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Snake graph calculus and cluster algebras from surfaces.(English)Zbl 1319.13012

Cluster algebras were introduced byS. Fomin andA. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002;Zbl 1021.16017)], motivated by combinatorial aspects of canonical bases in Lie theory. They are commutative algebras, whose generators and relations are constructed by a recursive process. This distinguished set of generators, called the cluster variables, are collected into groups of \(n\) elements calledclusters connected by local transition rules. By construction, cluster variables are Laurent polynomials with integer coefficients. Moreover, these coefficients are conjectured to be positive.
This positivity conjecture was proved for a large class of cluster algebras that are constructed from triangulations of surfaces, by Musiker, Shiffler and Williams [G. Musiker et al., Adv. Math. 227, No. 6, 2241–2308 (2011;Zbl 1331.13017)]. The proof uses a direct combinatorial formula for the cluster variables. This formula is parametrized by perfect matchings of so-calledsnake graphs, which are originally constructed from arcs on the surface. A snake graph is a connected graph consisting of a finite sequence of square tiles each sharing exactly one edge whose configuration depends on the arc. Hence, it makes sense to say that two snake graphs are crossing when the corresponding arcs are crossing. Also, one can define theresolution of two crossing snake graphs, as the pair of snake graphs associated with the resolution of the two crossing arcs.
The aim of this article is to develop the notion of abstract snake graphs which are not necessarily related to arcs on a surface. The authors give a combinatorial definition of snake graphs and define combinatorially what it means for two abstract snake graph to cross, and how to construct the resolution of this crossing as a new pair of snake graphs. When the abstract snake graphs are usual snake graphs and come from arcs on a surface, they prove that the notions of crossing and resolution coincide.

MSC:

13F60 Cluster algebras
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05E15 Combinatorial aspects of groups and algebras (MSC2010)
57Q15 Triangulating manifolds

Cite

References:

[1]Berenstein, A.; Fomin, S.; Zelevinsky, A., Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J., 126, 1, 1-52 (2005) ·Zbl 1135.16013
[2]Felikson, A.; Shapiro, M.; Tumarkin, P., Skew-symmetric cluster algebras of finite mutation type, J. Eur. Math. Soc. (JEMS), 14, 4, 1135-1180 (2012) ·Zbl 1262.13038
[3]Felikson, A.; Shapiro, M.; Tumarkin, P., Cluster algebras of finite mutation type via unfoldings, Int. Math. Res. Not. IMRN, 8, 1768-1804 (2012) ·Zbl 1283.13020
[4]Felikson, A.; Shapiro, M.; Tumarkin, P., Cluster algebras and triangulated orbifolds, Adv. Math., 231, 5, 2953-3002 (2012) ·Zbl 1256.13014
[5]Fock, V.; Goncharov, A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci., 103, 1-211 (2006) ·Zbl 1099.14025
[6]Fock, V.; Goncharov, A., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Ec. Norm. Super. (4), 42, 6, 865-930 (2009) ·Zbl 1180.53081
[7]Fomin, S.; Shapiro, M.; Thurston, D., Cluster algebras and triangulated surfaces. Part I: Cluster complexes, Acta Math., 201, 83-146 (2008) ·Zbl 1263.13023
[8]Fomin, S.; Thurston, D., Cluster algebras and triangulated surfaces. Part II: Lambda lengths (2008), preprint
[9]Fomin, S.; Zelevinsky, A., Cluster algebras I: Foundations, J. Amer. Math. Soc., 15, 497-529 (2002) ·Zbl 1021.16017
[10]Fomin, S.; Zelevinsky, A., Cluster algebras II. Finite type classification, Invent. Math., 154, 1, 63-121 (2003) ·Zbl 1054.17024
[11]Fomin, S.; Zelevinsky, A., Cluster algebras IV: Coefficients, Compos. Math., 143, 112-164 (2007) ·Zbl 1127.16023
[12]Gekhtman, M.; Shapiro, M.; Vainshtein, A., Cluster algebras and Weil-Petersson forms, Duke Math. J., 127, 291-311 (2005) ·Zbl 1079.53124
[13]Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., 3, 447-498 (1990) ·Zbl 0703.17008
[14]Lusztig, Introduction to Quantum Groups, Progr. Math., vol. 110 (1993), Birkhäuser ·Zbl 0788.17010
[15]Musiker, G.; Schiffler, R., Cluster expansion formulas and perfect matchings, J. Algebraic Combin., 32, 2, 187-209 (2010) ·Zbl 1246.13035
[16]Musiker, G.; Schiffler, R.; Williams, L., Positivity for cluster algebras from surfaces, Adv. Math., 227, 2241-2308 (2011) ·Zbl 1331.13017
[17]Musiker, G.; Schiffler, R.; Williams, L., Bases for cluster algebras from surfaces, Compos. Math., 149, 2, 217-263 (2013) ·Zbl 1263.13024
[18]Musiker, G.; Williams, L., Matrix formulae and skein relations for cluster algebras from surfaces, preprint ·Zbl 1320.13028
[19]Propp, J., The combinatorics of frieze patterns and Markoff numbers (2005), preprint
[20]Schiffler, R., A cluster expansion formula \((A_n\) case), Electron. J. Combin., 15 (2008), #R64 1 ·Zbl 1184.13064
[21]Schiffler, R., On cluster algebras arising from unpunctured surfaces II, Adv. Math., 223, 1885-1923 (2010) ·Zbl 1238.13029
[22]Schiffler, R.; Thomas, H., On cluster algebras arising from unpunctured surfaces, Int. Math. Res. Not. IMRN, 17, 3160-3189 (2009) ·Zbl 1171.30019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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