An expansion formula for type \(A\) and Kronecker quantum cluster algebras. (English) Zbl 1435.81115
Summary: We introduce an expansion formula for elements in quantum cluster algebras associated to type \(A\) and Kronecker quivers with principal quantization. Our formula is parametrized by perfect matchings of snake graphs as in the classical case. In the Kronecker type, the coefficients are \(q\)-powers whose exponents are given by a weight function induced by the lattice of perfect matchings. As an application, we prove that a reflectional symmetry on the set of perfect matchings satisfies Stembridge’s \(q = -1\) phenomenon with respect to the weight function. Furthermore, we discuss a relation of our expansion formula to generating functions of BPS states.
MSC:
| 81T10 | Model quantum field theories |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 13F60 | Cluster algebras |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 16G20 | Representations of quivers and partially ordered sets |
Keywords:
quantum cluster algebras; non-commutative rings; surface cluster algebras; triangulations; arcs; snake graphs; perfect matchings; lattices; Stembridge phenomenon; mathematical physicsSoftware:
CoulombHiggsReferences:
| [1] | Alim, M.; Cecotti, S.; Córdova, C.; Espahbodi, S.; Rastogi, A.; Vafa, C., BPS quivers and spectra of complete \(N = 2\) quantum field theories, Comm. Math. Phys., 323, 3, 1185-1227 (2013) · Zbl 1305.81118 |
| [2] | Alim, M.; Cecotti, S.; Córdova, C.; Espahbodi, S.; Rastogi, A.; Vafa, C., \(N = 2\) quantum field theories and their BPS quivers, Adv. Theor. Math. Phys., 18, 1, 27-127 (2014) · Zbl 1309.81142 |
| [3] | Allegretti, D., A duality map for the quantum symplectic double (2016), preprint |
| [4] | Allegretti, D., Categorified canonical bases and framed BPS states (2018), preprint · Zbl 1505.13032 |
| [5] | Allegretti, D.; Kim, H. K., A duality map for quantum cluster varieties from surfaces, Adv. Math., 306, 1164-1208 (2017) · Zbl 1433.13020 |
| [6] | Amiot, C.; Plamondon, P.-G., The cluster category of a surface with punctures via group actions (2017), preprint |
| [7] | 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 |
| [8] | Berenstein, A.; Retakh, V., Noncommutative marked surfaces, Adv. Math., 328, 1010-1087 (2018) · Zbl 1395.14001 |
| [9] | Berenstein, A.; Zelevinsky, A., Quantum cluster algebras, Adv. Math., 195, 405-455 (2005) · Zbl 1124.20028 |
| [10] | Brüstle, T.; Zhang, J., On the cluster category of a marked surface without punctures, Algebra Number Theory, 5, 4, 529-566 (2011) · Zbl 1250.16013 |
| [11] | Caldero, Ph.; Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv., 81, 3, 595-616 (2006) · Zbl 1119.16013 |
| [12] | Caldero, Ph.; Zelevinsky, A., Laurent expansions in cluster algebras via quiver representations, Mosc. Math. J., 6, 3, 411-429 (2006) · Zbl 1133.16012 |
| [13] | Çanakçı, İ.; Lee, K.; Schiffler, R., On cluster algebras for surfaces without punctures and one marked point, Proc. Amer. Math. Soc., Ser. B, 2, 35-49 (2015) · Zbl 1350.13019 |
| [14] | Çanakçı, İ.; Schiffler, R., Snake graph calculus and cluster algebras from surfaces, J. Algebra, 382, 240-281 (2013) · Zbl 1319.13012 |
| [15] | Çanakçı, İ.; Schiffler, R., Snake graph calculus and cluster algebras from surfaces II: self-crossing snake graphs, Math. Z., 281, 1, 55-102 (2015) · Zbl 1375.13029 |
| [16] | Çanakçı, İ.; Schiffler, R., Cluster algebras and continued fractions, Compos. Math., 154, 3, 565-593 (2018) · Zbl 1437.13033 |
| [17] | Çanakçı, İ.; Schiffler, R., Snake graph calculus and cluster algebras from surfaces III: band graphs and snake rings, Int. Math. Res. Not. IMRN, 2019, 4, 1145-1226 (2019) · Zbl 1436.13045 |
| [18] | Çanakçı, İ.; Schroll, S., With an appendix by C. Amiot: extensions in Jacobian algebras and cluster categories of unpunctured surfaces, Adv. Math., 313, 1-49 (2017) · Zbl 1401.13064 |
