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By Samiou E.

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B2 + . . b2 b1 = p a1 2 p 1 1 + σ(21) + σ(321) + . . p S= p + ... 9) GroupTheory December 10, 2002 51 PERMUTATIONS . ... + (p − 1) ... 1 p = ... which involves only p terms. This equation says, that if we start with the first index, we end up either with the first index, or the second index and so on. The remaining indices are fully symmetric. Multiplying by S23 . . 10) p-1 p-2 ... ... ... bp−1 . bp−1 ap = p = ... 11) ... For a contraction in (p − k) pairs of indices, we have ... p ... = ...

As the covariant and contravariant indices have to be permuted separately, it is sufficient to consider permutations of purely covariant tensors. 3) GroupTheory December 10, 2002 50 CHAPTER 6 Subscripts refer to the standard cycle notation. b2 b1 = δ b1 δ b2 . . δabpp + δab12 δab21 . . δabpp + . . p! a1 a2 + + ... + ... 1 p! = ... 4) yields the symmetrization operator S. In birdtrack notation, a white bar drawn across p lines will always denote symmetrization of the lines crossed. Factor 1/p!

The action of g ∈ G (G† )ab = δba − iDba . 24) D can be parametrized by N ≤ n2 real parameters. N , the maximal number of independent parameters, is called the dimension of the group (also the dimension of the Lie algebra, or the dimension of the adjoint rep). We shall consider only infinitesimal transformations, of form G = 1+iD, |Dba | 1. 8) connected to the identity. For example, we shall not consider invariances under coordinate reflections. 23) are hermitian matrices and belong to the Dba ∈ V ⊗ V¯ space.

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