$q{\rm RS}t$: A probabilistic Robinson--Schensted correspondence for Macdonald polynomials (extended abstract)

TL;DR

This paper introduces a probabilistic Robinson--Schensted correspondence parametrized by q and t, providing new insights into Macdonald polynomials and unifying various insertion algorithms through specialization.

Contribution

It develops a probabilistic generalization of the Robinson--Schensted correspondence that relates permutations to pairs of Young tableaux with probabilities depending on q and t, offering a new proof of a Macdonald polynomial identity.

Findings

Provides a probabilistic mapping from permutations to tableau pairs with q,t-dependent probabilities.

Recovers classical and deformed insertion algorithms through parameter specialization.

Connects combinatorial algorithms with Macdonald polynomial identities.

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. By specializing $q$ and $t$ in various ways, one recovers both the row and column insertion versions of the Robinson--Schensted correspondence, as well as several $q$- and $t$-deformations of row and column insertion which have been introduced in recent years in connection with integrable probability.