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Bases for cluster algebras from surfaces

Part of: Algebraic combinatorics Graph theory Arithmetic rings and other special rings

Published online by Cambridge University Press:  07 December 2012

Gregg Musiker ,
Ralf Schiffler  and
Lauren Williams
Gregg Musiker
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA (email: musiker@math.umn.edu)
Ralf Schiffler
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA (email: schiffler@math.uconn.edu)
Lauren Williams
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA (email: williams@math.berkeley.edu)
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Abstract

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, such as principal coefficients.

Keywords

cluster algebrabasistriangulated surfaces

MSC classification

Secondary: 13F60: Cluster algebras 05C70: Factorization, matching, partitioning, covering and packing 05E15: Combinatorial aspects of groups and algebras
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 2 , February 2013 , pp. 217 - 263
Copyright
© The Author(s) 2012

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References

[Ami09]Amiot, C., Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier 59 (2009), 2525–2590.Google Scholar
[ABCP10]Assem, I., Brüstle, T., Charbonneau-Jodoin, G. and Plamondon, P. G., Gentle algebras arising from surface triangulations, Algebra Number Theory 4 (2010), 201–229.CrossRefGoogle Scholar
[ASS12]Assem, I., Schiffler, R. and Shramchenko, V., Cluster automorphisms, Proc. Lond. Math. Soc. 104 (2012), 1271–1302.CrossRefGoogle Scholar
[BZ10]Brüstle, T. and Zhang, J., On the cluster category of a marked surface, Algebra Number Theory, to appear, arXiv:1005.2422.Google Scholar
[BMRRT06]Buan, A., Marsh, R., Reineke, M., Reiten, I. and Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–612.CrossRefGoogle Scholar
[CCS06]Caldero, P., Chapoton, F. and Schiffler, R., Quivers with relations arising from clusters (A n case), Trans. Amer. Math. Soc. 358 (2006), 1347–1364.CrossRefGoogle Scholar
[CK08]Caldero, P. and Keller, B., From triangulated categories to cluster algebras, Invent. Math. 172 (2008), 169–211.CrossRefGoogle Scholar
[Cer09]Cerulli Irelli, G., Cluster algebras of type A (1)2, Algebr. Represent. Theory, to appear, arXiv:0904.2543.Google Scholar
[Cer11]Cerulli Irelli, G., Positivity in skew-symmetric cluster algebras of finite type, Preprint (2011), arXiv:1102.3050.Google Scholar
[CL12]Cerulli Irelli, G. and Labardini-Fragoso, D., Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials, Compositio Math. 148 (2012), 1833–1866.CrossRefGoogle Scholar
[CKLP12]Cerulli Irelli, G., Keller, B., Labardini-Fragoso, D. and Plamondon, P.-G., Linear independence of cluster monomials for skew-symmetric cluster algebras, Preprint (2012), arXiv:1203:1307.CrossRefGoogle Scholar
[CL90]Conway, J. and Lagarias, J., Tiling with polyominoes and combinatorial group theory, J. Combin. Theory Ser. A 53 (1990), 183–208.Google Scholar
[DWZ10]Derksen, H., Weyman, J. and Zelevinsky, A., Quivers with potentials and their representations II: Applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749–790.CrossRefGoogle Scholar
[DXX09]Ding, M., Xiao, J. and Xu, F., Integral bases of cluster algebras and representations of tame quivers, Preprint (2009), arXiv:0901.1937.Google Scholar
[Dup08]Dupont, G., Generic variables in acyclic cluster algebras and bases in affine cluster algebras, Preprint (2008), arXiv:0811.2909.Google Scholar
[Dup10]Dupont, G., Transverse quiver Grassmannians and bases in affine cluster algebras, Algebra Number Theory 4 (2010), 599–624.CrossRefGoogle Scholar
[Dup11]Dupont, G., Generic variables in acyclic cluster algebras, J. Pure Appl. Algebra 215 (2011), 628–641.Google Scholar
[DT11]Dupont, G. and Thomas, H., Atomic bases in cluster algebras of types A and , Preprint, (2011), arXiv:1106.3758.Google Scholar
[EKLP92]Elkies, N., Kuperberg, G., Larsen, M. and Propp, J., Alternating-sign matrices and domino tilings I, J. Algebraic Combin. 1 (1992), 111–132.CrossRefGoogle Scholar
[FST12]Felikson, A., Shapiro, M. and Tumarkin, P., Skew-symmetric cluster algebras of finite mutation type, J. Eur. Math. Soc. 14 (2012), 1135–1180.Google Scholar
[FG06]Fock, V. and Goncharov, A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211.CrossRefGoogle Scholar
[FG07]Fock, V. and Goncharov, A., Dual Teichmüller and lamination spaces, in Handbook of Teichmüller theory. Volume I, IRMA Lectures in Mathematics and Theoretical Physics, vol. 11 (European Mathematical Society, Zürich, 2007), 647–684.Google Scholar
[FG09]Fock, V. and Goncharov, A., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865–930.CrossRefGoogle Scholar
[FST08]Fomin, S., Shapiro, M. and Thurston, D., Cluster algebras and triangulated surfaces. Part I: cluster complexes, Acta Math. 201 (2008), 83–146.Google Scholar
