Painlevé equations and complex reflections

Philip Boalch[1]

  • [1] Columbia University, Department of Mathematics, 2990 Broadway, New York NY 10027 (USA)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 4, page 1009-1022
  • ISSN: 0373-0956

Abstract

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We will explain how some new algebraic solutions of the sixth Painlevé equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-- Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.

How to cite

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Boalch, Philip. "Painlevé equations and complex reflections." Annales de l’institut Fourier 53.4 (2003): 1009-1022. <http://eudml.org/doc/116060>.

@article{Boalch2003,
abstract = {We will explain how some new algebraic solutions of the sixth Painlevé equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-- Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.},
affiliation = {Columbia University, Department of Mathematics, 2990 Broadway, New York NY 10027 (USA)},
author = {Boalch, Philip},
journal = {Annales de l’institut Fourier},
keywords = {Painlevé equations; isomonodromic deformations; non abelian cohomology; complex reflections; Painlevé equation; braid group},
language = {eng},
number = {4},
pages = {1009-1022},
publisher = {Association des Annales de l'Institut Fourier},
title = {Painlevé equations and complex reflections},
url = {http://eudml.org/doc/116060},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Boalch, Philip
TI - Painlevé equations and complex reflections
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 4
SP - 1009
EP - 1022
AB - We will explain how some new algebraic solutions of the sixth Painlevé equation arise from complex reflection groups, thereby extending some results of Hitchin and Dubrovin-- Mazzocco for real reflection groups. The problem of finding explicit formulae for these solutions will be addressed elsewhere.
LA - eng
KW - Painlevé equations; isomonodromic deformations; non abelian cohomology; complex reflections; Painlevé equation; braid group
UR - http://eudml.org/doc/116060
ER -

References

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  8. N. J. Hitchin, Frobenius manifolds, Gauge Theory and Symplectic Geometry, vol. 488 (1995), Kluwer Zbl0867.53027
  9. N. J. Hitchin, Poncelet polygons and the Painlevé equations, Geometry and analysis (Bombay, 1992) MR 97d:32042 (1995), 151-185 Zbl0893.32018
  10. N. J. Hitchin, Geometrical aspects of Schlesinger's equation, J. Geom. Phys. 23 (1997), 287-300 Zbl0896.53011MR1484592
  11. N. J. Hitchin, Quartic curves and icosahedra, talk at Edinburgh, September (1998) 
  12. M. Jimbo, Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982), 1137-1161 Zbl0535.34042MR688949
  13. M. Jimbo, T. Miwa, Monodromy preserving deformations of linear differential equations with rational coefficients II, Physica 2D (1981), 407-448 Zbl1194.34166MR625446
  14. L. Katzarkov, T. Pantev, C. Simpson, Density of monodromy actions on non-abelian cohomology Zbl1098.14012
  15. G. C. Shephard, J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274-304 Zbl0055.14305MR59914
  16. B. Totaro, Towards a Schubert calculus for complex reflection groups Zbl1045.05088MR1937794

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