## Electronic Journal of Probability

### A $q$-weighted version of the Robinson-Schensted algorithm

#### Abstract

We introduce a $q$-weighted version of the Robinson-Schensted (column insertion) algorithm which is closely connected to $q$ Whittaker functions (or Macdonald polynomials with $t=0)$ and reduces to the usual Robinson-Schensted algorithm when $q=0$. The $q$-insertion algorithm is randomised', or quantum', in the sense that when inserting a positive integer into a tableau, the output is a distribution of weights on a particular set of tableaux which includes the output which would have been obtained via the usual column insertion algorithm. There is also a notion of recording tableau in this setting. We show that the distribution of weights of the pair of tableaux obtained when one applies the $q$-insertion algorithm to a random word or permutation takes a particularly simple form and is closely related to $q$-Whittaker functions. In the case $0 \le$q$<1$, the $q$-insertion algorithm applied to a random word also provides a new framework for solving the $q$-TASEP interacting particle system introduced (in the language of $q$-bosons) by Sasamoto and Wadati (1998) and yields formulas which are equivalent to some of those recently obtained by Borodin and Corwin (2011) via a stochastic evolution on discrete Gelfand-Tsetlin patterns (or semistandard tableaux) which is coupled to the $q$-TASEP. We show that the sequence of $P$-tableaux obtained when one applies the $q$-insertion algorithm to a random word defines another, quite different, evolution on semistandard tableaux which is also coupled to the $q$-TASEP.

#### Article information

Source
Electron. J. Probab. Volume 18 (2013), paper no. 95, 25 pp.

Dates
Accepted: 29 October 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064320

Digital Object Identifier
doi:10.1214/EJP.v18-2930

Mathematical Reviews number (MathSciNet)
MR3126578

Zentralblatt MATH identifier
1278.05243

Subjects
Primary: 05E05: Symmetric functions and generalizations

Rights

#### Citation

O'Connell, Neil; Pei, Yuchen. A $q$-weighted version of the Robinson-Schensted algorithm. Electron. J. Probab. 18 (2013), paper no. 95, 25 pp. doi:10.1214/EJP.v18-2930. https://projecteuclid.org/euclid.ejp/1465064320

#### References

• 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.
• 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.
• 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.
• 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 arXiv:1205.3928
• Biane, Philippe; Bougerol, Philippe; O'Connell, Neil. Littelmann paths and Brownian paths. Duke Math. J. 130 (2005), no. 1, 127–167.
• Biane, Philippe; Bougerol, Philippe; O'Connell, Neil. Continuous crystal and Duistermaat-Heckman measure for Coxeter groups. Adv. Math. 221 (2009), no. 5, 1522–1583.
• 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, EtsurÅ; 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.
• 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.
• 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.
• 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
• 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.
• 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.
• Gerasimov, Anton; Lebedev, Dimitri; Oblezin, Sergey. On a classical limit of $q$-deformed Whittaker functions. Lett. Math. Phys. 100 (2012), no. 3, 279–290.
• Haglund, J.; Haiman, M.; Loehr, N. A combinatorial formula for Macdonald polynomials. J. Amer. Math. Soc. 18 (2005), no. 3, 735–761.
• Kirillov, Anatol N. Introduction to tropical combinatorics. Physics and combinatorics, 2000 (Nagoya), 82–150, World Sci. Publ., River Edge, NJ, 2001.
• Kirillov, Anatol N. Ubiquity of Kostka polynomials. Physics and combinatorics 1999 (Nagoya), 85–200, World Sci. Publ., River Edge, NJ, 2001.
• Knuth, Donald E. Permutations, matrices, and generalized Young tableaux. Pacific J. Math. 34 1970 709–727.
• Lecouvey, Cédric; Lesigne, Emmanuel; Peigné, Marc. Random walks in Weyl chambers and crystals. Proc. Lond. Math. Soc. (3) 104 (2012), no. 2, 323–358.
• Lenart, Cristian. On combinatorial formulas for Macdonald polynomials. Adv. Math. 220 (2009), no. 1, 324–340.
• 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.
• Lenart, Cristian; Schilling, Anne. Crystal energy functions via the charge in types $A$ and $C$. Math. Z. 273 (2013), no. 1-2, 401–426.
• 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
• O'Connell, Neil. Conditioned random walks and the RSK correspondence. Random matrix theory. J. Phys. A 36 (2003), no. 12, 3049–3066.
• O'Connell, Neil. Directed polymers and the quantum Toda lattice. Ann. Probab. 40 (2012), no. 2, 437–458.
• 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.
• Ram, Arun; Yip, Martha. A combinatorial formula for Macdonald polynomials. Adv. Math. 226 (2011), no. 1, 309–331.
• Robinson, G. de B. On the Representations of the Symmetric Group. Amer. J. Math. 60 (1938), no. 3, 745–760.
• Ruijsenaars, S. N. M. Relativistic Toda systems. Comm. Math. Phys. 133 (1990), no. 2, 217–247.
• 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.
• 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
• Schensted, C. Longest increasing and decreasing subsequences. Canad. J. Math. 13 1961 179–191.
• 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
• Sasamoto, Tomohiro; Wadati, Miki. Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A 31 (1998), no. 28, 6057–6071.
• 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.
• 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.