Unimodality and Cluster Algebras from Surfaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913097661284352 |
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| author | Kang, Wonwoo Lee, Kyeongjun Lim, Eunsung |
| author_facet | Kang, Wonwoo Lee, Kyeongjun Lim, Eunsung |
| contents | We prove that the rank polynomial of the lattice of order ideals of a loop fence poset is unimodal. This poset arises as the poset of join-irreducibles in the lattice of good matchings of loop graphs associated with notched arcs. Equivalently, such polynomials can be obtained by evaluating all coefficient variables in an F-polynomial at a single variable q. We also conclude that the rank polynomial of any tagged arc, whether plain or notched, is not only unimodal but also satisfies a symmetry condition known as almost interlacing. Furthermore, when the lamination consists of a single curve, the cluster expansion-evaluated by setting all cluster variables to 1 and all coefficient variables to q-is also unimodal. We conjecture that polynomials in this case are log-concave. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unimodality and Cluster Algebras from Surfaces Kang, Wonwoo Lee, Kyeongjun Lim, Eunsung Combinatorics 13F60, 06A07 We prove that the rank polynomial of the lattice of order ideals of a loop fence poset is unimodal. This poset arises as the poset of join-irreducibles in the lattice of good matchings of loop graphs associated with notched arcs. Equivalently, such polynomials can be obtained by evaluating all coefficient variables in an F-polynomial at a single variable q. We also conclude that the rank polynomial of any tagged arc, whether plain or notched, is not only unimodal but also satisfies a symmetry condition known as almost interlacing. Furthermore, when the lamination consists of a single curve, the cluster expansion-evaluated by setting all cluster variables to 1 and all coefficient variables to q-is also unimodal. We conjecture that polynomials in this case are log-concave. |
| title | Unimodality and Cluster Algebras from Surfaces |
| topic | Combinatorics 13F60, 06A07 |
| url | https://arxiv.org/abs/2508.04396 |