Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 82B23, 33C45

Retrieve articles in all journals with MSC (2000): 82B23, 33C45


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: http://dx.doi.org/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
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.