
(4.16) 
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 F_{n} is isotropic and <F_{i},F_{2n+1i}> = 0 for all i=1, 2, ..., 2n1. Given an isotropic flag F., the orthogonal Grassmannian has Schubert varieties W_{f} indexed by decreasing sequences f : n+1 > f_{1} > f_{2} > ... > f_{m} > 0 of positive integers, called strict partitions. (Here m can be any integer between 0 and n). The codimension of the Schubert variety W_{f} is f := f_{1} + f_{2} + ... + f_{m}. The orthogonal Schubert calculus asks for the number of points in a transverse zerodimensional intersection of Schubert varieties.
The simple Schubert variety W_{1}F. consists of those maximal isotropic subspaces meeting the subspace F_{n+1} nontrivially. The simple orthogonal Schubert calculus is fully real.
It remains an open problem whether the general Schubert calculus for the orthogonal Grassmannian is fully real.