A positive combinatorial formula for the double Edelman--Greene coefficients

TL;DR

This paper provides the first combinatorial proof of the positivity of double Edelman--Greene coefficients, using models like bumpless pipedreams and Bruhat order chains, advancing understanding in algebraic combinatorics.

Contribution

It introduces a combinatorial formula for double Edelman--Greene coefficients, demonstrating their positivity through new models and bijections.

Findings

First combinatorial proof of positivity

New formula using bumpless pipedreams and Bruhat chains

Establishes a symmetry property of increasing chains

Abstract

Lam, Lee, and Shimozono introduced the double Stanley symmetric functions in their study of the equivariant geometry of the affine Grassmannian. They proved that the associated double Edelman--Greene coefficients, the double Schur expansion coefficients of these functions, are positive, a result later refined by Anderson. They further asked for a combinatorial proof of this positivity. In this paper, we provide the first such proof, together with a combinatorial formula that manifests the finer positivity established by Anderson. Our formula is built from two combinatorial models: bumpless pipedreams and increasing chains in the Bruhat order. The proof relies on three key ingredients: a correspondence between these two models, a natural subdivision of bumpless pipedreams, and a symmetry property of increasing chains.