Milnor fiber complexes for Shephard groups

Abstract

The symmetry group of a regular polytope p is a finite Coxeter group W. The intersection of the unit sphere with the reflecting hyperplanes of W induces a simplicial triangulation of the sphere, called the Coxeter complex Γ W . In case p is convex, radial projection maps the barycentric subdivision of p homeomorphically onto Γ W . The symmetry group of a regular complex polytope p is a Shephard group G. In this paper we construct a simplicial complex Γ G in the Milnor fiber of a G-invariant polynomial of minimal positive degree, which shares many of the properties of the Coxeter complex. In case p is non-starry, we construct a linear complex B( p ) ⊂ p which is homeomorphic to Γ G . Thus the relation of B( p ) to Γ G is the same as the relation of the barycentric subdivision of a convex polytope to the Coxeter complex.