Reflection Groups Acting on Their Hyperplanes

Factorizations of Coxeter Elements in Complex Reflection Groups

In [CS12], Chapuy and Stump studied factorizations of Coxeter elements into products of reflections in well-generated, irreducible complex reflection groups, giving a simple closed form expression for their exponential generating function depending on certain natural parameters. In this work, their methods are used to consider a more general multivariate generating function FACW (u1, . . . , u`...

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

The volume product of convex bodies with many hyperplane symmetries

Mahler’s conjecture predicts a sharp lower bound on the volume of the polar of a convex body in terms of its volume. We confirm the conjecture for convex bodies with many hyperplane symmetries in the following sense: their hyperplanes of symmetries have a one-point intersection. Moreover, we obtain improved sharp lower bounds for classes of convex bodies which are invariant by certain reflectio...

A Refined Count of Coxeter Element Reflection Factorizations

For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter ...

Bases for certain cohomology representations of the symmetric group

We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we ...