Advertisement

Communications in Mathematical Physics

, Volume 341, Issue 3, pp 821–884 | Cite as

Twists of Plücker Coordinates as Dimer Partition Functions

  • R. J. MarshEmail author
  • J. S. Scott
Article

Abstract

The homogeneous coordinate ring of the Grassmannian Gr k,n has a cluster structure defined in terms of planar diagrams known as Postnikov diagrams. The cluster corresponding to such a diagram consists entirely of Plücker coordinates. We introduce a twist map on Gr k,n , related to the Berenstein–Fomin–Zelevinsky-twist, and give an explicit Laurent expansion for the twist of an arbitrary Plücker coordinate in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Plücker coordinate. We also relate the twist map to a maximal green sequence.

Keywords

Bipartite Graph Cluster Variable Cluster Algebra Boundary Vertex Black Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alim M., Cecotti S., Cordova 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)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Arkani-Hamed, N., Bourjaily, J.L., Cachazo, F., Goncharov, A.B., Postnikov, A., Trnka, J.: Scattering amplitudes and the positive Grassmannian (2012) (preprint). arXiv:1212.5605 [hep-th]
  3. 3.
    Assem I., Dupont G., Schiffler R.: On a category of cluster algebras. J. Pure Appl. Algebra 218(3), 553–582 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Berenstein A., Fomin S., Zelevinsky A.: Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1–52 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Berenstein A., Fomin S., Zelevinsky A.: Parametrizations of canonical bases and totally positive matrices. Adv. Math. 122(1), 49–149 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Berenstein A., Zelevinsky A.: Total positivity in Schubert varieties. Comm. Math. Helv. 72, 128–166 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Mathematics, vol. 126. Springer, New York (1991)Google Scholar
  8. 8.
    Brüstle T., Dupont G., Pérotin M.: On maximal green sequences. Int. Math. Res. Not. 16, 4547–4586 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    çanakçi I., Schiffler R.: Snake graph calculus and cluster algebras from surfaces. J. Algebra 382, 240–281 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Carroll, G., Price, G.: Two new combinatorial models for the Ptolemy recurrence. Unpublished memo (2003) (see [31] for details)Google Scholar
  11. 11.
    Cecotti, S., Córdova, C., Vafa, C.: Braids, walls and mirrors (2011) (preprint). arXiv:1110.2115 [hep-th]
  12. 12.
    Cecotti, S., Neitzke, A., Vafa, C.: R-twisting and 4d/2d correspondences (2010) (preprint). arXiv:1006.3435 [hep-th]
  13. 13.
    Ciucu M.: A complementation theorem for perfect matchings of graphs having a cellular completion. J. Combin. Theory Ser. A 81(1), 34–68 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Fomin S., Zelevinsky A.: Double Bruhat cells and total positivity. J. Am. Math. Soc. 12(2), 335–380 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Fomin S., Zelevinsky A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Fomin S., Zelevinsky A.: Cluster algebras. IV. Coefficients. Compos. Math. 143(1), 112–164 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3(3), 899–934, 1199 (2003)Google Scholar
  18. 18.
    Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster Algebras and Poisson Geometry. Mathematical Surveys and Monographs, vol. 167. American Mathematical Society, Providence (2010)Google Scholar
  19. 19.
    Geiss C., Leclerc B., Schröer J.: Partial flag varieties and preprojective algebras. Ann. Inst. Fourier (Grenoble) 58(3), 825–876 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Geiss C., Leclerc B., Schröer J.: Kac–Moody groups and cluster algebras. Adv. Math. 228(1), 329–433 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Geiss C., Leclerc B., Schröer J.: Generic bases for cluster algebras and the Chamber ansatz. J. Am. Math. Soc. 25(1), 21–76 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Goncharov, A.B.: Ideal webs and moduli spaces of local systems on surfaces (in progress)Google Scholar
  23. 23.
    Goncharov A.B., Kenyon R.: Dimers and cluster integrable systems. Ann. Sci. Éc. Norm. Supér. (4) 46(5), 747–813 (2013)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Keller, B.: On cluster theory and quantum dilogarithm identities. In: Representations of Algebras and Related Topics, pp. 85–116. EMS Series of Congress Reports. European Mathematical Society, Zürich (2011)Google Scholar
  25. 25.
    Keller B.: The periodicity conjecture for pairs of Dynkin diagrams. Ann. Math. (2) 177(1), 111–170 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Kenyon, R.W., Propp, J.G., Wilson, D.B.: Trees and matchings. Electron. J. Combin. 7 (2000). Research Paper 25, 34 pp.Google Scholar
  27. 27.
    Kuo E.H.: Applications of graphical condensation for enumerating matchings and tilings. Theor. Comput. Sci. 319(1–3), 29–57 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Leclerc B.: On identities satisfied by minors of a matrix. Adv. Math. 100(1), 101–132 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Lusztig G.: Total positivity in partial flag manifolds. Represent. Theory 2, 70–78 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Muller, G., Speyer, D.E.: Cluster algebras of Grassmannians are locally acyclic (2014) (preprint). arXiv:1401.5137v3 [math.CO]
  31. 31.
    Musiker G.: A graph theoretic expansion formula for cluster algebras of classical type. Ann. Comb. 15(1), 147–184 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Musiker G., Schiffler R.: Cluster expansion formulas and perfect matchings. J. Algebraic Comb. 32(2), 187–209 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Musiker G., Schiffler R., Williams L.: Positivity for cluster algebras from surfaces. Adv. Math. 227(6), 2241–2308 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Musiker G., Schiffler R., Williams L.: Bases for cluster algebras from surfaces. Compos. Math. 149(2), 217–263 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    Postnikov, A.: Total positivity, Grassmannians and networks (2006) (preprint). arXiv:math/0609764v1 [math.CO]
  36. 36.
    Ravanini F., Valleriani A., Tateo R.: Dynkin TBAs. Int. J. Mod. Phys. A 8, 1707–1727 (1993)CrossRefADSMathSciNetGoogle Scholar
  37. 37.
    Rietsch K., Williams L.: The totally nonnegative part of G/P is a CW complex. Transform. Groups 13(3–4), 839–853 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Schiffler, R.: A cluster expansion formula (A n case). Electron. J. Combin. 15(1) (2008). Research paper 64, 9 pp.Google Scholar
  39. 39.
    Scott J.: Grassmannians and cluster algebras. Proc. Lond. Math. Soc. (3) 92(2), 345–380 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    Talaska K.: Combinatorial formulas for Open image in new window -coordinates in a totally nonnegative Grassmannian. J. Combin. Theory Ser. A 118(1), 58–66 (2011)Google Scholar
  41. 41.
    Volkov A.Y.: On the periodicity conjecture for Y-systems. Commun. Math. Phys. 276(2), 509–517 (2007)CrossRefADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    Zamolodchikov A.B.: On the thermodynamic Bethe ansatz equations for reactionless ADE scattering theories. Phys. Lett. B 253, 391–394 (1991)CrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of MathematicsUniversity of LeedsLeedsUK
  2. 2.Departamento de MatemáticasUniversidad de los AndesBogotáColombia

Personalised recommendations