Neil O'Connell, Yuchen Pei
We introduce a generalisation of the dual RSK algorithm which is closely connected to q-deformed ${\mathfrak gl}_{l}$-Whittaker functions and reduces to the usual dual RSK algorithm when $q=0$. The insertion algorithm is `randomised' in the sense that when inserting a positive integer into a tableau, the output is a distribution of weights on a particular set of tableau 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 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 insertion algorithm applied to a random word also provides a framework for solving the $q$-TASEP interacting particle system introduced 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 which is coupled to the $q$-TASEP process. The sequence of $P$-tableaux obtained when one applies the randomised insertion algorithm to a random word defines another, quite different, evolution on discrete Gelfand-Tsetlin patterns which is also coupled to the $q$-TASEP process.
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http://arxiv.org/abs/1212.6716
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