By Salzmann H.
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Additional resources for 4-Dimensional projective planes of Lenz type III
Diagrammatic notation has several advantages over the tensor notation. Diagrams do not require dummy indices, so explicit labelling of such indices is unnecessary. More to the point, for a human eye it is easier to identify topologically identical diagrams than to recognize equivalence between the corresponding tensor expressions. The main disadvantage of diagrammatic notation is lack of standardization, especially in the case of Clebsch-Gordan coeﬃcients. Many of the diagrammatic notations [97, 98, 73] designed for atomic and nuclear spectroscopy, are complicated by various phase conventions.
16) yields the antisymmetrization projection operator A. In birdtrack notation, antisymmetrization of p lines will always be denoted by a black bar drawn across the lines. As in the previous section ... 17) ... ... ... = ... = ... ... A2 = A and in addition ... ... = ... = 0 ... ... SA = 0 = 0. 18) A transposition has eigenvalue −1 on the antisymmetric tensor space ... ... = − ... σ(i,i+1) A = −A . 19) Diagrammatically this means that legs can be crossed and uncrossed at will, but with a factor of −1 for a transposition of any two neighbouring legs.
Ap a2 = 1, 2, . . , d2 .. where . ap+q = 1, 2, . . , dp+q . 71) The action of transformation g on the index ak is given by the [dk × dk ] matrix representation Gk . The Clebsch-Gordan coeﬃcients are notoriously index-overpopulated as they require a representation label and a tensor index for each representation in the tensor product. 72) (an index, if written, is written at the end of a line; a representation label is written above the line); (ii) one can draw the propagators (Kronecker deltas) for diﬀerent representations with diﬀerent kinds of lines.