Elements of the geometry of 𝑆³, Hopf bundles and spin representations

10.1090/surv/151/02 â—˝  
2008 â—˝  
pp. 25-46
Author(s):  
Ernst Binz â—˝  
Sonja Pods
Keyword(s):  
Spin Representations

scholarly journals Massless finite and infinite spin representations of Poincaré group in six dimensions

Physics Letters B â—˝  
2021 â—˝  
pp. 136064
Author(s):  
I.L. Buchbinder â—˝  
S.A. Fedoruk â—˝  
A.P. Isaev â—˝  
M.A. Podoinitsyn
Keyword(s):  
Poincaré Group â—˝  
Poincare Group â—˝  
Six Dimensions â—˝  
Spin Representations

scholarly journals Spin representations with negative indices

2014 â—˝  
Vol 45 (4) â—˝  
pp. 367-374
Author(s):  
Madline Al-Tahan â—˝  
Mohammad N. Abdulrahim â—˝  
Samer S. Habre
Keyword(s):  
Braid Group â—˝  
Hermitian Matrix â—˝  
Sufficient Conditions â—˝  
Spin Representation â—˝  
Negative Index â—˝  
Spin Representations

We consider the spin representation of Artin's braid group, which has a negative index of one and was originally given by D. D. Long and explicitly computed by J.P.Tian. In our work, we find sufficient conditions under which the complex specialization of that representation, namely $\alpha :B_{n}\to GL_{n^{2}}(\mathbb C)$, is unitary relative to a nonsingular hermitian matrix.

scholarly journals ELEMENTARY PARTICLES AND SPIN REPRESENTATIONS

2004 â—˝  
Vol 19 (05) â—˝  
pp. 357-362 â—˝  
Author(s):  
PAOLO MARANER
Keyword(s):  
Matter Field â—˝  
Elementary Particles â—˝  
Spacetime Symmetries â—˝  
Spin Representation â—˝  
Dimensional Spacetime â—˝  
Higher Dimensional â—˝  
Spin Representations â—˝  
Theoretical Considerations â—˝  
Unified Description

We emphasize that the group-theoretical considerations leading to SO (10) unification of electroweak and strong matter field components naturally extend to spacetime components, providing a truly unified description of all generation degrees of freedoms in terms of a single chiral spin representation of one of the groups SO (13,1), SO (9,5), SO (7,7) or SO (3,11). The realization of these groups as higher-dimensional spacetime symmetries produces unification of all fundamental fermions is a single spacetime spinor.

Projective representations of rotation subgroups of Weyl groups

1987 â—˝  
Vol 101 (1) â—˝  
pp. 21-35 â—˝  
Author(s):  
D. Theo
Keyword(s):  
Recent Work â—˝  
Ad Hoc â—˝  
Weyl Groups â—˝  
Orthogonal Groups â—˝  
Central Extensions â—˝  
Projective Representations â—˝  
Rotation Groups â—˝  
Spin Representations â—˝  
Double Coverings

By exploiting the well known spin representations of the orthogonal groups O(l), Morris [12] was able to give a unified construction of some of the projective representations of Weyl groups W(Φ) which had previously only been available by ad hoc means [5]. The principal purpose of the present paper is to give a corresponding construction for projective representations of the rotation subgroups W+(Φ) of Weyl groups. Thus we construct non-trivial central extensions of W+(Φ) via the well-known double coverings of the rotation groups SO(l). This adaptation allows us to give a unified way of obtaining the basic projective representations of W+(Φ) from those of W(Φ) determined in [12]. Hence our work is a development of the recent work of Morris, and is an extension of Schur's work on the alternative groups [15].

scholarly journals MODULAR LOCALIZATION AND WIGNER PARTICLES

2002 â—˝  
Vol 14 (07n08) â—˝  
pp. 759-785 â—˝  
Author(s):  
R. BRUNETTI â—˝  
D. GUIDO â—˝  
R. LONGO
Keyword(s):  
Von Neumann Algebras â—˝  
Free Field â—˝  
De Sitter â—˝  
Geometric Transformations â—˝  
Von Neumann â—˝  
Sitter Spacetime â—˝  
Modular Localization â—˝  
Hilbert Subspace â—˝  
The One â—˝  
Spin Representations

We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano–Wichmann relations and a representation of the Poincaré group on the one-particle Hilbert space. The abstract real Hilbert subspace version of the Tomita–Takesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the Reeh–Schlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and of de Sitter spacetime.

The symmetric group: branching rules, products and plethysms for spin representations

1981 â—˝  
Vol 14 (2) â—˝  
pp. 327-348 â—˝  
Author(s):  
L Dehuai â—˝  
B G Wybourne
Keyword(s):  
Symmetric Group â—˝  
Branching Rules â—˝  
Spin Representations

scholarly journals Maximal Strings in the Crystal Graph of Spin Representations of the Symmetric and Alternating Groups

2009 â—˝  
Vol 37 (11) â—˝  
pp. 3779-3795
Author(s):  
Hussam Arisha â—˝  
Mary Schaps
Keyword(s):  
Alternating Groups â—˝  
Crystal Graph â—˝  
Spin Representations

scholarly journals A comment on continuous spin representations of the Poincaré group and perturbative string theory

2014 â—˝  
Vol 62 (11-12) â—˝  
pp. 975-980 â—˝  
Author(s):  
A. Font â—˝  
F. Quevedo â—˝  
S. Theisen
Keyword(s):  
String Theory â—˝  
Poincaré Group â—˝  
Continuous Spin â—˝  
Poincare Group â—˝  
Spin Representations
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