Positivity for cluster algebras

Pages 73-125 from Volume 182 (2015), Issue 1 by Kyungyong Lee, Ralf Schiffler

Abstract

We prove the positivity conjecture for all skew-symmetric cluster algebras.

  • [ADSS] Go to document I. Assem, G. Dupont, R. Schiffler, and D. Smith, "Friezes, strings and cluster variables," Glasg. Math. J., vol. 54, iss. 1, pp. 27-60, 2012.
    @article{ADSS, mrkey = {2862382},
      author = {Assem, Ibrahim and Dupont, Gr{é}goire and Schiffler, Ralf and Smith, David},
      title = {Friezes, strings and cluster variables},
      journal = {Glasg. Math. J.},
      fjournal = {Glasgow Mathematical Journal},
      volume = {54},
      year = {2012},
      number = {1},
      pages = {27--60},
      issn = {0017-0895},
      mrclass = {13F60 (16G20)},
      mrnumber = {2862382},
      mrreviewer = {Kyungyong Lee},
      doi = {10.1017/S0017089511000322},
      zblnumber = {1280.16015},
      }
  • [ARS] Go to document I. Assem, C. Reutenauer, and D. Smith, "Friezes," Adv. Math., vol. 225, iss. 6, pp. 3134-3165, 2010.
    @article{ARS, mrkey = {2729004},
      author = {Assem, Ibrahim and Reutenauer, Christophe and Smith, David},
      title = {Friezes},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {225},
      year = {2010},
      number = {6},
      pages = {3134--3165},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {13F60 (05A15 05C50 05C75 17B22)},
      mrnumber = {2729004},
      mrreviewer = {Kyungyong Lee},
      doi = {10.1016/j.aim.2010.05.019},
      zblnumber = {1275.13017},
      }
  • [BBH] Go to document A. Beineke, T. Brüstle, and L. Hille, "Cluster-cylic quivers with three vertices and the Markov equation," Algebr. Represent. Theory, vol. 14, iss. 1, pp. 97-112, 2011.
    @article{BBH, mrkey = {2763295},
      author = {Beineke, Andre and Br{ü}stle, Thomas and Hille, Lutz},
      title = {Cluster-cylic quivers with three vertices and the {M}arkov equation},
      journal = {Algebr. Represent. Theory},
      fjournal = {Algebras and Representation Theory},
      volume = {14},
      year = {2011},
      number = {1},
      pages = {97--112},
      issn = {1386-923X},
      coden = {ARTHF4},
      mrclass = {16G20 (13F60)},
      mrnumber = {2763295},
      mrreviewer = {Gregoire Dupont},
      doi = {10.1007/s10468-009-9179-9},
      zblnumber = {1239.16017},
      }
  • [BZ] Go to document A. Berenstein and A. Zelevinsky, "Quantum cluster algebras," Adv. Math., vol. 195, iss. 2, pp. 405-455, 2005.
    @article{BZ, mrkey = {2146350},
      author = {Berenstein, Arkady and Zelevinsky, Andrei},
      title = {Quantum cluster algebras},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {195},
      year = {2005},
      number = {2},
      pages = {405--455},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {20G42 (14M17 22E46)},
      mrnumber = {2146350},
      mrreviewer = {Oleg V. Ogievetski{\u\i}},
      doi = {10.1016/j.aim.2004.08.003},
      zblnumber = {1124.20028},
      }
  • [CC] Go to document P. Caldero and F. Chapoton, "Cluster algebras as Hall algebras of quiver representations," Comment. Math. Helv., vol. 81, iss. 3, pp. 595-616, 2006.
    @article{CC, mrkey = {2250855},
      author = {Caldero, Philippe and Chapoton, Fr{é}d{é}ric},
      title = {Cluster algebras as {H}all algebras of quiver representations},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
      volume = {81},
      year = {2006},
      number = {3},
      pages = {595--616},
      issn = {0010-2571},
      mrclass = {16G20 (18E30)},
      mrnumber = {2250855},
      doi = {10.4171/CMH/65},
      zblnumber = {1119.16013},
      }
  • [CK1] Go to document P. Caldero and B. Keller, "From triangulated categories to cluster algebras," Invent. Math., vol. 172, iss. 1, pp. 169-211, 2008.
    @article{CK1, mrkey = {2385670},
      author = {Caldero, Philippe and Keller, Bernhard},
      title = {From triangulated categories to cluster algebras},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {172},
      year = {2008},
      number = {1},
      pages = {169--211},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {16G20 (13A99 16G10 17B20 18E30)},
