Letters in Mathematical Physics

, Volume 96, Issue 1–3, pp 299–324 | Cite as

Discrete Integrable Systems, Positivity, and Continued Fraction Rearrangements



In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the Q- and T -systems based on A r . The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.

Mathematics Subject Classification (2010)

05E10 13F16 82B20 


cluster algebras Laurent phenomenon positivity integrable systems non-commutative continued fractions 


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  1. 1.
    Assem, I., Reutenauer, C., Smith, D.: Frises. Preprint (2009). arXiv:0906.2026 [math.RA]Google Scholar
  2. 2.
    Berenstein A., Zelevinsky A.: Quantum cluster algebras. Adv. Math. 195, 405–455 (2005) arXiv:math/0404446 [math.QA]CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Caldero P., Reineke M.: On the quiver Grassmannian in the acyclic case. J. Pure Appl. Algebra 212(11), 2369–2380 (2008) arXiv:math/0611074 [math.RT]CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Di Francesco P.: The solution of the A r T-system with arbitrary boundary. Electron. J. Comb. 17(1), R89 (2010) arXiv:1002.4427 [math.CO]MathSciNetGoogle Scholar
  5. 5.
    Di Francesco P., Kedem R.: Q-systems as cluster algebras II. Lett. Math. Phys. 89(3), 183–216 (2009) arXiv:0803.0362 [math.RT]CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Di Francesco P., Kedem R.: Q-systems, heaps, paths and cluster positivity. Comm. Math. Phys. 293(3), 727–802 (2009). doi: 10.1007/s00220-009-0947-5 arXiv:0811.3027 [math.CO]CrossRefMathSciNetGoogle Scholar
  7. 7.
    Di Francesco P., Kedem R.: Positivity of the T-system cluster algebra. Electron. J. Comb. 16(1), R140 (2009) Oberwolfach preprint OWP 2009-21. arXiv:0908.3122 [math.CO]MathSciNetGoogle Scholar
  8. 8.
    Di Francesco, P., Kedem, R.: Discrete non-commutative integrability: proof of a conjecture by M. Kontsevich. Int. Math. Res. Notices (2010). doi: 10.1093/imrn/rnq024. arXiv:0909.0615 [math-ph]
  9. 9.
    Di Francesco, P., Kedem, R.: Noncommutative integrability, paths and quasi-determinants. Preprint (2010). arXiv:1006.4774 [math-ph]Google Scholar
  10. 10.
    Dodgson C.: Condensation of determinants. Proc. R. Soc. Lond. 15, 150–155 (1866)CrossRefGoogle Scholar
  11. 11.
    Fomin S., Zelevinsky A.: Cluster algebras I. J. Am. Math. Soc. 15(2), 497–529 (2002) arXiv:math/0104151 [math.RT]CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Fomin S., Zelevinsky A.: The Laurent phenomenon. Adv. Appl. Math. 28(2), 119–144 (2002) arXiv:math/0104241 [math.CO]CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Fomin S., Zelevinsky A.: Cluster Algebras IV: coefficients. Compos. Math. 143, 112–164 (2007) arXiv:math/0602259 [math.RA]CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine algebras and deformations of W-algebras. In: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC 1998). Contemp. Math., vol. 248, pp. 163–205 (1999)Google Scholar
  15. 15.
    Gelfand I., Gelfand S., Retakh V., Wilson R.L.: Quasideterminants. Adv. Math. 193(1), 56–141 (2005) arXiv:math/0208146v4 [math.QA]CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Gelfand I., Krob D., Lascoux A., Leclerc B., Retakh V., Thibon J.-Y.: Noncommutative symmetric functions. Adv. Math. 112(2), 218–348 (1995) arXiv:hep-th/9407124CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Gessel I.M., Viennot X.: Binomial determinants, paths and hook formulae. Adv. Math. 58, 300–321 (1985)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Henriquès A.: A periodicity theorem for the octahedron recurrence. J. Algebraic Comb. 26(1), 1–26 (2007) arXiv:math/0604289 [math.CO]CrossRefMATHGoogle Scholar
  19. 19.
    Kedem, R.: Q-systems as cluster algebras. J. Phys. A: Math. Theor. 41, 194011 (14 pp.) (2008). arXiv:0712.2695 [math.RT]Google Scholar
  20. 20.
    Kirillov A.N., Reshetikhin N.Yu.: Representations of Yangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras. J. Sov. Math. 52, 3156–3164 (1990)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Knutson, A., Tao, T., Woodward, C.: A positive proof of the Littlewood-Richardson rule using the octahedron recurrence. Electron. J. Comb. 11, RP 61 (2004). arXiv:math/0306274 [math.CO]Google Scholar
  22. 22.
    Kuniba A., Nakanishi A., Suzuki J.: Functional relations in solvable lattice models. Part I: functional relations and representation theory. Int. J. Mod. Phys. A 9(30), 5215–5266 (1994)CrossRefMATHADSMathSciNetGoogle Scholar
  23. 23.
    Lindström B.: On the vector representations of induced matroids. Bull. Lond. Math. Soc. 5, 85–90 (1973)CrossRefMATHGoogle Scholar
  24. 24.
    Musiker, G., Schiffler, R., Williams, L.: Positivity for cluster algebras from surfaces. Preprint. arXiv:0906.0748 [math.CO]Google Scholar
  25. 25.
    Nakajima H.: t-Analogs of q-characters of Kirillov-Reshetikhin modules of quantum affine algebras. Represent. Theory 7, 259–274 (2003)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Speyer D.: Perfect matchings and the octahedron recurrence. J. Algebraic Comb. 25(3), 309–348 (2007) arXiv:math/0402452 [math.CO]CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Stieltjes, T.J.: Recherches sur les fractions continues. In: Oeuvres complètes de Thomas Jan Stieltjes, vol. II, no. LXXXI, pp. 402–566. P. Nordhoff, Groningen (1918)Google Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Institut de Physique Théorique du Commissariat à l’Energie AtomiqueUnité de Recherche associée du CNRSGif sur Yvette CedexFrance

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