Learn

Choose one of the three learning paths or test your understanding with the solved exercises. Each path connects a physical question to definitions, exact examples, atlas records, and bounded calculators.

Start with the object

• Groups, actions, and representations.
• Characters and reality types.
• Lie algebras and global groups.
• How to read a record.

Build the mathematical language

Begin with irreducible representations to distinguish ordinary complex representations, reality types, and noncompact categories. For finite symmetries, continue to characters and tensor products. For Lie groups, read roots, weights, and Dynkin labels, then covers and global forms to determine which labeled multiplets belong to the group you intend to use.

Follow the Lie and topology path

The Lie and topology path moves from roots and weights through real forms, global groups, specified branching maps, and the separate spaces G, G/H, and BG. Solve the global-form exercises to distinguish intermediate quotients, embedded product kernels, and sphere-bundle obstructions.

Follow the finite-symmetry path

The finite-symmetry path takes an S₃ doublet from a group action to its character, tensor powers, reality type, and scalar invariants. Solve the character exercise and use finite-character formulas as a calculation reference. The A₄ scalar model then turns invariant counts into explicit interactions and conditional residual symmetries.

Work through a calculation

• From fields to interactions: enumerate every allowed monomial through a dimension bound, impose charge selections, and interpret exact multiplicities.

• Standard Model operator sectors: reconcile the one- and three-generation counts with published operator classifications.

• Fields, Lorentz indices, and statistics: use the complete single-field representation to count local polynomial invariants.

• The Weinberg operator: derive the dimension-five field structure and symmetric flavor coefficients.

• Yukawa and four-fermion interactions: verify chirality, charges, and flavor counting in worked Standard Model sectors.

• Finite tensor products, with statistics: derive multiplicities from characters and distinguish tensor, symmetric, and exterior powers.

• Highest weights in practice: identify a representation from Dynkin labels, calculate its dimension and Casimir, and test global-form descent.

• The SU(3) tensor-product dictionary: keep conjugate irreps and repeated multiplets distinct in familiar color and flavor examples.

• SU(N) tensors for model building: SU(4), SU(5), and SU(6), repeated summands, and the limits of invariant counting.

• SU(5) to Standard Model multiplets: charge normalization and the embedded subgroup.

• The Standard Model global group: four matter-compatible global forms and the choices they leave open.

Each guide connects a derivation to a supported workbench and names the assumptions needed for a physical interpretation. The finite character reference, Weyl formula, and SU(3) branching maps provide the detailed mathematical contracts.

Follow the spacetime path

The spacetime and fields path distinguishes field components, particle states, conformal primaries and AdS modes. It connects spinor conventions, CFT unitarity, and AdS mass and scaling dimension to the curated real groups and covers.

Continue by a physics question

The physical setup library links every fixed-sector and inventory preset. The Standard Model walkthrough connects fields, quotients, statistics and exact counts. Which global forms admit the matter? derives the common-center-kernel criterion for SU(N). Further paths cover global gauge structure, invariant interactions, branching maps, spacetime representations, and topological defects.

Use the classification reference and atlas collections for systematic navigation. Read conventions when comparing two sources or importing an answer into a calculation.