The released catalogue and its capabilities
Know what is available.
Mathematical breadth, curated properties, executable calculations, and expert review are separate dimensions of coverage.
55classification records112reference records99executable records15bounded workbenches
A classification entry does not imply a character table or explicit matrix representation. A linked workbench supports only the named groups, representations, and input ranges.
Executable capabilities
| Workbench | Supported domain | Version |
|---|---|---|
| Global forms & representations | SU(2), SO(3), SU(3), PSU(3); each highest-weight input 0–30. Complete dimension-bounded browser: A1 up to 30, A2 up to 100. | groups-design-1.0.0 |
| Tensor products | Two SU(2) highest-weight inputs 0–30; output highest weights can reach 60. | groups-design-1.0.0 |
| Invariant couplings | The A4 real triplet, tensor or symmetric powers, degrees 0–12. No fermion or derivative operator engine. | groups-design-1.0.0 |
| Branching & embeddings | Only SU(3) representations 3 and 8 along the named regular and principal maps. | groups-design-1.0.0 |
| Symmetry breaking & topology | Only the specified SU(2)/U(1) and SO(3)/SO(2) fibrations with declared angular conventions. | groups-design-1.0.0 |
| Finite characters & tensor products | Seven curated groups; base representations of complex dimension ≤60; multiplicities ≤12; powers of degree 0–12. | finite-characters-1.0.0 |
| Highest weights & Weyl dimensions | A₁–A₈, B₂–B₈, C₂–C₈, D₄–D₈, E₆, E₇, E₈, F₄, G₂; Dynkin labels 0–30. No noncompact unitary-dual calculation. | lie-representations-1.0.0 |
| SU(3) tensor products | Each Dynkin label of either input is 0–6. Complete decomposition within that input domain; output labels may be larger. | lie-representations-1.0.0 |
| SU(3) branching by a chosen map | SU(3) labels p,q from 0–6. Complete SU(2) decomposition; the principal map has kernel Z₂. | lie-representations-1.0.0 |
| Matter & global gauge structure | 2≤N≤12; at most 16 multiplets; each Dynkin label 0–30. Kinematic descent only. | su-n-matter-kernel-1.0.0 |
| SU(N) tensor products | Each Dynkin label 0–12; at most 16 canonical Young boxes across the two inputs. Complete decomposition within the supported domain. | su-n-tensors-1.0.0 |
| Branching with U(1) charges | At most 12 canonical Young boxes; Dynkin labels 0–12. Primitive U(1) normalization and the finite map kernel are explicit. | su-n-block-branching-1.0.0 |
| Product gauge groups & matter | At most 16 multiplets; Dynkin labels 0–12; integer U(1) charges −24…24; at most 3 quotient generators with phase denominators ≤60; generated subgroup order ≤4096. | product-gauge-1.0.0 |
| Local operators & field statistics | At most 16 species, 3 flavors per species, 6 field insertions, and canonical dimension 6. Color labels p+q≤2, weak label 0–2, integer charges −24…24. Exact completion or an explicit resource-limit error. | local-operators-1.0.0 |
| Enumerate allowed interactions | At most 16 species, 3 flavors per species, canonical dimension 6, and 3 additional charge rules. Color labels p+q≤2, weak label 0–2, integer gauge and selection charges −24…24. Explicit resource bounds; complete results or an error. | operator-enumeration-1.0.0 |
Record-level matrix
| Record | Type | Coverage | Representation summary | Calculations |
|---|---|---|---|---|
| SU(2) | Group | executable | See stated category and scope | global-formstensor-productsbranchingtopologylie-representationsmatter-quotientssun-tensor-products |
| SO(3) | Group | executable | See stated category and scope | global-formstopology |
| SU(3) | Group | executable | See stated category and scope | global-formsbranchinglie-representationsmatter-quotientssu3-tensor-productssu3-branchingsun-tensor-productscharged-branching |
| PSU(3) | Group | executable | See stated category and scope | global-forms |
| U(1) | Group | reference | See stated category and scope | None attached |
| Spin(4) | Group | reference | See stated category and scope | None attached |
| SL(2,ℂ) | Group | reference | See stated category and scope | None attached |
| C₃ | Group | executable | Complete ordinary complex dimensions/types | finite-characters |
| S₃ | Group | executable | Complete ordinary complex dimensions/types | finite-characters |
| A₄ | Group | executable | Complete ordinary complex dimensions/types | invariantsfinite-characters |
| S₄ | Group | executable | Complete ordinary complex dimensions/types | finite-characters |
| A₅ | Group | executable | Complete ordinary complex dimensions/types | finite-characters |
