By Knebelman M. S.
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Vii') ~ operators belonging to the same diagram belong to the same IR. Those belonging to different diagrams belong to different IR's. Proo]: If two Y operators belong to the same diagram, we use (55) to show that ~ ' a Y # O. If Y, Y' belong to different diagrams with, say, n+ > n~-, then there is at least one site j" which is in t<+> and in t'C->. The product Y'Y then contains a factor (1--zj) (1 + z j ) = 0 . If n+=n~-, but the diagrams are nevertheless different, one shows as in the proof of (vii) that Y Y ' ~-0.
If the ligands are allowed to be chiral, we have ~ = ~n. In this case, a single scalar parameter is not sufficient to characterize a ligand, since that could not distinguish between the tigand and its mirror image. In addition to our scalar parameter ~, therefore, we need a pseudoscalar one, ~. In this case, one must use some care in defining what one means b y "lowest order", since it m a y be possible to obtain a decrease in the order of one of the parameters at the expense of an increase in the other.
Mead which are to go in first, then those which go in next, etc. The validity of the prescription for the smallest remainder now follows by interchanging rows and columns. F r o m the above discussion, we see immediately the criterion for satisfying the T-condition for ~7(~) into ~(r). It is satisfied if and only if 1) ~(r) D ~(p) and 2) ~m~,,(~'('+), 7(P+)) ~ ~,m~(~(P-), 7(~-)). We note that the T-condition in this case is not transitive. For example, referring to Fig. 7, we see that the T-condition is satisfied for ~(c) into ~(b), and for ~(b) into ~(a), but not for ~(e) into ~(a).
Conformal Geometry of Generalized Metric Spaces by Knebelman M. S.