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

Authors:
E. Mukhin and A. Varchenko

Journal:
Trans. Amer. Math. Soc. **359** (2007), 5383-5418

MSC (2000):
Primary 82B23, 33C45

Published electronically:
June 4, 2007

MathSciNet review:
2327035

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two irreducible modules. We study sequences of polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two highest weight 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.

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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

Email:
mukhin@math.iupui.edu

**A. Varchenko**

Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250

Email:
anv@email.unc.edu

DOI:
https://doi.org/10.1090/S0002-9947-07-04217-1

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.

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.