Chinese Physics C > 1980, Vol. 4 > Issue(2): 137-159

ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(II)

Sun Hong-zhou

Peking University

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Abstract：
In this paper, we analyse the commutation relations of the infinitesimal opera-tors of the group C ₂and find that the ten infinitesimal operators of the group canbe written as two mutually commuting sets of angular momentum operators( υ₁, υ₀,υ=1), ( τ₁, τ₀,τ_-1) ,and one set of dual irreducible tensor operators of rank (1/2 1/2), U _{±1∕2,±1∕2}. By means of their commutation relations, all irreducible representations ofthe group C ₂can be easily obtained. In this paper,the matrices corresponding to the irreducible representation ( λμ) are given; therefore the irreducible representation ( λμ) and its representation space R ^λμ）are completely defined. Besides, a method for calculating the scalar factors ofthe reduction coefficients and the symmetric relations of these factors are also given.As examples, the algebriac formulae of the scalar factors of the reduction coefficientsof ( λμ)×(10), ( λμ)×(01) and ( λμ)×(20) are derived. In the last part of this paper, we define the irreducible tensor operators of thegroup C ₂and prove the corresponding Wigner-Eckart Theory.

References

[1]	G.Racah, Group Theory and Spectroscopy, Princeton, 1951.[2] R.E. Behrends et al., Rev. mod. Phys., 34 (1962), 1.[3] A. Salam, Theoretical Physics, p, 173, Viena, (1963).[4] 杨国祯等,北京大学学报(自然科学版),10(1964),269.

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[20]	Sun Hung-zhou. ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(I). Chinese Physics C, 1980, 4(1): 73-92.

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Sun Hong-zhou. ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(II)[J]. Chinese Physics C, 1980, 4(2): 137-159.

Sun Hong-zhou. ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(II)[J]. Chinese Physics C, 1980, 4(2): 137-159. shu

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Received: 1978-12-08
Revised: 1900-01-01

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沈阳化工大学材料科学与工程学院沈阳 110142

ON THE IRREDUCIBLE REPRESENTATIONS OF THE COMPACT SIMPLE LIE GROUPS OF RANK 2(II)

Sun Hong-zhou

Peking University

Received Date:1978-12-08
Accepted Date:1900-01-01
Available Online:1980-04-05

Abstract:In this paper, we analyse the commutation relations of the infinitesimal opera-tors of the groupC₂and find that the ten infinitesimal operators of the group canbe written as two mutually commuting sets of angular momentum operators(υ₁, υ₀,υ=1), (τ₁, τ₀,τ_-1),and one set of dual irreducible tensor operators of rank (1/2 1/2),U_{±1∕2,±1∕2}. By means of their commutation relations, all irreducible representations ofthe groupC₂can be easily obtained. In this paper,the matrices corresponding to the irreducible representation (λμ) are given; therefore the irreducible representation (λμ) and its representation spaceR^λμ）are completely defined. Besides, a method for calculating the scalar factors ofthe reduction coefficients and the symmetric relations of these factors are also given.As examples, the algebriac formulae of the scalar factors of the reduction coefficientsof (λμ)×(10), (λμ)×(01) and (λμ)×(20) are derived. In the last part of this paper, we define the irreducible tensor operators of thegroupC₂and prove the corresponding Wigner-Eckart Theory.