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

Sun Hong-zhou

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2024年10月30日

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 C₂ and 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 C₂ can 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 C₂ and 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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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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1.
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 group C₂ and 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 C₂ can 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 C₂ and prove the corresponding Wigner-Eckart Theory.