Cluster expansion formulas and perfect matchings
Gregg Musiker
and Ralf Schiffler
DOI: 10.1007/s10801-009-0210-3
Abstract
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph G T, γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph G T, γ .
Pages: 187–209
Keywords: cluster algebra; triangulated surface; principal coefficients; F-polynomial; snake graph
Full Text: PDF
References
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2. Assem, I., Brüstle, T., Charbonneau-Jodoin, G., Plamondon, P.G.: Gentle algebras arising from surface triangulations. Preprint,
3. Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204, 572-612 (2006).
4. Buan, A.B., Marsh, R., Reiten, I.: Denominators of cluster variables. J. Lond. Math. Soc. 79(3), 589- 611 (2009)
5. Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81, 595-616 (2006).
6. Caldero, P., Keller, B.: From triangulated categories to cluster algebras II. Ann. Sci. École Norm. Sup. (4) 39(6), 983-1009 (2006).
7. Caldero, P., Keller, B.: From triangulated categories to cluster algebras. Invent. Math. 172, 169-211 (2008).
8. Caldero, P., Zelevinsky, A.: Laurent expansions in cluster algebras via quiver representations. Mosc. Math. J. 6(3), 411-429 (2006).
9. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (An case). Trans. Am. Math. Soc. 358(3), 1347-1364 (2006).
10. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster tilted algebras. Algebr. Represent. Theory 9(4), 359-376 (2006).
11. Carroll, G., Price, G.: Unpublished result
12. Elkies, N., Kuperberg, G., Larsen, M., Propp, J.: Alternating-sign matrices and domino tilings (Part I). J. Algebr. Comb. 1(2), 11-132 (1992).
13. Fock, V., Goncharov, A.: Cluster ensembles, quantization and the dilogarithm. Preprint (2003).
14. Fock, V., Goncharov, A.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. 103, 1-211 (2006)
15. Fock, V., Goncharov, A.: Dual Teichmüller and lamination spaces. In: Handbook of Teichmüller Theory. Vol. I. IRMA Lect. Math. Theor. Phys., vol. 11, pp. 647-684. Eur. Math. Soc., Zürich (2007)
16. Fomin, S., Thurston, D.: Cluster algebras and triangulated surfaces. Part II: Lambda lengths. Preprint (2008).
17. Fomin, S., Zelevinsky, A.: Cluster algebras I. Foundations. J. Am. Math. Soc. 15(2), 497-529 (2002) (electronic).
18. Fomin, S., Zelevinsky, A.: Cluster algebras IV: Coefficients. Comput. Math. 143, 112-164 (2007).
19. Fomin, S., Zelevinsky, A.: Unpublished result
20. Fomin, S., Shapiro, M., Thurston, D.: Cluster algebras and triangulated surfaces. Part I: Cluster complexes. Acta Math. 201, 83-146 (2008).
21. Fu, C., Keller, B.: On cluster algebras with coefficients and 2-Calabi-Yau categories. Trans. Am. Math. Soc. (to appear).
22. Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3(3), 899-934 (2003), 1199.
23. Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Weil-Petersson forms. Duke Math. J. 127(2), 291-311 (2005).
24. Labardini-Fragoso, D.: Quivers with potentials associated to triangulated surfaces. Proc. Lond. Math. Soc. (to appear).
25. Musiker, G.: A graph theoretic expansion formula for cluster algebras of classical type. Ann. Comb. (to appear).
26. Musiker, G., Propp, J.: Combinatorial interpretations for rank-two cluster algebras of affine type. Electron. J. Comb. 14(1) (2007), Research Paper 15, 23 pp. (electronic).
27. Musiker, G., Schiffler, R., Williams, L.: Positivity for cluster algebras from surfaces. Preprint.
28. Palu, Y.: Cluster characters for triangulated 2-Calabi-Yau triangulated categories. Ann. Inst. Fourier 58(6), 2221-2248 (2008)
29. Propp, J.: Lattice structure for orientations of graphs. Preprint (1993). [math]
30. Propp, J.: The combinatorics of frieze patterns and Markoff numbers. Preprint (2005).
31. Schiffler, R.: A cluster expansion formula (An case), Electron. J. Comb. 15 (2008), #R64 1.
32. Schiffler, R.: On cluster algebras arising from unpunctured surfaces II. Adv. Math. (to appear).
33. Schiffler, R., Thomas, H.: On cluster algebras arising from unpunctured surfaces. Int. Math. Res. Not. 17, 3160-3189 (2009).
34. Sherman, P., Zelevinsky, A.: Positivity and canonical bases in rank 2 cluster algebras of finite and affine types. Mosc. Math. J. 4(4), 947-974 (2004), 982. (1)
35. Zelevinsky, A.: Semicanonical basis generators of the cluster algebra of type A . Electron. J. Comb. 1 14(1) (2007), Note 4, 5 pp. (electronic).
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