Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams

The Cauchy identity gives a recipe for decomposing a double Grothendieck polynomial G^ (β) w (x;y) as a sum of products G^ (β) ᵥ (x) G^ (β) ᵤ (y) of single Grothendieck polynomials. Combinatorially, this identity suggests the existence of a weight-preserving bijection between certain families of diagrams called pipe dreams. In this paper, we provide such a bijection using an algorithm called pipe dream rectification. In turn, rectification is built from a new class of flow operators which themselves exhibit a surprising symmetry. Finally, we examine other applications of rectification including an insertion algorithm on pipe dreams which recovers a variant of the dual RSK correspondence.