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

**Sample text**

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 (−, +).