By W. G. Harter, C. W. Patterson

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I2i 2 ".. 39. There the factor D ~ (p) is an Sn representation using Fig. 7-a, and Ny constant to be evaluated shortly. ~,,,> = N~, 1s~ II 2 ... in~ of definite "order. " By order we mean that single particle states have been numbered (Recall the numbering 1-3 in Sec. 2B) and these numbers are ordered: i1~ i 2 ~ ... ~ in . 39 then give all other orderings. help explain the properties To of the permutation projectors (These properties were first discovered by Goddard 14. ) we shall make an analogy with the quantum theory of rigid rotators.

I . ,, ~ ... . - :: ~ "~ ~',-i ~- ~ • ~-~ . ~i~i~---~. SSs L~L~I~ ~ ..... ~ ..... ,--i II ~-4 L",I O,l . 25. 26a) and the desired interaction matrix (Eq26b). ,,31T> o -9IT> - (26c) v3. ) to label the repeated states. V ~ are multiples in the ([It , LII> ) representation,(M 1 of the unit matrix is the U s invariant, 1 while V -V is proportional to L 2) so eigenvectors of V3. 28 must also be eigenvectors ~I~'~ of the pairing operator. > (28) ~> [%

Biedenharn and Louck produced Gelfand bases from 25 polynomials of boson operators and thus avoided this repetition; 32 in effect they keep only the "first tableau"~l, which has all the numbers in sequence 123... in every row. Certainly one basis per ~ is all that ~s necessary as far as the mathematics of U m is concerned. However, the explicit Pauliantisymmetric states of orbit and spin (viz. 45. 46. The first term has an orbit factor with the "first tableau"~, (In~, and the spin factor with the "last tableau"~ i.