Kyoji Saito

Artin-Gruppen und Coxeter-Gruppen

Man erh~ilt also die Coxeter-Gruppen in nattirlicher Weise als gewisse Restklassengruppen der Artin-Gruppen. Ftir den Fall der Z6pfegruppen ergeben sich so gerade die symmetrischen Gruppen, was nattirlich schon lange bekannt ist. Die Coxeter-Gruppen sind seit ihrer Einftihrung durch Coxeter im Jahre 1935 eingehend studiert worden, und eine sch6ne Darstellung der dabei erhaltenen Resultate findet man bei Bourbaki [1]. Ftir die Artin- Gruppen gab es, von den freien Gruppen einmal abgesehen, nur ftir die Z6pfegruppen eine Reihe yon Untersuchungen, yon denen die wichtigste in letzter Zeit die L~Ssung des Konjugationsproblems durch Garside war. Ftir die anderen Artin-Gruppen gibt es einige vereinzelte Resultate in [2, 3 und 5]. Diese beziehen sich ebenso wie der gr6gte Teil der vorliegen- den Arbeit auf den Fall, dab die zu der Artin-Gruppe G geh~Srige Coxeter- Gruppe G endlich ist. Die endlichen Coxeter-Gruppen sind schon yon Coxeter selbstWoestimmt worden: Es sind die endlichen Spiegelungs- gruppen, also - im irreduziblen Fall - die Gruppen vom Typ A,, B,,

Limit Elements in the Configuration Algebra for a Cancellative Monoid

Associated with the Cayley graph ( ;G) of a cancellative monoid with a nite generating system G, we introduce two compact spaces: ( ;G) consisting of pre-partition functions and ( P ;G) consisting of series opposite to the growth function P ;G(t) := P 1 n=0 ] n t n (where

A new relation among Cartan matrix and Coxeter matrix

Here pqrfd is a numerical invariant concerning about the Cartan matrix of R defined in Section 1 and u/x/h is a numerical invariant concerning about a Coxeter iransformatioI1 of R defined in Section 2. We give a proof of the equality in Section 4 using the self-intersection number of a Weil divisor on a rational surface. The argument is a version of that for the strange duality on the 14 exceptional unimodular singularities of Arnold due to Pinkham (cf. [ 1, 61). It might be quite interesting to find a purely arithmetic proof of the equality. The author would like to express his glatitude to his colleages I. Naruki and M. Tomari for their interest and helpful1 discussions and also to K. Ueno, who has noticed the author on some works on anti-canonical divisors.

Uniformization of the orbifold of a finite reflection group

Let W be a finite reflection group of a real vector space V. If W is crystallographic, then the quotient space V*//W appears in several contexts in geometry: i) in Lie theory as the quotient space of a simple Lie algebra by the adjoint Lie group action [Ch1,2] and ii) in complex geometry as the base space of the universal unfolding of a simple singularity [Br1]. Having these backgrounds, V*//W carries some distinguished geometric properties and structures, which, fortunately and also amusingly, can be described only in terms of the reflection group regardless whether W is crystallographic or not. We recall two of them: 1. The complexified regular orbit space (V*//W) C reg is a K(π, 1)-space (Brieskorn [Br3], Deligne [De]). In other words, π 1((V*//W) C reg is an Artin group (i.e. a generalized braid group [B-S][De]) and the universal covering space of (V*//W) C reg is contractible (cf. also [Sa]). 2. The quotient space V*//W carries a flat structure (Saito [S3][S6])1. This means roughly that the tangent bundle of V*//W carries a flat metric J together with some additional structures. Nowadays, a flat structure without a primitive form is also called a Frobenius manifold structure with gravitational descendent (Dubrovin [Du], Manin [Ma1,2]).

Duality for Regular Systems of Weights: A Précis

The theory of a regular system of weights was originally developed in order to understand the flat structure in the period maps for primitive forms [S9]. In the present work, the theory is applied to understand the self-duality of ADE and the strange duality of Arnold. Beyond the original purpose, the theory yields further a class of dual weight systems, which seem to have close connection with the Conway group and to be interesting to be studied yet. On the other hand, the duality of weight systems has an interpretation in terms of the Dedekind eta function. But its meaning is not yet clear.

Primitive Automorphic Forms

There seems to be a marvellous interaction taking place between mathematical physics and mathematics in the area of geometry, demanding a greater contribution from non-commutative structures and higher cohomologies. It may require a revolutional extension of the concept of spaces in order to explain the dualities there. I am not the proper person to talk about the whole subject, but will restrict myself to that part of the topic where I have been involved from the view point of complex geometry, namely periods of integrals over vanishing cycles. In order to attack a big mathematical problem, there are two approaches: 1) to generalize the problem and to develop a new general framework and language within which the problem finds a natural place, or 2) to attempt to examine a cross-section of the problem and to give a precise solution of that part of the problem.

Moduli Space for Fuchsian Groups

Publisher Summary The field operators depend on complex parameters z that run over rational curves. The theory has been extended to nonrational curves, whose coordinate patchings depend on the moduli of the curves. Hence, one encounters a problem: the relation between coordinates and moduli for curves of higher genus. This problem has been approached by introducing an infinite-dimensional fiber space over the moduli of curves. It consists of all formal local coordinates for curves. The chapter explains a finite-dimensional space instead of the infinite-dimensional one. By considering all global coordinates for curves instead of infinitesimal coordinates, one obtains a real three-dimensional fiber space over the moduli.

Opposite power series

In order to analyze the singularities of a power series function P ( t ) on the boundary of its convergent disc, we introduced the space ? ( P ) of opposite power series in the opposite variable s = 1 / t , where P ( t ) was, mainly, the growth function (Poincare series) for a finitely generated group or a monoid Saito (2010) 10]. In the present paper, forgetting about that geometric or combinatorial background, we study the space ? ( P ) abstractly for any suitably tame power series P ( t ) ? C { t } . For the case when ? ( P ) is a finite set and P ( t ) is meromorphic in a neighborhood of the closure of its convergent disc, we show a duality between ? ( P ) and the highest order poles of P ( t ) on the boundary of its convergent disc.

Algebraic Surfaces for Regular Systems of Weights

Abstract We costruct the following families i)–iv) of algebraic surfaces. †) i) 49 families of K3 surfaces with certain curve configurations, most of which admit elliptic fibrations over P1. ii) 9 families of elliptic surfaces over P1 of Kodaira dimension κ = 1 such that the irregularity q = 0 and the geometric genus Pg = 1 or 2. iii) 6 families of surfaces of general type, satisfying the numerical equality Pg = [c12/2] + 2 for the Chern number c12 = 1, 1, 2, 2, 3, 5. ‡) iv) 4 families of rational elliptic surfaces for the types D4, E6, E7 and E8.

Coherence of Direct Images of the De Rham Complex

We show the coherence of the direct images of the De Rham complex relative to a flat holomorphic map with suitable boundary conditions. For this purpose, a notion of bi-dg-algebra called the Koszul-De Rham algebra is developed.