Learning paths

Choose a path by the question you need to answer. Each route orders the lessons, carries a concrete example forward, and ends with a calculation or category check. You can begin with the prerequisites listed below or join at a later step when the earlier distinctions are already familiar.

If you are new to the site, start with reading the atlas. It explains how group, algebra, representation, map, and space records fit together. The atlas collections provide thematic groups of records alongside these lesson sequences.

Path	Starting knowledge	Main calculation	What you should be able to explain at the end
Finite symmetries	Linear algebra and tensor products	Decompose an $S_3$ doublet square and compare $Q_8$	Why a tensor singlet can disappear for identical commuting fields
Lie groups and topology	Linear algebra, tensor products, and Lie-algebra basics	Compare $SU(2)$ and $SO(3)$ representations and sphere orbits	Why algebra, global group, subgroup map, orbit, and bundle topology give different answers
Spacetime fields and states	Basic Lorentz symmetry, quantum fields, and tensor products	Form a Weyl bilinear and count massive-vector polarizations	Why field components, particle states, and conformal multiplets use different representation data

Finite symmetries: from a table to a coupling

Begin the finite-symmetry path. It starts with $S_3$ acting on the plane $x_1+x_2+x_3=0$, obtains its character from the permutation action, and uses character orthogonality to reconstruct tensor products. The next steps add reality type and the symmetric-power projection appropriate to one commuting scalar multiplet.

The final exercise compares the real $S_3$ doublet with the quaternionic $Q_8$ doublet. Your output is a decomposition checked on every conjugacy class, a reality indicator using the square map, and an explicit allowed or vanishing scalar contraction. This route reaches a finite-character calculation without requiring EFT operator counting.

If you already know character orthogonality, join at finite tensor products or go directly to the worked exercise. The calculations use ordinary complex representations in characteristic zero; modular and projective categories require different inputs.

Lie groups: from infinitesimal data to topology

Begin the Lie-group and topology path. It separates real forms from global forms, turns highest weights into dimensions and Casimirs, and then tests descent through a central quotient. Branching introduces a second essential input: the actual subgroup homomorphism.

The route ends by comparing two sphere fibrations. You should distinguish the highest-weight parity allowed by $SO(3)$, the subgroup winding associated with a sphere in $G/H$, and the isomorphism class of a principal $G$-bundle on that sphere. The global-form exercises add an intermediate $SU(4)$ quotient and differently embedded product kernels.

For a research question about allowed matter, join at covers and global forms. For an order-parameter or defect question, join at groups, cosets, and classifying spaces. Highest-weight dimensions and elementary homotopy calculations do not constitute a classification of noncompact unitary representations or a stability analysis.

Spacetime symmetry: from indices to physical states

Begin the spacetime path. The first steps specify Lorentz chirality, gauge representation, flavor, and statistics before forming a local scalar. A gauge-singlet Weyl mass bilinear supplies the simplest check of the fermion exchange sign. The route then changes categories to the massive particle little group and compares a vector field’s four covariant components with its three physical polarizations.

The later steps lead to the conformal and AdS guides, where scaling dimension, null relations, spacetime covers, and boundary conditions become additional data. They build on the field-versus-state distinction; the earlier bilinear calculation stands on its own. The path states its metric convention, and the linked spinor guide states its gamma-matrix convention.

If your immediate task is an interaction, join at fields, Lorentz symmetry, and statistics, then use from fields to interactions and the local-operator workbench. Its supported calculation is a bounded derivative-free invariant count, not a complete EFT basis or a count of physical parameters after field redefinitions.

Keep a result you can check

For any route, record the object and category, the conventions or chosen map, and one independent check before consulting an output. The exercise index collects the two worked problem sets. The formula index gives shorter routes back to a definition, and reproducibility explains what to retain with a calculation link or export.