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arXiv · 2506.21052

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

Hugh Dennin
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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.

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Hugh Dennin. 2025-06-26. Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams. https://arxiv.org/abs/2506.21052

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