Representation Theory: Advanced Topics

The character table of a finite group is its 'genome': a small matrix that encodes all algebraic structure. Brauer's theorem turns this matrix into a tool of number theory: Artin L-functions are expressed through one-dimensional characters of abelian extensions.

• Number theory: Artin L-functions and the Artin conjecture on meromorphicity, a direct application of Brauer's theorem
• Particle physics: irreducible representations of a symmetry group classify particles (Wigner's theorem)
• Cryptography: representations of Galois groups feed into factoring algorithms (Schur-Seidle theorem)

Предварительные знания

• Tensor Decompositions in Algebra

Characters and the Character Table

Frobenius computed the first character table for a non-abelian group (S_4) in 1896, and Burnside used character orthogonality in 1904 to prove that every group of order p^a q^b is solvable. The ATLAS of Finite Groups (Conway et al., 1985) catalogues character tables for all 26 sporadic simple groups, including the Monster of order roughly 8.08 × 10^53.

**Character of a representation** ρ: G → GL(V): χ_V(g) = tr(ρ(g)), the trace of the matrix. Properties of characters: 1. χ_V(e) = dim(V) (trace of the identity matrix) 2. χ_V(hgh⁻¹) = χ_V(g), a character is a class function (constant on conjugacy classes) 3. χ_{V⊕W} = χ_V + χ_W (direct sum of representations) 4. χ_{V⊗W} = χ_V · χ_W (tensor product = pointwise product of characters) 5. χ_{V*}(g) = χ_V(g⁻¹) = conj(χ_V(g)) (for unitary representations) **Character table**: the matrix χᵢ(Cⱼ), where χᵢ are irreducible characters and Cⱼ are conjugacy classes. Its size is k×k, where k = the number of conjugacy classes = the number of irreducible representations. **Decomposition:** Any representation V decomposes into a direct sum of irreducibles: V ≅ ⊕ nᵢ·Vᵢ, where nᵢ = ⟨χ_V, χᵢ⟩ = (1/|G|) ∑_{g∈G} χ_V(g)·conj(χᵢ(g)). **Theorem:** The number of irreducible representations equals the number of conjugacy classes.

**Frobenius-Schur theorem:** The character table of a finite group G is a square k×k matrix (k = number of conjugacy classes). Rows are orthonormal: ⟨χᵢ,χⱼ⟩=δᵢⱼ. Columns are also orthogonal: ∑ᵢ χᵢ(g)·conj(χᵢ(h)) = (|G|/|Cg|)·δ_{Cg=Ch}. The character table determines the group up to isomorphism for many (but not all) groups.

Induced Representations and Frobenius Reciprocity

**Induced representation:** Let H ≤ G be a subgroup and σ: H → GL(W) a representation of H. Then Ind_H^G(σ) is a representation of G built from σ 'over cosets': Ind_H^G(W) = k[G] ⊗_{k[H]} W ≅ ⊕_{g ∈ G/H} gW (direct sum over cosets) Dimension: dim(Ind_H^G(W)) = [G:H] · dim(W). **Character of the induced representation:** χ_{Ind}(g) = (1/|H|) ∑_{x∈G, x⁻¹gx∈H} χ_σ(x⁻¹gx). **Frobenius reciprocity:** ⟨Ind_H^G(σ), ρ⟩_G = ⟨σ, Res_H^G(ρ)⟩_H Here Res is restriction (the representation of G restricted to H). This is an adjunction: Ind and Res are adjoint functors! **Examples:** The regular representation k[G] = Ind_1^G(k) (induced from the trivial representation on the trivial subgroup). Every irreducible appears in k[G] with multiplicity equal to its dimension.

**Mackey's formula:** Restricting an induced representation to a subgroup: Res_K^G ∘ Ind_H^G(σ) = ⊕_{g ∈ K\G/H} Ind_{H∩K^g}^K(Res_{H∩K^g}^{K^g}(σ^g)), where the sum runs over double cosets. This is the 'intermediate state rule', analogous to the transfer matrix method in physics.

Brauer's Theorem: All Characters Are Induced

**Brauer's theorem (1953):** Every character of a finite group G is an integer linear combination of characters induced from one-dimensional representations of elementary subgroups. An **elementary subgroup** has the form ⟨g⟩ × P, where ⟨g⟩ is cyclic and P is a p-group for some prime p. **Practical corollary:** To compute χ(g) it suffices to know one-dimensional characters on elementary subgroups, the 'building blocks' from which all characters are assembled. **Weaker Artin's theorem:** Every rational character χ is a ℚ-linear combination of characters induced from CYCLIC subgroups. (Brauer improved this to ℤ-coefficients and elementary subgroups.) **Application, L-functions:** Brauer's theorem is used to prove the meromorphicity of Artin L-functions: L(s, χ) = ∏ L(s, χᵢ)^{nᵢ}, where χᵢ are one-dimensional (abelian) and nᵢ ∈ ℤ.

**Artin L-functions:** L(s, χ) = ∏_p det(1 - Frob_p · p^{-s} | V^{I_p})^{-1}, the L-function attached to a representation χ of the Galois group Gal(L/Q). Brauer's theorem proves that L(s,χ) is meromorphic (decomposing χ into a ℤ-combination of one-dimensional characters reduces the L-function to a product of Hecke zeta functions). Analytic continuation of L(s,χ) to the entire complex plane remains the Artin conjecture, open for dim > 1.

Key Ideas

• Character χ_V(g) = tr(ρ(g)), a class function; ⟨χᵢ,χⱼ⟩ = δᵢⱼ (orthonormality)
• Number of irreducibles = number of conjugacy classes; ∑ dim(Vᵢ)² = |G|
• Ind_H^G, induction; Res_H^G, restriction; Frobenius reciprocity: ⟨Ind(σ),ρ⟩_G = ⟨σ,Res(ρ)⟩_H
• Brauer's theorem: every character = ℤ-combination of Ind from one-dimensional characters on elementary subgroups

Further Directions

Representation theory bridges abstract algebra and analysis. Characters of Lie groups (Weyl character formula), perverse sheaves, and the Langlands program are natural continuations. In ML, representation theory drives equivariant neural networks.

• Abstract Algebra in ML — Equivariant networks are architectures invariant to a group action; representation theory dictates which layers are admissible
• Tensor Decompositions — Characters of representations are special functions connected to symmetric and exterior powers of tensor products

Вопросы для размышления

• Compute the character table of Q₈ (quaternion group of order 8). Compare with the character table of D₄ (dihedral group of order 8), do they coincide?
• Prove Frobenius reciprocity: ⟨Ind_H^G(σ), ρ⟩_G = ⟨σ, Res_H^G(ρ)⟩_H. Use the explicit formula for the character of an induced representation.
• Find the decomposition of the regular representation k[S₃] into a direct sum of irreducibles. Verify: ∑ nᵢ·dim(Vᵢ) = |G| and nᵢ = dim(Vᵢ).

Связанные уроки

• la-01-vectors-intro