Painlevé equations and complex reflections

Boalch, Philip

doi:10.5802/aif.1972

Cited by 16 publications

(30 citation statements)

References 11 publications

(22 reference statements)

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“…In general, operating several additions and middle convolutions alternately, we get a system of any rank higher than 1. The argument developed in [1] may be extended to such systems.…”

Section: Note Thatmentioning

confidence: 99%

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Middle convolution and deformation for Fuchsian systems

Haraoka

Filipuk²

2007

Journal of the London Mathematical Society

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“…In general, operating several additions and middle convolutions alternately, we get a system of any rank higher than 1. The argument developed in [1] may be extended to such systems.…”

Section: Note Thatmentioning

confidence: 99%

“…It is also well known that this equation possesses the so-called Painlevé property: every solution may be analytically continued to a meromorphic function on the universal cover of CP 1 \ {0, 1, ∞}. Relations to the works by Boalch [1,2], Harnad [6], Mazzocco [10] and Novikov [11] are also clarified in Section 5.…”

Section: Introductionmentioning

confidence: 96%

Middle convolution and deformation for Fuchsian systems

Haraoka

Filipuk²

2007

Journal of the London Mathematical Society

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“…the isomonodromic deformation equations. This was used by Boalch to get information about finite Painlevé orbits in [14] [15]. It would be interesting to look further at the dynamics of the braid action and the isomonodromy equations.…”

Section: Further Questionsmentioning

confidence: 99%

Katz's Middle Convolution Algorithm

Simpson¹

2009

Pure and Applied Mathematics Quarterly

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This is an expository account of Katz's middle convolution operation on local systems over P 1 − {q 1 , . . . , q n }. We describe the Betti and de Rham versions, and point out that they give isomorphisms between different moduli spaces of local systems, following Völklein, Dettweiler-Reiter, Haraoka-Yokoyama. Kostov's program for applying the Katz algorithm is to say that in the range where middle convolution no longer reduces the rank, one should give a direct construction of local systems. This has been done by Kostov and Crawley-Boevey. We describe here an alternative construction using the notion of cyclotomic harmonic bundles: these are like variations of Hodge structure except that the Hodge decomposition can go around in a circle.

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“…It should be mentioned that, in the short paper [5], we previously showed by a dierent method that the equations for isomonodromic deformations of the 3 Â 3 Fuchsian systems mentioned above are equivalent to PVI; this was written before Jimbo's formula was xed and also does not give the relation between the rank 2 and rank 3 monodromy data. elements of SL 2 ðCÞ.…”

Section: Introductionmentioning

confidence: 99%

From Klein to Painlevé via Fourier, Laplace and Jimbo

Boalch

2004

Proc. London Math. Soc.

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103

193

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Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation, generalising that of Dubrovin-Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL 2 (C) out of triples of generators of three-dimensional complex reflection groups. (This involves the Fourier-Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painlevé VI solutions. (In particular this solves a Riemann-Hilbert problem explicitly.) Each step will be illustrated using the complex reflection group associated to Klein's simple group of order 168. This leads to a new algebraic solution with seven branches. We will also prove that, unlike the algebraic solutions of Dubrovin-Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of SL 2 (C).The results of this paper also yield a simple proof of a recent theorem of InabaIwasaki-Saito on the action of Okamoto's affine D 4 symmetry group as well as the correct connection formulae for generic Painlevé VI equations. IntroductionKlein's quartic curveis of genus three and has the maximum possible number 84(g − 1) = 168 of holomorphic automorphisms. Klein found these automorphisms explicitly (in terms of 3 × 3 matrices). They constitute Klein's simple group K ⊂ PGL 3 (C) which is isomorphic to PSL 2 (7). Lifting to GL 3 (C) there is a two-fold covering group K ⊂ GL 3 (C) of order 336 which is a complex reflection group-there are complex reflections (1) r 1 , r 2 , r 3 ∈ GL 3 (C) which generate K. (Recall a pseudo-reflection is an automorphism of the form "one plus rank one", a complex reflection is a pseudo-reflection of finite order and a complex reflection group is a finite group generated by complex reflections. Here, each generator r i has order two-as for real reflections.) Using the general tools to be described in this paper we will construct, starting from the Klein complex reflection group K, another algebraic curve with affine equation (2) F (t, y) = 0given by a polynomial F with integer coefficients. This curve will be a seven-fold cover of the t-line branched only at 0, 1, ∞ and such that the function y(t), defined implicitly by (2), solves the Painlevé VI differential equation. One upshot of this will be to construct an explicit rank three Fuchsian system of linear differential equations with four singularities (at 0, t, 1, ∞, for some t) on P 1 , and with 1 2 PHILIP BOALCH monodromy group equal to K in its natural representation (so the monodromy around each of the finite singularities 0, t, 1 is a generating reflection).In general the construction of linear differential equations with finite monodromy group is reasonably straightforward provided one works with rigid representations of the monodromy groups. In our situation the representation is minimally non-rigid; it lives in a complex two-dimensional moduli space, and this is the basic reason the (second order) Painlevé VI equati...

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Painlevé equations and complex reflections

Cited by 16 publications

References 11 publications

Middle convolution and deformation for Fuchsian systems

Middle convolution and deformation for Fuchsian systems

Katz's Middle Convolution Algorithm

From Klein to Painlevé via Fourier, Laplace and Jimbo

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