  Billey and Braden defined a geometric pattern map on flag manifolds which
extends the generalized pattern map of Billey and Postnikov on Weyl groups. The
interaction of this torus equivariant map with the Bruhat order and its action
on line bundles lead to formulas for its pullback on the equivariant cohomology
ring and on equivariant K-theory. These formulas are in terms of the Borel
presentation, the basis of Schubert classes, and localization at torus fixed
points.
