Cohomological consequences of the pattern map

TL;DR

This paper explores the cohomological effects of the pattern map on flag manifolds, providing explicit formulas for the induced map in cohomology and K-theory, revealing positivity and generalizing previous results.

Contribution

It introduces two new formulas for the cohomology map induced by the pattern map, connecting it to Schubert structure constants and extending known results to K-theory.

Findings

Coefficients are Schubert structure constants, ensuring positivity.

Formulas apply to both cohomology and K-theory.

Generalizes known type A formulas to broader contexts.

Abstract

Billey and Braden defined maps on flag manifolds 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. These formulas also hold for K-theory, and generalize known formulas in type A for cohomology and K-theory.