By R.D. Carmichael
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Extra resources for Introduction to the Theory of Groups of Finite Order
6) b b +1 b δab12 δab21 . . δaqq + . . δaqq +1 . . δapp 1 2 ... q = q . 7) ... Overall symmetrization also symmetrizes any subset of indices: ... = ... ... q = S . 8) Any permutation has eigenvalue 1 on the symmetric tensor space: ... = ... ... σS = S . 9) Diagrammatically this means that legs can be crossed and un-crossed at will. 2. 4) of the symmetrization operator as the sum of all p! b2 b1 = S = ... ... ... = ... b2 + . . p a1 2 p 1 1 + σ(21) + σ(321) + . . p p 1 + + + ... 10) 1 p p-1 + (p − 1) p-1 .
88) we obtain the invariance condition which the generators must satisfy: they annihilate invariant tensors Ti q = 0 . 89) To state the invariance condition for an arbitrary invariant tensor, we need to deﬁne the generators in the tensor representations. c2 c1 a d = (Ti )ac11 δca22 . . δcpp δbd11 . . b2 q a d a d +δca11 (Ti )ac22 . . δcpp δbd11 . . δbqq + . . + δca11 δca22 . . (Ti )cpp δbd11 . . δbqq a d a d − δca11 δca22 . . δcpp (Ti )db11 . . δbqq − . . − δca11 δca22 . . δcpp δbd11 .
56) ... ) can be written as 0 = + + = + + = p printed April 14, 2000 (p = number of indices) .