We introduce a probabilistic generalization of the dual Robinson--Schensted--Knuth correspondence, called \(qt{\rm RSK}^*\), depending on two parameters \(q\) and \(t\). This correspondence extends our recently introduced \(q{\rm RS}t\) correspondence and allows the first tableaux-theoretic proof of the dual Cauchy identity for Macdonald polynomials. By specializing \(q\) and \(t\), one recovers the row and column insertion version of the classical dual RSK correspondence as well as of \(q\)- and \(t\)-deformations thereof which are connected to \(q\)-Whittaker and Hall--Littlewood polynomials, but also a novel correspondence for Jack polynomials.
The talk is based on joint work with Gabriel Frieden.