Quantum Gravity 362.0: How Probable Causation/Influence (PI) Relates to S_4 in Section 361
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From: OsherD on 23 Mar 2010 00:05 From Osher Doctorow In addition to Ishimori et al (2010) referenced in Section 361, look at F. Bazzacchi and Stefano Morisi of U. de Valencia Spain in arXiv November 2008 (readers can look for their reference by name of author or first author). The latter authors point out that S_4 appears to be the minimal flavor symmetry compatible w tribimaximal neutrino mixing. S_4 is the group of permutations of 4 elements and has itself the number of different elements 4! = 4 times 3 times 2 times 1 = 24. In Probable Causation/Influence, the "arrow" expressions with 4 different sets/events A, B, C, D are: 1) A-->B-->C-->D, A-->C-->B-->D, A-->B-->D-->C, ..., D-->B-->C-->A which consists of 24 different such expressions. Likewise, the probabilities P(A-->B-->C-->D) and so on consist of 24 probabilities, although some may be equal. Both the S_4 permutations and the PI expressions like (1) are non- Abelian. The Symmetric Group S_n is the group of permutations of n different elements, that is to say, one to one transformations of a finite set onto itself (are permutations). For example: 2) (12345) = the permutation that maps 1 to 2, 2 to 3, 3 to 4, 4 to 5, 5 to 1. As Birkhoff and MacLane (1953) point out, it happens to equal (23451), (34512), (45123), (51234). Every finite group is isomorphic with one or more groups of permutations, and so all of finite group theory could be developed in the theory of permutation groups. Osher Doctorow |