
Ruth CorranAmerican University of Paris · Computer Science, Mathematics and Environmental Science
Ruth Corran
PhD Mathematics, University of Sydney
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13
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Introduction
Skills and Expertise
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April 2002 - June 2004
August 2017 - September 2017
August 2004 - August 2005
Publications
Publications (13)
We generalize the definition and properties of root systems to complex reflection groups — roots become rank one projective modules over the ring of integers of a number field k.
In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation.
In the case of spetsial reflection gr...
We generalize the definition and properties of root systems to complex reflection groups - roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation. In the case of spetsial reflection gr...
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...
Automaticity is an important concept in group theory as it yields an efficient solution to the word problem and provides other possibilities for effective computation. The concept of automaticity generalizes naturally from groups to monoids and semigroups and the efficiency of the solution of the word problem is preserved when we do this. Whilst th...
We consider the positive singular and the singular Artin monoids of finite type. These have been the subject of a great deal
of recent research and the main purpose of this paper is to prove that these monoids are automatic. In order to do this we
establish a new criterion for proving monoids automatic that may be of independent interest.
We describe a new presentation for the complex reflection groups of type (e, e, r) and their braid groups. A diagram for this presentation is proposed. The presentation is a monoid presentation that is shown
to give rise to a Garside structure. This structure has since been used in understanding periodic elements, calculating homology
and determini...
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....
We investigate a new lattice of generalised non-crossing partitions, constructed using the geometry of the complex reflection group G(e,e,r). For the particular case e=2 (resp. r=2), our lattice coincides with the lattice of simple elements for the type Dn (resp. I2(e)) dual braid monoid. Using this lattice, we construct a Garside structure for the...
We investigate a new lattice of generalised non-crossing partitions, constructed using the geometry of the complex reflection group $G(e,e,r)$. For the particular case $e=2$ (resp. $r=2$), our lattice coincides with the lattice of simple elements for the type $D_n$ (resp. $I_2(e)$) dual braid monoid. Using this lattice, we construct a Garside struc...
We give a new presentation of the braid group $B$ of the complex reflection group $G(e,e,r)$ which is positive and homogeneous, and for which the generators map to reflections in the corresponding complex reflection group. We show that this presentation gives rise to a Garside structure for $B$ with Garside element a kind of generalised Coxeter ele...
OUTLINE. In the first half of the thesis, we introduce the class of'chainable monoids' and describe general techniques for solving the word and division problems and for obtaining normal forms for monoids in this class. Their growth functions are shown to be rational, and calculable. We also show that a monoid with a certain central 'fundamental el...
The first part of this paper investigates a class of homogeneously presented monoids. Constructions which enable division and multiplication to be computed are described. The word problem and the division problem are solved, and a unique normal form is given for monoids in this class. The second part deals with the motivating examples of this class...
Introduction An Artin group is a group G with a presentation by a system of generators a i ; i 2 I, and relations a i a j a i Delta Delta Delta = a j a i a j Delta Delta Delta ; i; j 2 I where the words on each side of these relations are sequences of m ij letters where a i and a j alternate in the sequence. The matrix of values m ij is a Coxeter m...