Reference

Mathematics that travels with the result

Use the reference to identify the mathematical object, recover its conventions, and check the method behind a calculation. Classification, curated properties, executable operations, and independent review have separate coverage statements.

Mathematical orientation

Representation categories separates ordinary, modular, real, quaternionic, projective, and noncompact unitary problems. External resources explains what the major databases and software systems provide.

• Conventions and versioned convention profiles: coefficient fields, highest-weight ordering, root lengths, central charges, and specified homomorphisms.
• Notation: ambiguous group names and searches.
• Classification: finite simple families, complex Lie types, compact global forms, and the curated real-form index.

Formulas to use and check

Use the formula index to choose a method and verify its assumptions before substituting labels.

Finite-character formulas collect class-size weights, conjugation, power maps, tensor multiplicities, and the Frobenius–Schur indicator, with worked diagnostics for each. Lie invariant formulas fix dimensions, Casimirs, trace indices, and center characters. Group topology formulas state the hypotheses and maps behind covering and fibration calculations.

Methods for the research workbenches

• Operator enumeration: exhaustive multidegrees, charge selection, exact multiplicities, resource limits, and reproducible inventories.

• Local operator counting: graded characters, Lorentz and gauge projection, flavor multiplicities, and the precise meaning of a fixed-monomial count.

• Physical setups: scalar and Weyl fields, global gauge groups, strict schemas, and reproducible replay.

• Finite character methods: exact character fields, conjugacy classes, power maps, and decomposition.

• Weyl dimension formula: positive coroots, Dynkin labels, exceptional types, Casimirs, and global-form descent.

• The two SU(3) branching maps: regular SU(2) inclusion and the principal map with SO(3) image.

• Reproduce and cite a calculation: inputs, algorithms, conventions, sources, and downloadable records.

• Littlewood–Richardson method: bounded SU(N) products, determinant columns, and exact multiplicities.

• Block branching: the primitive U(1), finite kernel, and complete charged restriction.

• Product center lattices: embedded quotient generators, matter phases, continuous kernels, and fundamental groups.

Evidence and reusable data

• Sources and provenance: claim-level locators, derivations, and reverse dependencies.
• Coverage: exactly which operations and representation categories are supported.
• Downloads and data interface: catalogue snapshots, exact character tables, root systems, convention profiles, and method manifests.
• Data model: mathematical identities and typed relationships.
• Common questions.

Scientific review

This edition combines original exposition, AI-assisted source checking, and automated mathematical fixtures. Those activities do not supply independent human expert approval. The current review state is recorded by content fingerprint; see the method and review queue.