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An Introduction to Non-Abelian Discrete Symmetries for by Hajime Ishimori, Tatsuo Kobayashi, Hiroshi Ohki, Hiroshi

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

These lecture notes supply an educational evaluate of non-Abelian discrete teams and exhibit a few functions to matters in physics the place discrete symmetries represent an immense precept for version construction in particle physics. whereas Abelian discrete symmetries are frequently imposed with a view to keep an eye on couplings for particle physics - particularly version development past the traditional version - non-Abelian discrete symmetries were utilized to appreciate the three-generation taste constitution specifically.

certainly, non-Abelian discrete symmetries are thought of to be the main beautiful 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 potential origins of the non-Abelian discrete symmetry for flavors is one other subject of curiosity, as they could come up from an underlying idea - e.g. the string conception or compactification through orbifolding – thereby offering a potential bridge among the underlying conception and the corresponding low-energy zone of particle physics.

this article explicitly introduces and reports the group-theoretical facets of many concrete teams and exhibits the best way to derive conjugacy sessions, characters, representations, and tensor items for those teams (with a finite quantity) while algebraic family are given, thereby permitting readers to use this to different teams of curiosity.

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On the other hand, the doublet 2 of S3 decomposes into two singlets of Z2 . Since χ2 (a) = −1, the S3 doublet 2 decomposes into 10 and 11 of Z2 . 28) by a= 1 0 . 3) Then for the doublet (x1 , x2 ), the elements x1 and x2 correspond to x1 = 10 and x2 = 11 , respectively. , {e, b} and {e, aba}. In both cases, the same results are obtained when we choose a proper basis. These are examples of Abelian subgroups. For non-Abelian subgroups, the same situation arises. That is, different elements of a finite group G can generate the same subgroup.

6) where ρ = e2πi/N . 5) tell us that a k and a k(N/2−1) belong to the same conjugacy class and also that b and a m(N/2−2) b belong to the same conjugacy class. The group QD2N has the following (3 + N/2) conjugacy classes: C1 : C2[k] {e}, h = 1, a k , a k(N/2−1) , h = N/gcd(N, k), C1 : a N/2 , h = 2, CN/2 : b, a 2 b, . . , a N−2 b , h = 2, CN/2 : ab, a 3 b, . . 7) where [k] = k or k(N/2 − 1) mod N with k = 1, . . , N − 1 except N/2. We have also shown the order h of each element in the given conjugacy class.

That is, zk z¯ −k + z¯ −k zk and zk z¯ −k − z¯ −k zk correspond to 1++ and 1−− , respectively. Now, we study the tensor products of the doublets 2k and singlets, for example, 1−− × 2k . Here we denote the vector space for the singlet 1−− by w, where aw = w and bw = −w. 1 DN with N Even 55 is, 1−− × 2k = 2k . Similar results are obtained for other singlets. Furthermore, it is straightforward to study the tensor products among singlets. 22) 56 6 DN with si , si , si = ± (i = 1, 2), where si = + for (si , si ) = (+, +) and (−, −), and si = − for (si , si ) = (+, −) and (−, +).

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