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 ${\mathfrak{G}}_w^{(\beta )}(x;y)$ as a sum of products ${\mathfrak{G}}_v^{(\beta )}(x){\mathfrak{G}}_u^{(\beta )}(y)$ 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.