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A Combinatorial Formula for Certain Elements of Upper Cluster Algebras
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we show that each non-acyclic skew-symmetric cluster algebra of rank 3 is properly contained in its upper cluster algebra.
Keywords: cluster algebra; upper cluster algebra; Dyck path. 
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     author = {Kyungyong Lee and Li Li and Matthew R. Mills},
     title = {A {Combinatorial} {Formula} for {Certain} {Elements} of {Upper} {Cluster} {Algebras}},
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     year = {2015},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a48/}
}
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Kyungyong Lee; Li Li; Matthew R. Mills. A Combinatorial Formula for Certain Elements of Upper Cluster Algebras. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a48/

[1] Bayleran D. E., Finnigan D. J., Haj Ali A., Lee K., Locricchio C. M., Mills M. R., Stiefel D. P. P., Tran T. M., Urbaniuk R., A new combinatorial formula for cluster monomials of equioriented type $A$ quivers, Preprint, , 2014 http://math.wayne.edu/k̃lee/Cluster_A_public.pdf

[2] Beineke A., Brüstle T., Hille L., “Cluster-cylic quivers with three vertices and the Markov equation (with an appendix by Otto Kerner)”, Algebr. Represent. Theory, 14 (2011), 97–112, arXiv: math.RA/0612213 | DOI | MR | Zbl

[3] Benito A., Muller G., Rajchgot J., Smith K., Singularities of locally acyclic cluster algebras, arXiv: 1404.4399 | MR

[4] Berenstein A., Fomin S., Zelevinsky A., “Cluster algebras. III: Upper bounds and double Bruhat cells”, Duke Math. J., 126 (2005), 1–52, arXiv: math.RT/0305434 | DOI | MR | Zbl

[5] Fomin S., Zelevinsky A., “Cluster algebras. I: Foundations”, J. Amer. Math. Soc., 15 (2002), 497–529, arXiv: math.RT/0104151 | DOI | MR | Zbl

[6] Grabowski J., Graded cluster algebras, arXiv: 1309.6170

[7] Gross M., Hacking P., Keel S., “Birational geometry of cluster algebras”, Algebr. Geom., 2 (2015), 137–175, arXiv: 1309.2573 | DOI | MR

[8] Gross M., Hacking P., Keel S., Kontsevich M., Canonical bases for cluster algebras, arXiv: 1411.1394

[9] Lee K., Li L., Zelevinsky A., “Greedy elements in rank 2 cluster algebras”, Selecta Math. (N.S.), 20 (2014), 57–82, arXiv: 1208.2391 | DOI | MR | Zbl

[10] Lee K., Schiffler R., “Positivity of cluster algebras”, Ann. of Math., 182 (2015), 73–125, arXiv: 1306.2415 | DOI | Zbl

[11] Matherne J. P., Muller G., “Computing upper cluster algebras”, Int. Math. Res. Not., 2015 (2015), 3121–3149, arXiv: 1307.0579 | DOI | Zbl

[12] Muller G., “${\mathcal A}={\mathcal U}$ for locally acyclic cluster algebras”, SIGMA, 10 (2014), 094, 8 pp., arXiv: 1308.1141 | DOI | MR | Zbl

[13] Plamondon P. G., “Generic bases for cluster algebras from the cluster category”, Int. Math. Res. Not., 2013 (2013), 2368–2420, arXiv: 1308.1141 | DOI | MR | Zbl

[14] Plamondon P. G., Private communication, 2014

[15] Seven A. I., “Maximal green sequences of skew-symmetrizable $3\times3$ matrices”, Linear Algebra Appl., 440 (2014), 125–130, arXiv: 1207.6265 | DOI | MR | Zbl

[16] Speyer D., An infinitely generated upper cluster algebra, arXiv: 1305.6867