By Chen G.
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51) U (n), SU (n) : [Tba , Tdc ] = δbc Tda − Tbc δda . Another example is given by the orthogonal groups. The generators of rotations are antisymmetric matrices, and the adjoint projection operator antisymmetrizes generator indices: 1 gac Tbd − gad Tbc . 52) SO(n) : [Tab , Tcd ] = −g 2 bc Tad + gbd Tac Apart from the normalization convention, these are the familiar Lorentz group commutation relations (we shall return to this in chapter 10). 7 IRRELEVANCY OF CLEBSCHES As was emphasized in sect. 2, an explicit choice of clebsches is highly arbitrary; it corresponds to a particular coordinatization of the dλ -dimensional subspace Vλ .
1100 00 11 11 00 00 11 00 11 00 11 00 11 00 11 ... ···− 1 0 1 0 1 0 λ1010 ... 0 1 1 0 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 0 1 λ1010 ... λ 1010 00 11 11 00 00 11 00 11 00 11 00 11 00 11 00 11 + ··· . 39) The reality of the adjoint rep. For hermitian generators, the adjoint rep is real, and the upper and lower indices need not be distinguished; the “propagator” needs no arrow. For non-hermitian choices of generators, the adjoint rep is complex (“gluon” lines carry arrows), but A and A¯ are equivalent, as indices can be raised an lowered by the Cartan-Killing form gij = tr (Ti† Tj ).
Abpp + 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! has been introduced in order that S satisfies the projection operator normalization S2 = S . ... 5) A subset of indices a1 , a2 , . . b2 b1 = 1 b +1 δ b1 δ b2 . . δabqq + δab12 δab21 . . δabqq + . . δaqq +1 . . δabpp q! a1 a2 ... q. 6) ... q = 1 2 Overall symmetrization also symmetrizes any subset of indices: .
A Characterization of Alternating Groups by the Set of Orders of Maximal Abelian Subgroups by Chen G.