q\rm RSt: A probabilistic Robinson--Schensted correspondence for Macdonald polynomials

2020/09/08 by Aigner, Florian, Frieden, Gabriel
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2009.03526

Abstract

We present a probabilistic generalization of the Robinson--Schensted correspondence in which a permutation maps to several different pairs of standard Young tableaux with nonzero probability. The probabilities depend on two parameters q and t, and the correspondence gives a new proof of the squarefree part of the Cauchy identity for Macdonald polynomials (i.e., the equality of the coefficients of x₁ ⋯ x_n y₁ ⋯ y_n on either side, which are related to permutations and standard Young tableaux). By specializing q and t in various ways, one recovers the row and column insertion versions of the Robinson--Schensted correspondence, several q- and t-deformations of row and column insertion which have been introduced in recent years in connection with q-Whittaker and Hall--Littlewood processes, and the Plancherel measure on partitions. Our construction is based on Fomin's growth diagrams and the recently introduced notion of a probabilistic bijection between weighted sets.

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