Finite complex reflection groups

Finite Unitary Reflection Groups

Introduction. Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of unitary transformations (5, p. 257). Such a group may be thought of as a group of congruent transformations, keeping the origin fixed, in a unitary space Un of n dimensions, in which the p...

Semiinvariants of Finite Reflection Groups

Let G be a finite group of complex n× n unitary matrices generated by reflections acting on C. Let R be the ring of invariant polynomials, and χ be a multiplicative character of G. Let Ω be the R-module of χ-invariant differential forms. We define a multiplication in Ω and show that under this multiplication Ω has an exterior algebra structure. We also show how to extend the results to vector f...

Automorphisms of complex reflection groups

I. MARIN AND J. MICHEL Abstract. Let G ⊂ GL(Cr) be an irreducible finite complex reflection group. We show that (apart from the exception G = S6) any automorphism of G is the product of an automorphism induced by tensoring by a linear character, of an automorphism induced by an element of NGL(Cr)(G) and of what we call a “Galois” automorphism: we show that Gal(K/Q), where K is the field of defi...

On finite index reflection subgroups of discrete reflection groups

Let G be a discrete group generated by reflections in hyperbolic or Euclidean space, and H ⊂ G be a finite index subgroup generated by reflections. Suppose that the fundamental chamber of G is a finite volume polytope with k facets. In this paper, we prove that the fundamental chamber of H has at least k facets. 1. Let X be hyperbolic space H, Euclidean space E or spherical space S . A polytope...

Regular Elements of Finite Reflection Groups