By Mazurov V. D.

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N − p)! 24) ... = ... ... b b 2 1 has non-vanishing components only if all The antisymmetrization tensor Aap1 a2 ... lower (or upper) indices diﬀer from each other. If the deﬁning dimension is smaller than the number of indices, the tensor A has no non-vanishing components ... p ... 1 2 =0 if p > n . 25) This identity implies that for p > n, not all combinations of p Kronecker deltas are linearly independent. A typical relation is the p = n + 1 case ... 0= ... = ... − ... ... + − ... ... 26) 1 2 n+1 for example, for n = 2 we have f e d − n = 2; 0 = − + − + a b c 0 = δaf δbe δcd − δaf δce δbd − δbf δae δcd + δbf δce δad + δcf δae δbd − δcf δbe δad .

Clebsch-Gordan coeﬃcients are also invariant tensors. 75), we obtain Cλ G = Gλ Cλ , (no sum on λ) . 94) The Clebsch-Gordan matrix Cλ is a rectangular [dλ × d] matrix, hence g ∈ G acts on it with a [dλ × dλ ] representation from the left, and a [d × d] representation from the right. 27), the statement of invariance for square matrices: Cλ = G†λ Cλ G , C λ = G† C λ Gλ . 93), the invariance condition for any invariant tensor: (λ) 0 = −Ti Cλ + Cλ Ti 0 = . 75) now yields the generators in λ representation in terms of the deﬁning representation generators = printed April 14, 2000 .

75). 8 Zero- and one-dimensional subspaces If a projection operator projects onto a zero-dimensional subspace, it must vanish identically ⇒ dλ = 0 Pλ = = 0. 46); dλ is the number of 1’s on the diagonal on the right-hand side. For dλ = 0 the right-hand side vanishes. 79) k=1 where Mk are the invariant matrices used in construction of the projector operators, and ck are numerical coeﬃcients. Vanishing of Pλ therefore implies a relation among invariant matrices Mk . tex 14apr2000 32 CHAPTER 3. b2 q ...