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

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 sums with restrictions on the length of the first rows of the labels for both polynomials and prove an identity relating 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.