symmetry-group-identifier_skill

Overview

This skill maps identified symmetry transformations into precise mathematical groups to guide equivariant or invariant architecture design. It turns domain-level symmetry observations (rotations, reflections, permutations, translations) into formal group names, notations, and properties that directly inform model choice. Use it to produce a compact, implementable group specification for neural architecture design.

How this skill works

You provide candidate symmetries or I help elicit them from the domain. The skill classifies each symmetry as discrete or continuous, matches it to standard groups (C_n, D_n, S_n, SO(n), SE(n), E(n), etc.), determines combination structure (direct vs semidirect product), and verifies key group properties relevant for representation and implementation. The final output is a concise group specification with recommended architecture family and verification checklist.

When to use it

• You’ve discovered transformations that should leave outputs invariant or equivariant and need formal group names
• Designing equivariant neural networks for images, point clouds, molecules, or robots
• You need to decide whether symmetries combine independently or via twisting (direct vs semidirect product)
• Translating domain symmetries into constraints that affect layer design or representation choices
• Validating whether candidate symmetry hypotheses satisfy group axioms before engineering models

Best practices

• Start by listing concrete transformations and whether you require invariance or equivariance
• Classify each symmetry as discrete (finite) or continuous (Lie group) early — it drives implementation choices
• Prefer the simplest group that captures observed invariances to avoid over-constraining models
• When multiple symmetries exist, explicitly test if they commute to choose direct vs semidirect product
• Document group properties (compact, connected, abelian, finite) since they determine representation methods
• Empirically validate symmetry assumptions with controlled augmentations or ablation tests

Example use cases

• Image classifier with rotation-only symmetry → map to cyclic group C_n (or SO(2) if continuous)
• Square object recognition with flips → identify dihedral group D_4 and use equivariant convolutions
• 3D point-cloud pose-invariant model → select SO(3) × S_n (rotations × permutations)
• Molecular energy prediction where reflections are allowed → choose E(3) × S_n
• Robotics pose estimation with rigid motions → formalize as SE(3) and design SE(3)-equivariant layers