Reflection groups acting on their hyperplanes

Reflection Groups Acting on Their Hyperplanes

After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W ⊂ GL(V ) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural representations of W , through a ‘periodic’ family of representations of its braid group. We also prove that, when W is irreducible, then the squares of definin...

On Some Relationships Among the Association Schemes of Finite Orthogonal Groups Acting on Hyperplanes

On Complex Reflection Groups and Their Associated Braid Groups

Presentations \\ a la Coxeter" are given for all (irreducible) nite complex reeec-tion groups. They provide presentations for the corresponding generalized braid groups (still conjectural in some cases) which allow us to generalize some of the known properties of nite Coxeter groups (center of the braid group, construction of Hecke algebras). 1. Background from complex reflection groups For all...

Involutory Reflection Groups and Their Models

A finite subgroup of GL(n,C) is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups.

On Reflection Length in Reflection Groups

Let W be the Weyl group of a connected reductive group over a finite field. It is a consequence of the Borel-Tits rational conjugacy theorem for maximal split tori that for certain reflection subgroups W1 of W (including all parabolic subgroups), the elements of minimal reflection length in any coset wW1 are all conjugate, provided w normalises W1. We prove a sharper and more general result of ...