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Philippe Di Francesco1,2, Rinat Kedem1, Sergey Khoroshkin3,4

  • 1University of Illinois Urbana-Champaign, Champaign, IL USA.

Letters in Mathematical Physics
|December 2, 2024
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Summary
This summary is machine-generated.

We describe Hallnäs-Ruijsenaars eigenfunctions for the 2-particle hyperbolic Ruijsenaars system. These eigenfunctions relate to GL(2) quantum Teichmüller theory and yield GL(2) Macdonald polynomials.

Keywords:
Cluster varietiesRuijsenaars wavefunctionsSpherical DAHA

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Area of Science:

  • Mathematical Physics
  • Quantum Field Theory
  • Algebraic Geometry

Background:

  • The study concerns the 2-particle hyperbolic Ruijsenaars system and its connection to GL(2) quantum Teichmüller theory.
  • Understanding eigenfunctions and their relation to special polynomials is crucial in these fields.

Purpose of the Study:

  • To describe Hallnäs-Ruijsenaars eigenfunctions as matrix coefficients within GL(2) quantum Teichmüller theory.
  • To demonstrate how these coefficients lead to GL(2) Macdonald polynomials via analytic continuation.

Main Methods:

  • Utilizing the cluster structure on the moduli space of framed GL(2)-local systems on the punctured torus.
  • Employing an equivariant embedding of the GL(2) spherical DAHA into the quantized coordinate ring of a cluster Poisson variety.

Main Results:

  • Hallnäs-Ruijsenaars eigenfunctions are characterized as matrix coefficients of a specific operator.
  • GL(2) Macdonald polynomials are derived as special values of these analytically continued coefficients.

Conclusions:

  • The research establishes a novel connection between hyperbolic Ruijsenaars systems and GL(2) quantum Teichmüller theory.
  • The findings provide a new perspective on the structure and properties of GL(2) Macdonald polynomials.