Braid groups of imprimitive complex reflection groups

We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group $B(de,e,r)$ is a semidirect product of the braid group of affine type $\widetilde A_{r-1}$ and an infinite cyclic group. Elements of $B(de,e,r)$ are visualized as geometric braids on $r+1$ strings whose first string is pure and whose winding number is a multiple of $e$. We classify periodic elements, and show that the roots are unique up to conjugacy and that the braid group $B(de,e,r)$ is strongly translation discrete.Comment: published versio

Conjugacy in singular Artin monoids

We define a notion of conjugacy in singular Artin moniods, and solve the corresponding conjugacy problem for finite types. We sgiw that this definition is appropriate to describe type (1) singular Markov moves on singular braids. Parabolic submonoids of singular Artin monoids are defined and, in finite type, are shown to be singular Artin monoids. Solutions to conjugacy-type problems of parabolic submonoids are described. Geometric objects defined by Fenn, Rolfsen and Zhu, called (j, k)-bands, are algebraically characterised, and a procedure is given which determines when a word represents a (j, k)-ban

Lying and Perjury in Medieval Practical Thought

This is a study of medieval thought about dilemmas involving lying, justified concealment and broken promises. It argues that a distinctive way of thinking about the ethics of lying and perjury, which reasoned through cases of conscience and practical situations, first appeared in an academic context in late twelfth century scholasticism, most notably in the Summa de Sacramentis et Animae Consiliis of Peter the Chanter. It was a tradition which continued in pastoral writings of the thirteenth century, the practical moral questions addressed by theologians in universities in the second half of the thirteenth century, and in the Summae de Casibus Conscientiae of the late Middle Ages. These various genres all participated in a casuistical thought about lying and deception which centred on deciding the best course of action in non-ideal situations and offered responses that acknowledged the need to adjust one’s actions to a unique set of circumstances. In the light of this discovery, the thesis investigates the origins of the casuistical concepts of equivocation and mental reservation. These teachings, which attracted satire in the Early Modern period, first appeared in late twelfth-century cases of conscience. It has been assumed that these ideas could only earn their keep by permitting Catholics to evade the morality of lying and perjury: the medieval tradition paints a different picture. In this period, equivocation and mental reservation were part of an effort to explain how to follow the rules in ambiguous and perplexing cases. Instead of talking around the rules, these concepts were developed in order to make the rules work in exceptional situations. In Chapter 6 I show that assumptions made about early modern casuistical thought do not work for its medieval equivalent. A subsidiary argument will be that equivocation and mental reservation were not inherently academic ideas. I argue in chapter 1 that sustained thought about these questions was evinced in medieval vernacular literature quite independently from the scholastic tradition. Casuistical thought about lying and perjury existed at a deeper level in the culture