| [19] | Cirafici, M., Line defects and (framed) BPS quivers, J. High Energy Phys., 2013, 11, Article 141 pp. (2013) · Zbl 1342.81568 |
| [20] | Córdova, C.; Neitzke, A., Line defects, tropicalization, and multi-centered quiver quantum mechanics, J. High Energy Phys., 2014, 9, Article 099 pp. (2014) · Zbl 1333.81166 |
| [21] | Davison, B., Positivity for quantum cluster algebras, Ann. of Math. (2), 187, 1, 157-219 (2018) · Zbl 1408.13055 |
| [22] | Derksen, H.; Weyman, J.; Zelevinsky, A., Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc., 23, 3, 749-790 (2010) · Zbl 1208.16017 |
| [23] | Felikson, A.; Shapiro, M.; Tumarkin, P., Skew-symmetric cluster algebras of finite mutation type, J. Eur. Math. Soc., 14, 4, 1135-1180 (2012) · Zbl 1262.13038 |
| [24] | Felikson, A.; Shapiro, M.; Tumarkin, P., Cluster algebras of finite mutation type via unfoldings, Int. Math. Res. Not., 2012, 8, 1768-1804 (2012) · Zbl 1283.13020 |
| [25] | Fock, V.; Goncharov, A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci., 103, 1, 1-211 (2006) · Zbl 1099.14025 |
| [26] | Fock, V.; Goncharov, A., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér., 42, 6, 865-930 (2009) · Zbl 1180.53081 |
| [27] | Fomin, S.; Shapiro, M.; Thurston, D., Cluster algebras and triangulated surfaces. Part I: cluster complexes, Acta Math., 201, 1, 83-146 (2008) · Zbl 1263.13023 |
| [28] | Fomin, S.; Thurston, D., Cluster algebras and triangulated surfaces. Part II: lambda lengths, Mem. Amer. Math. Soc., 255, Article 1223 pp. (2018) · Zbl 07000309 |
| [29] | Fomin, S.; Zelevinsky, A., Cluster algebras I: foundations, J. Amer. Math. Soc., 15, 2, 497-529 (2002) · Zbl 1021.16017 |
| [30] | Fomin, S.; Zelevinsky, A., Cluster algebras II: finite type classification, Invent. Math., 154, 1, 63-121 (2003) · Zbl 1054.17024 |
| [31] | Fomin, S.; Zelevinsky, A., Cluster algebras IV: coefficients, Compos. Math., 143, 1, 112-164 (2007) · Zbl 1127.16023 |
| [32] | Gaiotto, D.; Moore, G.; Neitzke, A., Framed BPS states, Adv. Theor. Math. Phys., 17, 2, 241-397 (2013) · Zbl 1290.81146 |
| [33] | Geiß, Ch.; Leclerc, B.; Schröer, J., Cluster structures on quantum coordinate rings, Selecta Math., 19, 2, 337-397 (2013) · Zbl 1318.13038 |
| [34] | Gekhtman, M.; Shapiro, M.; Vainshtein, A., Cluster algebras and Weil-Petersson forms, Duke Math. J., 127, 2, 291-311 (2005) · Zbl 1079.53124 |
| [35] | Goodearl, K. R.; Yakimov, M., The Berenstein-Zelevinsky quantum cluster algebra conjecture (2016), preprint · Zbl 1471.13046 |
| [36] | Grabowski, J., Examples of quantum cluster algebras associated to partial flag varieties, J. Pure Appl. Algebra, 215, 7, 1582-1595 (2011) · Zbl 1255.16041 |
| [37] | Grabowski, J.; Launois, S., Graded quantum cluster algebras and an application to quantum Grassmannians, Proc. Lond. Math. Soc., 109, 3, 697-732 (2014) · Zbl 1315.13036 |
| [38] | Gross, M.; Hacking, P.; Keel, S.; Kontsevich, M., Canonical bases for cluster algebras, J. Amer. Math. Soc., 31, 497-608 (2018) · Zbl 1446.13015 |
| [39] | Huang, M., Quantum cluster algebras from unpunctured triangulated surfaces (2018), preprint |
| [40] | Kimura, Y.; Qin, F., Graded quiver varieties, quantum cluster algebras and dual canonical basis, Adv. Math., 262, 261-312 (2014) · Zbl 1331.13016 |
| [41] | Lampe, Ph., A quantum cluster algebra of Kronecker type and the dual canonical basis, Int. Math. Res. Not., 2011, 12, 2970-3005 (2011) · Zbl 1273.17020 |
| [42] | Lampe, Ph., Quantum cluster algebras of type A and the dual canonical basis, Proc. Lond. Math. Soc. (3), 108, 1, 1-43 (2014) · Zbl 1284.13031 |
| [43] | Manschot, J.; Pioline, B.; Sen, A., On the Coulomb branch and Higgs branch formulae for multi-centered black holes and quiver invariants, J. High Energy Phys., 2013, 5, Article 166 pp. (2013) · Zbl 1342.83401 |
