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), 53835418
MSC (2000):
Primary 82B23, 33C45
Published electronically:
June 4, 2007
MathSciNet review:
2327035
Fulltext PDF Free Access
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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 JacobiPiñeiro multiple orthogonal polynomial and others are given by Wronskiantype determinants of JacobiPiñeiro polynomials. As a byproduct we describe a counterexample to the Bethe Ansatz Conjecture for the Gaudin model.
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 H. Babujian, Offshell Bethe ansatz equations and point correlators in the WZNW theory, J. Phys. A 26 (1993), no. 23, 69816990 MR 1253889 (95a:82028)
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 H. Babujian and R. Flume, Offshell Bethe ansatz equation for Gaudin magnets and solutions of KnizhnikZamolodchikov equations, Modern Phys. Lett. A 9 (1994), n. 22, 20292039 MR 1290286 (95h:82007)
 [EH]
 D. Eisenbud, J. Harris, Limit Linear Series: Basic Theory, Inventiones Mathematicae, 85, 337371 MR 846932 (87k:14024)
 [FFR]
 B. Feigin, E. Frenkel, and N. Reshetikhin, Gaudin model, Bethe Ansatz and Critical Level, Commun. Math. Phys. 166 (1994), 2962 MR 1309540 (96e:82012)
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 M. Gaudin, Diagonalisation d'une classe d'Hamiltoniens de spin, J. Physique 37 (1976), no. 10, 10891098 MR 0421442 (54:9446)
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 A. Iserles, S.P. Norsett, On the theory of biorthogonal polynomials, Transactions of AMS, 306 (1988), 455474 MR 933301 (89c:42027)
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 V. Kac, Infinitedimensional Lie algebras, Cambridge University Press, 1990 MR 1104219 (92k:17038)
 [MV1]
 E. Mukhin, A. Varchenko, Critical points of master functions and flag varieties, Communications in Contemporary Mathematics (2004), vol. 6, no. 1, 111163 MR 2048778 (2005b:17052)
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 E. Mukhin, A. Varchenko, Norm of a Bethe Vector and the Hessian of the Master Function, Compos. Math. 141 (2005), no. 4, 10121028 MR 2148192 (2006d:82022)
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 E. Mukhin, A. Varchenko, Solutions to the XXX type Bethe Ansatz equations and flag varieties, Centr. Eur. J. Math, vol. 1, no.2 (2003), 238271 MR 1993451 (2004k:82026)
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 L. R. Piñeiro, On simultaneous Padé approximants for a collection of Markov functions, Vestnik Mosk. Univ. Ser., I, no. 2 (1987), 5255 (in Russian); translated in Moscow Univ. Math. Bull. 42, no. 2 (1987), 5255
 [PV]
 K. Postelmans, W. Van Assche, Multiple little qJacobi polynomials, math.CA/0403532, 115
 [RSV]
 R. Rimanyi, L. Stevens, and A. Varchenko, Combinatorics of rational functions and PoincaréBirkhoffWitt expansions of the canonical valued differential form, math.CO/0407101, 114
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 N. Reshetikhin, A. Varchenko, Quasiclassical asymptotics of solutions to the KZ equations, Geometry, topology physics, Conf. Proc. Lecture Notes Geom. Topology, VI, Internat. Press, Cambridge, MA (1995), 293322 MR 1358621 (96j:32025)
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 I. Scherbak, Intersections of Schubert varieties and highest weight vectors in tensor products of representations, math.RT/0409329, 123
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 I. Scherbak, A. Varchenko, Critical point of functions, representations and Fuchsian differential equations with only univalued solutions, Dedicated to Vladimir I. Arnold on the occasion of his 65th birthday. Mosc. Math. J. 3 (2003), no. 2, 621645 MR 2025276 (2004m:34204)
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Additional Information
E. Mukhin
Affiliation:
Department of Mathematics, Indiana UniversityPurdue UniversityIndianapolis, 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 275993250
Email:
anv@email.unc.edu
DOI:
http://dx.doi.org/10.1090/S0002994707042171
PII:
S 00029947(07)042171
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 DMS0140460.
The research of the second author was supported in part by NSF grant DMS0244579.
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
