scholarly journals Irreducibility of the Tensor Product of Specializations of the Burau Representation of the Braid Groups

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Mohammad N. Abdulrahim ◽  
Wiaam M. Zeid
Keyword(s):  
Tensor Product ◽  
Complex Number ◽  
Braid Group ◽  
Composition Factor ◽  
Braid Groups ◽  
Sufficient Condition ◽  
Burau Representation ◽  
Parameter Representation ◽  
Nonzero Complex Number

The reduced Burau representation is a one-parameter representation of , the braid group on strings. Specializing the parameter to nonzero complex number gives a representation : , which is either irreducible or has an irreducible composition factor : . In our paper, we let , and we determine a sufficient condition for the irreducibility of the tensor product of irreducible Burau representations. This is a generalization of our previous work concerning the cases and .

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

scholarly journals Attacks on the Faithfulness of the Burau Representation of the Braid Group $B_4$

2016 ◽  
Vol 8 (1) ◽  
pp. 5
Author(s):  
Mohammad Y. Chreif ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Unit Circle ◽  
Complex Number ◽  
Braid Group ◽  
Simple Algorithm ◽  
Necessary Condition ◽  
Burau Representation ◽  
Open Question

<p align="left">The faithfulness of the Burau representation of the 4-strand braid group, $B_4$, remains an open question.<br />In this work, there are two main results. First, we specialize the indeterminate $t$ to a complex number on the unit circle, and we find a necessary condition for a word of $B_4$ to belong to the kernel of the representation. Second, by using a simple algorithm,<br />we will be able to exclude a family of words in the generators from belonging to the kernel of the reduced Burau representation.</p>

scholarly journals On the irreducibility of linear representations of the pure braid group

2010 ◽  
Vol 41 (3) ◽  
pp. 283-292
Author(s):  
Mohammad N. Abdulrahim
Keyword(s):  
Tensor Product ◽  
Braid Group ◽  
Sufficient Condition ◽  
Pure Braid Group ◽  
Linear Representations

Following up on our result in [1], we find a milder sufficient condition for the tensor product of specializations of the reduced Gassner representation of the pure braid group to be irreducible. We prove that $G_n(x_1, \ldots, x_n) \otimes G_n(y_1, \ldots, y_n) : P_n \to GL(\mathbb{C}^{n-1} \otimes \mathbb{C}^{n-1})$ is irreducible if  $x_i \neq \pm y_i $ and $x_j \neq \pm {{y_j}^{-1}} $ for some $i$ and $j$.

scholarly journals Representations of the necklace braid group $${{\mathcal {N}}{\mathcal {B}}}_n$$ of dimension 4 ($$n=2,3,4$$)

2021 ◽  
Author(s):  
Taher I. Mayassi ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Tensor Product ◽  
Braid Group ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Irreducible Representations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hermitian Positive Definite Matrix ◽  
Dimension 2 ◽  
Necessary And Sufficient

AbstractWe consider the irreducible representations each of dimension 2 of the necklace braid group $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ). We then consider the tensor product of the representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) of degree 2 are unitary relative to a hermitian positive definite matrix.

scholarly journals Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

2018 ◽  
Vol 27 (06) ◽  
pp. 1850043 ◽  
Author(s):  
Paul P. Gustafson
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Braid Groups ◽  
Fusion Category ◽  
Mapping Class ◽  
Group Representations ◽  
Colored Graphs ◽  
Finite Image ◽  
Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

AUTOMORPHISMS OF BRAID GROUPS ON S2, T2, P2 AND THE KLEIN BOTTLE K

2008 ◽  
Vol 17 (01) ◽  
pp. 47-53 ◽  
Author(s):  
PING ZHANG
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Klein Bottle ◽  
Closed Surface ◽  
Group Extension ◽  
Braid Groups ◽  
Mapping Class ◽  
Group B ◽  
Central Automorphisms

It is shown that for the braid group Bn(M) on a closed surface M of nonnegative Euler characteristic, Out (Bn(M)) is isomorphic to a group extension of the group of central automorphisms of Bn(M) by the extended mapping class group of M, with an explicit and complete description of Aut (Bn(S2)), Aut (Bn(P2)), Out (Bn(S2)) and Out (Bn(P2)).

Braid lift representations of Artin's Braid Group

2000 ◽  
Vol 09 (08) ◽  
pp. 1005-1009
Author(s):  
Reinhard Häring-Oldenburg
Keyword(s):  
Braid Group ◽  
Braid Groups ◽  
Finite Dimensional ◽  
Braid Theory

We recast the braid-lift representation of Contantinescu, Lüdde and Toppan in the language of B-type braid theory. Composing with finite dimensional representations of these braid groups we obtain various sequences of finite dimensional multi-parameter representations.

(ANTI)COMMUTATIVITY OF THE HARMONIC CREATION AND ANNIHILATION OPERATORS

2006 ◽  
Vol 20 (11n13) ◽  
pp. 1808-1818
Author(s):  
S. KUWATA ◽  
A. MARUMOTO
Keyword(s):  
Irreducible Representation ◽  
Harmonic Oscillator ◽  
Linear Algebra ◽  
Commutation Relation ◽  
Complex Number ◽  
Single Mode ◽  
Equation Of Motion ◽  
Sufficient Condition ◽  
Special Linear ◽  
Creation And Annihilation Operators

It is known that para-particles, together with fermions and bosons, of a single mode can be described as an irreducible representation of the Lie (super) algebra 𝔰𝔩2(ℂ) (2-dimensional special linear algebra over the complex number ℂ), that is, they satisfy the equation of motion of a harmonic oscillator. Under the equation of motion of a harmonic oscillator, we obtain the set of the commutation relations which is isomorphic to the irreducible representation, to find that the equation of motion, conversely, can be derived from the commutation relation only for the case of either fermion or boson. If Nature admits of the existence of such a sufficient condition for the equation of motion of a harmonic oscillator, no para-particle can be allowed.

scholarly journals The Interplay between Linear Representations of the Braid Group

2007 ◽  
Vol 2007 ◽  
pp. 1-9
Author(s):  
Mohammad N. Abdulrahim ◽  
Nibal H. Kassem
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Burau Representation ◽  
Standard Representation ◽  
Linear Representations ◽  
Standard Action ◽  
Composition Factors

We consider Wada's representation as a twisted version of the standard action of the braid group,Bn, on the free group withngenerators. Constructing a free group,Gnm, of ranknm, we compose Cohen's mapBn→Bnmand the embeddingBnm→Aut(Gnm)via Wada's map. We prove that the composition factors of the obtained representation are one copy of Burau representation andm−1copies of the standard representation after changing the parameterttotkin the definitions of the Burau and standard representations. This is a generalization of our previous result concerning the standard Artin representation of the braid group.

scholarly journals Negative curvature in graph braid groups

2020 ◽  
pp. 1-36
Author(s):  
Anthony Genevois
Keyword(s):  
Braid Group ◽  
Negative Curvature ◽  
Braid Groups ◽  
Gromov Hyperbolic ◽  
Relatively Hyperbolic ◽  
Geometric Study

In this paper, we initiate a geometric study of graph braid groups. More precisely, by applying the formalism of special colorings introduced in a previous paper, we determine precisely when a graph braid group is Gromov-hyperbolic, toral relatively hyperbolic and acylindrically hyperbolic.

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