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

These lecture notes offer an academic assessment of non-Abelian discrete teams and convey a few functions to matters in physics the place discrete symmetries represent an incredible precept for version construction in particle physics. whereas Abelian discrete symmetries are frequently imposed for you to regulate 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 appealing selection for the flavour quarter: version developers have attempted to derive experimental values of quark and lepton plenty, and combining angles by means of 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 concept - e.g. the string idea 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 stories the group-theoretical points 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) whilst algebraic family are given, thereby permitting readers to use this to different teams of curiosity.

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

**Example text**

This implies that there are (3 + N/2) irreducible representations. 8) n for the mi , which also satisfy m1 + m2 + m3 + · · · = 3 + N/2. The solution is (m1 , m2 ) = (4, N/2 − 1), so there are four singlets and (N/2 − 1) doublets. 2 Characters and Representations We now turn to the characters and representations. The group QD2N has four singlets and (N/2 − 1) doublets. We denote the four singlets by 1ss , with s, s = ±. The characters of a and b are obtained as χ1±s (a) = ±1 for any s and χ1s± (b) = ±1 for any s.

For example, if k + k = 0 mod N , the doublet 2[k+k ] can be 72 8 QD2N reduced according to xk yk x(N/2−1)k y(N/2−1)k 2[k+k ] = (xk yk + x(N/2−1)k y(N/2−1)k )1++ ⊕ (xk yk − x(N/2−1)k y(N/2−1)k )1+− . 13) Similarly, when k + k = N/2 mod N , the doublet 2[k+k ] can be reduced according to xk yk x(N/2−1)k y(N/2−1)k 2[k+k ] = (xk yk + x(N/2−1)k y(N/2−1)k )1−+ ⊕ (xk yk − x(N/2−1)k y(N/2−1)k )1−− . 17) , 2[k] . 19) where si = si si (i = 1, 2). , QD16 . This group is generated by the 2π/8 rotation a and the reflection b.

Similarly, for a generic value of N , the element aa a commutes with all the elements. When we define a˜ = a N −1 a and a˜ = a a N −1 , the closed algebra generated by a, ˜ a˜ , and b corresponds to Δ(3N 2 ). For the case N/3 = integer, the element aa a is not included in Δ(3N 2 ). Thus, we find that this group is isomorphic to ZN × Δ(3N 2 ). On the other hand, when N/3 is an integer, the element (aa a )N k/3 with k = 0, 1, 2, is included in Δ(3N 2 ). That is, the group Σ(3N 3 ) cannot be ZN × Δ(3N 2 ).