3:00 p.m., Friday (March 18, 2005)
Math Annex 1100
University of Washington
Intersecting Schubert Varieties
Abstract: Using a blend of combinatorics and geometry, we give an
algorithm for algebraically finding all flags in any zero-dimensional
intersection of Schubert varieties with respect to three transverse
flags, and more generally, any number of flags. In particular, the
number of flags in a triple intersection is also a structure constant
for the cohomology ring of the flag manifold. Our algorithm is based
on solving a limited number of determinantal equations for each
intersection (far fewer than the naive approach). These equations are
also sufficient for computing Galois groups and monodromy of intersections
of Schubert varieties. We are able to limit the number of equations by
using the permutation arrays of Eriksson and Linusson.
We show that there exists a unique permutation array corresponding to
each realizable Schubert problem and give a simple recurrence relation
to compute the corresponding rank table. We describe pathologies of
Eriksson and Linusson's permutation array varieties, and define the
more natural permutation array schemes. In particular, we give several
counterexamples to the Realizability Conjecture based on classical projective
geometry. Finally, we give examples where Galois/monodromy groups
experimentally appear to be smaller than expected.
This is joint work with Ravil Vakil at Stanford University.
Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).