The PDF file you selected should load here if your Web browser has a PDF reader plug-in installed (for example, a recent version of Adobe Acrobat Reader).

Alternatively, you can also download the PDF file directly to your computer, from where it can be opened using a PDF reader. To download the PDF, click the Download link below.

If you would like more information about how to print, save, and work with PDFs, Highwire Press provides a helpful Frequently Asked Questions about PDFs.

Download this PDF file Fullscreen Fullscreen Off


  • Aldous, David; Diaconis, Persi. Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem. Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 4, 413--432. MR1694204
  • Baik, Jinho; Deift, Percy; Johansson, Kurt. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999), no. 4, 1119--1178. MR1682248
  • Balázs, Márton; Komjáthy, Júlia; Seppäläinen, Timo. Microscopic concavity and fluctuation bounds in a class of deposition processes. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 1, 151--187. MR2919202
  • Berg, Sonya J. A quantum algorithm for the quantum Schur-Weyl transform. Thesis (Ph.D.)–University of California, Davis. ProQuest LLC, Ann Arbor, MI, 2012. 68 pp. ISBN: 978-1267-65637-7 MR3094056 arXiv:1205.3928
  • Biane, Philippe; Bougerol, Philippe; O'Connell, Neil. Littelmann paths and Brownian paths. Duke Math. J. 130 (2005), no. 1, 127--167. MR2176549
  • Biane, Philippe; Bougerol, Philippe; O'Connell, Neil. Continuous crystal and Duistermaat-Heckman measure for Coxeter groups. Adv. Math. 221 (2009), no. 5, 1522--1583. MR2522427
  • A. Borodin and I. Corwin. Macdonald processes. Probab. Th. Rel. Fields, to appear. arXiv:1111.4408.
  • A. Borodin, I. Corwin and T. Sasamoto. From duality to determinants for q-TASEP and ASEP. Ann. Prob., to appear. arXiv:1207.5035.
  • A. Braverman and M. Finkelberg. Weyl modules and q-Whittaker functions. arXiv:1203.1583.
  • Ben Brubaker, Daniel Bump, Anthony Licata. Whittaker Functions and Demazure Operators. arXiv:1111.4230.
  • R. Chhaibi. Modèle de Littelmann pour cristaux géométriques, fonctions de Whittaker sur des groupes de Lie et mouvement brownien. PhD thesis, Université Paris VI - Pierre et Marie Curie., 2012.
  • I. Corwin, N. O'Connell, T. Seppäläinen and N. Zygouras. Tropical combinatorics and Whittaker functions. Duke Math. J., to appear. arXiv:1110.3489.
  • Date, Etsurō; Jimbo, Michio; Miwa, Tetsuji. Representations of $U_ q(\mathfrak{gl}(n,{\bf C}))$ at $q=0$ and the Robinson-Shensted [Schensted] correspondence. Physics and mathematics of strings, 185--211, World Sci. Publ., Teaneck, NJ, 1990. MR1104259
  • Etingof, Pavel. Whittaker functions on quantum groups and $q$-deformed Toda operators. Differential topology, infinite-dimensional Lie algebras, and applications, 9--25, Amer. Math. Soc. Transl. Ser. 2, 194, Amer. Math. Soc., Providence, RI, 1999. MR1729357
  • Forrester, Peter J.; Rains, Eric M. Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Related Fields 131 (2005), no. 1, 1--61. MR2105043
  • Fulton, William. Young tableaux. With applications to representation theory and geometry. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997. x+260 pp. ISBN: 0-521-56144-2; 0-521-56724-6 MR1464693
  • Gerasimov, Anton; Lebedev, Dimitri; Oblezin, Sergey. On $q$-deformed ${\mathfrak{gl}}_ {l+1}$-Whittaker function. I. Comm. Math. Phys. 294 (2010), no. 1, 97--119. MR2575477
  • Gerasimov, Anton; Lebedev, Dimitri; Oblezin, Sergey. On $q$-deformed $\mathfrak{gl}_ {\ell+1}$-Whittaker function III. Lett. Math. Phys. 97 (2011), no. 1, 1--24. MR2802312
  • Gerasimov, Anton; Lebedev, Dimitri; Oblezin, Sergey. On a classical limit of $q$-deformed Whittaker functions. Lett. Math. Phys. 100 (2012), no. 3, 279--290. MR2923976
