Résumé: The Robinson–Schensted (RS) correspondence is a bijection between permutations and pairs of standard Young tableaux which plays a central role in the theory of Schur polynomials. In recent years, several probabilistic q-RS and t-RS algorithms have been introduced; these are probabilistic deformations of Robinson–Schensted in which a permutation maps to several different pairs of tableaux, with probabilities depending on the parameter q or t. These algorithms are related to q-Whittaker and Hall–Littlewood polynomials, and they have applications to probabilistic models such as the TASEP and stochastic six-vertex model.
In this talk, I will present a (q,t)-dependent probabilistic deformation of Robinson–Schensted which is related to the Cauchy identity for Macdonald polynomials. By specializing q and t in various ways, one recovers the above-mentioned q-RS and t-RS maps, as well as both the row and column insertion versions of RS itself. I will also explain how part of the construction can be understood in terms of a (q,t)-generalization of the Greene–Nijenhuis–Wilf random hook walk.
This is joint work with Florian Aigner.