qRSt: A Probabilistic Robinson–Schensted Correspondence for Macdonald Polynomials

DOI10.1093/IMRN/RNAB083zbMATH Open1504.05290arXiv2009.03526OpenAlexW3165302789MaRDI QIDQ5102304

Publication date: 6 September 2022

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

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

x_{1} c d o t s x_{n} y_{1} c d o t s 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.

Full work available at URL: https://arxiv.org/abs/2009.03526

zbMATH Keywords

Macdonald polynomials Cauchy's identity probabilistic Robinson-Schensted correspondence

Mathematics Subject Classification ID

Combinatorial identities, bijective combinatorics (05A19) Symmetric functions and generalizations (05E05)