Spectre des syst\`emes int\'egrables quantiques et repr\'esentations lin\'eaires

TL;DR

This paper reviews the spectral structure of quantum integrable systems, highlighting how generalized Baxter polynomials, arising from representation theory, describe their spectra and confirming a longstanding conjecture.

Contribution

It demonstrates how representation theory naturally leads to generalized Baxter polynomials that describe spectra of quantum integrable systems, confirming a key conjecture.

Findings

Spectral polynomials are linked to representation theory.

The conjecture about polynomial spectra has been proven.

Representation theory provides a natural framework for these polynomials.

Abstract

We review arXiv:1308.3444 and arXiv:1104.1891. The structure of the spectrum of a quantum integrable system is crucial to understand its properties. In his seminar 1971 paper, Baxter observed that the spectrum of the "ice model" has a very remarkable form involving polynomials. Then it was conjectured that analog polynomials can be used to describe the spectra of more general quantum integrable systems (arXiv:math/9810055). We discuss how these (generalized Baxter's) polynomials arise naturally in terms of representation theory. This result lead recently to the proof of the conjecture.