no. 4
Painlevé equations and complex reflections
[Équations de Painlevé et réflexions complexes]
Boalch, Philip1
1 Columbia University, Department of Mathematics, 2990 Broadway, New York NY 10027 (USA)
Annales de l'Institut Fourier, Tome 53 (2003) no. 4, pp. 1009-1022.

Nous expliquerons comment de nouvelles solutions algébriques de la sixième équation de Painlevé proviennent des groupes complexes de réflexion, prolongeant les résultats de Hitchin et de Dubrovin--Mazzocco pour les groupes réels de réflexion. Le problème de trouver des formules explicites pour ces solutions sera traité ailleurs.

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.

DOI : 10.5802/aif.1972
Classification : 34M55, 20H15, 55N99
Keywords: Painlevé equations, isomonodromic deformations, non abelian cohomology, complex reflections
Mot clés : équations de Painlevé, déformations isomonodromiques, cohomologie non abélienne, réflections complexes

Boalch, Philip 1

1 Columbia University, Department of Mathematics, 2990 Broadway, New York NY 10027 (USA) 
@article{AIF_2003__53_4_1009_0,
     author = {Boalch, Philip},
     title = {Painlev\'e equations and complex reflections},
     journal = {Annales de l'Institut Fourier},
     pages = {1009--1022},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {4},
     year = {2003},
     doi = {10.5802/aif.1972},
     zbl = {1081.34086},
     mrnumber = {2033508},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1972/}
}
TY  - JOUR
AU  - Boalch, Philip
TI  - Painlevé equations and complex reflections
JO  - Annales de l'Institut Fourier
PY  - 2003
SP  - 1009
EP  - 1022
VL  - 53
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1972/
DO  - 10.5802/aif.1972
LA  - en
ID  - AIF_2003__53_4_1009_0
ER  - 
%0 Journal Article
%A Boalch, Philip
%T Painlevé equations and complex reflections
%J Annales de l'Institut Fourier
%D 2003
%P 1009-1022
%V 53
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1972/
%R 10.5802/aif.1972
%G en
%F AIF_2003__53_4_1009_0
Boalch, Philip. Painlevé equations and complex reflections. Annales de l'Institut Fourier, Tome 53 (2003) no. 4, pp. 1009-1022. doi : 10.5802/aif.1972. https://aif.centre-mersenne.org/articles/10.5802/aif.1972/

[1] P.P. Boalch Symplectic manifolds and isomonodromic deformations, Adv. in Math, Volume 163 (2001), pp. 137-205 | DOI | MR | Zbl

[2] P.P. Boalch G-bundles, isomonodromy and quantum Weyl groups, Int. Math. Res. Not. (2002) no. 22, pp. 1129-1166 | DOI | MR | Zbl

[3] M. Broué; G. Malle; J. Michel Towards spetses. I, Dedicated to the memory of Claude Chevalley (Transform. Groups), Volume 4, no 2-3 (1999), pp. 157-218 | Zbl

[4] C. De Concini; V. G. Kac; C. Procesi Quantum coadjoint action, J. Amer. Math. Soc, Volume 5 (1992) no. 1, pp. 151-189 | DOI | MR | Zbl

[5] C. F. Doran Algebraic and geometric isomonodromic deformations, J. Differential Geom., Volume 59 (2001) no. 1, pp. 33-85 | MR | Zbl

[6] B. Dubrovin Painlevé transcendents in two-dimensional topological field theory, The Painlevé property (1999), pp. 287-412 | Zbl

[7] B. Dubrovin; M. Mazzocco Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math., Volume 141 (2000) no. 1, pp. 55-147 | DOI | MR | Zbl

[8] N. J. Hitchin Frobenius manifolds, Gauge Theory and Symplectic Geometry, NATO ASI Series C: Maths \& Phys, vol. 488, Kluwer, 1995 | Zbl

[9] N. J. Hitchin Poncelet polygons and the Painlevé equations, Geometry and analysis (Bombay, 1992) (Tata Inst. Fund. Res., Bombay), Volume MR 97d:32042 (1995), pp. 151-185 | Zbl

[10] N. J. Hitchin Geometrical aspects of Schlesinger's equation, J. Geom. Phys., Volume 23 (1997) no. 3-4, pp. 287-300 | DOI | MR | Zbl

[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., Volume 18 (1982) no. 3, pp. 1137-1161 | DOI | MR | Zbl

[13] M. Jimbo; T. Miwa Monodromy preserving deformations of linear differential equations with rational coefficients II, Physica 2D (1981), pp. 407-448 | MR

[14] L. Katzarkov; T. Pantev; C. Simpson Density of monodromy actions on non-abelian cohomology (e-print, math.AG/0101223) | Zbl

[15] G. C. Shephard; J. A. Todd Finite unitary reflection groups, Canadian J. Math., Volume 6 (1954), pp. 274-304 | DOI | MR | Zbl

[16] B. Totaro Towards a Schubert calculus for complex reflection groups (Math. Proc. Camb. Phil. Soc., to appear, www.dpmms.cam.ac.uk/ bt219/hall.dvi.gz) | MR | Zbl

Cité par Sources :