[FT08]Fomin, S. and Thurston, D., Cluster algebras and triangulated surfaces. Part II: lambda lengths, Preprint (2008), http://www.math.lsa.umich.edu/∼fomin/Papers/cats2.ps.Google Scholar
[FZ02]Fomin, S. and Zelevinsky, A., Cluster algebras I: foundations, J. Amer. Math. Soc. 15 (2002), 497–529.CrossRefGoogle Scholar
[FZ07]Fomin, S. and Zelevinsky, A., Cluster algebras IV: coefficients, Compositio Math. 143 (2007), 112–164.CrossRefGoogle Scholar
[FG00]Frohman, C. and Gelca, R., Skein modules and the noncommutative torus, Trans. Amer. Math. Soc. 352 (2000), 4877–4888.Google Scholar
[GLS11a]Geiss, C., Leclerc, B. and Schröer, J., Kac–Moody groups and cluster algebras, Adv. Math. 228 (2011), 329–433.Google Scholar
[GLS11b]Geiss, C., Leclerc, B. and Schröer, J., Cluster structures on quantum coordinate rings, Selecta Math., to appear, arXiv:1104.0531.Google Scholar
[GLS12]Geiss, C., Leclerc, B. and Schröer, J., Generic bases for cluster algebras and the Chamber Ansatz, J. Amer. Math. Soc. 25 (2012), 21–76.Google Scholar
[GSV05]Gekhtman, M., Shapiro, M. and Vainshtein, A., Cluster algebras and Weil–Petersson forms, Duke Math. J. 127 (2005), 291–311.Google Scholar
[GSV10]Gekhtman, M., Shapiro, M. and Vainshtein, A., Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, vol. 167 (American Mathematical Society, Providence, RI, 2010).Google Scholar
[HL11]Hernandez, D. and Leclerc, B., Quantum Grothendieck rings and derived Hall algebras, Preprint (2011), arXiv:1109.0862.Google Scholar
[Lab09]Labardini-Fragoso, D., Quivers with potentials associated to triangulated surfaces, Proc. Lond. Math. Soc. (3) 98 (2009), 797–839.Google Scholar
[Lam11a]Lampe, P., A quantum cluster algebra of Kronecker type and the dual canonical basis, Int. Math. Res. Not. 2011 (2011), 2970–3005.Google Scholar
[Lam11b]Lampe, P., Quantum cluster algebras of type A and the dual canonical basis, Preprint (2011), arXiv:1101.0580.Google Scholar
[Lus90]Lusztig, , Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc 3 (1990), 447–498.Google Scholar
[Lus93]Lusztig, , Introduction to quantum groups, Progress in Mathematics, vol. 110 (Birkhäuser, Basel, 1993).Google Scholar
[Lus94]Lusztig, , Total positivity in reductive groups, in Lie theory and geometry: in honor of Bertram Kostant, Progress in Mathematics, vol. 123 (Birkhäuser, Basel, 1994).Google Scholar
[MS10]Musiker, G. and Schiffler, R., Cluster expansion formulas and perfect matchings, J. Algebraic Combin. 32 (2010), 187–209.Google Scholar
[MSW11]Musiker, G., Schiffler, R. and Williams, L., Positivity for cluster algebras from surfaces, Adv. Math. 227 (2011), 2241–2308.CrossRefGoogle Scholar
[MW11]Musiker, G. and Williams, L., Matrix formulae and skein relations for cluster algebras from surfaces, Int. Math. Res. Not., to appear, arXiv:1108.3382.Google Scholar
[Pen87]Penner, R., The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), 299–339.Google Scholar
[Pen06]Penner, R., Lambda lengths, Preprint (2006),http://www.ctqm.au.dk/research/MCS/lambdalengths.pdf.Google Scholar
[Pla11a]Plamondon, P. G., Catégories amassées aux espaces de morphismes de dimension infinie, applications, PhD thesis, Université Paris Diderot (2011).http://people.math.jussieu.fr/∼plamondon/plamondonthese.pdf.Google Scholar
[Pla11b]Plamondon, P. G., Cluster algebras via cluster categories with infinite-dimensional morphism spaces, Compositio Math. 147 (2011), 1921–1934.Google Scholar
[Pro02]Propp, J., Lattice structure for orientations of graphs, Preprint (2002), arXiv:math/0209005.Google Scholar
[Sch08]Schiffler, R., A geometric model for cluster categories of type D n, J. Algebraic Combin. 27 (2008), 1–21.Google Scholar
[Sch08]Schiffler, R., A cluster expansion formula (A n case), Electron. J. Combin. 15 (2008), #R64.Google Scholar
[Sch10]Schiffler, R., On cluster algebras arising from unpunctured surfaces II, Adv. Math. 223 (2010), 1885–1923.Google Scholar
[ST09]Schiffler, R. and Thomas, H., On cluster algebras arising from unpunctured surfaces, Int. Math. Res. Not. 2009 (2009), 3160–3189.Google Scholar
[SZ04]Sherman, P. and Zelevinsky, A., Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Mosc. Math. J. 4 (2004), 947–974, 982.Google Scholar
[Thu]Thurston, D., Geometric intersection of curves in surfaces, Preprint,http://www.math.columbia.edu/∼dpt/DehnCoordinates.ps.Google Scholar
[Thu88]Thurston, W., On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417–431.CrossRefGoogle Scholar
[Thu90]Thurston, W., Conway’s tiling groups, Amer. Math. Monthly 97 (1990), 757–773.Google Scholar
[Thu08]Thurston, D., Lectures at the International Conference on Cluster Algebras and Related Topics, Mexico, December 8–20, 2008, based on joint work with Sergey Fomin and Michael Shapiro.Google Scholar
[Zel07]Zelevinsky, A., Semicanonical basis generators of the cluster algebra of type A (1)1, Electron. J. Combin. 14 (2007), #N4.CrossRefGoogle Scholar