Group theory for unified model building

The results gathered here on simple Lie algebras have been selected with attention to the needs of unified model builders who study Yang-Mills theories based on simple, local-symmetry groups that contain as a subgroup the SU^w₂ × U^w₁ × SU^c₃ symmetry of the standard theory of electromagnetic, weak, and strong interactions. The major topics include, after a brief review of the standard model and its unification into a simple group, the use of Dynkin diagrams to analyze the structure of the group generators and to keep track of the weights (quantum numbers) of the representation vectors; an analysis of the subgroup structure of simple groups, including explicit coordinatizations of the projections in weight space; lists of representations, tensor products and branching rules for a number of simple groups; and other details about groups and their representations that are often helpful for surveying unified models, including vector-coupling coefficient calculations. Tabulations of representations, tensor products, and branching rules for E ₆, SO ₁₀, SU ₆, F ₄, SO ₉, SU ₅, SO ₈, SO ₇, SU ₄, E ₇, E ₈, SU ₈, SO ₁₈, SO ₂₂, and for completeness, SU ₃ are included. (These tables may have other applications.) Group-theoretical techniques for analyzing symmetry breaking are described in detail and many examples are reviewed, including explicit parameterizations of mass matrices.