On the solutions of Knizhnik-Zamolodchikov system

Lev Sakhnovich

Open Mathematics (2009)

  • Volume: 7, Issue: 1, page 145-162
  • ISSN: 2391-5455

Abstract

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We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of the KZ system is rational too. We give the method of constructing the corresponding rational solution. We deduce the asymptotic formulas for the solution of the KZ system when the parameter ρ is an integer.

How to cite

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Lev Sakhnovich. "On the solutions of Knizhnik-Zamolodchikov system." Open Mathematics 7.1 (2009): 145-162. <http://eudml.org/doc/269315>.

@article{LevSakhnovich2009,
abstract = {We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of the KZ system is rational too. We give the method of constructing the corresponding rational solution. We deduce the asymptotic formulas for the solution of the KZ system when the parameter ρ is an integer.},
author = {Lev Sakhnovich},
journal = {Open Mathematics},
keywords = {Symmetric group; Natural representation; Linear differential system; Rational fundamental solution; symmetric group; natural representation; linear differential system; rational fundamental solution; Knizhnik-Zamolodchikov system},
language = {eng},
number = {1},
pages = {145-162},
title = {On the solutions of Knizhnik-Zamolodchikov system},
url = {http://eudml.org/doc/269315},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Lev Sakhnovich
TI - On the solutions of Knizhnik-Zamolodchikov system
JO - Open Mathematics
PY - 2009
VL - 7
IS - 1
SP - 145
EP - 162
AB - We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of the KZ system is rational too. We give the method of constructing the corresponding rational solution. We deduce the asymptotic formulas for the solution of the KZ system when the parameter ρ is an integer.
LA - eng
KW - Symmetric group; Natural representation; Linear differential system; Rational fundamental solution; symmetric group; natural representation; linear differential system; rational fundamental solution; Knizhnik-Zamolodchikov system
UR - http://eudml.org/doc/269315
ER -

References

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  1. [1] Bateman H., Erdely A., Higher transcendental functions 1, New York, 1953 
  2. [2] Burrow M., Representation theory of finite groups, Academic Press, New York-London, 1965 
  3. [3] Etingof P.I., Frenkel I.B., Kirillov A.A.(Jr.), Lectures on representation theory and Knizhnik-Zamolodchikov equations, Mathematical Surveys and Monographs 58, American Mathematical Society, Providence, RI, 1998 Zbl0903.17006
  4. [4] Felder G., Veselov A., Polynomial solutions of the Knizhnik-Zamolodchikov equations and Schur-Weyl duality, Int. Math. Res. Not. IMRN, 2007, 15, 1–27 Zbl1130.35024
  5. [5] Matsuo A., An application of Aomoto-Gelfand hypergeometric functions to the SU(n) Knizhnik-Zamolodchikov equation. Comm. Math. Phys., 1990, 134, 65–77 http://dx.doi.org/10.1007/BF02102089[Crossref] Zbl0714.33012
  6. [6] Sakhnovich L.A., Spectral theory of canonical differential systems. Method of operator identities, Operator Theory: Advances and Applications 107, Birkhäuser Verlag, Basel, 1999 Zbl0918.47003
  7. [7] Sakhnovich L.A., Meromorphic solutions of linear differential systems, Painleve type functions, Oper. Matrices, 2007, 1, 87–111 Zbl1114.34068
  8. [8] Sakhnovich L.A., Explicit rational solutions of Knizhnik-Zamolodchikov equation, Cent. Eur. J. Math., 2008, 6, 179–187 http://dx.doi.org/10.2478/s11533-008-0013-0[Crossref] Zbl1153.34054
  9. [9] Sakhnovich L.A., Rational solutions of KZ equation, Existence and construction, preprint available at arXiv.math:- ph/0609067 
  10. [10] Sakhnovich L.A., Rational solutions of KZ equation, Case S4, preprint available at arXiv:math.CA/0702404 
  11. [11] Tydnyuk A., Rational solution of KZ equation, preprint available at arXiv:math/0162153 
  12. [12] Tydnyuk A., Explicit rational solution of KZ equation, preprint available at arXiv:math/07091141 
  13. [13] Varchenko A., Asymptotic solutions of Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys., 1995, 171, 99–137 http://dx.doi.org/10.1007/BF02103772[Crossref] Zbl0893.17009

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