Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 16 (2020), 122, 22 pages      arXiv:2004.12466      https://doi.org/10.3842/SIGMA.2020.122
Contribution to the Special Issue on Cluster Algebras

An Analog of Leclerc's Conjecture for Bases of Quantum Cluster Algebras

Fan Qin
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, People's Republic of China

Received May 14, 2020, in final form November 13, 2020; Published online November 27, 2020

Abstract
Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the analogous conjecture is true. Our result applies to the dual canonical bases of quantum unipotent subgroups. It also applies to the $t$-analogs of $q$-characters of simple modules of quantum affine algebras.

Key words: dual canonical bases; cluster algebras; Leclerc's conjecture.

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References

  1. Berenstein A., Fomin S., Zelevinsky A., Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 1-52, arXiv:math.RT/0305434.
  2. Berenstein A., Zelevinsky A., Quantum cluster algebras, Adv. Math. 195 (2005), 405-455, arXiv:math.QA/0404446.
  3. Cautis S., Williams H., Cluster theory of the coherent Satake category, J. Amer. Math. Soc. 32 (2019), 709-778, arXiv:1801.08111.
  4. Derksen H., Weyman J., Zelevinsky A., Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749-790, arXiv:0904.0676.
  5. Fock V.V., Goncharov A.B., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006), 1-211, arXiv:math.AG/0311149.
  6. Fock V.V., Goncharov A.B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 865-930, arXiv:math.AG/0311245.
  7. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, arXiv:math.RT/0104151.
  8. Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112-164, arXiv:math.RA/0602259.
  9. Geiß C., Leclerc B., Schröer J., Kac-Moody groups and cluster algebras, Adv. Math. 228 (2011), 329-433, arXiv:1001.3545.
  10. Geiß C., Leclerc B., Schröer J., Cluster structures on quantum coordinate rings, Selecta Math. (N.S.) 19 (2013), 337-397, arXiv:1104.0531.
  11. Goodearl K., Yakimov M., Integral quantum cluster structures, arXiv:2003.04434.
  12. Goodearl K.R., Yakimov M.T., Quantum cluster algebra structures on quantum nilpotent algebras, Mem. Amer. Math. Soc. 247 (2017), vii+119 pages.
  13. Gross M., Hacking P., Keel S., Birational geometry of cluster algebras, Algebr. Geom. 2 (2015), 137-175, arXiv:1309.2573.
  14. Gross M., Hacking P., Keel S., Kontsevich M., Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), 497-608, arXiv:1411.1394.
  15. Hernandez D., Algebraic approach to $q,t$-characters, Adv. Math. 187 (2004), 1-52, arXiv:math.QA/0212257.
  16. Hernandez D., Leclerc B., Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2010), 265-341, arXiv:0903.1452.
  17. Kang S.J., Kashiwara M., Kim M., Oh S.-J., Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349-426, arXiv:1801.05145.
  18. Kashiwara M., Bases cristallines, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 277-280.
  19. Kashiwara M., On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465-516.
  20. Kashiwara M., Kim M., Laurent phenomenon and simple modules of quiver Hecke algebras, Compos. Math. 155 (2019), 2263-2295, arXiv:1811.02237.
  21. Kashiwara M., Kim M., Oh S.-J., Park E., Monoidal categorification and quantum affine algebras, Compos. Math. 156 (2020), 1039-1077, arXiv:1910.08307.
  22. Keller B., Cluster algebras, quiver representations and triangulated categories, in Triangulated Categories, London Math. Soc. Lecture Note Ser., Vol. 375, Cambridge Univetsity Press, Cambridge, 2010, 76-160, arXiv:0807.1960.
  23. Khovanov M., Lauda A.D., A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347, arXiv:0803.4121.
  24. Khovanov M., Lauda A.D., A categorification of quantum ${\rm sl}(n)$, Quantum Topol. 1 (2010), 1-92, arXiv:0807.3250.
  25. Kimura Y., Quantum unipotent subgroup and dual canonical basis, Kyoto J. Math. 52 (2012), 277-331, arXiv:1010.4242.
  26. Kimura Y., Qin F., Graded quiver varieties, quantum cluster algebras and dual canonical basis, Adv. Math. 262 (2014), 261-312, arXiv:1205.2066.
  27. Leclerc B., Imaginary vectors in the dual canonical basis of $U_q({\mathfrak n})$, Transform. Groups 8 (2003), 95-104, arXiv:math.QA/0202148.
  28. Lusztig G., Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498.
  29. Lusztig G., Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365-421.
  30. Lusztig G., Total positivity in reductive groups, in Lie Theory and Geometry, Progr. Math., Vol. 123, Birkhäuser Boston, Boston, MA, 1994, 531-568.
  31. Nakajima H., Quiver varieties and $t$-analogs of $q$-characters of quantum affine algebras, Ann. of Math. 160 (2004), 1057-1097, arXiv:math.QA/0105173.
  32. Nakajima H., Quiver varieties and cluster algebras, Kyoto J. Math. 51 (2011), 71-126, arXiv:0905.0002.
  33. Qin F., Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. J. 166 (2017), 2337-2442, arXiv:1501.04085.
  34. Qin F., Bases for upper cluster algebras and tropical points, arXiv:1902.09507.
  35. Qin F., Dual canonical bases and quantum cluster algebras, arXiv:2003.13674.
  36. Rouquier R., 2-Kac-Moody algebras, arXiv:0812.5023.
  37. Tran T., $F$-polynomials in quantum cluster algebras, Algebr. Represent. Theory 14 (2011), 1025-1061, arXiv:0904.3291.
  38. Varagnolo M., Vasserot E., Perverse sheaves and quantum Grothendieck rings, in Studies in Memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math., Vol. 210, Birkhäuser Boston, Boston, MA, 2003, 345-365, arXiv:math.QA/0103182.

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