×

zbMATH — the first resource for mathematics

Canonical bases for cluster algebras. (English) Zbl 1446.13015
In the study of cluster algebras, an important question dating back to the introduction of the theory [S. Fomin and A. Zelevinsky, J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)] concerns the description of various well-behaved bases. Good properties of a basis for a cluster algebra are that it should contain cluster monomials, and have nonnegative structure constants.
In this paper, the authors construct a set of linearly independent elements of a completion of an upper cluster algebra (of type \(\mathcal{A}\) or \(\mathcal{X}\)). This set consists of theta-functions, and is indexed by a subset \(\Theta\) of the tropical points of the mirror dual cluster variety. The authors christen the subspace spanned by these functions the middle cluster algebra – in many situations, such as in type \(\mathcal{X}\) or in type \(\mathcal{A}\) with principal coefficients, this space is contained in the upper cluster algebra and contains the ordinary one.
A conjecture of V. V. Fock and A. B. Goncharov [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865–930 (2009; Zbl 1180.53081)], false in general [M. Gross et al., Algebr. Geom. 2, No. 2, 137–175 (2015; Zbl 1322.14032)], is that the set of all tropical points of the mirror dual cluster variety should index a basis of the upper cluster algebra, and the details of the preceding paragraph are presented here as a corrected version of this conjecture. The authors also give sufficient conditions under which \(\Theta\) does consist of all tropical points, and the middle cluster algebra coincides with the upper one, so that the conjecture is true in its original form. These conditions are technical but implied by cluster-theoretically familiar ones, including acyclicity of the initial quiver, or the existence of a maximal green sequence.
The techniques used are geometric, strongly motivated by log-Calabi-Yau geometry. These include scattering diagrams and broken lines, concepts introduced in earlier work involving the authors [M. Kontsevich and Y. Soibelman, Prog. Math. 244, 321–385 (2006; Zbl 1114.14027); M. Gross and B. Siebert, Ann. Math. (2) 174, No. 3, 1301–1428 (2011; Zbl 1266.53074); M. Gross, Adv. Math. 224, No. 1, 169–245 (2010; Zbl 1190.14038)]. Indeed, the structure constants for the theta functions are computed by counting broken lines, the obstruction to the Fock-Goncharov conjecture in general being that these counts can be infinite (either in the calculation of an individual structure constant, or in the sense that infinitely many structure constants involved in computing a product may be non-zero). These geometric techniques turn out to give an extremely powerful perspective on cluster algebras, and the authors show how they can be used to prove deep results such as positivity of the Laurent phenomenon [K. Lee and R. Schiffler, Ann. Math. (2) 182, No. 1, 73–125 (2015; Zbl 1350.13024)]. A precursor to this may be found in work of the first three authors, who gave a proof of the Laurent phenomenon using geometric methods [M. Gross et al., Algebr. Geom. 2, No. 2, 137–175 (2015; Zbl 1322.14032)].

MSC:
13F60 Cluster algebras
14J33 Mirror symmetry (algebro-geometric aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alexeev, Valery; Brion, Michel, Toric degenerations of spherical varieties, Selecta Math. (N.S.), 10, 4, 453-478, (2004) · Zbl 1078.14075
[2] Auroux, Denis, Mirror symmetry and \(T\)-duality in the complement of an anticanonical divisor, J. G\"okova Geom. Topol. GGT, 1, 51-91, (2007) · Zbl 1181.53076
[3] Berenstein, Arkady; Kazhdan, David, Geometric and unipotent crystals, Geom. Funct. Anal., Special Volume, Part I, (2000), 188-236 · Zbl 1044.17006
[4] Berenstein, Arkady; Kazhdan, David, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases. Quantum groups, Contemp. Math. 433, 13-88, (2007), Amer. Math. Soc., Providence, RI · Zbl 1154.14035
[5] Berenstein, Arkady; Zelevinsky, Andrei, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math., 143, 1, 77-128, (2001) · Zbl 1061.17006
[6] Berenstein, Arkady; Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J., 126, 1, 1-52, (2005) · Zbl 1135.16013
[7] Bern\v ste\u\i n, I. N.; Gel\cprime fand, I. M.; Ponomarev, V. A., Coxeter functors, and Gabriel’s theorem, Uspehi Mat. Nauk, 28, 2(170), 19-33, (1973) · Zbl 0269.08001
[8] Bondal, A. I., Helices, representations of quivers and Koszul algebras. Helices and vector bundles, London Math. Soc. Lecture Note Ser. 148, 75-95, (1990), Cambridge Univ. Press, Cambridge · Zbl 0742.14010
[9] [Bri]Bridge T. Bridgeland, \emph Scattering diagrams, Hall algebras and stability conditions, preprint, 2016.