      mrnumber = {2385670},
      mrreviewer = {Dirk Kussin},
      doi = {10.1007/s00222-008-0111-4},
      zblnumber = {1141.18012},
      }
  • [DMSS] B. Davison, D. Maulik, J. Schürmann, and B. SzenrHoi, Purity for graded potentials and quantum cluster positivity.
    @misc{DMSS,
      author = {Davison, B. and Maulik, D. and Schürmann, J. and Szenrői, B.},
      title = {Purity for graded potentials and quantum cluster positivity},
      arxiv = {1307.3379},
      note = {{\em Compositio Math.},
      to appear},
     }
  • [DWZ2] Go to document H. Derksen, J. Weyman, and A. Zelevinsky, "Quivers with potentials and their representations II: applications to cluster algebras," J. Amer. Math. Soc., vol. 23, iss. 3, pp. 749-790, 2010.
    @article{DWZ2, mrkey = {2629987},
      author = {Derksen, Harm and Weyman, Jerzy and Zelevinsky, Andrei},
      title = {Quivers with potentials and their representations {II}: applications to cluster algebras},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {23},
      year = {2010},
      number = {3},
      pages = {749--790},
      issn = {0894-0347},
      mrclass = {16G20 (13F60)},
      mrnumber = {2629987},
      mrreviewer = {M{á}ty{á}s Domokos},
      doi = {10.1090/S0894-0347-10-00662-4},
      zblnumber = {1208.16017},
      }
  • [Ef] A. I. Efimov, Quantum cluster variables via vanishing cycles.
    @misc{Ef,
      author = {Efimov, A. I.},
      title = {Quantum cluster variables via vanishing cycles},
      arxiv = {1112.3601},
      }
  • [FeShTu] Go to document A. Felikson, M. Shapiro, and P. Tumarkin, "Cluster algebras and triangulated orbifolds," Adv. Math., vol. 231, iss. 5, pp. 2953-3002, 2012.
    @article{FeShTu, mrkey = {2970470},
      author = {Felikson, Anna and Shapiro, Michael and Tumarkin, Pavel},
      title = {Cluster algebras and triangulated orbifolds},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {231},
      year = {2012},
      number = {5},
      pages = {2953--3002},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {13F60},
      mrnumber = {2970470},
      mrreviewer = {Kyungyong Lee},
      doi = {10.1016/j.aim.2012.07.032},
      zblnumber = {1256.13014},
      }
  • [FG] Go to document V. Fock and A. Goncharov, "Moduli spaces of local systems and higher Teichmüller theory," Publ. Math. Inst. Hautes Études Sci., vol. 103, pp. 1-211, 2006.
    @article{FG, mrkey = {2233852},
      author = {Fock, Vladimir and Goncharov, Alexander},
      title = {Moduli spaces of local systems and higher {T}eichmüller theory},
      journal = {Publ. Math. Inst. Hautes Études Sci.},
      fjournal = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      volume = {103},
      year = {2006},
      pages = {1--211},
      issn = {0073-8301},
      mrclass = {32G15 (14D22 20F34 57M50)},
      mrnumber = {2233852},
      mrreviewer = {William Goldman},
      doi = {10.1007/s10240-006-0039-4},
      zblnumber = {1099.14025},
      }
  • [FST] Go to document S. Fomin, M. Shapiro, and D. Thurston, "Cluster algebras and triangulated surfaces. Part I. Cluster complexes," Acta Math., vol. 201, iss. 1, pp. 83-146, 2008.
    @article{FST, mrkey = {2448067},
      author = {Fomin, Sergey and Shapiro, Michael and Thurston, Dylan},
      title = {Cluster algebras and triangulated surfaces. {P}art {I}. {C}luster complexes},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {201},
      year = {2008},
      number = {1},
      pages = {83--146},
      issn = {0001-5962},
      coden = {ACMAA8},
      mrclass = {57Q15 (13F60 32G15 52B70)},
      mrnumber = {2448067},
      mrreviewer = {Christof Geiß},
      doi = {10.1007/s11511-008-0030-7},
      zblnumber = {1263.13023},
      }
  • [FZ] Go to document S. Fomin and A. Zelevinsky, "Cluster algebras. I. Foundations," J. Amer. Math. Soc., vol. 15, iss. 2, pp. 497-529, 2002.
    @article{FZ, mrkey = {1887642},
      author = {Fomin, Sergey and Zelevinsky, Andrei},
      title = {Cluster algebras. {I}. {F}oundations},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {15},
      year = {2002},
      number = {2},
      pages = {497--529},
      issn = {0894-0347},
      mrclass = {16S99 (14M99 17B99)},
