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

Takashi Imamura, Matteo Mucciconi, Tomohiro Sasamoto

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.

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 -Whittaker and the skew Schur polynomials produce the same factorized expressions modulo a -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 -Whittaker measure and the periodic Schur measure. The relation follows by comparing their Fredholm determinant formulas.

Paper Structure

This paper contains 19 sections, 14 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The identity and connections to integrable probability
  3. Background from integrable probability
  4. Outline
  5. Fredholm determinants for $\boldsymbol{q}$-Whittaker and shift-mixed periodic Schur measures
  6. Fredholm determinant formula for the $\boldsymbol{q}$-Whittaker measure
  7. Fredholm determinant formula for the shift-mixed periodic Schur measure
  8. Equivalence of Theorems \ref{['t1']} and \ref{['t12']}
  9. Equivalence of two Fredholm determinants by shift of contours
  10. Restating the main theorem as a determinant identity
  11. Proof of Lemma \ref{['l32']} (i)
  12. Proof of Lemma \ref{['l32']} (ii)
  13. Proof of Lemma \ref{['l32']} (iii)
  14. Comparison with Borodin's matching
  15. Notations and formulas
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $n\in\mathbb{Z}_{\ge 0}$, we have

Figures (3)

  • Figure 1: The contours $C$, $D$, and $\bar{D}$ appearing in \ref{['p222']} and \ref{['t252']}. "$\bullet$"s and ""s represent the poles of $w$ and $z$ respectively.
  • Figure 2: Contours $D_{\ell}$, $\tilde{D}_\ell$, and $D_{\infty}$ are illustrated. We set $c_{\ell}$ to be the center of the interval $[\frac{1}{a_{N}q^\ell}, \frac{1}{a_{1} q^{\ell+1}}]$, i.e., $c_{\ell}=\frac{1}{2q^\ell}(\frac{1}{a_{1}q}+\frac{1}{a_{N}} )$.
  • Figure 3: All of the contours appearing in the proof of Lemma \ref{['l32']}. We set $c>0$, $c_-<0$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 6 more