# Download e-book for kindle: 4-Homogeneous groups by Kantor W.M.

By Kantor W.M.

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5 will enable us to exploit this fact to decompose the V q ⊗ V¯ p space into r irreducible subspaces. This can be said in another way; the choice of basis {e, t 1 , t2 · · · tr−1 } is arbitrary, the only requirement being that the basis elements are linearly independent. 5. 11. 4 INVARIANCE GROUPS So far we have defined invariant tensors as the tensors invariant under transformations of a given group. Now we proceed in reverse: given a set of tensors, what is the group of transformations that leaves them invariant?

16) The action of the transformation g on the index a k is given by the [d k × dk ] matrix rep Gk . Clebsches are notoriously index overpopulated, as they require a rep label and a tensor index for each rep in the tensor product. Diagrammatic notation alleviates this index plague in either of two ways: 1. One can indicate a rep label on each line: Caaλμ aν , aσ aλ = aμ aν λ 00 11 00 μ 11 11 00 00 11 00 11 00 ν 11 σ 111 000 000 111 aσ . ) 2. One can draw the propagators (Kronecker deltas) for different reps with different kinds of lines.

27) ¯q Definition. c 2 p = Ga1 c1 Ga2 c2 . . Gb1 d1 . . d . 25) xα = Gα β xβ . 29) as a vector in [d×d]-dimensional space, d = n p+q . 30) e is an invariant matrix. 13), we find that the invariant matrices commute with all transformations g ∈ G: [G, M] = 0 . 31). 6)). Definition. 33) H(x, y, z¯, r¯, s¯) = hab cde xb ya se rd z c , as an invariant in V q ⊗ V¯ p (repeated indices, as always, summed over). 28). 34) a b c e d makes it easier to distinguish different types of invariant tensors. 1; sketching a few simple examples should suffice for the time being.