scholarly journals On the Unitary Representations of the Braid Group B6

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1080
Author(s):  
Malak M. Dally ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Braid Group ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Dimensional Representation ◽  
One Dimensional ◽  
Low Degree ◽  
Hermitian Positive Definite Matrix ◽  
Group B

We consider a non-abelian leakage-free qudit system that consists of two qubits each composed of three anyons. For this system, we need to have a non-abelian four dimensional unitary representation of the braid group B 6 to obtain a totally leakage-free braiding. The obtained representation is denoted by ρ . We first prove that ρ is irreducible. Next, we find the points y ∈ C * at which the representation ρ is equivalent to the tensor product of a one dimensional representation χ ( y ) and μ ^ 6 ( ± i ) , an irreducible four dimensional representation of the braid group B 6 . The representation μ ^ 6 ( ± i ) was constructed by E. Formanek to classify the irreducible representations of the braid group B n of low degree. Finally, we prove that the representation χ ( y ) ⊗ μ ^ 6 ( ± i ) is a unitary relative to a hermitian positive definite matrix.

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.

The Genuine Omega-regular Unitary Dual of the Metaplectic Group

2012 ◽  
Vol 64 (3) ◽  
pp. 669-704
Author(s):  
Alessandra Pantano ◽  
Annegret Paul ◽  
Susana A. Salamanca-Riba
Keyword(s):  
Tensor Product ◽  
Unitary Representations ◽  
Dimensional Representation ◽  
Oscillator Representation ◽  
One Dimensional ◽  
Metaplectic Group ◽  
Unitary Dual ◽  
Cohomological Induction

Abstract We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group.

scholarly journals Inequalities for the eigenvalues of the positive definite solutions of the nonlinear matrix equation

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2667-2671
Author(s):  
Guoxing Wu ◽  
Ting Xing ◽  
Duanmei Zhou
Keyword(s):  
Matrix Equation ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
The Matrix ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Positive Integers ◽  
Equivalent Conditions

In this paper, the Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q are considered, where Q is a Hermitian positive definite matrix, s and t are positive integers. Bounds for the sum of eigenvalues of the solutions to the equation are given. The equivalent conditions for solutions of the equation are obtained. The eigenvalues of the solutions of the equation with the case AQ = QA are investigated.

Composite Fermions in Braid Group Terms

2010 ◽  
Vol 17 (01) ◽  
pp. 53-71
Author(s):  
J. Jacak ◽  
I. Jóźwiak ◽  
L. Jacak
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Charged Particle ◽  
Braid Group ◽  
Quantum Information Processing ◽  
Particle Systems ◽  
Unitary Representations ◽  
Composite Fermions ◽  
Cyclotron Motion ◽  
One Dimensional

A new implementation of composite fermions, and more generally — of composite anyons is formulated, exploiting one-dimensional unitary representations of appropriately constructed subgroups of the full braid group, in accordance with a cyclotron motion of 2D charged particle systems. The nature of hypothetical fluxes attached to the Jain's composite fermions is explained via additional cyclotron trajectory loops consistently with braid subgroup structure. It is demonstrated that composite fermions and composite anyons are rightful 2D particles (not an auxiliary construction) associated with cyclotron braid subgroups instead of the full braid group, which may open a new opportunity for non-Abelian composite anyons for quantum information processing applications.

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