Tensor Product of Representations

Core Idea

Given representations ρ: G → GL(V) and σ: G → GL(W), their tensor product ρ ⊗ σ: G → GL(V ⊗ W) is defined by (ρ ⊗ σ)(g)(v ⊗ w) = ρ(g)v ⊗ σ(g)w. The character of the tensor product is the pointwise product of characters: χ_{ρ⊗σ}(g) = χ_ρ(g)·χ_σ(g). Decomposing tensor products of irreducibles into irreducible summands (the Clebsch-Gordan problem) is fundamental in both mathematics and physics.

Explainer

The tensor product gives a way to combine two representations into a new, larger one. If G acts on V via ρ and on W via σ, the tensor product representation acts on V ⊗ W by the diagonal action: g sends v ⊗ w to ρ(g)v ⊗ σ(g)w, extended linearly to all of V ⊗ W. This is not the same as acting on V and W independently (which would be the direct sum ρ ⊕ σ) — in the tensor product, the same group element g acts simultaneously on both factors.

The dimension of V ⊗ W is dim(V) · dim(W), and in a chosen basis the representing matrix is the Kronecker product of the two individual matrices. The character has a beautiful form: χ_{ρ⊗σ}(g) = χ_ρ(g) · χ_σ(g), a pointwise product. This follows from the fact that the eigenvalues of A ⊗ B are all pairwise products of eigenvalues of A and B, so tr(A ⊗ B) = tr(A) · tr(B). This multiplicativity converts the tensor product decomposition problem into arithmetic with characters.

The Clebsch-Gordan problem asks: given irreducible representations V_i and V_j, decompose V_i ⊗ V_j into irreducibles. The answer is V_i ⊗ V_j ≅ ⊕_k N_{ij}^k V_k, where the multiplicities N_{ij}^k (called Clebsch-Gordan coefficients or Kronecker coefficients for symmetric groups) are computed by N_{ij}^k = ⟨χ_i · χ_j, χ_k⟩. For finite groups over ℂ, this is always a finite computation. For SU(2) in physics, the Clebsch-Gordan decomposition governs angular momentum addition: coupling spin-j₁ and spin-j₂ gives all spins from |j₁−j₂| to j₁+j₂.

Tensor products interact with direct sums distributively: (V₁ ⊕ V₂) ⊗ W ≅ (V₁ ⊗ W) ⊕ (V₂ ⊗ W). This, combined with the tensor product of irreducibles, means the entire tensor product structure is determined by the Clebsch-Gordan coefficients for irreducible pairs. These coefficients encode deep information about the group and are the subject of ongoing research, particularly for symmetric groups (where computing Kronecker coefficients is a major open problem in algebraic combinatorics).