3.0.0 — Research workbench I

Research workbench I turns the reference foundation into a broader calculation workspace. Five new exact workbenches connect group identities, conventions, source evidence, and reproducible results. This delivers a defined subset of Phase C; the full research-workbench roadmap remains unfinished.

Release field	Value
Date	2026-11-08
Application	3.0.0
Dataset	2026.11-workbench.1
Schema	2.0.0
Release algorithm umbrella	groups-workbench-1.0.0

The catalogue now contains 256 mathematical records, 2,886 claims, and 99 sources. The 194 foundation records are preserved. The 62 added representation records comprise all 30 ordinary complex irreducible characters across the seven executable finite groups, plus 32 selected compact Lie representations. They provide concrete entry points into the calculations; they do not enumerate the infinitely many dominant integral weights of a compact simple type.

Five new workbenches

Workbench	Released domain
Finite characters and tensor products	C₃, S₃, D₈, Q₈, A₄, S₄, A₅; ordinary complex representations, base dimension ≤60, each multiplicity ≤12; tensor products and tensor/symmetric/exterior powers of degree 0–12
Highest weights and Weyl dimensions	A₁–A₈, B₂–B₈, C₂–C₈, D₄–D₈, E₆, E₇, E₈, F₄, G₂; each Dynkin label 0–30
SU(3) tensor products	Each input Dynkin label 0–6; complete decomposition, retaining conjugates and multiplicities
SU(3) branching by a chosen map	Input labels 0–6; complete restriction along the regular or principal SU(2) map
Matter and global gauge structure	One SU(N) factor, 2≤N≤12; at most 16 multiplets, labels 0–30; all central quotients compatible with their action

The five original calculators retain their original algorithms and limits, bringing the total to ten. Classification and reference coverage remain broader than any individual tool’s domain.

Mathematics that remains exact

The finite engine keeps algebraic character values exact, records the class power maps, and decomposes products or powers by character inner products. The Weyl engine generates positive coroots from the declared Cartan matrix and evaluates the dimension product with arbitrary-precision integers. Quadratic Casimirs and trace indices are reduced exact rationals in the long-root-length-two convention. Conjugation, reality type, and full-center adjoint descent are reported separately.

SU(3) tensors and restriction use complete Gelfand–Tsetlin weight characters. For example,

$$8\otimes8=1\oplus8\oplus8\oplus10\oplus\overline{10}\oplus27.$$

Both octets remain visible as multiplicity two, and the two ten-dimensional irreps remain distinct. The worked tensor guide explains the labels and physical interpretation. For branching, the principal SU(2) map has kernel $\{\pm I\}$ and image $SO(3)$; every restricted summand has integer spin. The same SU(3) sextet restricts as $3\oplus2\oplus1$ under the regular map and $5\oplus1$ under the principal map. The map reference derives these results and the specified torus actions.

The matter workbench computes the common central kernel by the gcd of $N$ and the matter’s $N$-alities, then lists all subgroup orders that can be divided out. A two-index antisymmetric $6$ of SU(4), for example, admits the quotient $SU(4)/\mathbb Z_2$ but obstructs the full-center quotient. This condition is kinematic: it does not choose line operators, discrete theta data, anomalies, or a vacuum. See matter and global form.

KaTeX and mathematical conventions

KaTeX renders the mathematical guides and the new calculation displays, including exact character entries, Cartan matrices, rational Casimirs, decompositions, and map formulas. Six versioned convention profiles make the coefficient field, root numbering, Casimir scale, center generator, and selected homomorphism part of the result.

The highest-weight tool integrates to the simply connected compact group by default. Its adjoint-descent test concerns the full center; a representation that fails it can still descend to an intermediate quotient. The profile and the Weyl formula reference explain that distinction and the non-simply-laced normalization.

Reuse and reproduction

The downloads now include all seven exact character tables, 32 root-system profiles with positive coroot coordinates, the six convention profiles, and a workbench-method manifest. JSON exports preserve exact values, normalized inputs, source references, and specific mathematical versions:

Implementation	Version
Original five calculators	groups-design-1.0.0
Finite character calculations	finite-characters-1.0.0
Weyl dimensions and SU(3) tensors/branching	lie-representations-1.0.0
SU(N) matter kernel	su-n-matter-kernel-1.0.0

Four of the new tools also generate verification scripts for an independent environment: GAP for finite characters, SageMath for Weyl invariants and SU(3) tensor products, and standard-library Python for matter quotients. The finite script verifies the serialized table and its operations; it does not independently identify a group or assume an upstream class ordering. There is no branching-script export in this edition.

The site executes its own local JavaScript engines and does not invoke GAP or SageMath. Python verification scripts have been executed in development; GAP and SageMath are unavailable in that environment, so their scripts have source serialization or syntax checks but runtime execution remains unverified. Read reproducibility for the distinction between a link that replays inputs and an export that preserves a specific result.

New reading paths

• Finite tensor products, with statistics.
• Highest weights in practice.
• The SU(3) tensor-product dictionary.
• Matter and global form.
• Finite character methods.
• Weyl dimension formula.
• SU(3) branching maps.

The methods were checked against primary materials including the GAP Reference Manual, Sage’s Weyl-character documentation, Etingof’s Lie theory notes, Conrad’s center tables, and the papers and precise locators on the individual method pages. The prose and implementations are original; source links do not claim an upstream endorsement or a blanket data-reuse license.

Verification and independent review

The aggregate automated suite passes 86 verification sections plus catalogue validation. The new finite and Lie engines have 24 mathematical sections, supplemented by six matter-quotient sections, six KaTeX-rendering sections, reproduction-script checks, and integration checks. Fixtures cover exact orthogonality and character reconstruction, exceptional fundamental and adjoint dimensions, non-simply-laced conventions, spinor reality, dimensions beyond machine integer precision, full-center descent, SU(3) tensor identities, and both branching maps.

Source checking, automated mathematics, and browser validation do not constitute independent human scientific review. Named mathematics and physics approvals remain pending and are tracked against current content fingerprints in the review queue. The deployment retains its private review audience; this release does not establish the public groups.qft.org domain.

Remaining steps

1. Complete independent mathematics/physics review, comprehensive accessibility assessment, final reuse-license decisions, and a staffed correction/release process; then complete the custom-domain and public-audience transition.
2. Finish Phase C through reviewed external adapters, wider compact-group tensors, more explicit maps and charge-refined branching, product/abelian gauge factors, explicit invariant bases, and reusable physical setups.
3. Extend internal invariant counts to complete field/Lorentz/derivative-aware operator workflows with declared equations-of-motion and integration-by-parts treatment.
4. Develop specialist topology, noncompact representation categories, central extensions, finite/continuous bridges, and carefully scoped anomaly questions under the Phase D review requirements.
5. Build notebook interoperability, collaborative contribution, and shared research projects in Phase E after the core workflows are dependable.

The roadmap and blueprint retain the complete specification and distinguish these remaining steps from released capabilities.