Groups, actions, and representations
Begin with a symmetry action
Section titled “Begin with a symmetry action”A group collects transformations with composition, identity, and inverses. To use it in physics, specify what it acts on: coordinates, configurations, fields, states, or operators. The same abstract group can act in physically different ways.
Representations linearize the action
Section titled “Representations linearize the action”A representation is a homomorphism from the group to invertible linear maps on a vector space. Its kernel consists of elements acting trivially. A faithful representation has trivial kernel.
Decompose the space
Section titled “Decompose the space”For finite groups over ℂ and compact groups with continuous finite-dimensional representations, complete reducibility organizes representations into irreducibles. Tensor products describe combined systems and products of fields.
Track the assumptions
Section titled “Track the assumptions”The coefficient field, characteristic, continuity, dimensionality, and unitarity requirements belong to the problem. Ordinary complex character tables answer a particular representation-theoretic question.
Continue with a calculation
Section titled “Continue with a calculation”Open an atlas record, inspect its representation data, then use a linked tool. The SU(2) tensor-product calculator gives an immediate connection with addition of angular momentum.
Work through an example
Section titled “Work through an example”The irreducible representation lesson decomposes the three-dimensional permutation action of S₃ into an invariant line and a doublet, then proves the character projection used by the tools. Follow the finite-symmetry path to carry that same action into a complete calculation.
Continue in the atlas
Section titled “Continue in the atlas”Sources
Section titled “Sources”Source curation and automated validation are separate from independent expert review.
First public edition · mathematical baseline 6.0.0. Source curation and automated validation are distinct from independent expert review.
About · Contributors · Review status · Suggest a correction · Data and reproducibility · Articles CC BY 4.0 · Code MIT