The stable braid group and the determinant of the Burau representation (original) (raw)

On the Linear Representation of Braid Groups

Transactions of the American Mathematical Society, 1989

We give a new derivative of the Burau and Gassner representations of the braid and pure braid groups. Various applications are explored.

A Basic n Dimensional Representation of Artin Braid Group B n , and a General Burau Representation

Mathematics and Statistics, 2022

Braid groups and their representations are at the center of study, not only in low-dimensional topology, but also in many other branches of mathematics and theoretical physics. Burau representation of the Artin braid group which has two versions, reduced and unreduced, has been the focus of extensive study and research since its discovery in 1930’s. It remains as one of the very important representations for the braid group. Partly, because of its connections to the Alexander polynomial which is one of the first and most useful invariants for knots and links. In the present work, we show that interesting representations of braid group could be achieved using a simple and intuitive approach, where we simply analyse the path of strands in a braid and encode the over-crossings, under-crossings or no-crossings into some parameters. More precisely, at each crossing, where, for example, the strand i crosses over the strand i + 1 we assign t to the top strand and b to the bottom strand. We consider the parameter t as a relative weight given to strand i relative to i + 1, hence the position i i + 1 for t in the matrix representation. Similarly, the parameter b is a relative weight given to strand i + 1 relative to i, hence the position i + 1 i for b in the matrix representation. We show this simple path analyzing approach leads us to an interesting simple representation. Next, we show that following the same intuitive approach, only by introducing an additional parameter, we can greatly improve the representation into the one with much smaller kernel. This more general representation includes the unreduced Burau representation, as a special case. Our new path analyzing approach has the advantage that it applies a very simple and intuitive method capturing the fundamental interactions of the strands in a braid. In this approach we intuitively follow each strand in a braid and create a history for the strand as it interacts with other strands via over-crossings, under-crossings or no-crossings. This, directly, leads us to the desired representations.

A Family of representations of braid groups on surfaces

2010

We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.

Some monodromy representations of generalized braid groups

Communications in Mathematical Physics, 1994

A flat connection on the trivial bundle over the complement in C" of the complexification of the system of the reflecting hyperplanes of the B n , D n Coxeter groups is built from a simple Lie algebra and its representation. The corresponding monodromy representations of the generalized braid groups XB n , XD n are computed in the simplest case.

Representations of Braid Groups via Determinantal Rings

2000

Abstract. We construct representations for braid groups B n via actions of B n on a deter-minantal ring, thus mirroring the setting of the classical representation theory for GL n. The representations that we construct fix a certain unitary form.

Configuration Spaces and Braid Groups

2011

Abstract. The main thrust of these notes is 3-fold: (1) An analysis of certain K(π,1)’s that arise from the connections between configuration spaces, braid groups, and mapping class groups, (2) a function space interpretation of these results, and (3) a homological analysis of the cohomology of some of these groups for genus zero, one, and two surfaces possibly with marked points, as well as the cohomology of certain associated function spaces. An example of the type of results given here is an analysis of the space k particles moving on a punctured torus up to equivalence by the natural SL(2, Z) action. 1.

Representations of the loop braid group

We study representations of the loop braid group LBnLB_nLBn from the perspective of extending representations of the braid group BnB_nBn. We also pursue a generalization of the braid/Hecke/Temperlely-Lieb paradigm---uniform finite dimensional quotient algebras of the loop braid group algebras.

Monodromy representations of the braid group

Physics of Atomic Nuclei, 2001

Chiral conformal blocks in a rational conformal field theory are a far going extension of Gauss hypergeometric functions. The associated monodromy representations of Artin's braid group B n capture the essence of the modern view on the subject, which originates in ideas of Riemann and Schwarz.

A Garside-theoretic analysis of the Burau representations

Journal of Knot Theory and Its Ramifications, 2017

We establish relations between both the classical and the dual Garside structures of the braid group and the Burau representation. Using the classical structure, we formulate a non-vanishing criterion for the Burau representation of the 4-strand braid group. In the dual context, it is shown that the Burau representation for arbitrary braid index is injective when restricted to the set of simply-nested braids.

Low-dimensional representations of the three component loop braid group

Journal of Mathematical Physics, 2015

Motivated by physical and topological applications, we study representations of the group LB 3 of motions of 3 unlinked oriented circles in R 3 . Our point of view is to regard the three strand braid group B 3 as a subgroup of LB 3 and study the problem of extending B 3 representations. We introduce the notion of a standard extension and characterize B 3 representations admiting such an extension. In particular we show, using a classification result of Tuba and Wenzl, that every irreducible B 3 representation of dimension at most 5 has a (standard) extension. We show that this result is sharp by exhibiting an irreducible 6-dimensional B 3 representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible 2-dimensional LB 3 representations (2) extensions of irreducible 3-dimensional B 3 representations and irreducible LB 3 representations whose restriction to B 3 has abelian image.