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no. 3
Answering two OPAC problems involving Banff quivers
Ervin, Tucker J.1; Jackson, Blake2
1 University of Alabama Dept. of Mathematics Tuscaloosa AL 35487 (USA)
2 University of Connecticut Dept. of Mathematics Storrs CT 06269 (USA)
Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 853-860.

In a post on the Open Problems in Algebraic Combinatorics (OPAC) blog, E. Bucher and J. Machacek posed three open problems: OPAC-033, OPAC-034, and OPAC-035. These three problems deal with the relationships between three infinite classes of quivers: the Banff, Louise, and 𝒫 quivers. OPAC-034 asks whether or not every Banff quiver can be verified to be Banff by only considering sources and sinks, and OPAC-035 asks whether or not every Banff quiver is contained in the class 𝒫. We answer both questions in the affirmative, showing that every Banff quiver can be verified to be Banff by using sources and sinks, and therefore that every Banff quiver lives in the class 𝒫. We also make some progress on OPAC-033, showing a result similar to our result OPAC-034 for Louise quivers.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.352
Classification: 13F60, 05E40
Keywords: Cluster algebras, Banff, Louise, triangular extension, quiver, mutation

Ervin, Tucker J. 1; Jackson, Blake 2

1 University of Alabama Dept. of Mathematics Tuscaloosa AL 35487 (USA)
2 University of Connecticut Dept. of Mathematics Storrs CT 06269 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Ervin, Tucker J.; Jackson, Blake. Answering two OPAC problems involving Banff quivers. Algebraic Combinatorics, Volume 7 (2024) no. 3, pp. 853-860. doi : 10.5802/alco.352. https://alco.centre-mersenne.org/articles/10.5802/alco.352/

[1] Classes of directed graphs (Bang-Jensen, Jørgen; Gutin, Gregory, eds.), Springer Monographs in Mathematics, Springer, Cham, 2018, xxii+636 pages | DOI | MR

[2] Bernšteĭn, I. N.; Gel’fand, I. M.; Ponomarev, V. A. Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk, Volume 28 (1973) no. 2(170), pp. 19-33 | MR | Zbl

[3] Bucher, Eric; Machacek, John Reddening sequences for Banff quivers and the class 𝒫, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 16 (2020), Paper no. 049, 11 pages | DOI | MR | Zbl

[4] Bucher, Eric; Machacek, John Banff, Louise, and class 𝒫 quivers, 2021 https://realopacblog.wordpress.com/2021/01/10/banff-louise-and-class-p-quivers/ (Accessed 2023-04-03)

[5] Gabriel, Peter Unzerlegbare Darstellungen. I, Manuscripta Math., Volume 6 (1972), pp. 71-103 correction, ibid. 6 (1972), 309 | DOI | MR | Zbl

[6] Kontsevich, Maxim; Soibelman, Yan Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, 2008 | arXiv

[7] Ladkani, Sefi On cluster algebras from once punctured closed surfaces, 2013 | arXiv

[8] Lam, Thomas; Speyer, David E. Cohomology of cluster varieties I: Locally acyclic case, Algebra Number Theory, Volume 16 (2022) no. 1, pp. 179-230 | DOI | MR | Zbl

[9] Muller, Greg Locally acyclic cluster algebras, Adv. Math., Volume 233 (2013), pp. 207-247 | DOI | MR | Zbl

[10] Muller, Greg Skein and cluster algebras of marked surfaces, Quantum Topol., Volume 7 (2016) no. 3, pp. 435-503 | DOI | MR | Zbl

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