Combinatorial proofs of the type A quiver component formulas

The K-theoretic quiver component formula expresses the K-polynomial of a type A quiver locus as an alternating sum of products of double Grothendieck polynomials. This formula was conjectured by A. Buch and R. Rimányi and later proved by R. Kinser, A. Knutson, and the second author. We provide a new proof of this formula which replaces Gröbner degenerations by combinatorics. Along the way, we obtain a new proof of A. Buch and R. Rimányi’s cohomological quiver component formula. Again, our proof replaces geometric techniques by combinatorics.