| Q₈ | Group | executable | Complete ordinary complex dimensions/types | finite-characters |
| D₈ | Group | executable | Complete ordinary complex dimensions/types | finite-characters |
| Cₚ | Family | classification | See stated category and scope | None attached |
| Aₙ | Family | classification | See stated category and scope | None attached |
| PSL(n,q) | Family | classification | See stated category and scope | None attached |
| PSU(n,q) | Family | classification | See stated category and scope | None attached |
| PSp(2n,q) | Family | classification | See stated category and scope | None attached |
| PΩ(2n+1,q) | Family | classification | See stated category and scope | None attached |
| PΩ⁺(2n,q) | Family | classification | See stated category and scope | None attached |
| PΩ⁻(2n,q) | Family | classification | See stated category and scope | None attached |
| G₂(q) | Family | classification | See stated category and scope | None attached |
| F₄(q) | Family | classification | See stated category and scope | None attached |
| E₆(q) | Family | classification | See stated category and scope | None attached |
| E₇(q) | Family | classification | See stated category and scope | None attached |
| E₈(q) | Family | classification | See stated category and scope | None attached |
| ²E₆(q) | Family | classification | See stated category and scope | None attached |
| ³D₄(q) | Family | classification | See stated category and scope | None attached |
| ²B₂(q) | Family | classification | See stated category and scope | None attached |
| ²G₂(q) | Family | classification | See stated category and scope | None attached |
| ²F₄(q) | Family | classification | See stated category and scope | None attached |
| Sporadic | Family | classification | See stated category and scope | None attached |
| Cₙ | Family | reference | See stated category and scope | None attached |
| D₂ₙ | Family | reference | See stated category and scope | None attached |
| Dicₙ | Family | reference | See stated category and scope | None attached |
| 2T · 2O · 2I | Family | reference | See stated category and scope | None attached |
| Sₙ | Family | reference | See stated category and scope | None attached |
| C2 | Group | reference | Complete ordinary complex dimensions/types | None attached |
| C4 | Group | reference | Complete ordinary complex dimensions/types | None attached |
| C5 | Group | reference | Complete ordinary complex dimensions/types | None attached |
| C6 | Group | reference | Complete ordinary complex dimensions/types | None attached |
| V₄ | Group | reference | Complete ordinary complex dimensions/types | None attached |
| D₁₀ | Group | reference | Complete ordinary complex dimensions/types | None attached |
| D₁₂ | Group | reference | Complete ordinary complex dimensions/types | None attached |
| Dic₃ | Group | reference | Complete ordinary complex dimensions/types | None attached |
| Q₁₆ | Group | reference | Complete ordinary complex dimensions/types | None attached |
| 2T | Group | reference | Complete ordinary complex dimensions/types | None attached |
| 2O | Group | reference | Complete ordinary complex dimensions/types | None attached |
| 2I | Group | reference | Complete ordinary complex dimensions/types | None attached |
| PSL(2,7) | Group | reference | Complete ordinary complex dimensions/types | None attached |
| Δ(27) | Group | reference | Complete ordinary complex dimensions/types | None attached |
| A₆ | Group | reference | Complete ordinary complex dimensions/types | None attached |
| M₁₁ | Group | classification | Not curated | None attached |
| M₁₂ | Group | classification | Not curated | None attached |
| M₂₂ | Group | classification | Not curated | None attached |
| M₂₃ | Group | classification | Not curated | None attached |
| M₂₄ | Group | classification | Not curated | None attached |
| J₁ | Group | classification | Not curated | None attached |
| J₂ | Group | classification | Not curated | None attached |
| J₃ | Group | classification | Not curated | None attached |
| J₄ | Group | classification | Not curated | None attached |
| Co₁ | Group | classification | Not curated | None attached |
| Co₂ | Group | classification | Not curated | None attached |
| Co₃ | Group | classification | Not curated | None attached |
| Fi₂₂ | Group | classification | Not curated | None attached |
| Fi₂₃ | Group | classification | Not curated | None attached |
| Fi₂₄′ | Group | classification | Not curated | None attached |
| HS | Group | classification | Not curated | None attached |
| McL | Group | classification | Not curated | None attached |
