Embeddings of the Euclidean algebra e(3) into sl(4,C) and restrictions of irreducible representations of sl(4,C)

2011 ◽  
Vol 52 (1) ◽  
pp. 013504 ◽  
Author(s):  
Andrew Douglas ◽  
Joe Repka
Keyword(s):  
Irreducible Representations ◽  
Euclidean Algebra

scholarly journals INDECOMPOSABLE REPRESENTATIONS OF THE EUCLIDEAN ALGEBRA 𝔢(3) FROM IRREDUCIBLE REPRESENTATIONS OF

2011 ◽  
Vol 83 (3) ◽  
pp. 439-449 ◽  
Author(s):  
ANDREW DOUGLAS ◽  
JOE REPKA
Keyword(s):  
Euclidean Space ◽  
Lie Algebra ◽  
Semidirect Product ◽  
Lie Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Dimensional Euclidean Space ◽  
Euclidean Group ◽  
New Class ◽  
Euclidean Algebra

AbstractThe Euclidean group E(3) is the noncompact, semidirect product group E(3)≅ℝ3⋊SO(3). It is the Lie group of orientation-preserving isometries of three-dimensional Euclidean space. The Euclidean algebra 𝔢(3) is the complexification of the Lie algebra of E(3). We embed the Euclidean algebra 𝔢(3) into the simple Lie algebra $\mathfrak {sl}(4,\mathbb {C})$ and show that the irreducible representations V (m,0,0) and V (0,0,m) of $\mathfrak {sl}(4,\mathbb {C})$ are 𝔢(3)-indecomposable, thus creating a new class of indecomposable 𝔢(3) -modules. We then show that V (0,m,0) may decompose.

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].

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