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.
Neil O'Connell. Yuchen Pei. "A $q$-weighted version of the Robinson-Schensted algorithm." Electron. J. Probab. 18 1 - 25, 2013. https://doi.org/10.1214/EJP.v18-2930