| He | Group | classification | Not curated | None attached |
| Ru | Group | classification | Not curated | None attached |
| Suz | Group | classification | Not curated | None attached |
| O’N | Group | classification | Not curated | None attached |
| HN | Group | classification | Not curated | None attached |
| Ly | Group | classification | Not curated | None attached |
| Th | Group | classification | Not curated | None attached |
| B | Group | classification | Not curated | None attached |
| M | Group | classification | Not curated | None attached |
| ²F₄(2)′ | Group | classification | Not curated | None attached |
| Aᵣ | Family | classification | See stated category and scope | None attached |
| Bᵣ | Family | classification | See stated category and scope | None attached |
| Cᵣ | Family | classification | See stated category and scope | None attached |
| Dᵣ | Family | classification | See stated category and scope | None attached |
| E₆ | Family | classification | See stated category and scope | None attached |
| E₇ | Family | classification | See stated category and scope | None attached |
| E₈ | Family | classification | See stated category and scope | None attached |
| F₄ | Family | classification | See stated category and scope | None attached |
| G₂ | Family | classification | See stated category and scope | None attached |
| sl(2, C) | Lie algebra | reference | See stated category and scope | None attached |
| sl(3, C) | Lie algebra | reference | See stated category and scope | None attached |
| sl(4, C) | Lie algebra | reference | See stated category and scope | None attached |
| sl(5, C) | Lie algebra | reference | See stated category and scope | None attached |
| so(5, C) ≅ sp(4, C) | Lie algebra | reference | See stated category and scope | None attached |
| sp(6, C) | Lie algebra | reference | See stated category and scope | None attached |
| so(8, C) | Lie algebra | reference | See stated category and scope | None attached |
| so(10, C) | Lie algebra | reference | See stated category and scope | None attached |
| e₆(C) | Lie algebra | reference | See stated category and scope | None attached |
| e₇(C) | Lie algebra | reference | See stated category and scope | None attached |
| e₈(C) | Lie algebra | reference | See stated category and scope | None attached |
| f₄(C) | Lie algebra | reference | See stated category and scope | None attached |
| g₂(C) | Lie algebra | reference | See stated category and scope | None attached |
| sl(2, C) ⊕ sl(2, C) | Lie algebra | reference | See stated category and scope | None attached |
| C (zero bracket) | Lie algebra | reference | See stated category and scope | None attached |
| sl(2, C) ⊕ C | Lie algebra | reference | See stated category and scope | None attached |
| su(2) | Lie algebra | reference | See stated category and scope | None attached |
| su(3) | Lie algebra | reference | See stated category and scope | None attached |
| su(4) ≅ so(6) | Lie algebra | reference | See stated category and scope | None attached |
| su(5) | Lie algebra | reference | See stated category and scope | None attached |
| sp(2) ≅ so(5) | Lie algebra | reference | See stated category and scope | None attached |
| sp(3) | Lie algebra | reference | See stated category and scope | None attached |
| so(8) | Lie algebra | reference | See stated category and scope | None attached |
| so(10) | Lie algebra | reference | See stated category and scope | None attached |
| e₆ compact | Lie algebra | reference | See stated category and scope | None attached |
| e₇ compact | Lie algebra | reference | See stated category and scope | None attached |
| e₈ compact | Lie algebra | reference | See stated category and scope | None attached |
| f₄ compact | Lie algebra | reference | See stated category and scope | None attached |
| g₂ compact | Lie algebra | reference | See stated category and scope | None attached |
| su(2) ⊕ su(2) | Lie algebra | reference | See stated category and scope | None attached |
| u(1) | Lie algebra | reference | See stated category and scope | None attached |
| u(2) | Lie algebra | reference | See stated category and scope | None attached |
| sl(2, R) | Lie algebra | reference | See stated category and scope | None attached |
| sl(3, R) | Lie algebra | reference | See stated category and scope | None attached |
| su(2,2) ≅ so(2,4) | Lie algebra | reference | See stated category and scope | None attached |
| sp(4, R) ≅ so(2,3) | Lie algebra | reference | See stated category and scope | None attached |
| sl(2, C) viewed over R | Lie algebra | reference | See stated category and scope | None attached |
