By Hajime Ishimori, Tatsuo Kobayashi, Hiroshi Ohki, Hiroshi Okada, Yusuke Shimizu, Morimitsu Tanimoto

These lecture notes supply an instructional overview of non-Abelian discrete teams and convey a few purposes to matters in physics the place discrete symmetries represent an enormous precept for version development in particle physics. whereas Abelian discrete symmetries are usually imposed so as to keep watch over couplings for particle physics - particularly version development past the traditional version - non-Abelian discrete symmetries were utilized to appreciate the three-generation style constitution particularly.

certainly, non-Abelian discrete symmetries are thought of to be the main appealing selection for the flavour quarter: version developers have attempted to derive experimental values of quark and lepton lots, and combining angles via assuming non-Abelian discrete style symmetries of quarks and leptons, but, lepton blending has already been intensively mentioned during this context, in addition. the prospective origins of the non-Abelian discrete symmetry for flavors is one other subject of curiosity, as they could come up from an underlying thought - e.g. the string concept or compactification through orbifolding – thereby offering a potential bridge among the underlying idea and the corresponding low-energy area of particle physics.

this article explicitly introduces and experiences the group-theoretical points of many concrete teams and exhibits tips on how to derive conjugacy periods, characters, representations, and tensor items for those teams (with a finite quantity) while algebraic relatives are given, thereby allowing readers to use this to different teams of curiosity.

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**An Introduction to Non-Abelian Discrete Symmetries for Particle Physicists**

Those lecture notes offer an academic assessment of non-Abelian discrete teams and exhibit a few purposes to matters in physics the place discrete symmetries represent a major precept for version development in particle physics. whereas Abelian discrete symmetries are usually imposed with a view to keep an eye on couplings for particle physics - specifically version development past the traditional version - non-Abelian discrete symmetries were utilized to appreciate the three-generation taste constitution particularly.

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We exemplify this by considering D6 . All elements of D6 can be written in the form a m bk for m = 0, 1, . . , 5, and k = 0, 1, where a 6 = e and bab = a −1 . Denoting a˜ = a 2 , we find that the elements a˜ m bk for m = 0, 1, 2, and k = 0, 1, correspond to the subgroup D3 S3 . On the other hand, denoting b˜ = ab, we find that the elements a˜ m b˜ k for m = 0, 1, 2, and k = 0, 1, correspond to another D3 subgroup. The decompositions of D6 multiplets into D3 multiplets are the same for both D3 subgroups when we move to a proper basis.

28) h = 2, where h is the order of each element in the conjugacy class. D4 has four singlets 1++ , 1+− , 1−+ , and 1−− , and one doublet 2. 3. 3 Characters of D4 representations h χ1++ C1 1 1 C2 4 1 χ1+− χ1−+ χ1−− χ2 1 1 1 2 −1 −1 1 0 −2 C1 2 1 1 1 1 C2 2 1 1 −1 −1 0 C2 2 1 −1 1 −1 0 Fig. 31) where s1 = s1 s1 and s2 = s2 s2 . 4 D5 The group D5 is the symmetry group of a regular pentagon. This is generated by the 2π/5 rotation a and the reflection b (see Fig. 2). The generators satisfy a 5 = e, b2 = e, and bab = a −1 .

1. Now consider the doublet representations, that is, (2 × 2) matrix representations. 4), these (2 × 2) matrices correspond to one of the doublet representations. The (2 × 2) matrix representations for the generic doublet 2k are obtained by replacing a → ak . 1 Characters of DN even representations h χ1++ χ1+− χ1−+ C1 1 1 1 1 1 2 C2(1) .. 8) where k = 1, . . , N/2 − 1 for N even and k = 1, . . , (N − 1)/2 for N = odd. , yk → −yk . These transformations can be represented on the complex coordinate zk and its conjugate z¯ −k .