Tableau formula for vexillary double Edelman--Greene coefficients

TL;DR

This paper provides a tableau formula for vexillary double β-Edelman--Greene coefficients, revealing their positivity properties in the context of double Grothendieck polynomials and K-theory of flag varieties.

Contribution

It introduces a new tableau formula for vexillary double β-Edelman--Greene coefficients that explicitly shows their β-Graham positivity.

Findings

The formula is manifestly β-Graham positive.

It demonstrates a finer positivity notion than previously known.

Provides combinatorial tools for K-theory classes of flag varieties.

Abstract

Lam, Lee and Shimozono recently introduced backstable double Grothendieck polynomials to represent $K$-theory classes of the infinite flag variety. They used them to define double $\beta$-Stanley symmetric functions, which expand into double stable Grothendieck functions with polynomial coefficients called double $\beta$-Edelman--Greene coefficients. Anderson proved these coefficients are $\beta$-Graham positive. For vexillary permutations, this is equivalent to a statement for skew flagged double $\beta$-Grothendieck functions. Working in this setting, we give a tableau formula for vexillary double $\beta$-Edelman--Greene coefficients that is manifestly $\beta$-Graham positive. Our formula demonstrates a finer notion of positivity than was previously known.