Katz's middle convolution and Yokoyama's extending operation

Oshima, Toshio

doi:10.7494/opmath.2015.35.5.665

“…One of its turning points is the introduction of the notions of middle convolution and rigidity index by N.M. Katz [35]. With the help of his idea, T. Oshima [50,51,52] developed a classification theory of Fuchsian differential equations in terms of their rigidity indices and spectral types; [21] is a nice introductory text on the "Katz-Oshima theory". (From the viewpoint of the theory of isomonodromic deformations, the number of accessory parameters of an irreducible linear differential equation is equal to two plus the index of rigidity.)…”

Section: Introductionmentioning

confidence: 99%

“…Okubo system (which was systematically studied by K. Okubo in [46]) plays a central role in these developments (cf. [15,16,52,65]): a matrix system of linear differential equations with the form (z − T ) dY dz = BY, (1.1)…”

Section: Introductionmentioning

confidence: 99%

Flat Structure on the Space of Isomonodromic Deformations

Kato¹,

Mano

²

,

Sekiguchi

³

2020

SIGMA

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Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of systems of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of Frobenius manifold. As its consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation.

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“…One of its turning points is the introduction of the notions of middle convolution and rigidity index by N. M. Katz [34]. With the help of his idea, T. Oshima developed a classification theory of Fuchsian differential equations in terms of their rigidity indices and spectral types [51,52,50]. ([20] is a nice introductory text on the "Katz-Oshima theory".)…”

Section: Introductionmentioning

confidence: 99%

“…Okubo system (which was systematically studied by K. Okubo in [46]) plays a central role in these developments (cf. [14,15,52,63]): a matrix system of linear differential equations with the form (z − T )…”

Section: Introductionmentioning

confidence: 99%

Flat Structure on the Space of Isomonodromic Deformations

Kato,

Mano,

Sekiguchi

2015

Preprint

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Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of systems of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of Frobenius manifold. As its consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation.

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“…Middle convolution is related to the Euler transformation of solutions of the Fuchsian systems. There have been numerous studies on middle convolution in recent years, including applications to special functions (e.g., [17,19,20,21,43,48]), extensions to irregular systems (e.g., [28,35,49]) and various others (see, for instance, [2,11,12,13,14,18,26,39,40,41,47,50,51]).…”

Section: Introductionmentioning

confidence: 99%

Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

Bibilo

¹

,

Filipuk

²

2015

SIGMA

1

0

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Abstract. The paper is devoted to non-Schlesinger isomonodromic deformations for resonant Fuchsian systems. There are very few explicit examples of such deformations in the literature. In this paper we construct a new example of the non-Schlesinger isomonodromic deformation for a resonant Fuchsian system of order 5 by using middle convolution for a resonant Fuchsian system of order 2. Moreover, it is known that middle convolution is an operation that preserves Schlesinger's deformation equations for non-resonant Fuchsian systems. In this paper we show that Bolibruch's non-Schlesinger deformations of resonant Fuchsian systems are, in general, not preserved by middle convolution.

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Katz's middle convolution and Yokoyama's extending operation

Abstract: Abstract. We give a concrete relation between Katz's middle convolution and Yokoyama's extension and show the equivalence of both algorithms using these operations for the reduction of Fuchsian systems on the Riemann sphere.

Cited by 10 publications

References 16 publications

Flat Structure on the Space of Isomonodromic Deformations

Flat Structure on the Space of Isomonodromic Deformations

Flat Structure on the Space of Isomonodromic Deformations

Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution

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