symmetry-group-identifier

This skill helps you map identified symmetries to mathematical groups. Once you know what transformations should leave your predictions unchanged, this skill formalizes them into the language of group theory.

Why groups matter: Neural network architectures are built around specific symmetry groups. Knowing your group tells you exactly which architecture patterns to use.

Copy this checklist and track your progress:

```

Group Identification Progress:

• [ ] Step 1: List symmetries from discovery phase
• [ ] Step 2: Classify each as discrete or continuous
• [ ] Step 3: Match to specific groups using taxonomy
• [ ] Step 4: Determine how groups combine
• [ ] Step 5: Verify group properties
• [ ] Step 6: Document final group specification

```

Step 1: List symmetries from discovery phase

Gather the identified symmetries from the discovery phase. List each identified transformation and whether it requires invariance or equivariance. Note confidence levels. If symmetries haven't been discovered yet, work with user to identify them through domain analysis first.

Step 2: Classify each as discrete or continuous

For each symmetry, determine: Is the transformation set finite (discrete) or infinite (continuous)? Discrete examples: 90° rotations (4 elements), permutations of n items (n! elements). Continuous examples: rotation by any angle, translation by any distance. Use [Group Taxonomy](#group-taxonomy) to guide classification. For mathematical foundations, see [Group Theory Primer](./resources/group-theory-primer.md).

Step 3: Match to specific groups using taxonomy

Use the [Discrete Groups](#discrete-groups) and [Continuous Groups](#continuous-groups-lie-groups) reference sections. Identify the specific group name and notation for each symmetry. Common matches: n-fold rotation → Cₙ, rotation+reflection → Dₙ, permutation → Sₙ, 3D rotation → SO(3), rigid motion → SE(3), full Euclidean → E(3). For detailed Lie group information (SO(3), SE(3), E(3)), consult [Lie Groups Reference](./resources/lie-groups.md).

Step 4: Determine how groups combine

If multiple symmetries are present, determine how they combine. Direct product (G × H): symmetries act independently. Semidirect product (G ⋊ H): one symmetry "twists" the other (e.g., SE(3) = SO(3) ⋊ ℝ³). Use [Combining Groups](#combining-groups) reference.

Step 5: Verify group properties

Check that identified structure satisfies group axioms: closure, associativity, identity, inverses. Verify important properties: Is it compact? (affects representation theory). Is it abelian? (commutative or not). Is it connected? (affects implementation). Use [Group Properties Checklist](#group-properties-checklist). For detailed verification methodology, see [Methodology](./resources/methodology.md).

Step 6: Document final group specification

Create specification using [Output Template](#output-template). Include: group name/notation, dimension/size, key properties, invariance vs equivariance requirements, and recommended architecture family. This specification provides the foundation for architecture design. Quality criteria for this output are defined in [Quality Rubric](./resources/evaluators/rubric_group_identification.json).

Overview Diagram

```

SYMMETRY GROUPS

│

┌───────────────┴───────────────┐

│ │

DISCRETE CONTINUOUS

│ (Lie Groups)

│ │

┌─────┼─────┐ ┌────────┼────────┐

│ │ │ │ │ │

Cyclic Dihedral Symmetric SO(n) SE(n) E(n)

Cₙ Dₙ Sₙ rotations rigid Euclidean

only motions (w/ reflect)

```

Quick Reference Table

| Symmetry Type | Group | Notation | Elements | Common Use |

|---------------|-------|----------|----------|------------|

| n-fold rotation | Cyclic | Cₙ | n | Image rotation (90°, 60°) |

| Rotation + reflection | Dihedral | Dₙ | 2n | Regular polygons |

| Permutation | Symmetric | Sₙ | n! | Sets, graphs |

| 2D rotation (continuous) | Special orthogonal | SO(2) | ∞ | Continuous rotation |

| 3D rotation | Special orthogonal | SO(3) | ∞ | 3D orientation |

| 3D rigid motion | Special Euclidean | SE(3) | ∞ | Robotics, molecules |

| 3D with reflections | Euclidean | E(3) | ∞ | Chemistry, physics |

Cyclic Groups (Cₙ)

What they represent: Rotations by multiples of 360°/n

Elements: {e, r, r², ..., rⁿ⁻¹} where rⁿ = e (identity)

| Group | Rotations | Example |

|-------|-----------|---------|

| C₂ | 0°, 180° | Playing cards |

| C₄ | 0°, 90°, 180°, 270° | Square images |

| C₆ | 60° increments | Hexagonal patterns |

Use when: Rotation symmetry present but NOT reflection symmetry.

