# 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 offer an instructional evaluation of non-Abelian discrete teams and express a few functions to concerns in physics the place discrete symmetries represent a huge precept for version development in particle physics. whereas Abelian discrete symmetries are frequently imposed with a purpose to keep an eye on couplings for particle physics - particularly version construction past the normal version - non-Abelian discrete symmetries were utilized to appreciate the three-generation style constitution specifically.

certainly, non-Abelian discrete symmetries are thought of to be the main appealing selection for the flavour region: version developers have attempted to derive experimental values of quark and lepton plenty, and combining angles by means of assuming non-Abelian discrete taste 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 idea - e.g. the string thought 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 experiences the group-theoretical facets of many concrete teams and indicates how you can derive conjugacy periods, characters, representations, and tensor items for those teams (with a finite quantity) while algebraic kinfolk are given, thereby allowing readers to use this to different teams of curiosity.

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**Example text**

6) h = 3N/ gcd(N, p). 7) +q−p , where we have used a p a q a r b = ba q a r a p , a p a q a r b2 = b2 a r a p a q . Note that the sum of the factors of a, a , and a is the same. Since the numbers of each of (p) (p) ( ) ( ,m,n) , CN 2 , and CN 2 are (N − 1), (N 3 − N )/3, N , and N , the classes C1 , C3 respectively, the total number of conjugacy classes is 1 1 + (N − 1) + N 3 − N /3 + N + N = N N 2 + 8 . 8) The number of irreducible representations can be determined by using the relations m1 + 4m2 + 9m3 + · · · = 3N 3 , 1 m1 + m2 + m3 + · · · = N N 2 + 8 .

27) References 1. 2. 3. 4. : J. Phys. Soc. Jpn. : J. Math. Phys. : J. High Energy Phys. 1103, 101 (2011). : Phys. Rev. D 79, 085005 (2009). 1057 [hep-ph] Chapter 6 DN In this chapter, we discuss the dihedral group, which is denoted by DN . It is the symmetry group of the regular polygon with N sides. This group is isomorphic to ZN Z2 and is also denoted by Δ(2N ). It consists of cyclic rotations ZN and reflections. That is, it is generated by two generators a and b, which act on the N edges xi (i = 1, .

On the other hand, the doublet 2 of S3 decomposes into two singlets of Z3 . Since χ2 (ab) = −1, the S3 doublet 2 decomposes into 11 and 12 of Z3 . , √ −1/2 − 3/2 ab = √ . , 11 : x1 − ix2 , 12 : x1 + ix2 . , {e, a}. It has two singlet representations 1k , for k = 0, 1, that is, a = (−1)k on 1k . Recall that χ1 (a) = 1 and χ1 (a) = −1 for 1 and 1 of S3 . Thus, 1 and 1 of S3 correspond to 10 and 11 of Z2 , respectively. 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 .