Tableau formula for vexillary double Edelman--Greene coefficients

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 ${\hat{I}}^2$-Stanley symmetric functions, which expand into double stable Grothendieck functions with polynomial coefficients called double ${\hat{I}}^2$-Edelman--Greene coefficients. Anderson proved these coefficients are ${\hat{I}}^2$-Graham positive. For vexillary permutations, this is equivalent to a statement for skew flagged double ${\hat{I}}^2$-Grothendieck functions. Working in this setting, we give a tableau formula for vexillary double ${\hat{I}}^2$-Edelman--Greene coefficients that is manifestly ${\hat{I}}^2$-Graham positive. Our formula demonstrates a finer notion of positivity than was previously known.