      mrnumber = {1887642},
      mrreviewer = {Eric N. Sommers},
      doi = {10.1090/S0894-0347-01-00385-X},
      zblnumber = {1021.16017},
      }
  • [FZ4] Go to document S. Fomin and A. Zelevinsky, "Cluster algebras. IV. Coefficients," Compos. Math., vol. 143, iss. 1, pp. 112-164, 2007.
    @article{FZ4, mrkey = {2295199},
      author = {Fomin, Sergey and Zelevinsky, Andrei},
      title = {Cluster algebras. {IV}. {C}oefficients},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {143},
      year = {2007},
      number = {1},
      pages = {112--164},
      issn = {0010-437X},
      mrclass = {16S99 (05E15 14M17 22E46)},
      mrnumber = {2295199},
      mrreviewer = {Christof Geiß},
      doi = {10.1112/S0010437X06002521},
      zblnumber = {1127.16023},
      }
  • [FK] Go to document C. Fu and B. Keller, "On cluster algebras with coefficients and 2-Calabi-Yau categories," Trans. Amer. Math. Soc., vol. 362, iss. 2, pp. 859-895, 2010.
    @article{FK, mrkey = {2551509},
      author = {Fu, Changjian and Keller, Bernhard},
      title = {On cluster algebras with coefficients and 2-{C}alabi-{Y}au categories},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {362},
      year = {2010},
      number = {2},
      pages = {859--895},
      issn = {0002-9947},
      coden = {TAMTAM},
      mrclass = {13F60 (18E99)},
      mrnumber = {2551509},
      mrreviewer = {Calin Chindris},
      doi = {10.1090/S0002-9947-09-04979-4},
      zblnumber = {1201.18007},
      }
  • [KQ] Go to document Y. Kimura and F. Qin, "Graded quiver varieties, quantum cluster algebras and dual canonical basis," Adv. Math., vol. 262, pp. 261-312, 2014.
    @article{KQ, mrkey = {3228430},
      author = {Kimura, Yoshiyuki and Qin, Fan},
      title = {Graded quiver varieties, quantum cluster algebras and dual canonical basis},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {262},
      year = {2014},
      pages = {261--312},
      issn = {0001-8708},
      mrclass = {13F60 (16G20 18D10)},
      mrnumber = {3228430},
      doi = {10.1016/j.aim.2014.05.014},
      zblnumber = {06317147},
      }
  • [HL] Go to document D. Hernandez and B. Leclerc, "Cluster algebras and quantum affine algebras," Duke Math. J., vol. 154, iss. 2, pp. 265-341, 2010.
    @article{HL, mrkey = {2682185},
      author = {Hernandez, David and Leclerc, Bernard},
      title = {Cluster algebras and quantum affine algebras},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {154},
      year = {2010},
      number = {2},
      pages = {265--341},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {17B37 (13F60)},
      mrnumber = {2682185},
      mrreviewer = {Jan E. Grabowski},
      doi = {10.1215/00127094-2010-040},
      zblnumber = {1284.17010},
      }
  • [L] Go to document K. Lee, "On cluster variables of rank two acyclic cluster algebras," Ann. Comb., vol. 16, iss. 2, pp. 305-317, 2012.
    @article{L, mrkey = {2927610},
      author = {Lee, Kyungyong},
      title = {On cluster variables of rank two acyclic cluster algebras},
      journal = {Ann. Comb.},
      fjournal = {Annals of Combinatorics},
      volume = {16},
      year = {2012},
      number = {2},
      pages = {305--317},
      issn = {0218-0006},
      mrclass = {13F60 (16G20)},
      mrnumber = {2927610},
      mrreviewer = {Yu Zhou},
      doi = {10.1007/s00026-012-0134-9},
      zblnumber = {1266.13016},
      }
  • [LLZ] Go to document K. Lee, L. Li, and A. Zelevinsky, "Greedy elements in rank 2 cluster algebras," Selecta Math., vol. 20, iss. 1, pp. 57-82, 2014.
    @article{LLZ, mrkey = {3147413},
      author = {Lee, Kyungyong and Li, Li and Zelevinsky, Andrei},
      title = {Greedy elements in rank 2 cluster algebras},
      journal = {Selecta Math.},
      fjournal = {Selecta Mathematica. New Series},
      volume = {20},
      year = {2014},
      number = {1},
      pages = {57--82},
      issn = {1022-1824},
      mrclass = {13F60},
      mrnumber = {3147413},
      mrreviewer = {Calin Chindris},
      doi = {10.1007/s00029-012-0115-1},
      zblnumber = {1295.13031},
      }
