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 constructed by M. Feigin, D. Valeri and J. Wright . 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 .

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.

This paper studies the basic invariants, constructed by Conway and Sloane, of the complex reflection group numbered as 34 in the list of Shephard-Todd \cite{ST}.

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 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.

Formal and Analytic Solutions of Differential Equations, pp. 23-85 (2022) No AccessChapter 2: The Algebraic Potentials Having Tri-Hamiltonian Structures of the Reflection Groups of Types D4, F4, H4Jiro SekiguchiJiro SekiguchiDepartment of Mathematics, Tokyo University of Agriculture and Technology, Tokyo, Japanhttps://doi.org/10.1142/9781800611368_0002Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: The purpose of this chapter is to study the relationship between the algebraic potential having tri-Hamiltonian structure of the reflection group of type D4 (resp., F4, H4) and the polynomial potential of type A3 (resp., B3, H3). The present work is based on the paper by S. Romano [12] on algebraic Frobenius manifolds having tri-Hamiltonian structure. We will construct a 4-parameter family of algebraic potentials having tri-hamiltonian structure in the D4 case (resp., F4 case) and define an ordinary differential equation from a member of the 4-parameter family and show the coincidence of the differential equation in question and an ordinary differential equation constructed from the polynomial potential of type A3 (resp., B3). We will also treat the correspondence between H4 case and H3 case and show a partial answer to the case (H4, H3). FiguresReferencesRelatedDetails Formal and Analytic Solutions of Differential EquationsMetrics History PDF download

We calculate explicitly the quadratic solution to the WDVV equations that corresponds to the primitive conjugacy class E 8 ( a 1 ) by two different ways. The first one is using the associated classical W -algebra. The second one is solving the WDVV equation directly. Last we identify them by showing the coordinate transformation between them.

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 Painleve equation.

Potential vector fields are solutions to extended WDVV equation and play an important role in the theory of fiat structures. There is a (C* )(n)-action on the set of potential vector fields of n-variables with a same weight system. It is a question whether under this action, the set of polynomial potential vector fields is a unique orbit or not. Among others, it is complicated to solve the question for the two cases: the real reflection group of type E-8 and the complex reflection group ST34. The purpose of this paper is to give an answer to this question for these two groups.

The notion of the extended WDVV equations is a generalization of the WDVV equations. Their solutions are vector-valued functions. In the three-dimensional case, there is a correspondence between the extended WDVV equations and the family of the Painleve VI equations. It is expected that potential vector fields corresponding to algebraic solutions to the Painleve VI equation can be written by using algebraic functions explicitly. The purpose of this paper is to establish a method of constructing potential vector fields corresponding to algebraic solutions. The idea is based on the argument by Jimbo and Miwa inducing the Painleve VI equation from a Pfaffian system and on middle convolution.

§ O. Introduction. § 1. Semisimple symmetric pairs. § 2. The restricted root system of a symmetric pair. § 3. The (e, a)-system of roots. § 4. A reduction to the case of rank 1. § 5. The irreducible symmetric pairs of split rank I. § 6. Determination of the restricted root system. § 7. The Weyl group of a symmetric pair. § 8. A parabolic subalgebra connected with a symmetric pair. Appendix A. A lemma on the root systems. Appendix B. The Levi part of a parabolic subalgebra.

A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painleve VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this paper is to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions.

The aim of this paper is first to formulate the definition of Frobenius manifolds and its generalization. Then we study the algebraic solutions to Painlevé VI obtained by Dubrovin-Mazzocco related with the reflection group of type H3.

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.

Flat structures are formulated by K. Saito in the course of the study of moduli spaces of isolated singularities. The purpose of this paper is to study flat structures without potentials, to formulate one of generalisations of ordinary differential equations of Okubo type to several variables case and to give examples of potential vector fields related with algebraic solutions of Painleve VI, free divisors arising from 1-parameter deformations of singularities on plane curves and discriminants of complex reflection groups.

K. Saito [RIMS Kokyuroku Bessatsu 287 (1977), 117-137] introduced uniformization systems of equations with singularities along free divisors. The purpose of this paper is to construct such systems for the case of finite irreducible complex reflection groups of rank three. Each of them is essentially a system of linear differential equations of Fuchsian type in two variables with three linearly independent solutions. As a special case, uniformization systems of equations whose monodoromy groups are the finite irreducible complex reflection groups themselves are obtained in a closed form. This case has already been treated by Haraoka and the first author in [Funkcial. Ekvac. 53 (2010), 435-488] and our study is a generalization of their result.

The article is devoted to the study of the classification problem for Saito free divisors making use of the deformation theory of varieties. In particular, in the quasihomogeneous case, we describe an approach for computation of free deformations of quasicones over quasismooth varieties based on properties of deformations of varieties with \( {\mathbb{G}_m} \)-action. We also discuss some applications including the problem of compactification of modular spaces and computation of free deformations for certain simple, unimodal, and unimodular singularities.

The purpose of this paper is to study systems of uniformization equations with respect to Saito free divisors which have solutions expressed in terms of hyperelliptic integrals. There are two such divisors. Both are hypersurfaces in a three-dimensional affine space defined by weighted homogeneous polynomials. One is constructed by the discriminant of a dihedral group of order 2(2 n +1). The other is the discriminant of the reflection group of type H 3 . In the former case, we construct fundamental solutions by Gaussian hypergeometric functions in addition to a solution expressed by a hyperelliptic integral.

Assume the system of differential equations E-4(a, b, c, c'; X, Y) satisfied by Appell's hypergeometric function F4(a, b, c, c'; X, Y) has a finite irreducible monodromy group M-4(a, 13, c, The monodromy matrix Gamma(3*) derived from a loop Gamma(3) surrounding once the irreducible component C = {(X, Y) vertical bar (X - Y)(2) - 2(X + Y) + 1 = 0} of the singular locus of E-4 is a complex reflection. The minimal normal subgroup N-C of M-4 containing Gamma 3(*) is, by definition, a finite complex reflection group of rank four. Let P(G) be the projective monodromy group of the Gauss hypergeometric differential equation E-2(1)(a, b, c). It is known that N-C is reducible if epsilon := c + c' a b-1 is not an element of Z or if epsilon is an element of Z and P(G) is a dihedral group. We prove that, if epsilon is an element of Z, then N-C is the (irreducible) Coxeter group W (D-4), W(F-4) and W (H-4) according as P(G) is the tetrahedral, octahedral and icosahedral group, respectively.