Equivariant cohomology theories and the pattern map

TL;DR

This paper explores the geometric pattern map on flag manifolds, analyzing its effects on equivariant cohomology and K-theory, and providing explicit formulas for its pullback actions in algebraic and combinatorial terms.

Contribution

It extends the pattern map to flag manifolds and derives explicit formulas for its pullback on equivariant cohomology and K-theory, linking geometric and algebraic structures.

Findings

Formulas for the pullback on equivariant cohomology in Borel presentation

Formulas for the pullback on equivariant K-theory

Analysis of the pattern map's interaction with Bruhat order

Abstract

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.