirivì and Maffei contains the exact relations (phrased in terms of pfaffians of skew-symmetric matrices) you're looking for.

There is one small complication in that Chirivì and Maffei work with $So(2n+2)$ (Type D) and you're interested in $So(2n+1)$ (Type B), but if I remember correctly we actually have an isomorphism of the relevant varieties $OG(n+1,2n+2)\simeq OG(n,2n+1)$, so this makes no difference.

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  • $\begingroup$ Does this mean that in this particular case the Plucker relation is same as those in $G(3,7)$ ? This is sitting inside $G(3,7)$ as a family of isotropic $3$ dimensional subspaces. $\endgroup$ – icmes imrf May 15 at 2:06
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    $\begingroup$ @icmesimrf: see my edited answer for a paper with the exact relations you're looking for. $\endgroup$ – Sam Hopkins May 15 at 2:23

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