From Klein to Painleve Via Fourier, Laplace and Jimbo |
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Authors: | Boalch Philip |
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Affiliation: | École Normale Supérieure (DMA) 45 rue d'Ulm, 75005 Paris, France |
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Abstract: | We describe a method for constructing explicit algebraic solutionsto the sixth Painlevé equation, generalising that ofDubrovin and Mazzocco. There are basically two steps. Firstwe explain how to construct finite braid group orbits of triplesof elements of SL2(C) out of triples of generators of three-dimensionalcomplex reflection groups. (This involves the FourierLaplacetransform for certain irregular connections.) Then we adapta result of Jimbo to produce the Painlevé VI solutions.(In particular, this solves a RiemannHilbert problemexplicitly.) Each step is illustrated using the complex reflection groupassociated to Klein's simple group of order 168. This leadsto a new algebraic solution with seven branches. We also provethat, unlike the algebraic solutions of Dubrovin and Mazzoccoand Hitchin, this solution is not equivalent to any solutioncoming from a finite subgroup of SL2(C). The results of this paper also yield a simple proof of a recenttheorem of Inaba, Iwasaki and Saito on the action of Okamoto'saffine D4 symmetry group as well as the correct connection formulaefor generic Painlevé VI equations. 2000 Mathematics SubjectClassification 34M55, 34M40, 20F55. |
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Keywords: | Painlevé VI isomonodromy complex reflection groups |
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