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**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.

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 |