Unitary Irreducible Representations of SU(2, 2). III. Reduction with Respect to an Iso‐Poincaré Subgroup

1971 ◽  
Vol 12 (3) ◽  
pp. 315-342 ◽  
Author(s):  
Tsu Yao
Keyword(s):  
Irreducible Representations

An Intuitive Method for the Determination of Crystal-Field Parameters in Laser Crystals

1993 ◽  
Vol 329 ◽  
Author(s):  
Frederick G. Anderson ◽  
H. Weidner ◽  
P. L. Summers ◽  
R. E. Peale ◽  
B. H. T. Chai
Keyword(s):  
Crystal Field ◽  
Irreducible Representations ◽  
Laser Crystals ◽  
Intuitive Method ◽  
Crystal Field Parameters ◽  
Intuitive Interpretation

AbstractExpanding the crystal field in terms of operators that transform as the irreducible representations of the Td group leads to an intuitive interpretation of the crystal-field parameters. We apply this method to the crystal field experienced by Nd3+ dopants in the laser crystals YLiF4, YVO4, and KLiYF5.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

scholarly journals Clebsch–Gordan coefficients of discrete groups in subgroup bases

2018 ◽  
Vol 33 (10) ◽  
pp. 1850055
Author(s):  
Gaoli Chen
Keyword(s):  
Discrete Group ◽  
Irreducible Representations ◽  
Complex Conjugate ◽  
Discrete Groups

We express each Clebsch–Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase ambiguities. Particularly, they are invariant under basis transformations of irreducible representations of both the group and its subgroup. We then impose on the embedding factors constraints, which relate them to their counterparts under complex conjugate and therefore restrict the phases of embedding factors. In some cases, the phase ambiguities are reduced to sign ambiguities. We describe the procedure of obtaining embedding factors and then calculate CG coefficients of the group [Formula: see text] in terms of embedding factors of its subgroups [Formula: see text] and [Formula: see text].

L'application des representations au calcul explicite sur certaines chaînes de Markov finies

1984 ◽  
Vol 16 (3) ◽  
pp. 656-666 ◽  
Author(s):  
Bernard Ycart
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
Irreducible Representations ◽  
Permutation Groups ◽  
A Finite Group

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.

Vertex Subgroups of Irreducible Representations of Solvable Groups

1971 ◽  
Vol 23 (1) ◽  
pp. 12-21
Author(s):  
J. Malzan
Keyword(s):  
Vector Space ◽  
Unit Sphere ◽  
Convex Subset ◽  
Orthogonal Group ◽  
Solvable Groups ◽  
Fundamental Region ◽  
Irreducible Representations ◽  
Theoretical Problem ◽  
Real Vector ◽  
The Real

If ρ(G) is a finite, real, orthogonal group of matrices acting on the real vector space V, then there is defined [5], by the action of ρ(G), a convex subset of the unit sphere in V called a fundamental region. When the unit sphere is covered by the images under ρ(G) of a fundamental region, we obtain a semi-regular figure.The group-theoretical problem in this kind of geometry is to find when the fundamental region is unique. In this paper we examine the subgroups, ρ(H), of ρ(G) with a view of finding what subspace, W of V consists of vectors held fixed by all the matrices of ρ(H). Any such subspace lies between two copies of a fundamental region and so contributes to a boundary of both. If enough of these boundaries might be found, the fundamental region would be completely described.

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