Related papers: Braid groups and quiver mutation
We describe presentations of braid groups of type $A$ arising from coloured quivers of mutation type $A$. We show that these can be interpreted geometrically as generalised triangulations of regular polygons.
We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the $G$-gaugings of a pointed…
Braid groups are an important and flexible tool used in several areas of science, such as Knot Theory (Alexander's theorem), Mathematical Physics (Yang-Baxter's equation) and Algebraic Geometry (monodromy invariants). In this note we will…
Let $C$ be a simply laced generalized Cartan matrix. Given an element $b$ of the generalized braid semigroup related to $C$, we construct a collection of mutation-equivalent quivers with potentials. A quiver with potential in such a…
The paper deals with braided Clifford algebras, understood as Chevalley-Kahler deformations of braided exterior algebras. It is shown that Clifford algebras based on involutive braids can be naturally endowed with a braided quantum group…
We explicitly describe the derived Picard groups of symmetric representation-finite algebras of type $D$. In particular, we prove that these groups are generated by spherical twists along collections of $0$-spherical objects, the shift and…
We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is…
In a recent paper, Barot and Marsh presented an explicit construction of presentation of a finite Weyl group by any seed of corresponding cluster algebra, i.e. by any diagram mutation-equivalent to an orientation of a Dynkin diagram with…
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding…
The descent algebra of a finite Coxeter group $W$ is a basic algebra, and as such it has a presentation as quiver with relations. In recent work, we have developed a combinatorial framework which allows us to systematically compute such a…
In 2011, Barot and Marsh provided an explicit construction of presentation of a finite Weyl group $W$ by any quiver mutation-equivalent to an orientation of a Dynkin diagram with Weyl group $W$. The construction was extended by the authors…
We study new symmetries between A and D type quiver gauge theories with different numbers of colors. We realize these gauge theories with twisted masses via a brane construction that reproduces all the parameters of the Gauge/Bethe…
Each quiver appearing in a seed of a cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in seeds of cluster algebras of…
In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.
This article is a survey on the braid groups, the Artin groups, and the Garside groups. It is a presentation, accessible to non-experts, of various topological and algebraic aspects of these groups. It is also a report on three points of…
We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the…
We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite…
Braid groups may be defined for every Coxeter diagram. Artin's braid group is of type A. Analogs of Temperley-Lieb, Hecke and Birman-Wenzl algebras exist for B-type. Our general hypothethis is that the braid group of B-type replaces Artin's…
A Hecke endomorphism algebra is a natural generalisation of the $q$-Schur algebra associated with the symmetric group to a Coxeter group. For Weyl groups, B. Parshall, L. Scott and the first author \cite{DPS,DPS4} investigated the…
We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of…