Finite Braid group orbits in $\mathbf{Aff}\boldsymbol{(\mathbb{C})}$-character varieties of the punctured sphere

Cousin, Gaël; Moussard, Delphine

doi:10.1093/imrn/rnw283

Cited by 8 publications

(7 citation statements)

References 21 publications

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“…A similar simplification holds for each inhomogeneous system (5). Note that N i=1 d b k,k+1 i (a) ≡ 0, and thus system (7) is equivalent to…”

Section: A Particular Case Of the Exponents And Solutions Via Periodsmentioning

confidence: 92%

“…With respect to this correspondence, in the case of a non-degenerate linear monodromy (that is, neither finite, nor dihedral, nor triangular), algebraic solutions were partially classified by K. Diarra [9] for an arbitrary M and by P. Calligaris, M. Mazzocco [4] for M = 2. For a non-abelian triangular linear monodromy, the classification of Schlesinger isomonodromic (2 × 2)families leading to algebraic solutions of Garnier systems, was done by G. Cousin, D. Moussard [7]. For a dihedral linear monodromy, there are families of algebraic solutions obtained by A. Girand [17], for M = 2 and by A. Komyo [28] for an arbitrary even M .…”

Section: Families Of Algebraic Solutions Of Garnier Systems: a Sphere...mentioning

confidence: 99%

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Triangular Schlesinger systems and superelliptic curves

Dragović¹,

Gontsov²,

Shramchenko³

2018

Preprint

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We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size (p×p) are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference q, the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference q, for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the (2 × 2)-case, we obtain explicit sequences of rational solutions and one-parameter families of rational solutions of Painlevé VI equations as well as a class of Liouvillian solutions for various Painlevé VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems.

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“…A similar simplification holds for each inhomogeneous system (5). Note that N i=1 d b k,k+1 i (a) ≡ 0, and thus system (7) is equivalent to…”

Section: A Particular Case Of the Exponents And Solutions Via Periodsmentioning

confidence: 92%

Section: Families Of Algebraic Solutions Of Garnier Systems: a Sphere...mentioning

confidence: 99%

Triangular Schlesinger systems and superelliptic curves

Dragović¹,

Gontsov²,

Shramchenko³

2018

Preprint

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“…The reader will find a more precise statement in [10]. Let us just mention in the first case the works of Girand [19] and Komyo [33] for deformations of equation ( 1) with dihedral monodromy, and Cousin and Moussard [13] in the reducible case. Deformations of equation ( 1) having a finite group are algebraic and provide an algebraic Garnier solution in a systematic way, but computations can be very tedious as it has been in the works of Boalch for the Painlevé case N = 1.…”

Section: Himentioning

confidence: 99%

“…(and a twist) to reduce the matrix A in the companion form (12): set F dx = γ and G = 0. One can further reduce (12), or accordingly (11), by gauge transformation (13) with F ≡ 1, or equivalently setting u := u/ exp( G); by this way, we can arrive to the unique SL-form ( 14)…”

Section: Monodromy and Stokes Matricesmentioning

confidence: 99%

Classification of algebraic solutions of irregular Garnier systems

Diarra¹,

Loray²

2020

Compositio Math.

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We prove that algebraic solutions of Garnier systems in the irregular case are of two types. The classical ones come from isomonodromic deformations of linear equations with diagonal or dihedral differential Galois group; we give a complete list in the rank N = 2 case (two indeterminates).The pull-back ones come from deformations of coverings over a fixed degenerate hypergeometric equation; we provide a complete list when the differential Galois group is SL 2 (C). By the way, we have a complete list of algebraic solutions for the rank N = 2 irregular Garnier systems.

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“…We then apply this explicit description of the mapping class group action to the specific study of finite g,n -orbits on χ g,n (GL 2 C) that correspond to reducible representations. For g = 0, this study has been completely carried out in [8]. In this special case, the study can be reduced to linear dynamics.…”

Section: B Dynamical Study Of Finite Orbits In the Reducible Rank 2 Casementioning

confidence: 99%

Algebraic Isomonodromic Deformations and the Mapping Class Group

Cousin¹,

Heu²

2019

J. Inst. Math. Jussieu

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The germ of the universal isomonodromic deformation of a logarithmic connection on a stable $n$ -pointed genus $g$ curve always exists in the analytic category. The first part of this article investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this work studies the dynamics of this action in the particular case of reducible rank 2 representations and genus $g>0$ , allowing to classify all finite orbits. Both of these results extend recent ones concerning the genus 0 case.

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Finite Braid group orbits in $\mathbf{Aff}\boldsymbol{(\mathbb{C})}$-character varieties of the punctured sphere

Cited by 8 publications

References 21 publications

Triangular Schlesinger systems and superelliptic curves

Triangular Schlesinger systems and superelliptic curves

Classification of algebraic solutions of irregular Garnier systems

Algebraic Isomonodromic Deformations and the Mapping Class Group

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