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

These lecture notes supply an academic assessment of non-Abelian discrete teams and convey a few functions to matters in physics the place discrete symmetries represent an enormous precept for version development in particle physics. whereas Abelian discrete symmetries are frequently imposed to be able to keep watch over couplings for particle physics - specifically version development past the normal version - non-Abelian discrete symmetries were utilized to appreciate the three-generation taste constitution particularly.

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 through assuming non-Abelian discrete taste symmetries of quarks and leptons, but, lepton blending has already been intensively mentioned during this context, to boot. 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 concept - e.g. the string concept or compactification through orbifolding – thereby supplying a potential bridge among the underlying thought and the corresponding low-energy quarter of particle physics.

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

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1 Characters of QD2N representations, where ρ = e2π i/N . Note that ρ mk + ρ mk(N/2−1) = 2 cos(2πmk/N) when mk is even, and ρ mk + ρ mk(N/2−1) = 2i sin(2πmk/N) when mk is odd h χ1++ χ1−+ χ1−− χ1+− χ2[k] C1 1 1 1 1 1 2 C2[m] N/gcd(N, m) 1 (−1)m (−1)m 1 ρ mk + ρ mk(N/2−1) =1 (−1)N/2 =1 −2 C1 2 1 (−1)N/2 CN/2 2 1 1 −1 −1 0 CN/2 4 1 −1 1 −1 0 1 on the doublet 2[k] . 10) , where each of the up and down components xk and x(N/2−1)k , respectively, has definite ZN charge. 1. 3 Tensor Products We now discuss the tensor products of representations of the QD2N group.

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 . 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.

Thus, both 1 and 1 of S3 correspond to 10 of Z3 . 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 .