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Group Representations

Source lecture(s): PHY622 Lec8-9

Intuition

A representation realizes an abstract group as concrete matrices acting on a vector space.

Formal Definition

A representation \(D(g)\) is a homomorphism \(G\to GL(V)\) preserving multiplication: \(D(g_1g_2)=D(g_1)D(g_2)\).

Mathematical Formulation

Character: \(\chi(g)=\text{Tr}[D(g)]\). Orthogonality: \(\sum_g \chi^{(r)}(g)\chi^{(s)}(g)^* = |G|\delta_{rs}\).

Derivation

Choose a basis for \(V\); each group element becomes a matrix. Irreducible representations cannot be block-diagonalized further.

Worked Example

The Pauli matrices realize an SU(2) spin-1/2 representation.

Common Mistakes

  • Confusing reducible and irreducible representations.
  • Using characters of different dimensions in orthogonality.

Quiz

Q1. What does irreducible mean?

Answer

No nontrivial invariant subspace.