Next: 4.iii.e Unreality in the Lagrangian Schubert calculus
Previous: 4.iii.c. Schubert Calculus of Flags

### 4.iii.d. The Orthogonal Schubert Calculus

Let V be a (2n+1)-dimensional vector space with basis e1, e2, ..., e2n+1. We equip V with a split nondegenerate symmetric form, which we may take to be
   := xi y2n+2-i   ,
(4.16)
the sum over all i from 1 to 2n+1. Here, x and y are vectors in V with 2n+1 components in the given basis.

A subspace H of V is isotropic if the restriction of the form to H is identically zero, <H,H> = 0. The dimension of an isotropic subspace is at most n. The orthogonal Grassmannian OG(n) is the set of isotropic subspaces of V with this maximal dimension. This is an algebraic manifold of dimension N := n(n+1)/2.

A complete flag F. in V is isotropic if Fn is isotropic and <Fi,F2n+1-i> = 0 for all i=1, 2, ..., 2n-1. Given an isotropic flag F., the orthogonal Grassmannian has Schubert varieties Wf indexed by decreasing sequences f : n+1 > f1 > f2 > ... > fm > 0 of positive integers, called strict partitions. (Here m can be any integer between 0 and n). The codimension of the Schubert variety Wf is |f| := f1 + f2 + ... + fm. The orthogonal Schubert calculus asks for the number of points in a transverse zero-dimensional intersection of Schubert varieties.

The simple Schubert variety W1F. consists of those maximal isotropic subspaces meeting the subspace Fn+1 non-trivially. The simple orthogonal Schubert calculus is fully real.

Theorem 4.11   There exist isotropic real flags F.1, F.2, ..., F.N, (where N = n(n+1)/2) such that the intersection of simple orthogonal Schubert varieties

W1 F.1,   W1 F.2,   ...,   W1 F.N

is transverse with all points real.

It remains an open problem whether the general Schubert calculus for the orthogonal Grassmannian is fully real.

Next: 4.iii.e Unreality in the Lagrangian Schubert calculus