Towards generalized cohmology Schubert calculus via formal root polynomials

An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham–Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. In this paper we define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We focus on the case of the hyperbolic formal group law (related to elliptic cohomology). We study some of the properties of formal root polynomials.We give applications to the efficient computation of the transition matrix between two natural bases of the formal Demazure algebra in the hyperbolic case. As a corollary, we rederive in a transparent and uniform manner the formulas of Billey and Graham–Willems. We also prove the corresponding formula in connective $K$-theory, which seems new, and a duality result in this case. Other applications, including some related to the computation of Bott–Samelson classes in elliptic cohomology, are also discussed.

Keywords

Schubert calculus, equivariant oriented cohomology, flag variety, root polynomial, hyperbolic formal group law

2010 Mathematics Subject Classification

05E99, 14F43, 14M15, 19L47, 55N20, 55N22

Full Text (PDF format)

Received 19 July 2015

Accepted 2 May 2016

Published 1 September 2017