4.iii.d. The Orthogonal Schubert Calculus

<x, y>   :=   xi y2n+2-i   ,

(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+1-i> = 0 for all i=1, 2, ..., 2n-1. Given an isotropic flag F., the orthogonal Grassmannian has Schubert varieties W_f indexed by decreasing sequences f : n+1 > f₁ > f₂ > ... > 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| := f₁ + f₂ + ... + f_m. 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 F_n+1 non-trivially. 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.