2024/03/24 by Gabriel Frieden, Frieden, Gabriel, Florian Aigner +1
Mathematics · #05E05 #33C52 #60C05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2403.16243
openalex publication_date 2024/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a probabilistic generalization of the dual Robinson--Schensted--Knuth correspondence, called qtRSK^*, depending on two parameters q and t. This correspondence extends the qRSt correspondence, recently introduced by the authors, 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. When restricting to Jack polynomials and \0,1\-matrices corresponding to words, we prove that the insertion tableaux obtained by qtRSK^* are invariant under swapping letters in the input word. Our approach is based on Fomin's growth diagrams and the notion of probabilistic bijections.