Non-Abelian Discrete Symmetries in Particle Physics

2010/01/01 by Hajime Ishimori, Tatsuo Kobayashi, Hiroshi Ohki +3 · 1 voice · 11 citations
Mathematics · Physics and Astronomy · #Abelian group #Algebra over a field #Black Holes and Theoretical Physics #Conjugacy class #Discrete group #Discrete symmetry #Geometry #Group (periodic table) #Homogeneous space #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Particle physics theoretical and experimental studies #Physics #Pure mathematics #Quantum mechanics #Sigma #Symmetry (geometry) #Symmetry group #Tensor (intrinsic definition) #Tensor product #Theoretical physics #hep-ph #hep-th

paper · pdf · doi:10.1143/ptps.183.1

published as Prog.Theor.Phys.Suppl.183:1-163,2010 · 179 pages, 8 figures, section 15 is changed, some references are added

openalex publication_date 2010/01/01 · arxiv published 2010/03/18 · arxiv created 2010/04/15 · arxiv updated 2010/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We review pedagogically non-Abelian discrete groups, which play an important role in particle physics. We show group-theoretical aspects for many concrete groups, such as representations and their tensor products. We explain how to derive, conjugacy classes, characters, representations, and tensor products for these groups (with a finite number). We discuss them explicitly for itSinitN, itAinitN, itT', itDntitN, itQinitN, itΣ(2itNsu2), Δ(3itNsu2), itTin7, Σ(3itNsu3), and Δ(6itNsu2), which have been applied for model building in particle physics. We also present typical flavor models by using itAin4, itSin4, and Δ(54) groups. Breaking patterns of discrete groups and decompositions of multiplets are important for applications of the non-Abelian discrete symmetry. We discuss these breaking patterns of the non-Abelian discrete group, which are a powerful tool for model buildings. We also review briefly anomalies of non-Abelian discrete symmetries by using the path integral approach.

Citations

• Lie Algebras in Particle Physics: From Isospin to Unified Theories

Cited by

• 125 GeV Higgs decay with lepton flavor violation in the μ ν SSM
• New mixing structures of chiral generations in a model with noncompact horizontal symmetry
• A unified model of quarks and leptons with a universal texture zero
• Ameliorating the popular lepton mixings with A4 symmetry: A seesaw model for realistic neutrino masses and mixing
• Neutrino CP violation and sign of baryon asymmetry in the minimal seesaw model
• Discrete flavour symmetries from the Heisenberg group
• A Further Study of the Frampton-Glashow-Yanagida Model for Neutrino Masses, Flavor Mixing and Baryon Number Asymmetry
• Common Origin of Fermion Mixing and GeometricalCPViolation, and Its Test Through Higgs Physics at the LHC
• Finite modular groups and lepton mixing
• A4flavor symmetry model for Dirac neutrinos and sizableUe3
• Diluting quark flavor hierarchies using dihedral symmetry

Discussions

• i spent a lot of time reading this, and have met all of the authors, which starts with the symmetric groups of 3 and 4 elements arxiv.org/abs/1003.3552 [bsky, 1 points, 0 comments]

Related

• Non-Abelian Discrete Flavor Symmetries
• Gauge extension of non-Abelian discrete flavor symmetry
• Non-Abelian Discrete Family Symmetries of Leptons and Quarks
• Discrete Symmetries in the MSSM
• A Scan for Models of Neutrino Mixing from Non-Abelian Discrete Symmetries