Cohmological Consequences of the Pattern Map


Billey and Braden defined maps on flag manifolds of any Lie 
type that are the geometric counterpart of permutation patterns. 
A section of their pattern map is an embedding of the flag 
manifold of a Levi subgroup into the full flag manifold.
We give two expressions for the induced map on cohomology.
One is in terms of generators and the other is in terms of 
the Schubert basis.  We show that the coefficients in the 
second expression are naturally Schubert structure constants 
and therefore positive.  Similar results hold for K-theory, 
generalizing known formulas in type A for cohomology and
K-theory.   This is joint work with Praise Adeyemo of the
University of Ibadan in Nigeria.