Multiple orthogonal polynomials and a counterexample to Gaudin Bethe Ansatz Conjecture

2005/01/10 by E. Mukhin, Mukhin, E., Alexander Varchenko +2 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math.QA

paper · pdf · doi:10.48550/arxiv.math/0501144

Latex, 36 pages. Final version. To appear in Transaction of AMS

openalex publication_date 2005/01/10 · arxiv created 2005/11/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl₂ irreducible modules. We study sequences of r polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two highest weight sl_r+1 irreducible modules, with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight. We describe the recursion which can be used to compute these polynomials. Moreover, we show that the first polynomial in the sequence coincides with the Jacobi-Piñeiro multiple orthogonal polynomial and others are given by Wronskian type determinants of Jacobi-Piñeiro polynomials. As a byproduct we obtain a counterexample to the Bethe Ansatz Conjecture for the Gaudin model.

Cited by

• Differential equation for Jacobi-Pineiro polynomials
• On separation of variables and completeness of the Bethe ansatz for quantum gl_N Gaudin model
• Bispectral and (\glN,\glM) Dualities

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