The lattice of normal reflection subgroups of an irreducible reflection group

Abstract

The reflection subgroups of a reflection group have a natural lattice structure given by the reflections that they contain. By considering the conjugation action orbits of the reflection subgroups for a given root line, we are able to give an essentially combinatorial way to calculate the lattice of all the normal reflection subgroups of a given (finite irreducible) reflection group, and natural generators for them. Moreover, we observe that every complex reflection group is a normal subgroup of the unique maximal reflection group which shares its collineation group. Hence, we are able to present the Shephard-Todd classification of the complex reflection groups as a collection of maximal reflection groups, together with appropriate (collineation preserving) normal reflection subgroups. We investigate the quotients by the normal reflection subgroups, which are known to be reflection groups. We also consider the action of the collineation group on some appropriate small systems of lines, and how these results extend to quaternionic reflection groups. Some novel techniques are introduced, including the notion of a "hidden reflection", a combinatorial-geometric description of the reflection subgroups and the size of their conjugacy class, and the role played by the abelianisation of the reflection group.