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

TL;DR

This paper establishes a combinatorial bijection for Grothendieck polynomials using pipe dream rectification, introduces flow operators with symmetry, and connects to a dual RSK correspondence.

Contribution

It introduces a new bijection via pipe dream rectification, develops flow operators with symmetry, and links these to a dual RSK algorithm for Grothendieck polynomials.

Findings

Established a weight-preserving bijection for Grothendieck polynomials.

Developed flow operators exhibiting symmetry.

Connected rectification to a dual RSK correspondence.

Abstract

The Cauchy identity gives a recipe for decomposing a double Grothendieck polynomial $\mathfrak{G}^{(\beta)}_w(x;y)$ as a sum of products $\mathfrak{G}^{(\beta)}_v(x)\mathfrak{G}^{(\beta)}_u(y)$ 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.