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5.iii.c. The Lagrangian Grassmannian

We use the definitions concerning the Lagrangian Grassmannian LG(n) from Section 4.iii.e. Consider the rational normal curve defined by

g(t)   =   1,  t,  t2/2,  ...,  tn/n!,  -tn+1/(n+1)!,  tn+2/(n+2)!,  ...,  (-1)n-1t2n-1/(2n-1)!, (5.13)

The flag F.(t) of subspaces osculating this rational normal curve is isotropic for all t in C.

Given a strict partition f : n+1 > f1 > f2 > ... > fl > 0 and an isotropic flag F., Define the Schubert variety Wf F. by

Wf F.   :=   {H in LG(n)  |  H meets Fn+1-fi in a subspace of dimension at least i, for i= 1, ..., l} .

The codimension of this Schubert variety is |f| = f1 + f1 + ... + fl. For example,

Wm F. = { H in LG(n)  |  H meets Fn+1-m non-trivially } .

With these definitions, we state the version of Theorem 4.12 proven in [So8].

Theorem 5.14 ([So8, Theorem 4.2])   Given a strict partition f, let N:=n(n+1)/2 - |f|, which is the dimension of Wf F.. If N>1, then there exist distinct real numbers t1, t2, ..., tN such that the intersection of Schubert varieties

Wf F.(0),   W1 F.(t1),   W1 F.(t2),   ...,   W1 F.(tN)

is zero-dimensional with no points real.

Thus the generalization of Conjecture 5.1 is badly false for the Lagrangian Grassmannian. On the other hand it holds for some enumerative problems in the Lagrangian Grassmannian.

Theorem 5.15 ([So11])   For any distinct real numbers t1, t2, t3, t4, the intersection of Schubert varieties in LG(3)

W1F.(t1),   W1F.(t2),   W2F.(t3),   W2F.(t4)

is transverse with all points real.

These two results illustrate a dichotomy that is emerging from experimentation. Call a list of strict partitions f1, f2, ..., fs with |f1|+|f2|+ ... +|fs| = n(n+1)/2, the dimension of LG(n) Lagrangian Schubert data. In every instance we have computed of a zero-dimensional intersection of Lagrangian Schubert varieties whose flags osculate the rational normal curve (5.13), the intersection has been transverse with either all points real or no points real. Most interestingly, the outcome--all real or no real--has depended only upon the Lagrangian Schubert data of the intersection.

Conjecture 5.16   Given Lagrangian Schubert data f1, f2, ..., fs and distinct real numbers t1, t2, ..., ts, the intersection of Lagrangian Schubert varieties

Wf1 F.(t1),   Wf2 F.(t2),   , ...,   Wfs F.(ts)

is transverse with either
(a)
all points real, or
(b)
no points real,
and the outcome (a) or (b) depends only upon the list f1, f2, ..., fs.

We do not have a good idea what distinguishes the Lagrangian Schubert data giving all points real from the data giving no points real. Further experimentation is needed.


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