| su(2) | Real form | reference | See stated category and scope | None attached |
| su(3) | Real form | reference | See stated category and scope | None attached |
| su(4) ≅ so(6) | Real form | reference | See stated category and scope | None attached |
| su(5) | Real form | reference | See stated category and scope | None attached |
| sp(2) ≅ so(5) | Real form | reference | See stated category and scope | None attached |
| sp(3) | Real form | reference | See stated category and scope | None attached |
| so(8) | Real form | reference | See stated category and scope | None attached |
| so(10) | Real form | reference | See stated category and scope | None attached |
| e₆ compact | Real form | reference | See stated category and scope | None attached |
| e₇ compact | Real form | reference | See stated category and scope | None attached |
| e₈ compact | Real form | reference | See stated category and scope | None attached |
| f₄ compact | Real form | reference | See stated category and scope | None attached |
| g₂ compact | Real form | reference | See stated category and scope | None attached |
| su(2) ⊕ su(2) | Real form | reference | See stated category and scope | None attached |
| sl(2, R) | Real form | reference | See stated category and scope | None attached |
| sl(3, R) | Real form | reference | See stated category and scope | None attached |
| su(2,2) ≅ so(2,4) | Real form | reference | See stated category and scope | None attached |
| sp(4, R) ≅ so(2,3) | Real form | reference | See stated category and scope | None attached |
| sl(2, C) viewed over R | Real form | reference | See stated category and scope | None attached |
| SU(4) | Group | executable | See stated category and scope | lie-representationsmatter-quotientssun-tensor-productscharged-branching |
| PSU(4) | Group | reference | See stated category and scope | None attached |
| SU(5) | Group | executable | See stated category and scope | lie-representationsmatter-quotientssun-tensor-productscharged-branching |
| U(2) | Group | reference | See stated category and scope | None attached |
| SO(4) | Group | reference | See stated category and scope | None attached |
| SO(5) | Group | reference | See stated category and scope | None attached |
| SO(6) | Group | reference | See stated category and scope | None attached |
| Sp(2) | Group | executable | See stated category and scope | lie-representations |
| Sp(3) | Group | reference | See stated category and scope | None attached |
| Spin(8) | Group | executable | See stated category and scope | lie-representations |
| SO(8) | Group | reference | See stated category and scope | None attached |
| PSO(8) | Group | reference | See stated category and scope | None attached |
| Spin(10) | Group | executable | See stated category and scope | lie-representations |
| SO(10) | Group | reference | See stated category and scope | None attached |
| G₂ compact | Group | executable | See stated category and scope | lie-representations |
| F₄ compact | Group | executable | See stated category and scope | lie-representations |
| E₆ compact, simply connected | Group | executable | See stated category and scope | lie-representations |
| E₇ compact, simply connected | Group | executable | See stated category and scope | lie-representations |
| E₈ compact | Group | executable | See stated category and scope | lie-representations |
| E₆ compact, adjoint | Group | reference | See stated category and scope | None attached |
| E₇ compact, adjoint | Group | reference | See stated category and scope | None attached |
| SL(2, R) | Group | reference | See stated category and scope | None attached |
| PSL(2, R) | Group | reference | See stated category and scope | None attached |
| Universal cover of SL(2, R) | Group | reference | See stated category and scope | None attached |
| SL(3, R) | Group | reference | See stated category and scope | None attached |
| SU(2,2) | Group | reference | See stated category and scope | None attached |
| Sp(4, R) | Group | reference | See stated category and scope | None attached |
| SO₀(1,3) | Group | reference | See stated category and scope | None attached |
| SO₀(2,4) | Group | reference | See stated category and scope | None attached |
| SO₀(2,3) | Group | reference | See stated category and scope | None attached |
| SU(2) → SO(3) | Map | executable | See stated category and scope | global-forms |
| SU(3) → PSU(3) | Map | executable | See stated category and scope | global-forms |
| SU(2) ↪ SU(3) | Map | executable | See stated category and scope | branchingsu3-branching |
