COMBINATORICS AND INTEGRABLE GEOMETRY

  • Pierre Van Moerbeke
Conference paper
Part of the NATO Science Series book series (NAII, volume 201)

Abstract

Since Russel's horse back journey along the canal from Glasgow to Edinburg in 1834, since the birth of the Korteweg-de Vries equation in 1895 and since the remarkable renaissance initiated by M. Kruskal and coworkers in the late 60's, the field of integrable systems has emerged as being at the crossroads of important new developments in the sciences.

Keywords

Vertex Operator Young Diagram Integrable Geometry Weyl Chamber Fredholm Determinant 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    van Moerbeke, P. (2001) Integrable lattices: random matrices and random permutations, in: Random Matrices and their Applications, Mathematical Sciences research Institute Publications #40, Cambridge University Press, pp. 321–406. http://www.msri.org/publications/books/Book40/files/moerbeke.pdfGoogle Scholar
  2. 2.
    Adler, M. and van Moerbeke, P. (2000) Integrals over classical groups, random permutations, Toda and Toeplitz lattices, Comm. Pure Appl. Math. 54, pp. 153–205, (arXiv: math.CO/9912143).CrossRefGoogle Scholar
  3. 3.
    Gessel, I. M. (1990) Symmetric functions and P-recursiveness, J. of Comb. Theory, Ser. A, 53, pp. 257–285.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Tracy, C. A. and Widom, H. (2001) On the distributions of the lengths of the longest monotone subsequences in random words. Probab. Theory Related Fields, 119(3), pp. 350–380. (arXiv:math.CO/9904042).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Rains, E. M. (1998) Increasing subsequences and the classical groups, Elect. J. of Combinatorics, 5, R12.MathSciNetGoogle Scholar
  6. 6.
    Adler, M. and van Moerbeke, P. (2004) Integrals over Grassmannians and Random permutations, Adv. in Math. 181, pp. 190–249. (arXiv: math.CO/0110281).MATHCrossRefGoogle Scholar
  7. 7.
    MacDonald, I. G. (1995) “Symmetric functions and Hall polynomials”, Clarendon Press.Google Scholar
  8. 8.
    Sagan, B. E. (1991) The Symmetry Group, Wadsworth & Brooks, Pacific Grove, California.Google Scholar
  9. 9.
    Stanley, R. S. (1999) Enumerative Combinatorics, Vol. 2, Cambridge studies in advanced mathematics 62, Cambridge University Press.Google Scholar
  10. 10.
    Stanton, D. and White, D. (1986) Constructive Combinatorics, Springer-Verlag, NY.MATHGoogle Scholar
  11. 11.
    Borodin, A. and Olshanski, G. (2001) Z-measures on partitions, Robinson- Shensted-Knuth correspondence, and β = 2 random matrix ensembles, Random matrix models and their applications, pp. 71–94, Math. Sci. Res. Inst. Publ., 40, Cambridge University Press, Cambridge.Google Scholar
  12. 12.
    Okounkov, A. (2001) Infinite wedge and measures on partitions, Selecta Math. 7, pp. 57–81. (arXiv: math.RT/9907127)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Adler, M. and van Moerbeke, P. (2004) Virasoro action on Schur function expansions, skew Young tableaux and random walks, Comm. Pure Appl. Math. (ArXiv: math.PR/0309202)Google Scholar
  14. 14.
    Forrester, P. J. (2001) Random walks and random permutations, J. Phys. A 34(31), pp. L417–L423. (arXiv:math.CO/9907037).MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Grabiner, D. J. and Magyar, Peter (1993) Random walks inWeyl chambers and the decomposition of tensor powers. J. Algebraic Combin. 2(3), pp. 239–260.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Grabiner, D. J. (2002) Random walk in an alcove of an affine Weyl group, and non-colliding random walks on an interval, J. Combin. Theory, Ser. A 97(2), pp. 285–306.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Baik, J. (2000) Random vicious walks and random matrices. Comm. Pure Appl. Math. 53, pp. 1385–1410.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    McMillan, E. M. (1971) A problem in the stability of periodic systems, Topics in Modern Physics, A tribute to E. U. Condon, eds. W. E. Brittin and H. Odabasi (Colorado Ass. Univ. Press, Boulder), pp. 219–244.Google Scholar
  19. 19.
    Borodin, A. (2003) Discrete gap probabilities and discrete Painlevé equations, Duke Math. J. 117, pp. 489–542. (arXiv:math-ph/0111008).MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Baik, J. (2003) Riemann-Hilbert problems for last passage percolation. Recent developments in integrable systems and Riemann-Hilbert problems (Birmingham, AL, 2000), pp. 1–21, Contemp. Math., 326, Amer. Math. Soc., Providence, RI. (arXiv:math.PR/0107079).Google Scholar
  21. 21.
    Suris, Yu. B. (1987) Integrable mappings of standard type, Funct. Anal. Appl. 23, pp. 74–76.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Grammaticos, B. Nijhoff, F., and Ramani, A. (1999) Discrete Painlevé equations, The Painlevé property, CRM series in Math. Phys., Chapter 7, Springer, New York, 1999, pp. 413–516.Google Scholar
  23. 23.
    Tracy, C. A. and Widom, H. (1999) Random unitary matrices, permutations and Painlevé, Commun. Math. Phys. 207, pp. 665–685.MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Adler, M. and van Moerbeke, P. (2003) Recursion relations for Unitary integrals of Combinatorial nature and the Toeplitz lattice, Comm. Math. Phys. 237, pp. 397–440. (arXiv: math-ph/0201063).MATHADSMathSciNetGoogle Scholar
  25. 25.
    Borodin, A. and Okounkov, A. (2000) A Fredholm determinant formula for Toeplitz determinants, Integral Equations Operator Theory 37, pp. 386–396. (math.CA/9907165).MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Borodin, A. and Olshanski, G. (2000) Distributions on partitions, point processes, and the hypergeometric kernel, (Comm. Math. Phys. 211(2), pp. 335–358. (math.RT/9904010).MATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    Borodin, A. and Deift, P. (2002) Fredholm determinants, Jimbo-Miwa-Ueno taufunctions, and representation theory, Comm. Pure Appl. Math. 55, pp. 1160–1230. (arXiv:math-ph/0111007).MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Gessel, I. M. (1990) Symmetric functions and P-recursiveness, J. of Comb. Theory, Ser. A, 53, pp. 257–285.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Gessel, I. M. and Zeilberger, Doron (1992) Random walk in aWeyl chamber, Proc. Amer. Math. Soc. 115(1), pp. 27–31.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Pierre Van Moerbeke
    • 1
    • 2
  1. 1.Department of MathematicsUniversit de LouvainLouvain-la-NeuveBelgium
  2. 2.Brandeis UniversityWaltham, MassUSA

Personalised recommendations