Towards generalized cohomology Schubert calculus via formal root polynomials

TL;DR

This paper introduces formal root polynomials for arbitrary formal group laws, enabling uniform computation in generalized cohomology theories, and rederives key formulas in Schubert calculus with new insights for elliptic and connective K-theory.

Contribution

It defines formal root polynomials for any formal group law, extending Schubert calculus formulas to generalized cohomology theories, including elliptic and connective K-theory.

Findings

Rederived Billey and Graham-Willems formulas uniformly

Proved new formulas in connective K-theory

Discussed applications to Bott-Samelson classes in elliptic cohomology

Abstract

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 (corresponding 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.