Scope and roadmap

Groups for Physics connects a finite reference catalogue to lessons, physical applications and exact calculations. Its first edition covers the questions described below. A classification entry, a curated property, an executable operation and an independent human review are separate promises; coverage records those distinctions in detail.

The connected reference

The catalogue contains 266 mathematical records, including finite and Lie groups, algebras, selected real forms, specified maps, spaces and 62 representation records. Its 2,998 claims retain source or derivation information, supported by 114 source records. Selected representation records illustrate the category and conventions; they are not a finite list of all Lie-group irreducibles.

Three learning paths connect this data to worked questions:

• Finite symmetries: characters, tensor and symmetric powers, reality types, and scalar invariants, with S3, Q8 and A4 examples.
• Lie groups and topology: roots and weights, real and global forms, specified branching maps, and the distinct topology of groups, vacuum orbits and classifying spaces.
• Spacetime fields and states: Weyl bilinears, particle polarizations, conformal primary states and AdS scalar modes, with covers and boundary conditions retained.

The solved exercises and formula reference provide shorter routes to a calculation. Physics guides connect representations to gauge groups, matter, symmetry breaking and interactions without treating a necessary selection rule as proof that an invariant or stable solution exists.

The calculation domain

Fifteen browser-local workbenches implement bounded operations:

Question	Implemented scope
Finite characters and invariant powers	Seven ordinary complex character tables, plus the original A4 scalar example
Compact Lie representations	Selected simple-type Weyl dimensions and invariants; bounded SU(2), SU(3) and SU(N) tensor products
Branching	The two specified SU(3) maps and primitive SU(N) block maps with abelian charges
Global gauge groups	Matter descent for supported single and product groups, retaining embedded central kernels
Local fields	Fixed derivative-free scalar/Weyl monomials with Lorentz, gauge, flavor and exchange statistics
Operator inventories	All admitted monomials through a selected canonical dimension at most six, with optional abelian selection rules
Elementary topology	Two specified sphere fibrations with their subgroup and boundary maps

Every interface states its numerical limits. Eleven fixed-monomial physical setups and six field inventories retain reproducible examples, including Standard Model sectors. Exact counts concern complex invariant-polynomial spaces before imposing Hermiticity or identifying couplings under flavor-basis changes. Derivatives, gauge-field strengths and a complete EFT basis are outside the current operator domain.

Use coverage for precise ranges and failure contracts, and reproducibility for the supported exports, imports and calculation links. The website is static and the calculators run locally; it is not a remote GAP, SageMath or general computer-algebra service.

Evidence and version history

Scientific source checking, exact mathematical fixtures and software verification have specific scopes. The local verification recorded on 2026-11-09 executed 39 generated scripts with SageMath 9.7 and 12 generated standard-library Python scripts. The Sage fixtures cover all 32 supported root systems, rank-one cases, a large exact E8 dimension and three SU(3) tensor products. Ninety-three GAP fixtures were generated and source-checked; native GAP execution remains unverified. These checks concern those fixtures, not arbitrary future calculations or a live external backend.

Independent human scientific review remains a separate recorded status. The current review queue has zero records with completed independent expert review. Sources, AI-assisted cross-checks and passing tests do not create that approval. The editorial method explains how a review is tied to the content actually inspected.

The numbered edition records preserve earlier internal software and dataset milestones. Their dates and counts are historical; they do not establish a public deployment. Application, dataset, schema and algorithm versions remain separate so that a saved calculation can identify its mathematical inputs and implementation. The technical blueprint snapshot preserves the broader design history.

Later mathematical and computational extensions

The next stages widen coverage only when their conventions, examples and verification can be retained with the result:

• Representation theory: more specified embeddings and branching maps, modular and projective categories, explicit matrices and intertwining tensors, and carefully delimited noncompact unitary representations.
• Topology: more group manifolds and homogeneous spaces, central extensions, and bundle calculations with explicit maps and base-space hypotheses.
• Physical setups: several gauge U(1) factors, wider charge lattices, other spacetime dimensions, and additional field representations.
• Operator bases: derivatives and field strengths, followed by equations of motion and integration by parts. Explicit normalized contractions, Hermiticity and flavor-basis identifications need their own conventions.
• Reproducible research: broader versioned GAP/SageMath comparisons, notebook interchange, and contributor workflows that preserve provenance.

These directions are a roadmap, not implemented calculators or promised complete classifications. The external-resource guide helps identify an appropriate database or system for work beyond the present scope. Original articles use CC BY 4.0; original code uses MIT, with third-party terms retained separately.