## Multiple orthogonal polynomials and a counterexample to the Gaudin Bethe Ansatz Conjecture

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- by E. Mukhin and A. Varchenko PDF
- Trans. Amer. Math. Soc.
**359**(2007), 5383-5418 Request permission

## Abstract:

Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two $sl_2$ 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 describe a counterexample to the Bethe Ansatz Conjecture for the Gaudin model.## References

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## Additional Information

**E. Mukhin**- Affiliation: Department of Mathematics, Indiana University-Purdue University-Indianapolis, 402 N. Blackford St., LD 270, Indianapolis, Indiana 46202
- MR Author ID: 317134
- Email: mukhin@math.iupui.edu
**A. Varchenko**- Affiliation: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250
- MR Author ID: 190269
- Email: anv@email.unc.edu
- Received by editor(s): May 17, 2005
- Received by editor(s) in revised form: September 15, 2005
- Published electronically: June 4, 2007
- Additional Notes: The research of the first author was supported in part by NSF grant DMS-0140460.

The research of the second author was supported in part by NSF grant DMS-0244579. - © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**359**(2007), 5383-5418 - MSC (2000): Primary 82B23, 33C45
- DOI: https://doi.org/10.1090/S0002-9947-07-04217-1
- MathSciNet review: 2327035