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Jiro Sekiguchi

Publications and source records attributed to Jiro Sekiguchi.

6 recordsLinked to original sources
arXiv2026

Flat coordinates of Frobenius prepotentials related with the reflection groups of types $H_3$ and $H_4$

In this article, we first explain a group theoretic interpretation of the derivation of the relation between the flat coordinates of the polynomial prepotential $(H_3)$ and those of the algebraic prepotential $(H_3)'$ given in \cite{KMS2} constructed by M. Feigin, D. Valeri and J. Wright \cite{FVW}. By the same idea explained in the case of $(H_3)$, we will show a relation between the flat coordinates of the polynomial prepotential $(H_4)$ and those of the algebraic prepotential $H_4(9)$ given in \cite{Se}.

math.AC↗
arXiv2023

The Construction Problem of Algebraic Potentials and Reflection Groups

This paper has two aims. The first one is the construction problem of algebraic potentials of Frobenius manifolds. We show examples of such potentials for the cases of reflection groups of types $H_4,E_6,E_7,E_8$ and also include those which are already known. The second one is an application of such potentials to singularity theory. We introduce families of hypersurfaces of ${\bf C}^3$ which are deformations of $E_n$-singularities $(n=6,7,8)$ but are not the versal families of $E_n$-singularities. We study the properties of the families. In particular we show the correspondence between such families and the algebraic potentials constructed in the first aim. Moreover we discuss the relationship between the complex reflection groups $ST33$ and $ST34$ and the two families corresponding to the $E_6$-singularity and the $E_7$-singularity.

math.AG↗
arXiv2023

Simple singularity of type $E_7$ and the complex reflection group ST34

This paper studies a family of surfaces of ${\bf C}^3$ which is a deformation of a simple singularity of type $E_7$. This family has six parameters which are regarded as basic invariants of the complex reflection group No.34 in the list of the paper of Shephard and Todd \cite{ST}. We compute 1-parameter subfamilies of the family in question corresponding to corank one reflection subgroups of No.34 group. In particular, we determine the types of simple singularities on the surfaces appeared in this manner.

math.AG↗
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↗
arXiv2018

Differential relations for almost Belyi maps

Several kinds of differential relations for polynomial components of almost Belyi maps are presented. Saito's theory of free divisors give particularly interesting (yet conjectural) logarithmic action of vector fields. The differential relations implied by Kitaev's construction of algebraic Painleve VI solutions through pull-back transformations are used to compute almost Belyi maps for the pull-backs giving all genus 0 and 1 Painleve VI solutions in the Lisovyy-Tykhyy classification.

math.AG↗