Strong equivariant positivity for homogeneous varieties and back-stable coproduct coefficients

TL;DR

This paper proves enhanced positivity results for coefficients in the equivariant cohomology and K-theory of flag manifolds, confirming conjectures and providing new formulas and methods for these calculations.

Contribution

It strengthens existing equivariant positivity theorems by constraining roots and introduces new formulas and computational methods for coproduct coefficients in infinite flag manifolds.

Findings

Positivity of coefficients in equivariant cohomology and K-theory established

Confirmation of conjectures on structure constants in infinite flag manifolds

New formulas and methods for computing coproduct coefficients

Abstract

Using a transversality argument, we demonstrate the positivity of certain coefficients in the equivariant cohomology and K-theory of a generalized flag manifold. This strengthens earlier equivariant positivity theorems (of Graham and Anderson-Griffeth-Miller) by further constraining the roots which can appear in these coefficients. As an application, we deduce that structure constants for comultiplication in the equivariant K-theory of an infinite flag manifold exhibit an unusual positivity property, establishing conjectures of Lam-Lee-Shimozono. Along the way, we present alternative formulas for the back stable Grothendieck polynomials defined by those authors, as well as a new method for computing the coproduct coefficients.