Module representation 

Representation theory for Lie groups and Lie algebras.

This module provides tools for working with irreducible representations (irreps), tensor products, characters, and Casimir operators.

Overview

Core Concepts

• Irreducible Representations: Classified by highest weights
• Casimir Operators: Invariants that label representations
• Characters: Traces of representation matrices (Weyl character formula)
• Tensor Products: Clebsch-Gordan / Littlewood-Richardson decomposition

Implemented Groups

• SU(2): Complete (see crate::representation module)
• SU(3): In progress (Casimir operators implemented)
• SU(N): Planned (generic implementation)

Physical Applications

Representation theory is essential for:

• Classifying particle states in QCD (quarks, gluons, hadrons)
• Computing scattering amplitudes (Clebsch-Gordan coefficients)
• Understanding symmetry breaking (Higgs mechanism)
• Gauge fixing via Peter-Weyl decomposition

References

• Georgi: “Lie Algebras in Particle Physics” (1999)
• Slansky: “Group Theory for Unified Model Building” (1981)
• Weyl: “The Theory of Groups and Quantum Mechanics” (1928)

Re-exports

pub use casimir::Casimir;

pub use su3_irrep::Su3Irrep;

Modules

casimir

Casimir operators for Lie algebras.

su3_irrep

SU(3) irreducible representations.

Structs

Spin

Spin quantum number j (half-integer)

Functions

character

Character of a representation: χⱼ(g) = Tr(D^j(g))

character_orthogonality_delta

Character orthogonality relation (analytical)

character_su2

Character of an SU(2) group element

clebsch_gordan_decomposition

Clebsch-Gordan decomposition: Vⱼ₁ ⊗ Vⱼ₂ = ⨁ₖ Vₖ

representation_dimension

Dimension formula: dim(Vⱼ) = 2j + 1