| SU(2) → SU(3), spin 1 | Map | executable | See stated category and scope | branchingsu3-branching |
| U(1) ↪ SU(2) | Map | executable | See stated category and scope | topology |
| U(1) ≅ SO(2) ↪ SO(3) | Map | executable | See stated category and scope | topology |
| Spin(4) → SO(4) | Map | reference | See stated category and scope | None attached |
| SU(4) → SO(6) | Map | reference | See stated category and scope | None attached |
| SU(4) → PSU(4) | Map | reference | See stated category and scope | None attached |
| Sp(2) → SO(5) | Map | reference | See stated category and scope | None attached |
| Spin(8) → SO(8) | Map | reference | See stated category and scope | None attached |
| Spin(10) → SO(10) | Map | reference | See stated category and scope | None attached |
| SL(2, C) → SO₀(1,3) | Map | reference | See stated category and scope | None attached |
| SL(2, R) → PSL(2, R) | Map | reference | See stated category and scope | None attached |
| Universal SL(2,R) cover → SL(2,R) | Map | reference | See stated category and scope | None attached |
| SU(2,2) → SO₀(2,4) | Map | reference | See stated category and scope | None attached |
| Sp(4, R) → SO₀(2,3) | Map | reference | See stated category and scope | None attached |
| SU(2)/U(1) ≅ S² | Space | executable | See stated category and scope | topology |
| SO(3)/SO(2) ≅ S² | Space | executable | See stated category and scope | topology |
| C₃ · 1 | Representation | executable | See stated category and scope | finite-characters |
| C₃ · χ | Representation | executable | See stated category and scope | finite-characters |
| C₃ · χ² | Representation | executable | See stated category and scope | finite-characters |
| S₃ · 1 | Representation | executable | See stated category and scope | finite-characters |
| S₃ · sgn | Representation | executable | See stated category and scope | finite-characters |
| S₃ · 2 | Representation | executable | See stated category and scope | finite-characters |
| D₈ · 1 | Representation | executable | See stated category and scope | finite-characters |
| D₈ · a | Representation | executable | See stated category and scope | finite-characters |
| D₈ · b | Representation | executable | See stated category and scope | finite-characters |
| D₈ · c | Representation | executable | See stated category and scope | finite-characters |
| D₈ · 2 | Representation | executable | See stated category and scope | finite-characters |
| Q₈ · 1 | Representation | executable | See stated category and scope | finite-characters |
| Q₈ · a | Representation | executable | See stated category and scope | finite-characters |
| Q₈ · b | Representation | executable | See stated category and scope | finite-characters |
| Q₈ · c | Representation | executable | See stated category and scope | finite-characters |
| Q₈ · 2 | Representation | executable | See stated category and scope | finite-characters |
| A₄ · 1 | Representation | executable | See stated category and scope | finite-characters |
| A₄ · 1′ | Representation | executable | See stated category and scope | finite-characters |
| A₄ · 1″ | Representation | executable | See stated category and scope | finite-characters |
| A₄ · 3 | Representation | executable | See stated category and scope | finite-characters |
| S₄ · 1 | Representation | executable | See stated category and scope | finite-characters |
| S₄ · sgn | Representation | executable | See stated category and scope | finite-characters |
| S₄ · 2 | Representation | executable | See stated category and scope | finite-characters |
| S₄ · 3 | Representation | executable | See stated category and scope | finite-characters |
| S₄ · 3′ | Representation | executable | See stated category and scope | finite-characters |
| A₅ · 1 | Representation | executable | See stated category and scope | finite-characters |
| A₅ · 3 | Representation | executable | See stated category and scope | finite-characters |
| A₅ · 3′ | Representation | executable | See stated category and scope | finite-characters |
| A₅ · 4 | Representation | executable | See stated category and scope | finite-characters |
| A₅ · 5 | Representation | executable | See stated category and scope | finite-characters |
| A1 · (1) | Representation | executable | See stated category and scope | lie-representationssun-tensor-products |
| A1 · (2) | Representation | executable | See stated category and scope | lie-representationssun-tensor-products |
| A2 · (1,0) | Representation | executable | See stated category and scope | lie-representationssu3-tensor-productssu3-branchingsun-tensor-productscharged-branching |
