$Z_n$ Baxter Models and Quantum Symmetric Spaces

TL;DR

This paper explores the connection between $Z_n$ Baxter models, quantum symmetric spaces, and Macdonald polynomials, revealing how these mathematical structures describe excitations in magnetic models and interpolate between different symmetric spaces.

Contribution

It establishes a link between $Z_n$ Baxter models and Macdonald polynomials as zonal spherical functions for quantum symmetric spaces, extending understanding of their mathematical and physical significance.

Findings

Macdonald polynomials describe scattering in the $Z_n$ Baxter model as $n o fty$

Quantum symmetric spaces interpolate between $p$-adic and real symmetric spaces

The framework connects integrable models with harmonic analysis on quantum groups

Abstract

The scattering of two excitations (both of the simplest kind) in the magnetic model related to the $Z_n$\--Baxter model is naturally described for $n \rightarrow \infty$ in terms of the Macdonald polynomials for root system $A_1$. These polynomials play the role of zonal spherical functions for a two parameter family of quantum symmetric spaces. These spaces ``interpolate'' between various $p$\--adic and real symmetric spaces.