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Matchings and Clusters on Plabic Fences — arXiv:2609.08694

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.08694
License (as archived): CC0-1.0
Archived: 2026-10-16T14:49:03.094Z
SHA-256 of the archived text: 83e22dd7044682a23f956e8052b7ef982974bdcc4a5e5b2acf155a3e0b77637e
Stable URL of this page: https://metraipolis.com/documento/dc_e8aeb12442da

How to cite

Matchings and Clusters on Plabic Fences — arXiv:2609.08694 — arXiv (en), https://arxiv.org/abs/2609.08694 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e8aeb12442da (Metraipolis archive dc_e8aeb12442da, sha256 83e22dd7044682a23f956e8052b7ef982974bdcc4a5e5b2acf155a3e0b77637e)

BibTeX

@misc{dce8aeb12442da,
  title = {Matchings and Clusters on Plabic Fences — arXiv:2609.08694},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.08694},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e8aeb12442da; sha256 83e22dd7044682a23f956e8052b7ef982974bdcc4a5e5b2acf155a3e0b77637e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.08694. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Fix two positive braid words $\beta_+,\beta_-$, and let $\text{Conf}(\beta_+,\beta_-)$ be the corresponding (type A) double Bott-Samelson variety. Let $\beta$ be a double braid word containing $\beta_+,\beta_-$ as the top, bottom words. We consider the open cluster torus $T(C_\beta)$ associated to a triangulation $C_\beta$ in $\text{Conf}(\beta_+,\beta_-)$ from arXiv:1904.07992 , and we identify these with weighted plabic fences, where the usual local moves on planar bipartite graphs naturally correspond to change of torus coordinates in double Bott-Samelson variety. Moreover, $T(C_\beta)$ can be parametrized explicitly by certain matrix products, and also has monomial coordinates given by the cluster variables, which are matrix minors. We interpret these minors as a generalized notion of perfect matchings on weighted plabic fences, which may not be reduced plabic graphs. Using this interpretation, we show that the cluster variables are given by "generalized minimal matchings", extending the minimal matchings from arXiv:1606.08383 . Lastly, we derive a Chamber Ansatz formula for double Bott-Samelson varieties via face alternating products from dimer theory. In general, plabic fences are not reduced plabic graphs, yet we are able to extend and apply the standard tools such as local moves on plabic graphs, trips, and minimal matchings to them.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-09 in math.CO).
Categorie arXiv: Combinatorics (math.CO) ; Algebraic Geometry (math.AG).
Pagina del lavoro: https://arxiv.org/abs/2609.08694 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T14:32:29.868Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-09?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).