[10] Br\"ustle, Thomas; Dupont, Gr\'egoire; P\'erotin, Matthieu, On maximal green sequences, Int. Math. Res. Not. IMRN, 16, 4547-4586, (2014) · Zbl 1346.16009
[11] Caldero, Philippe, Toric degenerations of Schubert varieties, Transform. Groups, 7, 1, 51-60, (2002) · Zbl 1050.14040
[12] Canakci, Ilke; Lee, Kyungyong; Schiffler, Ralf, On cluster algebras from unpunctured surfaces with one marked point, Proc. Amer. Math. Soc. Ser. B, 2, 35-49, (2015) · Zbl 1350.13019
[13] [CPS]CPS M. Carl, M. Pumperla, and B. Siebert, \emph A tropical view of Landau-Ginzburg models, available at \urlhttp://www.math.uni-hamburg.de/home/siebert/preprints/LGtrop.pdf
[14] Cerulli Irelli, Giovanni; Keller, Bernhard; Labardini-Fragoso, Daniel; Plamondon, Pierre-Guy, Linear independence of cluster monomials for skew-symmetric cluster algebras, Compos. Math., 149, 10, 1753-1764, (2013) · Zbl 1288.18011
[15] Cheung, Man Wai; Gross, Mark; Muller, Greg; Musiker, Gregg; Rupel, Dylan; Stella, Salvatore; Williams, Harold, The greedy basis equals the theta basis: a rank two haiku, J. Combin. Theory Ser. A, 145, 150-171, (2017) · Zbl 1403.13036
[16] Cho, Cheol-Hyun; Oh, Yong-Geun, Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds, Asian J. Math., 10, 4, 773-814, (2006) · Zbl 1130.53055
[17] Fock, Vladimir; Goncharov, Alexander, Moduli spaces of local systems and higher Teichm\"uller theory, Publ. Math. Inst. Hautes \'Etudes Sci., 103, 1-211, (2006) · Zbl 1099.14025
[18] Fock, Vladimir V.; Goncharov, Alexander B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. \'Ec. Norm. Sup\'er. (4), 42, 6, 865-930, (2009) · Zbl 1180.53081
[19] [FG11]FG11 V. Fock and A. Goncharov, \emph Cluster \(X\)-varieties at infinity, preprint, 2011.
[20] Fomin, Sergey; Shapiro, Michael; Thurston, Dylan, Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math., 201, 1, 83-146, (2008) · Zbl 1263.13023
[21] Fomin, Sergey; Zelevinsky, Andrei, Double Bruhat cells and total positivity, J. Amer. Math. Soc., 12, 2, 335-380, (1999) · Zbl 0913.22011
[22] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15, 2, 497-529, (2002) · Zbl 1021.16017
[23] Fomin, Sergey; Zelevinsky, Andrei, The Laurent phenomenon, Adv. in Appl. Math., 28, 2, 119-144, (2002) · Zbl 1012.05012
[24] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. II. Finite type classification, Invent. Math., 154, 1, 63-121, (2003) · Zbl 1054.17024
[25] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. IV. Coefficients, Compos. Math., 143, 1, 112-164, (2007) · Zbl 1127.16023
[26] Geiss, Christof; Leclerc, Bernard; Schr\"oer, Jan, Partial flag varieties and preprojective algebras, Ann. Inst. Fourier (Grenoble), 58, 3, 825-876, (2008) · Zbl 1151.16009
[27] Gekhtman, Michael; Shapiro, Michael; Vainshtein, Alek, Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs 167, xvi+246 pp., (2010), American Mathematical Society, Providence, RI · Zbl 1217.13001
[28] Goncharov, Alexander; Shen, Linhui, Geometry of canonical bases and mirror symmetry, Invent. Math., 202, 2, 487-633, (2015) · Zbl 1355.14030
[29] [GS16]GS16 A.  Goncharov and L.  Shen, \emph Donaldson-Thomas transformations of moduli spaces of \(G\)-local systems, preprint 2016, arXiv:1602.06479
[30] Goodearl, Kenneth R.; Yakimov, Milen T., Quantum cluster algebras and quantum nilpotent algebras, Proc. Natl. Acad. Sci. USA, 111, 27, 9696-9703, (2014) · Zbl 1355.16037
[31] Gross, Mark, Mirror symmetry for \(\mathbb{P}^2\) and tropical geometry, Adv. Math., 224, 1, 169-245, (2010) · Zbl 1190.14038
[32] Gross, Mark, Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics 114, xvi+317 pp., (2011), Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI · Zbl 1215.14061
[33] Gross, Mark; Hacking, Paul; Keel, Sean, Mirror symmetry for log Calabi-Yau surfaces I, Publ. Math. Inst. Hautes \'Etudes Sci., 122, 65-168, (2015) · Zbl 1351.14024
[34] Gross, Mark; Hacking, Paul; Keel, Sean, Moduli of surfaces with an anti-canonical cycle, Compos. Math., 151, 2, 265-291, (2015) · Zbl 1330.14062
[35] Gross, Mark; Hacking, Paul; Keel, Sean, Birational geometry of cluster algebras, Algebr. Geom., 2, 2, 137-175, (2015) · Zbl 1322.14032
[36] [GHKII]GHKII M. Gross, P. Hacking and S. Keel, \emph Mirror symmetry for log Calabi-Yau surfaces II, in preparation.
