Racah’s lemma

The irreducible representations of a group are generally reducible as representations of a sub-group. In Racah s lemma this statement is considered by introducing the concept of a group which is reduced with respect to a sub-group. By this is meant that the irreducible representa-... 

In order to apply Racah s lemma to the group R3 reduced with respect to Cm (most reduced representations with a real basis), one simply uses the classification of irreducible representations according to Dm as explained in Table 13. As the two sigma terms in A X A cannot be distinguished in the group Cm, the Racah lemma matrices whose elements are characterized by (/A lA l, will appear in block-diagonal form, since the E can be either Ei or Zz (Table 13). 

The group D3 has the irreducible representations Ai, A2, and E (Table 5), where E is the conventional abbreviation for Ei. Racah s lemma is fulfilled for these representations when they are associated with the first three irreducible representations of Da> (Table 11, compare Table 5) and the 3-F symbols can be directly read out of Table 13, except for the EEE symbols, which require additional considerations. The basis functions for E2 of Dm span the E representation of D3 in the standard manner when taken as Hence the 3-F symbols... 

In order to use Racah s lemma to the group Rsi reduced with respect to Coot), it is noted that the standard basis functions of the irreducible representations in Eq. (25) [which are the functions of Eq. (15)] are symmetry-adapted to Coo in such a way that sigma and cosine functions... 

Racah s Lemma Applied to the Group Rzi Reduced with Respect to the Group Coot,. 

In Sect. 8a Racah s lemma was illustrated for the group R reduced with respect to Doo- It is valuable also to consider the group Rzi reduced with respect to Coo , since this will exemplify the standardizations made in section 8c. 

In this expression the coefficients in brackets < > are the isoscalar factors (Clebsch-Gordan coefficients) for coupling two 0(4) and two 0(3) representations, respectively. They can be evaluated either analytically using Racah s factorization lemma (Section B.14) or numerically using subroutines explicitly written for this purpose.2... 

A lemma by Racah (5), a consequence of Schur s lemma, provides a relationship between the coupling coefficients of a group and those of a sub-group. This relationship is here illustrated and standardized by introducing differential operator equivalents for the real standard bases of the three-dimensional rotation group Rs. 

It is noted that among the Rg bases used in Eq. (65) the function and the set ( nc. S ) transform in the standard way according to the Ai and Ei = representations of Doo (Table 11), whereas the set is not standard under C. The function is symmetrical under Cl and the function antisymmetrical. However, in order to transform by the standard matrices, i. e. by those of [ c. ]. one of the p-functions will have to change its sign. We take the set [— as the standard choice so that the set ( , — sc) = (z. —y, x) forms a right-handed coordinate system. By this choice, the two last columns of the Racah lemma matrix are completely specified. One has, returning to the middle element,... 

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