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. We define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We usethese polynomials to simplify the approach of Billey and Graham-Willems, as well as to generalize it to connective $K$-theory and elliptic cohomology. Another result is concerned with defining a Schubert basis in elliptic cohomology (i.e., classes independent of a reduced word), using the Kazhdan-Lusztig basis of the corresponding Hecke algebra. Un résultat combinatoire important dans le calcul de Schubert pour la cohomologie et la $K$-théorie équivariante est représenté par les formules de Billey et Graham-Willems pour la localisation des classes de Schubert aux points fixes du tore. Ces formules sont uniformes pour tous les types de Lie, et sont basés sur le concept d’un polynôme de racines. Nous définissons les polynômes formels de racines associées à une loi arbitraire de groupe formel (et donc à une théorie de cohomologie généralisée). Nous utilisons ces polynômes pour simplifier les preuves de Billey et Graham-Willems, et aussi pour généraliser leurs résultats à la $K$-théorie connective et la cohomologie elliptique. Un autre résultat concerne la définition d’une base de Schubert dans cohomologie elliptique (c’est à dire, des classes indépendantes d’un mot réduit), en utilisant la base de Kazhdan-Lusztig de l’algèbre de Hecke correspondant.
Highlights
Modern Schubert calculus has been mostly concerned with the cohomology and K-theory of generalized flag manifolds G/B, where G is a connected complex semisimple Lie group and B a Borel subgroup; Kac-Moody flag manifolds have been studied, but we will restrict ourselves here to the finite case
We focus on elliptic cohomology, more precisely, on the associated hyperbolic formal group law, which we view as the first case after K-theory in terms of complexity. (The correspondence between generalized cohomology theories and formal group laws is explained below.) The main difficulty beyond K-theory is the fact that the naturally defined cohomology classes corresponding to a Schubert variety depend on its chosen Bott-Samelson desingularization, and on a reduced word for the given Weyl group element
We extend the combinatorial formulas for localizations of Schubert classes in cohomology and K-theory, which are due to Billey [Bi99] and Graham-Willems [Gr02, Wil04], respectively
Summary
Modern Schubert calculus has been mostly concerned with the cohomology and K-theory (as well as their quantum deformations) of generalized flag manifolds G/B, where G is a connected complex semisimple Lie group and B a Borel subgroup; Kac-Moody flag manifolds have been studied, but we will restrict ourselves here to the finite case. We extend the combinatorial formulas for localizations of Schubert classes in (torus equivariant) cohomology and K-theory, which are due to Billey [Bi99] (cf [AJS94]) and Graham-Willems [Gr02, Wil04], respectively. These formulas are based on the concept of a root polynomial. We derive a duality result in connective K-theory, as well as other related results, and state a positivity conjecture Another result is concerned with defining a Schubert basis in elliptic cohomology (i.e., classes independent of a reduced word), using the Kazhdan-Lusztig basis of the corresponding Hecke algebra
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