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

Let V be a 2n-dimensional vector space equipped with a nondegenerate alternating bilinear form, which we may take to be
   := xi y2n+1-i - yi x2n+1-i   ,
(4.17)
the sum over all i from 1 to n. Here, x and y are vectors in V and so have 2n components. 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 and Lagrangian subspaces are isotropic subspaces with this maximal dimension. The Lagrangian Grassmannian LG(n) is the set of Lagrangian subspaces of V, an algebraic manifold of dimension n(n+1)/2.

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.

Theorem 4.12   There exist isotropic real flags F.1, F.2, ..., F.N, (where N = n(n+1)/2) such that the intersection of simple Lagrangian Schubert varieties

W1 F.1,   W1 F.2,   ...,   W1 F.N

is transverse with no real points.

Remark 4.13   In [So8, Theorem 4.2] flags were given so that the intersection had no real points, and it was not known if that intersection was transverse. Perturbing those flags slightly (so that the intersection becomes transverse) gives the above result.

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.

Table 1: Frequency with given number of real solutions.
 Number of real solutions 0 2 4 6 8 10 12 14 16 Frequency 4983 8176 9314 5027 1978 445 67 8 2

§This will be reported in the forthcoming article [So11].

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