Combinatorics of Double Grothendieck Polynomials

We give a proof of the generalized Cauchy identity for double Grothendieck polynomials, a combinatorial interpretation of the stable double Grothendieck polynomials in terms of triples of tableaux, and an interpolation between the stable double Grothendieck polynomial and the weak stable double Grothendieck polynomial. This so-called half weak stable double Grothendieck polynomial evaluated at $x=y$ generalizes the type $B$ Stanley symmetric function of Billey and Haiman and is $Q$-Schur positive by degree. We conclude with two open problems as well as a conjecture regarding the $K$-theoretic analogues of factorial Schur $Q$-functions defined by Ikeda and Naruse. The conjecture is supported by code given in the appendices.