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Multiple orthogonal polynomials and a counterexample to the Gaudin Bethe Ansatz Conjecture
Author(s):
E.
Mukhin;
A.
Varchenko
Journal:
Trans. Amer. Math. Soc.
359
(2007),
5383-5418.
MSC (2000):
Primary 82B23, 33C45
Posted:
June 4, 2007
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Additional information
Abstract:
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:
10.1090/S0002-9947-07-04217-1
PII:
S 0002-9947(07)04217-1
Received by editor(s):
May 17, 2005
Received by editor(s) in revised form:
September 15, 2005
Posted:
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 of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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