  • [LS] Go to document K. Lee and R. Schiffler, "A combinatorial formula for rank 2 cluster variables," J. Algebraic Combin., vol. 37, iss. 1, pp. 67-85, 2013.
    @article{LS, mrkey = {3016302},
      author = {Lee, Kyungyong and Schiffler, Ralf},
      title = {A combinatorial formula for rank 2 cluster variables},
      journal = {J. Algebraic Combin.},
      fjournal = {Journal of Algebraic Combinatorics. An International Journal},
      volume = {37},
      year = {2013},
      number = {1},
      pages = {67--85},
      issn = {0925-9899},
      coden = {JAOME7},
      mrclass = {13F60 (05Exx)},
      mrnumber = {3016302},
      mrreviewer = {Xian Neng Du},
      doi = {10.1007/s10801-012-0359-z},
      zblnumber = {1266.13017},
      }
  • [LS2] Go to document K. Lee and R. Schiffler, "Proof of a positivity conjecture of M. Kontsevich on non-commutative cluster variables," Compos. Math., vol. 148, iss. 6, pp. 1821-1832, 2012.
    @article{LS2, mrkey = {2999306},
      author = {Lee, Kyungyong and Schiffler, Ralf},
      title = {Proof of a positivity conjecture of {M}. {K}ontsevich on non-commutative cluster variables},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {148},
      year = {2012},
      number = {6},
      pages = {1821--1832},
      issn = {0010-437X},
      mrclass = {16S38 (05E99 13F60)},
      mrnumber = {2999306},
      mrreviewer = {Xueqing Chen},
      doi = {10.1112/S0010437X12000425},
      zblnumber = {1266.16027},
      }
  • [LS3] Go to document K. Lee and R. Schiffler, "Positivity for cluster algebras of rank 3," Publ. Res. Inst. Math. Sci., vol. 49, iss. 3, pp. 601-649, 2013.
    @article{LS3, mrkey = {3097017},
      author = {Lee, Kyungyong and Schiffler, Ralf},
      title = {Positivity for cluster algebras of rank 3},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Publications of the Research Institute for Mathematical Sciences},
      volume = {49},
      year = {2013},
      number = {3},
      pages = {601--649},
      issn = {0034-5318},
      mrclass = {13F60},
      mrnumber = {3097017},
      mrreviewer = {Li Li},
      doi = {10.4171/PRIMS/114},
      zblnumber = {06218353},
      }
  • [LS3v1] K. Lee and R. Schiffler, Positivity for cluster algebras of rank 3.
    @misc{LS3v1, mrkey = {3097017},
      author = {Lee, Kyungyong and Schiffler, Ralf},
      title = {Positivity for cluster algebras of rank 3},
      arxiv = {1205.5466v1},
      NOTE={first version},
      }
  • [MSW] Go to document G. Musiker, R. Schiffler, and L. Williams, "Positivity for cluster algebras from surfaces," Adv. Math., vol. 227, iss. 6, pp. 2241-2308, 2011.
    @article{MSW, mrkey = {2807089},
      author = {Musiker, Gregg and Schiffler, Ralf and Williams, Lauren},
      title = {Positivity for cluster algebras from surfaces},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {227},
      year = {2011},
      number = {6},
      pages = {2241--2308},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {13F60 (05C70 05E15)},
      mrnumber = {2807089},
      mrreviewer = {Gregoire Dupont},
      doi = {10.1016/j.aim.2011.04.018},
      zblnumber = {05919159},
      }
  • [N] Go to document H. Nakajima, "Quiver varieties and cluster algebras," Kyoto J. Math., vol. 51, iss. 1, pp. 71-126, 2011.
    @article{N, mrkey = {2784748},
      author = {Nakajima, Hiraku},
      title = {Quiver varieties and cluster algebras},
      journal = {Kyoto J. Math.},
      fjournal = {Kyoto Journal of Mathematics},
      volume = {51},
      year = {2011},
      number = {1},
      pages = {71--126},
      issn = {2156-2261},
      mrclass = {13F60 (14D21 16G20 17B37)},
      mrnumber = {2784748},
      mrreviewer = {Kyungyong Lee},
      doi = {10.1215/0023608X-2010-021},
      zblnumber = {1223.13013},
      }
  • [Q] Go to document F. Qin, "Quantum cluster variables via Serre polynomials," J. Reine Angew. Math., vol. 668, pp. 149-190, 2012.
    @article{Q, mrkey = {2948875},
      author = {Qin, Fan},
      title = {Quantum cluster variables via {S}erre polynomials},
      titlenote={(with an appendix by B. Keller)},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {668},
      year = {2012},
      pages = {149--190},
