4.i. Totally positivity and the conjecture of Shapiro and Shapiro
A real upper triangular matrix g is totally positive if every minor
of g is positive, except those which vanish for all upper triangular
matrices.
Let TP be the set of all totally
positive matrices.
We remark that TP forms a
multiplicative semigroup.
Boris Shapiro and Michael Shapiro also have a conjecture concerning
totally positive matrices.
Here is a copy of a letter to Sottile
explaining that conjecture.
While dated October 1997, they made the conjecture public at least a year
previously.
Define a partial order on flags by
F. < G.
if G. = gF.
for some totally positive matrix g in
TP.
Unfortunately, this conjecture has been found to be false, and so this is
no longer relevant.
Conjecture 3. (Shapiro-Shapiro)
Let a1, a2,
..., an
be Schubert conditions with
(|a1| + |a2|
+ ... + |an| =
mp).
Then, given any real flags F.1,
F.2, ...,
F.n
with
F.1 <
F.2 < ... <
F.n,
the intersection of Schubert varieties
|
Ya1F.1,
Ya2F.2,
...,
YanF.n,
|
consists solely of real p-planes H.
To see this generalizes
Conjecture 2, let g(s)
be the matrix exp(sN), where N
is the nilpotent matrix whose only non-zero entries are
1, 2, ..., m+p-1,
just above the diagonal.
Then the ith row is the
(i-1)st derivative of the vector
(1, s, s2, s3, ...,
sm+p-1).
Hence if we define the flag F.(s) by
Fi(s)
is the row space of the first i rows of g(s),
then F.(s) is the flag of subspaces osculating the
standard rational normal curve at the point s
(cf. Section 3.iii).
Either by direct calculation, or using the fact the
TP is generated by exponentials of
upper diagonal matrices with positive entries
[Lö], we see that for
s > 0, the matrix g(s) is totally positive.
Thus, if
s1 < s2 < ... <
sn are real numbers, then we have
F.(s1) <
F.(s2) < ... <
F.(sn),
and so we see that
Conjecture 2
is a special case of Conjecture 3.