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Further generalizations of Conjecture 5.1 for the orthogonal Grassmannian

As we shall see, the generalization of Conjecture 5.1 to most other flag varieties is likely false, but in a very interesting way. For the orthogonal Grassmannian however, there is a fair amount of evidence in its favor.

We use the definitions from Section 4.iii.d. Consider the rational normal curve defined by

g(t)   =   1,  t,  t2/2,  ...,  tn/n!,  -tn+1/(n+1)!,  tn+2/(n+2)!,  ...,  (-1)nt2n/(2n)! (5.8)

This has the property that g(t) is isotropic for all t. In fact, the flag F.(t) of subspaces osculating the curve at the point g(t) is an isotropic flag. The analog of Theorem 5.3 (involving codimension 1 Schubert varieties given by isotropic flags osculating the rational normal curve) holds for the Orthogonal Grassmannian. Let N := n(n+1)/2, the dimension of the orthogonal Grassmannian.

Theorem 5.9 ([So5, Corollary 3.3])   There exist t1, t2, ..., tN in R such that the simple Schubert varieties
W1 F.(t1),   W2 F.(t2),   ...,   WN F.(tN) (5.9)

intersect transversally in the complex orthogonal Grassmannian with all points real.

Based on this asymptotic result and many calculations, we make the following conjecture, the obvious generalization of Conjecture 5.1 to the orthogonal Grassmannian

Conjecture 5.10   Let f1, f2, ..., fs be strict partitions with |f1|+|f2|+...+|fs|. Then, for every distinct t1, t2, ..., ts in R, the intersection of the orthogonal Schubert varieties

Wf1 F.(t1),   Wf2 F.(t2),   ...,   Wfs F.(ts),

is (a) transverse, and (b) consists only of real points.


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