Tag - Braid groups

Claus Hertling: Upper triangular matrices and induced structures: vanishing cycles, monodromy groups, distinguished bases, braid group orbits, moduli spaces

13th March, 2023

Upper triangular matrices with ones on the diagonal and entries which are integers (or algebraic integers) arise in many contexts, e.g. as Stokes matrices in the theory of meromorphic connections with irregular poles, in many situations in algebraic geometry (often related to Stokes matrices), especially in quantum cohomology and the theory of isolated hypersurface singularities, but also in the theory of Coxeter groups.

Concepts from singularity theory like vanishing cycles, monodromy groups, Seifert forms, tuples of (pseudo-)reflections and distinguished bases can be derived from upper triangular
matrices in cases beyond singularity theory and are interesting to study.

Additionally, always braid group actions on the matrices and on the distinguished bases are in the background. They give rise to certain covering spaces of the classifying space of the braid group. These are interesting natural global manifolds. Some are well known, others are new.

The talk presents concepts and old and new results. It puts emphasis on some cases from singularity theory and some 3x3 cases.

Stephen Doty: Schur-Weyl duality for braid and twin groups via the Burau representation

15th February, 2022

The natural permutation representation of the symmetric group admits a q-analogue known as the Burau representation. The symmetric group admits two natural covering groups: the braid group of Artin and the twin group of Khovanov, obtained respectively by forgetting the cubic and quadratic relations in the Coxeter presentation of the symmetric group. By computing centralizers of tensor powers of the Burau representation, we obtain new instances of Schur-Weyl duality for braid groups and twin groups, in terms of the partial permutation and partial Brauer algebras. The method produces many representations of each group that can be understood combinatorially.

Jingyin Huang: The Helly geometry of some Garside and Artin groups

15th November, 2021

Artin groups emerged from the study of braid groups and complex hyperplane arrangements. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required.

Bert Wiest: Pseudo-Anosov braids are generic

2nd July, 2013

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n≥3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to infinity. Moreover, with a similar notion of genericity, we prove that for generic pairs of elements of the braid group, the conjugacy search problem can be solved in quadratic time. The idea behind both results is that generic braids can be conjugated ''easily'' into a rigid braid.

Fred Cohen: Connections Between Braid Groups, Homotopy Theory, and Low Dimensional Topology

6th October, 2004

An elementary homomorphism from a free group to the pure braid group yields interesting connections between braid groups, homotopy theory, and low dimensional topology. This map induces a map on the Lie algebra obtained from the descending central series. Further, this map induces a morphism of simplicial groups. All of these maps are shown to be injective.

Brunnian braids are discussed. The analogous maps of Lie algebras induced on the filtration quotients of the mod-p descending central series is again an injection. Using these facts it turns out that the homotopy groups of this simplicial group, those of the 2-sphere, are isomorphic to natural subquotients of the pure braid group. In addition, the mod-p analogues give a connection between the classical unstable Adams spectral sequence, and the mod-p analogues of Vassiliev invariants of pure braids.