[37] [GHKS]GHKS M. Gross, P. Hacking, S. Keel, and B. Siebert, \emph Theta functions on varieties with effective anti-canonical class, preprint, 2016.
[38] Gross, Mark; Pandharipande, Rahul, Quivers, curves, and the tropical vertex, Port. Math., 67, 2, 211-259, (2010) · Zbl 1227.14049
[39] Gross, Mark; Pandharipande, Rahul; Siebert, Bernd, The tropical vertex, Duke Math. J., 153, 2, 297-362, (2010) · Zbl 1205.14069
[40] Gross, Mark; Siebert, Bernd, From real affine geometry to complex geometry, Ann. of Math. (2), 174, 3, 1301-1428, (2011) · Zbl 1266.53074
[41] Gross, Mark; Siebert, Bernd, Theta functions and mirror symmetry. Surveys in differential geometry 2016. Advances in geometry and mathematical physics, Surv. Differ. Geom. 21, 95-138, (2016), Int. Press, Somerville, MA · Zbl 1354.14062
[42] Inaba, Michi-aki; Iwasaki, Katsunori; Saito, Masa-Hiko, Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlev\'e equation of type VI. I, Publ. Res. Inst. Math. Sci., 42, 4, 987-1089, (2006) · Zbl 1127.34055
[43] Kac, V. G., Infinite root systems, representations of graphs and invariant theory, Invent. Math., 56, 1, 57-92, (1980) · Zbl 0427.17001
[44] Kac, V. G., Infinite root systems, representations of graphs and invariant theory. II, J. Algebra, 78, 1, 141-162, (1982) · Zbl 0497.17007
[45] King, A. D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2), 45, 180, 515-530, (1994) · Zbl 0837.16005
[46] Knutson, Allen; Tao, Terence, The honeycomb model of \({\rm GL}_n(\textbf{C})\) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc., 12, 4, 1055-1090, (1999) · Zbl 0944.05097
[47] Kogan, Mikhail; Miller, Ezra, Toric degeneration of Schubert varieties and Gelfand-Tsetlin polytopes, Adv. Math., 193, 1, 1-17, (2005) · Zbl 1084.14049
[48] Koll\'ar, J\'anos, Singularities of the minimal model program, Cambridge Tracts in Mathematics 200, x+370 pp., (2013), Cambridge University Press, Cambridge · Zbl 1282.14028
[49] Kontsevich, Maxim; Soibelman, Yan, Affine structures and non-Archimedean analytic spaces. in The unity of mathematics, Progr. Math. 244, 321-385, (2006), Birkh\"auser Boston, Boston, MA · Zbl 1114.14027
[50] Kontsevich, Maxim; Soibelman, Yan, Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry. in Homological mirror symmetry and tropical geometry, Lect. Notes Unione Mat. Ital. 15, 197-308, (2014), Springer, Cham · Zbl 1326.14042
[51] Lee, Kyungyong; Schiffler, Ralf, Positivity for cluster algebras, Ann. of Math. (2), 182, 1, 73-125, (2015) · Zbl 1350.13024
[52] Lee, Kyungyong; Li, Li; Zelevinsky, Andrei, Greedy elements in rank 2 cluster algebras, Selecta Math. (N.S.), 20, 1, 57-82, (2014) · Zbl 1295.13031
[53] [M14]MandelThesis T. Mandel, \emph Tropical theta functions and cluster varieties, Ph.D. thesis, UT Austin, 2014.
[54] [Ma15]Magee T. Magee, \emph Fock–Goncharov conjecture and polyhedral cones for \(U⊂ SL_n\) and base affine space \(SL_n/U\), preprint, 2015.
[55] [Ma17]TimThesis T. Magee, \emph GHK mirror symmetry, the Knutson-Tao hive cone, and Littlewood–Richardson coefficients, preprint, 2017.
[56] Matherne, Jacob P.; Muller, Greg, Computing upper cluster algebras, Int. Math. Res. Not. IMRN, 11, 3121-3149, (2015) · Zbl 1350.13026
[57] Matsumura, Hideyuki, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, xiv+320 pp., (1989), Cambridge University Press, Cambridge · Zbl 0666.13002
[58] Muller, Greg, The existence of a maximal green sequence is not invariant under quiver mutation, Electron. J. Combin., 23, 2, Paper 2.47, 23 pp., (2016) · Zbl 1339.05163
[59] Nakanishi, Tomoki; Zelevinsky, Andrei, On tropical dualities in cluster algebras. Algebraic groups and quantum groups, Contemp. Math. 565, 217-226, (2012), Amer. Math. Soc., Providence, RI · Zbl 1317.13054
[60] Reineke, Markus, Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu, 9, 3, 653-667, (2010) · Zbl 1232.53072
[61] Reineke, Markus, Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants, Compos. Math., 147, 3, 943-964, (2011) · Zbl 1266.16013
[62] [R14]R14 M. Reineke, Personal communcation, 2014.
[63] Schofield, Aidan, General representations of quivers, Proc. London Math. Soc. (3), 65, 1, 46-64, (1992) · Zbl 0795.16008
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.