Electronic Journal of Probability
- Electron. J. Probab.
- Volume 18 (2013), paper no. 95, 25 pp.
A $q$-weighted version of the Robinson-Schensted algorithm
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.
Electron. J. Probab. Volume 18 (2013), paper no. 95, 25 pp.
Accepted: 29 October 2013
First available in Project Euclid: 4 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 05E05: Symmetric functions and generalizations
Secondary: 15A52 82C22: Interacting particle systems [See also 60K35]
This work is licensed under a Creative Commons Attribution 3.0 License.
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