Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 064, 28 pages      arXiv:2106.11913      https://doi.org/10.3842/SIGMA.2024.064

Identity between Restricted Cauchy Sums for the $q$-Whittaker and Skew Schur Polynomials

Takashi Imamura a, Matteo Mucciconi b and Tomohiro Sasamoto c
a) Department of Mathematics and Informatics, Chiba University, Chiba, 263-8522 Japan
b) Department of Mathematics, University of Warwick, Coventry, CV4 7HP, UK
c) Department of Physics, Tokyo Institute of Technology, Tokyo, 152-8551 Japan

Received December 20, 2023, in final form July 02, 2024; Published online July 16, 2024

Abstract
The Cauchy identities play an important role in the theory of symmetric functions. It is known that Cauchy sums for the $q$-Whittaker and the skew Schur polynomials produce the same factorized expressions modulo a $q$-Pochhammer symbol. We consider the sums with restrictions on the length of the first rows for labels of both polynomials and prove an identity which relates them. The proof is based on techniques from integrable probability: we rewrite the identity in terms of two probability measures: the $q$-Whittaker measure and the periodic Schur measure. The relation follows by comparing their Fredholm determinant formulas.

Key words: integrable probability; Kardar-Parisi-Zhang class; stochastic processes; Macdonald polynomials.

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