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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.15531 |
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| _version_ | 1866909184055836672 |
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| author | Elkin, Moriah Musiker, Gregg Wright, Kayla |
| author_facet | Elkin, Moriah Musiker, Gregg Wright, Kayla |
| contents | We give a combinatorial interpretation for certain cluster variables in Grassmannian cluster algebras in terms of double and triple dimer configurations. More specifically, we examine several Gr(3,n) cluster variables that may be written as degree two or degree three polynomials in terms of Plücker coordinates, and give generating functions for their images under the twist map - a cluster algebra automorphism introduced in work of Berenstein-Fomin-Zelevinsky. The generating functions range over certain double or triple dimer configurations on an associated plabic graph, which we describe using particular non-crossing matchings or webs (as defined by Kuperberg), respectively. These connections shed light on a recent conjecture of Cheung et al., extend the concept of web duality introduced in a paper of Fraser-Lam-Le, and more broadly make headway on understanding Grassmannian cluster algebras for Gr(3,n). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_15531 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Twists of Gr(3,n) Cluster Variables as Double and Triple Dimer Partition Functions Elkin, Moriah Musiker, Gregg Wright, Kayla Combinatorics Representation Theory We give a combinatorial interpretation for certain cluster variables in Grassmannian cluster algebras in terms of double and triple dimer configurations. More specifically, we examine several Gr(3,n) cluster variables that may be written as degree two or degree three polynomials in terms of Plücker coordinates, and give generating functions for their images under the twist map - a cluster algebra automorphism introduced in work of Berenstein-Fomin-Zelevinsky. The generating functions range over certain double or triple dimer configurations on an associated plabic graph, which we describe using particular non-crossing matchings or webs (as defined by Kuperberg), respectively. These connections shed light on a recent conjecture of Cheung et al., extend the concept of web duality introduced in a paper of Fraser-Lam-Le, and more broadly make headway on understanding Grassmannian cluster algebras for Gr(3,n). |
| title | Twists of Gr(3,n) Cluster Variables as Double and Triple Dimer Partition Functions |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2305.15531 |