      issn = {0075-4102},
      coden = {JRMAA8},
      mrclass = {13F60 (81R50)},
      mrnumber = {2948875},
      mrreviewer = {Kyungyong Lee},
      zblnumber = {1252.13013},
      DOI = {10.1515/CRELLE.2011.129},
     }
  • [R] Go to document D. Rupel, "Proof of the Kontsevich non-commutative cluster positivity conjecture," C. R. Math. Acad. Sci. Paris, vol. 350, iss. 21-22, pp. 929-932, 2012.
    @article{R, mrkey = {2996767},
      author = {Rupel, Dylan},
      title = {Proof of the {K}ontsevich non-commutative cluster positivity conjecture},
      journal = {C. R. Math. Acad. Sci. Paris},
      fjournal = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
      volume = {350},
      year = {2012},
      number = {21-22},
      pages = {929--932},
      issn = {1631-073X},
      mrclass = {16S38 (05E05 13F60)},
      mrnumber = {2996767},
      mrreviewer = {Xueqing Chen},
      doi = {10.1016/j.crma.2012.10.034},
      zblnumber = {1266.16028},
      }
  • [S3] Go to document R. Schiffler, "On cluster algebras arising from unpunctured surfaces. II," Adv. Math., vol. 223, iss. 6, pp. 1885-1923, 2010.
    @article{S3, mrkey = {2601004},
      author = {Schiffler, Ralf},
      title = {On cluster algebras arising from unpunctured surfaces. {II}},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {223},
      year = {2010},
      number = {6},
      pages = {1885--1923},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {13F60 (05E15 16G20)},
      mrnumber = {2601004},
      mrreviewer = {Gregoire Dupont},
      doi = {10.1016/j.aim.2009.10.015},
      zblnumber = {1238.13029},
      }
  • [S2] Go to document R. Schiffler, "A cluster expansion formula ($A_n$ case)," Electron. J. Combin., vol. 15, iss. 1, p. 64, 2008.
    @article{S2, mrkey = {2398856},
      author = {Schiffler, Ralf},
      title = {A cluster expansion formula ({$A\sb n$} case)},
      journal = {Electron. J. Combin.},
      fjournal = {Electronic Journal of Combinatorics},
      volume = {15},
      year = {2008},
      number = {1},
      pages = {Research paper 64, 9},
      issn = {1077-8926},
      mrclass = {13F99 (05E15)},
      mrnumber = {2398856},
      mrreviewer = {Pavlo Pylyavskyy},
      url = {http://www.combinatorics.org/Volume_15/Abstracts/v15i1r64.html},
      zblnumber = {1184.13064},
      }
  • [ST] Go to document R. Schiffler and H. Thomas, "On cluster algebras arising from unpunctured surfaces," Int. Math. Res. Not., vol. 2009, p. no. 17, 3160-3189.
    @article{ST, mrkey = {2534994},
      author = {Schiffler, Ralf and Thomas, Hugh},
      title = {On cluster algebras arising from unpunctured surfaces},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices. IMRN},
      pages = {no.~17, 3160--3189},
      issn = {1073-7928},
      mrclass = {13F60 (05E99 16G20 16S99)},
      mrnumber = {2534994},
      mrreviewer = {Gregoire Dupont},
      doi = {10.1093/imrn/rnp047},
      volume = {2009},
      zblnumber = {1171.30019},
      }
  • [SZ] Go to document P. Sherman and A. Zelevinsky, "Positivity and canonical bases in rank 2 cluster algebras of finite and affine types," Mosc. Math. J., vol. 4, iss. 4, pp. 947-974, 982, 2004.
    @article{SZ, mrkey = {2124174},
      author = {Sherman, Paul and Zelevinsky, Andrei},
      title = {Positivity and canonical bases in rank 2 cluster algebras of finite and affine types},
      journal = {Mosc. Math. J.},
      fjournal = {Moscow Mathematical Journal},
      volume = {4},
      year = {2004},
      number = {4},
      pages = {947--974, 982},
      issn = {1609-3321},
      mrclass = {16S99 (05E15 22E46)},
      mrnumber = {2124174},
      zblnumber = {1103.16018},
      url = {http://mi.mathnet.ru/eng/mmj/v4/i4/p947},
     

Keywords
,
Mathematical Subject Classification
Primary:
DOI
https://doi.org/10.4007/annals.2015.182.1.2
MR
3374957
zbMATH
06456006
Milestones
Received: 23 July 2013
Revised: 6 July 2014
Accepted: 6 October 2014

Authors

Kyungyong Lee

Department of Mathematics, Wayne State University, Detroit, MI 48202 and Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, Republic of Korea

Ralf Schiffler

University of Connecticut, Storrs, CT