Representation Theory of Groups

How can an abstract group be studied when only a multiplication table is given? Representation theory's answer: 'animate' the group - turn its elements into concrete matrices ready for multiplication and computation. And it turns out that the humble character table (just traces of matrices!) completely encodes the entire structure of representations.

• Quantum mechanics: the electron's spin-½ is a two-dimensional representation of SU(2); spectroscopic selection rules in chemistry follow from representation theory of symmetry groups
• Signal processing: the Fast Fourier Transform (FFT) is representation theory of cyclic groups; generalized FFTs use representations of arbitrary groups

Предварительные знания

• Subgroups and Cosets
• Quotient Groups and Noether's Theorem

What is a Group Representation

A **representation** of a group G on a vector space V over a field k is a group homomorphism ρ: G → GL(V), where GL(V) is the group of invertible linear transformations of V. In other words, each g ∈ G is assigned an invertible matrix ρ(g), satisfying ρ(gh) = ρ(g)ρ(h) and ρ(e) = I. The **dimension** of a representation is dim(V). A two-dimensional representation gives 2×2 matrices, a three-dimensional one gives 3×3 matrices.

**Why representations?** An abstract group is a complex object. Its representation turns abstract symmetry into concrete matrices ready for computation. Representation theory bridges algebra and linear algebra. Physicists use representations of symmetry groups to classify elementary particles: the electron's spin is a representation of SU(2).

A **subrepresentation** is a subspace W ⊆ V invariant under all ρ(g): if w ∈ W, then ρ(g)w ∈ W for all g ∈ G. An **irreducible** representation (irrep) is one with no nontrivial invariant subspaces. Irreducible representations are the atoms of the theory; all others are built from them.

Characters of Representations

The **character** of a representation ρ is the function χ_ρ: G → k defined as the trace of the matrix: χ_ρ(g) = tr(ρ(g)). The character is a class function (constant on conjugacy classes), since tr(ABA⁻¹) = tr(B). Key properties: - χ(e) = dim(V) (trace of the identity matrix) - χ(g⁻¹) = χ̄(g) (complex conjugate, for unitary representations) - χ_{ρ₁ ⊕ ρ₂} = χ_{ρ₁} + χ_{ρ₂} (character of a direct sum is sum of characters)

**Character Orthogonality Theorem:** Characters of irreducible representations form an orthonormal basis in the space of class functions on G. This means the character table is a unitary matrix (with appropriate weights). The number of irreducible representations = the number of conjugacy classes of G.

Schur's Lemma and Irreducible Representations

**Schur's Lemma** - a fundamental result in representation theory: 1. If ρ₁: G → GL(V₁) and ρ₂: G → GL(V₂) are irreducible representations, and φ: V₁ → V₂ is a linear map commuting with all ρ(g) (i.e., φ∘ρ₁(g) = ρ₂(g)∘φ), then φ = 0 or φ is an isomorphism. 2. If ρ is an irreducible representation over an algebraically closed field (e.g., ℂ), then the only operators commuting with all ρ(g) are scalars: φ = λI.

**Applications in physics:** Classifying molecular orbitals by irreducible representations of the molecule's symmetry group determines which spectral transitions are allowed (selection rules). The water molecule H₂O has symmetry group C₂ᵥ; its four irreducible representations (A₁, A₂, B₁, B₂) classify orbitals and vibrational modes.

Key Ideas

• A representation ρ: G → GL(V) is a homomorphism from the group to invertible operators
• An irreducible representation has no nontrivial invariant subspaces
• The character χ(g) = tr(ρ(g)) is a class function that completely determines the representation
• Number of irreducible representations = number of conjugacy classes
• Schur's Lemma: intertwining operators between irreducibles are 0 or an isomorphism
• Character table is a unitary matrix; orthogonality of characters

Further Directions

Representation theory of finite groups (Maschke's theorem, Burnside's theorem) leads to representations of Lie groups, which govern particle physics. Character tables are computed in GAP and SageMath.

• Simple Groups — Classification of simple groups is deeply linked to their irreducible representations
• Lie Algebras — Representations of Lie groups are studied via representations of the corresponding Lie algebras

Вопросы для размышления

• Why does the orthogonality theorem for characters guarantee that the character table is square (number of rows = number of columns)?
• The sign representation of S_n: sgn(σ) = ±1. What is the character of the tensor product of the sign representation with itself?
• How does representation theory explain why the hydrogen atom has energy levels with degeneracy n²?

Связанные уроки

• la-01-vectors-intro