| [44] | Muller, G., Skein and cluster algebras of marked surfaces, Quantum Topol., 7, 3, 435-503 (2016) · Zbl 1375.13038 |
| [45] | Musiker, G.; Propp, J., Combinatorial interpretations for rank-two cluster algebras of affine type, Electron. J. Combin., 14, 1, Article 15 pp. (2007) · Zbl 1140.05053 |
| [46] | Musiker, G.; Schiffler, R., Cluster expansion formulas and perfect matchings, J. Algebraic Combin., 32, 2, 187-209 (2010) · Zbl 1246.13035 |
| [47] | Musiker, G.; Schiffler, R.; Williams, L., Positivity for cluster algebras from surfaces, Adv. Math., 227, 6, 2241-2308 (2011) · Zbl 1331.13017 |
| [48] | Musiker, G.; Schiffler, R.; Williams, L., Bases for cluster algebras from surfaces, Compos. Math., 149, 2, 217-263 (2013) · Zbl 1263.13024 |
| [49] | Nagao, K., Donaldson-Thomas theory and cluster algebras, Duke Math. J., 162, 7, 1313-1367 (2013) · Zbl 1375.14150 |
| [50] | Nakanishi, T., Periodicities in cluster algebras and dilogarithm identities, (Representations of Algebras and Related Topics, EMS Series of Congress Reports (2011)), 407-443 · Zbl 1320.13029 |
| [51] | Nakanishi, T.; Zelevinsky, A., On tropical dualities in cluster algebras, Contemp. Math., 565, 217-226 (2012) · Zbl 1317.13054 |
| [52] | Penner, R. C., The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys., 113, 299-339 (1987) · Zbl 0642.32012 |
| [53] | Plamondon, P.-G., Cluster algebras via cluster categories with infinite-dimensional morphism spaces, Compos. Math., 147, 6, 1921-1954 (2011) · Zbl 1244.13017 |
| [54] | Propp, J., Lattice structure for orientations of graphs (2002), preprint |
| [55] | Qin, F., Quantum cluster variables via Serre polynomials, J. Reine Angew. Math., 668, 149-190 (2012), with an appendix by Bernhard Keller · Zbl 1252.13013 |
| [56] | Qin, F., t-analog of q-characters, bases of quantum cluster algebras, and a correction technique, Int. Math. Res. Not. IMRN, 2014, 22, 6175-6232 (2014) · Zbl 1328.13033 |
| [57] | Qin, F., Quantum groups via cyclic quiver varieties I, Compos. Math., 152, 2, 299-326 (2016) · Zbl 1377.16009 |
| [58] | Qin, F., Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. J., 166, 12, 2337-2442 (2017) · Zbl 1454.13037 |
| [59] | Qiu, Y.; Zhou, Y., Cluster categories for marked surfaces: punctured case, Compos. Math., 153, 9, 1779-1819 (2017) · Zbl 1405.16024 |
| [60] | Rupel, D., On a quantum analog of the Caldero-Chapoton formula, Int. Math. Res. Not. IMRN, 2011, 14, 3207-3236 (2011) · Zbl 1237.16013 |
| [61] | Rupel, D., Quantum cluster characters for valued quivers, Trans. Amer. Math. Soc., 367, 7061-7102 (2015) · Zbl 1371.16014 |
| [62] | Schiffler, R.; Thomas, H., On cluster algebras arising from unpunctured surfaces, Int. Math. Res. Not. IMRN, 2019, 17, 3160-3189 (2009) · Zbl 1171.30019 |
| [63] | Stembridge, J., Canonical bases and self-evacuating tableaux, Duke Math. J., 82, 3, 585-606 (1996) · Zbl 0869.17011 |
| [64] | Szántó, C., On the cardinalities of Kronecker quiver Grassmannians, Math. Z., 269, 833-846 (2011) · Zbl 1266.16014 |
| [65] | Tran, T., F-polynomials in quantum cluster algebras, Algebr. Represent. Theory, 14, 1025-1061 (2011) · Zbl 1253.16016 |
| [66] | Williams, H., Toda systems, cluster characters, and spectral networks, preprint · Zbl 1360.37150 |
| [67] | Zelevinsky, A., Quantum cluster algebras: Oberwolfach talk, February 2005 (2005), preprint |
| [68] | Zelevinsky, A., Semicanonical basis generators of the cluster algebra of type \(A_1^{(1)}\), Electron. J. Combin., 14, 1, Article 4 pp. (2007) · Zbl 1144.16015 |
| [69] | Zhang, J.; Zhou, Y.; Zhu, B., Cotorsion pairs in the cluster category of a marked surface, J. Algebra, 391, 209-226 (2013) · Zbl 1297.13028 |
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