Geodesic
$T$-Path Formula and Atomic Bases for Cluster Algebras of Type $D$
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 extend a $T$-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type $D$.
Keywords: cluster algebra; triangulated surface; atomic basis. 
@article{SIGMA_2015_11_a59,
     author = {Emily Gunawan and Gregg Musiker},
     title = {$T${-Path} {Formula} and {Atomic} {Bases} for {Cluster} {Algebras} of {Type~}$D$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2015},
     volume = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a59/}
}
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%0 Journal Article
%A Emily Gunawan
%A Gregg Musiker
%T $T$-Path Formula and Atomic Bases for Cluster Algebras of Type $D$
%J Symmetry, integrability and geometry: methods and applications
%D 2015
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%F SIGMA_2015_11_a59
Emily Gunawan; Gregg Musiker. $T$-Path Formula and Atomic Bases for Cluster Algebras of Type $D$. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a59/

[1] Baur K., Marsh R. J., “Frieze patterns for punctured discs”, J. Algebraic Combin., 30 (2009), 349–379, arXiv: 0711.1443 | DOI | MR | Zbl

[2] Caldero P., Keller B., “From triangulated categories to cluster algebras”, Invent. Math., 172 (2008), 169–211, arXiv: math.RT/0506018 | DOI | MR | Zbl

[3] Ceballos C., Pilaud V., Cluster algebras of type $D$: Pseudotriangulations approach, arXiv: 1504.06377

[4] Cerulli Irelli G., Positivity in skew-symmetric cluster algebras of finite type, arXiv: 1102.3050

[5] Cerulli Irelli G., Keller B., Labardini-Fragoso D., Plamondon P. G., “Linear independence of cluster monomials for skew-symmetric cluster algebras”, Compos. Math., 149 (2013), 1753–1764, arXiv: 1203.1307 | DOI | MR | Zbl

[6] Cerulli Irelli G., Labardini-Fragoso D., “Quivers with potentials associated to triangulated surfaces. Part III: Tagged triangulations and cluster monomials”, Compos. Math., 148 (2012), 1833–1866, arXiv: 1108.1774 | DOI | MR | Zbl

[7] Dupont G., Thomas H., “Atomic bases of cluster algebras of types $A$ and $\tilde{A}$”, Proc. Lond. Math. Soc., 107 (2013), 825–850, arXiv: 1106.3758 | DOI | MR | Zbl

[8] Fock V., Goncharov A., “Moduli spaces of local systems and higher Teichmüller theory”, Publ. Math. Inst. Hautes Études Sci., 2006, 1–211, arXiv: math.AG/0311149 | DOI | MR | Zbl

[9] Fock V. V., Goncharov A. B., “Cluster ensembles, quantization and the dilogarithm”, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 865–930, arXiv: math.AG/0311245 | MR | Zbl

[10] Fomin S., Shapiro M., Thurston D., “Cluster algebras and triangulated surfaces. I: Cluster complexes”, Acta Math., 201 (2008), 83–146, arXiv: math.RA/0608367 | DOI | MR | Zbl

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

[12] Fomin S., Zelevinsky A., “Cluster algebras. II: Finite type classification”, Invent. Math., 154 (2003), 63–121, arXiv: math.RA/0208229 | DOI | MR | Zbl

[13] Gekhtman M., Shapiro M., Vainshtein A., “Cluster algebras and Weil–Petersson forms”, Duke Math. J., 127 (2005), 291–311, arXiv: math.QA/0309138 | DOI | MR | Zbl

[14] Hernandez D., Leclerc B., “Cluster algebras and quantum affine algebras”, Duke Math. J., 154 (2010), 265–341, arXiv: 0903.1452 | DOI | MR | Zbl

[15] Lee K., Li L., Zelevinsky A., “Positivity and tameness in rank 2 cluster algebras”, J. Algebraic Combin., 40 (2014), 823–840, arXiv: 1303.5806 | DOI | MR | Zbl

[16] Musiker G., Schiffler R., “Cluster expansion formulas and perfect matchings”, J. Algebraic Combin., 32 (2010), 187–209, arXiv: 0810.3638 | DOI | MR | Zbl

[17] Musiker G., Schiffler R., Williams L., “Positivity for cluster algebras from surfaces”, Adv. Math., 227 (2011), 2241–2308, arXiv: 0906.0748 | DOI | MR | Zbl

[18] Musiker G., Schiffler R., Williams L., “Bases for cluster algebras from surfaces”, Compos. Math., 149 (2013), 217–263, arXiv: 1110.4364 | DOI | MR | Zbl

[19] Nakajima H., “Quiver varieties and cluster algebras”, Kyoto J. Math., 51 (2011), 71–126, arXiv: 0905.0002 | DOI | MR | Zbl

[20] Schiffler R., “A geometric model for cluster categories of type $D_n$”, J. Algebraic Combin., 27 (2008), 1–21, arXiv: math.RT/0608264 | DOI | MR | Zbl

[21] Schiffler R., “On cluster algebras arising from unpunctured surfaces, II”, Adv. Math., 223 (2010), 1885–1923, arXiv: 0809.2593 | DOI | MR | Zbl

[22] Schiffler R., Thomas H., “On cluster algebras arising from unpunctured surfaces”, Int. Math. Res. Not., 2009 (2009), 3160–3189, arXiv: 0712.4131 | DOI | MR | Zbl

[23] Sherman P., Zelevinsky A., “Positivity and canonical bases in rank 2 cluster algebras of finite and affine types”, Mosc. Math. J., 4 (2004), 947–974, arXiv: math.RT/0307082 | MR | Zbl