| A2 · (0,1) | Representation | executable | See stated category and scope | lie-representationssu3-tensor-productssu3-branchingsun-tensor-productscharged-branching |
| A2 · (1,1) | Representation | executable | See stated category and scope | lie-representationssu3-tensor-productssu3-branchingsun-tensor-productscharged-branching |
| A2 · (2,0) | Representation | executable | See stated category and scope | lie-representationssu3-tensor-productssu3-branchingsun-tensor-productscharged-branching |
| A2 · (3,0) | Representation | executable | See stated category and scope | lie-representationssu3-tensor-productssu3-branchingsun-tensor-productscharged-branching |
| A2 · (2,2) | Representation | executable | See stated category and scope | lie-representationssu3-tensor-productssu3-branchingsun-tensor-productscharged-branching |
| A3 · (1,0,0) | Representation | executable | See stated category and scope | lie-representationssun-tensor-productscharged-branching |
| A3 · (0,1,0) | Representation | executable | See stated category and scope | lie-representationssun-tensor-productscharged-branching |
| A3 · (1,0,1) | Representation | executable | See stated category and scope | lie-representationssun-tensor-productscharged-branching |
| A4 · (1,0,0,0) | Representation | executable | See stated category and scope | lie-representationssun-tensor-productscharged-branching |
| A4 · (0,1,0,0) | Representation | executable | See stated category and scope | lie-representationssun-tensor-productscharged-branching |
| A4 · (1,0,0,1) | Representation | executable | See stated category and scope | lie-representationssun-tensor-productscharged-branching |
| C2 · (1,0) | Representation | executable | See stated category and scope | lie-representations |
| C2 · (0,1) | Representation | executable | See stated category and scope | lie-representations |
| D4 · (1,0,0,0) | Representation | executable | See stated category and scope | lie-representations |
| D4 · (0,0,1,0) | Representation | executable | See stated category and scope | lie-representations |
| D4 · (0,0,0,1) | Representation | executable | See stated category and scope | lie-representations |
| D5 · (1,0,0,0,0) | Representation | executable | See stated category and scope | lie-representations |
| D5 · (0,0,0,0,1) | Representation | executable | See stated category and scope | lie-representations |
| D5 · (0,0,0,1,0) | Representation | executable | See stated category and scope | lie-representations |
| D5 · (0,1,0,0,0) | Representation | executable | See stated category and scope | lie-representations |
| G2 · (1,0) | Representation | executable | See stated category and scope | lie-representations |
| G2 · (0,1) | Representation | executable | See stated category and scope | lie-representations |
| F4 · (0,0,0,1) | Representation | executable | See stated category and scope | lie-representations |
| F4 · (1,0,0,0) | Representation | executable | See stated category and scope | lie-representations |
| E6 · (1,0,0,0,0,0) | Representation | executable | See stated category and scope | lie-representations |
| E6 · (0,1,0,0,0,0) | Representation | executable | See stated category and scope | lie-representations |
| E7 · (0,0,0,0,0,0,1) | Representation | executable | See stated category and scope | lie-representations |
| E7 · (1,0,0,0,0,0,0) | Representation | executable | See stated category and scope | lie-representations |
| E8 · (0,0,0,0,0,0,0,1) | Representation | executable | See stated category and scope | lie-representations |
| su(3) ⊕ su(2) ⊕ u(1) | Lie algebra | reference | See stated category and scope | None attached |
| SU(3) × SU(2) × U(1) | Group | executable | See stated category and scope | product-gauge-groupslocal-operatorsoperator-enumeration |
| G_SM^(2) | Group | executable | See stated category and scope | product-gauge-groupslocal-operatorsoperator-enumeration |
| G_SM^(3) | Group | executable | See stated category and scope | product-gauge-groupslocal-operatorsoperator-enumeration |
| G_SM^(6) | Group | executable | See stated category and scope | product-gauge-groupslocal-operatorsoperator-enumeration |
| G_SM^(1) → G_SM^(2) | Map | executable | See stated category and scope | product-gauge-groups |
| G_SM^(1) → G_SM^(3) | Map | executable | See stated category and scope | product-gauge-groups |
| G_SM^(1) → G_SM^(6) | Map | executable | See stated category and scope | product-gauge-groups |
| SU(3) × SU(2) × U(1)_X → SU(5) | Map | executable | See stated category and scope | charged-branching |
| G_SM^(6) ↪ SU(5) | Map | reference | See stated category and scope | None attached |