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flag manifold
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.
Next: 6. Lower bounds in the real Schubert
calculus
Up: 5.iii. Generalizations of the conjecture of Shapiro and Shapiro: Table of Contents
Next: 5.ii.b. Shapiros' conjecture for the
flag manifold