Finite irreducible groups, generated by reflections, are monodromy groups of suitable singularities
type: Item
instance of: scholarly article
title: Finite irreducible groups, generated by reflections, are monodromy groups of suitable singularities
zbMATH Open document ID: 0573.32012
DOI: 10.1007/BF01086155
publication date: 1984-01-01T00:00:00Z
review text: There is a well known correspondence between simple singularities (of a function of several complex variables) and finite reflection groups (of type A, D and E). A fundamental property of this is the existence of an isomorphism from the orbit space of the complexified action of the group to the miniversal deformation space of the singularity which (1) takes the variety of non-regular orbits to the discriminant, (2) induces an isomorphism of the reflection group with the monodromy group of the singularity, and (3) is induced by the period map. The object of this paper is to find analogues of these results for all finite reflection groups. For types B, C and F such analogues were already given by \textit{V. I. Arnol'd} [Usp. Mat. Nauk 33, No.5(203), 91-105 (1978; Zbl 0408.58009)]: the structure here is of a singularity of a function invariant under a group of order 2 acting by a reflection. For the rest, we have \(G_ 2(x^ 3+y^ 3,{\mathbb{Z}}_ 3);\) \(I_ 2(n)(x^ n+x^ 2y^ 2+y^ n,{\mathbb{Z}}_ n);\) \(H_ 3(x^ 5+y^ 5,{\mathbb{Z}}_ 5)\) where in each case \({\mathbb{Z}}_ n\) acts by \(\zeta (x,y)=(\zeta x,\zeta^{-1}y)\) \((\zeta^ n=1)\). The case of \(H_ 4\) is not treated here. The proofs of the assertions involve extensions of the techniques used in the cases previously known. A new feature is that as well as restricting the unfolding to one by invariant functions, the homology of the Milnor fibre is decomposed into eigenspaces for the group \({\mathbb{Z}}_ n\). Then for \(X=B_{\mu}\), \(C_{\mu}\), \(F_ 4\), \(I_ 2(n)\) and \(H_ 3\) the images of the vanishing cycles in the eigenspace belonging to \(\zeta (=\exp (2\pi i)/n)\) form a root lattice of type X, and the image of the monodromy group is the corresponding reflection group. (For the \(+1\) eigenspace we have a simpler reflection group in the cases B, C, F but an infinite one for \(I_ 2(n)\), \(H_ 3\). For \(G_ 2\) these cases are reversed.)
Mathematics Subject Classification ID: 32S05
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 32S30
Mathematics Subject Classification ID: 32Sxx
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 3914701
zbMATH Keywords: monodromy groups of singularities
zbMATH Keywords: simple singularities
zbMATH Keywords: finite reflection groups
zbMATH Keywords: period map
zbMATH Keywords: homology of the Milnor fibre
zbMATH Keywords: root lattice
author: A. Varchenko
author: S. Chmutov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01086155
OpenAlex ID: W2086477361
published in: Functional Analysis and Its Applications
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description: scientific article; zbMATH DE number 3914701