Combinatorics of Double Grothendieck Polynomials

We give a thorough combinatorial treatment of double Grothendieck polynomials that is accessible to anyone with a basic knowledge of algebra and combinatorics. The text includes many new combinatorial models for these and related polynomials and provides detailed combinatorial proofs. We generalize the Giambelli formula for double Schubert polynomials to a $k$-theoretic version and our proof specializes to a new combinatorial proof of the former.