  • Haglund, J.; Haiman, M.; Loehr, N. A combinatorial formula for Macdonald polynomials. J. Amer. Math. Soc. 18 (2005), no. 3, 735--761. MR2138143
  • Kirillov, Anatol N. Introduction to tropical combinatorics. Physics and combinatorics, 2000 (Nagoya), 82--150, World Sci. Publ., River Edge, NJ, 2001. MR1872253
  • Kirillov, Anatol N. Ubiquity of Kostka polynomials. Physics and combinatorics 1999 (Nagoya), 85--200, World Sci. Publ., River Edge, NJ, 2001. MR1865038
  • Knuth, Donald E. Permutations, matrices, and generalized Young tableaux. Pacific J. Math. 34 1970 709--727. MR0272654
  • Lecouvey, Cédric; Lesigne, Emmanuel; Peigné, Marc. Random walks in Weyl chambers and crystals. Proc. Lond. Math. Soc. (3) 104 (2012), no. 2, 323--358. MR2880243
  • Lenart, Cristian. On combinatorial formulas for Macdonald polynomials. Adv. Math. 220 (2009), no. 1, 324--340. MR2462843
  • Lenart, Cristian; Lubovsky, Arthur. A generalization of the alcove model and its applications. 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 875--886, Discrete Math. Theor. Comput. Sci. Proc., AR, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012. MR2958056
  • Lenart, Cristian; Schilling, Anne. Crystal energy functions via the charge in types $A$ and $C$. Math. Z. 273 (2013), no. 1-2, 401--426. MR3010167
  • Macdonald, I. G. Symmetric functions and Hall polynomials. Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. ISBN: 0-19-853489-2 MR1354144
  • O'Connell, Neil. Conditioned random walks and the RSK correspondence. Random matrix theory. J. Phys. A 36 (2003), no. 12, 3049--3066. MR1986407
  • O'Connell, Neil. Directed polymers and the quantum Toda lattice. Ann. Probab. 40 (2012), no. 2, 437--458. MR2952082
  • N. O'Connell. Whittaker functions and related stochastic processes. To appear in proceedings of Fall 2010 MSRI semester Random matrices, interacting particle systems and integrable systems. arXiv:1201.4849.
  • N. O'Connell, T. Seppäläinen and N. Zygouras. Geometric RSK correspondence, Whittaker functions and symmetrized random polymers. Invent. Math., October 2013. arXiv:1210.5126.
  • Okounkov, Andrei. Infinite wedge and random partitions. Selecta Math. (N.S.) 7 (2001), no. 1, 57--81. MR1856553
  • Ram, Arun; Yip, Martha. A combinatorial formula for Macdonald polynomials. Adv. Math. 226 (2011), no. 1, 309--331. MR2735761
  • Robinson, G. de B. On the Representations of the Symmetric Group. Amer. J. Math. 60 (1938), no. 3, 745--760. MR1507943
  • Ruijsenaars, S. N. M. Relativistic Toda systems. Comm. Math. Phys. 133 (1990), no. 2, 217--247. MR1090424
  • Ruijsenaars, S. N. M. Systems of Calogero-Moser type. Particles and fields (Banff, AB, 1994), 251--352, CRM Ser. Math. Phys., Springer, New York, 1999. MR1668150
  • Sagan, Bruce E. The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, 2001. xvi+238 pp. ISBN: 0-387-95067-2 MR1824028
  • Schensted, C. Longest increasing and decreasing subsequences. Canad. J. Math. 13 1961 179--191. MR0121305
  • Stanley, Richard P. Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. xii+581 pp. ISBN: 0-521-56069-1; 0-521-78987-7 MR1676282
  • Sasamoto, Tomohiro; Wadati, Miki. Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31 (1998), no. 28, 6057--6071. MR1633078
  • Schilling, Anne; Tingely, Peter. Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function. [Second author's name now "Tingley'' on article]. Electron. J. Combin. 19 (2012), no. 2, Paper 4, 42 pp. MR2923717
  • Tracy, Craig A.; Widom, Harold. On the distributions of the lengths of the longest monotone subsequences in random words. Probab. Theory Related Fields 119 (2001), no. 3, 350--380. MR1821139

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.