Walls in Milnor fiber complexes

TL;DR

This paper extends Abramenko's classification of hyperplanes intersecting Coxeter complexes to Milnor fiber complexes in complex reflection groups, introducing Milnor walls and characterizing when walls are Milnor walls.

Contribution

It generalizes Abramenko's results to a broader class of complex reflection groups and introduces the concept of Milnor walls for better understanding of the complex's structure.

Findings

Classifies groups with hyperplanes meeting Milnor fiber complexes in another Milnor fiber complex.

Introduces Milnor walls to analyze walls not forming Milnor fiber complexes.

Shows walls are Milnor walls iff the diagram lacks certain subdiagrams.

Abstract

For a real reflection group the reflecting hyperplanes cut out on the unit sphere a simplicial complex called the Coxeter complex. Abramenko showed that each reflecting hyperplane meets the Coxeter complex in another Coxeter complex if and only if the Coxeter diagram contains no subdiagram of type $D_4$, $F_4$, or $H_4$. The present paper extends Abramenko's result to a wider class of complex reflection groups. These groups have a Coxeter-like presentation and a Coxeter-like complex called the Milnor fiber complex. Our first main theorem classifies the groups whose reflecting hyperplanes meet the Milnor fiber complex in another Milnor fiber complex. To understand better the walls that fail to be Milnor fiber complexes we introduce Milnor walls. Our second main theorem generalizes Abramenko's result in a second way. It says that each wall of a Milnor fiber complex is a Milnor wall if and only if the diagram contains no subdiagram of type $D_4$, $F_4$, or $H_4$.