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The $q$-Racah polynomials from scalar products of Bethe states

Pascal Baseilhac, Rodrigo A. Pimenta

TL;DR

This work addresses representing the $q$-Racah polynomials within an integrable-model framework by expressing them as ratios of scalar products of Bethe states associated with homogeneous or inhomogeneous Bethe equations. The authors leverage the Askey-Wilson algebra and Leonard-pair structure to connect eigenbases via a transition matrix, and realize these bases through the modified algebraic Bethe ansatz applied to a reflection algebra. They show that Bethe states corresponding to homogeneous or inhomogeneous Bethe equations provide two complementary realizations of the Leonard-pair eigenbases, yielding multiple equivalent presentations of the polynomials. Consequently, the $q$-Racah polynomials arise as ratios of scalar products, highlighting a bridge between the discrete Askey-scheme and integrable systems with $q$-deformed symmetry.

Abstract

The $q$-Racah polynomials are expressed in terms of certain ratios of scalar products of Bethe states associated with Bethe equations of either homogeneous or inhomogeneous type. This result is obtained by combining the theory of Leonard pairs and the modified algebraic Bethe ansatz.

The $q$-Racah polynomials from scalar products of Bethe states

TL;DR

This work addresses representing the -Racah polynomials within an integrable-model framework by expressing them as ratios of scalar products of Bethe states associated with homogeneous or inhomogeneous Bethe equations. The authors leverage the Askey-Wilson algebra and Leonard-pair structure to connect eigenbases via a transition matrix, and realize these bases through the modified algebraic Bethe ansatz applied to a reflection algebra. They show that Bethe states corresponding to homogeneous or inhomogeneous Bethe equations provide two complementary realizations of the Leonard-pair eigenbases, yielding multiple equivalent presentations of the polynomials. Consequently, the -Racah polynomials arise as ratios of scalar products, highlighting a bridge between the discrete Askey-scheme and integrable systems with -deformed symmetry.

Abstract

The -Racah polynomials are expressed in terms of certain ratios of scalar products of Bethe states associated with Bethe equations of either homogeneous or inhomogeneous type. This result is obtained by combining the theory of Leonard pairs and the modified algebraic Bethe ansatz.
Paper Structure (15 sections, 9 theorems, 94 equations)

This paper contains 15 sections, 9 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. The Askey-Wilson algebra, Leonard pairs and transition matrix
  3. Askey-Wilson algebra and Leonard pairs
  4. Transition matrix and the $q$-Racah polynomials
  5. Eigenbases for Leonard pairs from Bethe states
  6. Bethe states and dual Bethe states
  7. Eigenbases of homogenous type for the Leonard pair
  8. Eigenbases of inhomogenous type for the Leonard pair
  9. Relating homogeneous and inhomogeneous Bethe states
  10. The $q$-Racah polynomials
  11. Concluding remarks
  12. Askey-Wilson generators and the dynamical operators
  13. The dynamical operators and exchange relations
  14. Products of dynamical operators and exchange relations
  15. Expressions of $\textsf{A},\textsf{A}^*$ in terms of the dynamical operators

Key Result

Lemma 3.1

If the parameter $\alpha$ is such that: then

Theorems & Definitions (14)

  • Lemma 3.1
  • Lemma 3.2
  • Definition 3.1
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • Lemma 3.7
  • proof
  • ...and 4 more