Milnor fiber complexes for Shephard groups

AbstractThe 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 ΓW. In case p is convex, radial projection maps the barycentric subdivision of p homeomorphically onto ΓW. The symmetry group of a regular complex polytope p is a Shephard group G. In this paper we construct a simplicial complex ΓG 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 ΓG. Thus the relation of B(p) to ΓG is the same as the relation of the barycentric subdivision of a convex polytope to the Coxeter complex