A flag F. in V is isotropic if Fn is Lagrangian and <Fi,F2n-i> = 0 for all i=1, 2, ..., 2n-1. Given an isotropic flag F., the Lagrangian 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 Wf is |f| := f1 + f2 + ... + fm. The Lagrangian Schubert calculus asks for the number of points in a transverse zero-dimensional intersection of Schubert varieties.
We remark that, while the orthogonal Grassmannian and the Lagrangian Grassmannian have combinatorially identical decompositions into Schubert varieties and also have the same dimension, they are quite different as spaces.
The simple Schubert variety W1F. consists of those Lagrangian subspaces meeting the Lagrangian subspace Fn non-trivially. The simple Lagrangian Schubert calculus is fully unreal.
We do not know if these (or many other) enumerative problems in the Lagrangian Schubert calculus are fully real. Experimentation§ suggests that the situation is complicated. Briefly, while many other enumerative problems in the Lagrangian Schubert calculus are fully unreal, there are a few which are fully real. For example, there exist 2 real isotropic 2-planes and 2 real isotropic 3-planes such that all 4 Lagrangian subspaces meeting each of these are real (see Theorem 5.15).
The problems of Theorem 4.12 may give examples
of enumerative problems that are not fully real.
Experimental evidence suggests however that this may be unlikely.
For example, (case n=3 of
Theorem 4.12) there are 16
Lagrangian subspaces in C6 having
non-trivial intersection with 6 general Lagrangian subspaces.
We computed 30,000 random instances of this enumerative problem,
and found several examples of 6 real Lagrangian subspaces such that all 16
Lagrangian subspaces meeting them are real.
Table 1 shows the number of these
30,000 systems having a given number of real solutions.