Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909660608462848 |
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| author | Dennin, Hugh |
| author_facet | Dennin, Hugh |
| contents | The Cauchy identity gives a recipe for decomposing a double Grothendieck polynomial $\mathfrak{G}^{(β)}_w(x;y)$ as a sum of products $\mathfrak{G}^{(β)}_v(x)\mathfrak{G}^{(β)}_u(y)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21052 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams Dennin, Hugh Combinatorics 05A19 05E05 14N15 The Cauchy identity gives a recipe for decomposing a double Grothendieck polynomial $\mathfrak{G}^{(β)}_w(x;y)$ as a sum of products $\mathfrak{G}^{(β)}_v(x)\mathfrak{G}^{(β)}_u(y)$ 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. |
| title | Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams |
| topic | Combinatorics 05A19 05E05 14N15 |
| url | https://arxiv.org/abs/2506.21052 |