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

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