arXiv subjects

Toshiyuki Mano

Publications and source records attributed to Toshiyuki Mano.

5 recordsLinked to original sources

arXiv2020

Flat Structure on the Space of Isomonodromic Deformations

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.

math.CA↗

arXiv2019

Freeness of multi-reflection arrangements via primitive vector fields

In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. We then extend our results further to all imprimitive irreducible unitary reflection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reflection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees.

math.GR↗

arXiv2018

Regular flat structure and generalized Okubo system

We study a relationship between regular flat structures and generalized Okubo systems. We show that the space of variables of isomonodromic deformations of a regular generalized Okubo system can be equipped with a flat structure. As its consequence, we introduce flat structures on the spaces of independent variables of generic solutions to (classical) Painlevé equations (except for PI). In our framework, the Painlevé equations PVI-PII can be treated uniformly as just one system of differential equations called the four-dimensional extended WDVV equation. Then the well-known coalescence cascade of the Painlevé equations corresponds to the degeneration scheme of the Jordan normal forms of a square matrix of rank four.

math.CA↗

arXiv2018

Determinant structure for tau-function of holonomic deformation of linear differential equations

In our previous works, a relationship between Hermite's two approximation problems and Schlesinger transformations of linear differential equations has been clarified. In this paper, we study tau-functions associated with holonomic deformations of linear differential equations by using Hermite's two approximation problems. As a result, we present a determinant formula for the ratio of tau-functions (tau-quotient).

math.CA↗

arXiv2016

Hermite-Pade approximation, isomonodromic deformation and hypergeometric integral

We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite's two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitzs determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painleve equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions.

math.CA↗