Let (W, S) be a Coxeter system, let S = I ∪ J be a partition of S such that no element of I is conjugate to an element of J, let J be the set of W I-conjugates of elements of J and let W be the subgroup of W generated by J. We show that W = W ⋊ W I and that J is the canonical set of Coxeter generators of the reflection subgroup W of W. We also provide… (More)
Lecture notes by Matthew Dyer for lectures at the workshop " Coxeter groups and convex geometry. " 1. Root labelled Bruhat graph 1.1. Bruhat order. Let (W, S) be a Coxeter group. Bruhat order is the partial order defined by the following proposition; see  and  for more details.
BACKGROUND Ciclesonide is a new inhaled corticosteroids licensed for the prophylactic treatment of persistent asthma in adults. Currently beclomethasone dipropionate, budesonide and fluticasone propionate are the most commonly prescribed inhaled corticosteroids for the treatment of asthma but there has been no systematic review comparing the effectiveness… (More)
The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of a reflection subgroup of W is contained in that of W , and that for irreducible infinite W of finite rank, the closed… (More)
We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection subgroups using completed Dynkin diagrams, for which there seems to be no convenient source in the literature. This is… (More)
OBJECTIVES To pilot using a panel of members of the public to provide preference data via the Internet METHODS A stratified random sample of members of the general public was recruited and familiarized with the standard gamble procedure using an Internet based tool. Health states were periodically presented in "sets" corresponding to different conditions,… (More)
In this note, we provide a short and self-contained proof that the braid group on n strands acts transitively on the set of reduced factorizations of a Coxeter element in a Coxeter group of finite rank n into products of reflections. We moreover use the same argument to also show that all factorizations of an element in a parabolic subgroup of W also lie in… (More)
A b s t r a c t O b j e c t i v e s : The objective of this study was to evaluate the long-term cost-effectiveness of germline BRCA1 and BRCA2 (collectively termed 'BRCA') testing in women with epithelial ovarian cancer, and testing for the relevant mutation in first and second degree relatives of BRCA mutation-positive individuals, compared with no… (More)