## A $$q$$-weighted version of the Robinson-Schensted algorithm.(English)Zbl 1278.05243

Summary: 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 $$0\leq q<1$$, 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 T. Sasamoto and M. Wadati [J. Phys. A, Math. Gen. 31, No. 28, 6057–6071 (1998; Zbl 1085.83501)] and yields formulas which are equivalent to some of those recently obtained by A. Borodin and I. Corwin [“Macdonald processes”, Probab. Theory Relat. Fields (to appear)] 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.

### MSC:

 05E05 Symmetric functions and generalizations 15B52 Random matrices (algebraic aspects) 82C22 Interacting particle systems in time-dependent statistical mechanics

### Keywords:

$$q$$-Whittaker functions; Macdonald polynomials

Zbl 1085.83501
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