Dihedral Groups (Dₙ)

What they represent: Rotations + reflections of regular n-gon

Elements: n rotations + n reflections = 2n total

| Group | Elements | Example |

|-------|----------|---------|

| D₄ | 8 | Square with diagonals (p4m group) |

| D₆ | 12 | Regular hexagon |

Use when: Both rotation AND reflection symmetry present.

Symmetric Groups (Sₙ)

What they represent: All permutations of n elements

Elements: n! permutations

Use when: Element ordering is arbitrary (sets, graphs, point clouds).

SO(2) - 2D Rotations

Elements: Rotation by any angle θ ∈ [0, 2π)

Matrix form: R(θ) = [[cos θ, -sin θ], [sin θ, cos θ]]

Use when: Continuous rotation symmetry in 2D.

SO(3) - 3D Rotations

Elements: All rotations in 3D (3 degrees of freedom)

Representations: Rotation matrices, quaternions, Euler angles, axis-angle

Use when: 3D orientation doesn't matter, but handedness does.

SE(3) - 3D Rigid Motions

Elements: Rotations + Translations in 3D

Structure: SE(3) = SO(3) ⋊ ℝ³ (semidirect product)

Use when: Objects can be anywhere and in any orientation, handedness matters.

E(3) - Full Euclidean Group

Elements: SE(3) + Reflections

Structure: E(3) = O(3) ⋊ ℝ³

Use when: SE(3) symmetry PLUS reflection symmetry (most molecules).

Group Hierarchy

```

E(3) = O(3) ⋊ ℝ³

│ exclude reflections

▼

SE(3) = SO(3) ⋊ ℝ³

│ exclude translations

▼

SO(3)

│ 2D restriction

▼

SO(2)

```

Direct Product (G × H)

When to use: Symmetries act independently (neither affects the other).

Example: Image with separate translation and color permutation → SE(2) × S₃

Property: (g₁, h₁) · (g₂, h₂) = (g₁g₂, h₁h₂)

Semidirect Product (G ⋊ H)

When to use: One symmetry "twists" the other (don't commute).

Example: SE(3) = SO(3) ⋊ ℝ³ (rotating then translating ≠ translating then rotating)

Common cases: SE(n) = SO(n) ⋊ ℝⁿ, E(n) = O(n) ⋊ ℝⁿ, Dₙ = Cₙ ⋊ C₂

For your identified group, verify:

| Property | Question | Why It Matters |

|----------|----------|----------------|

| Compact | Is the group "bounded"? | Affects representation theory |

| Abelian | Does order matter? (g₁g₂ = g₂g₁?) | Simplifies architecture |

| Connected | Is group in one piece? | Affects irreducible representations |

| Finite | Finite number of elements? | Discrete vs continuous architecture |

| Domain | Typical Group | Notes |

|--------|--------------|-------|

| 2D Image Classification | C₄ or D₄ | p4 or p4m groups |

| 3D Molecular Energy | E(3) × Sₙ | Full Euclidean + atom permutation |

| 3D Molecular Chirality | SE(3) × Sₙ | No reflections |

| Point Cloud Classification | SO(3) × Sₙ | Rotation + permutation |

| Graph Classification | Sₙ | Permutation invariant |

| Robotics | SE(3) | Sometimes with gravity constraint |

```

SYMMETRY GROUP SPECIFICATION

============================

Identified Symmetries:

1. [Symmetry] → Group: [name] ([notation])
2. [Symmetry] → Group: [name] ([notation])

Combined Group Structure:

• Full group: [G₁ × G₂] or [G₁ ⋊ G₂]
• Size: [# elements] or [continuous]

Group Properties:

• Compact: [Yes/No]
• Abelian: [Yes/No]
• Connected: [Yes/No]

Symmetry Requirements:

• [Group]: [Invariant/Equivariant] for [task type]

Recommended Architecture Family:

• [Architecture] supporting [group]

NEXT STEPS:

• Empirically validate symmetry hypotheses if not yet confirmed
• Design equivariant architecture based on group specification

```