Questions: Tensor Product of Representations

Question 1

If ρ has dimension m and σ has dimension n, what is the dimension of ρ ⊗ σ?

• A. m + n
• B. m · n
• C. max(m, n)
• D. m² + n²

Answer: (B)

dim(V ⊗ W) = dim(V) · dim(W) = mn. In terms of matrices, if ρ(g) is m×m and σ(g) is n×n, then (ρ ⊗ σ)(g) is the mn×mn Kronecker product ρ(g) ⊗ σ(g). This multiplicativity of dimension under tensor products contrasts with the additivity under direct sums: dim(V ⊕ W) = m + n.

Question 2

The character of ρ ⊗ σ equals the product χ_ρ · χ_σ. Why?

Model answer: If the eigenvalues of ρ(g) are α₁, ..., αₘ and of σ(g) are β₁, ..., βₙ, then the eigenvalues of ρ(g) ⊗ σ(g) are all products αᵢβⱼ. The trace is Σᵢ,ⱼ αᵢβⱼ = (Σ αᵢ)(Σ βⱼ) = tr(ρ(g)) · tr(σ(g)) = χ_ρ(g) · χ_σ(g).

This multiplicativity of characters under tensor products makes decomposition computationally tractable. To find the irreducible decomposition of ρ ⊗ σ, compute the pointwise product χ_ρ · χ_σ, then take inner products with each irreducible character. This avoids ever constructing the tensor product space explicitly.

Question 3

The tensor product of two irreducible representations is always irreducible.

• T. True
• F. False

Answer: False

Tensor products of irreducibles are generally reducible. For example, for S₃: the standard 2-dimensional representation ρ has χ_ρ = (2, 0, −1) on the three conjugacy classes. Then χ_{ρ⊗ρ}(g) = χ_ρ(g)² = (4, 0, 1). Decomposing: ⟨χ_{ρ⊗ρ}, χ_triv⟩ = (4+0+2)/6 = 1, ⟨χ_{ρ⊗ρ}, χ_sign⟩ = (4+0+2)/6 = 1, ⟨χ_{ρ⊗ρ}, χ_ρ⟩ = (8+0−2)/6 = 1. So ρ ⊗ ρ ≅ triv ⊕ sign ⊕ ρ — three irreducible summands.

Question 4

In physics, the decomposition of tensor products of representations of SU(2) is governed by Clebsch-Gordan coefficients. The result for spin-j₁ ⊗ spin-j₂ is:

• A. A single irreducible of spin j₁ + j₂
• B. Direct sum of irreducibles with spins from |j₁−j₂| to j₁+j₂ in integer steps
• C. A direct sum of j₁·j₂ copies of the trivial representation
• D. An irreducible of spin j₁·j₂

Answer: (B)

The Clebsch-Gordan decomposition for SU(2) is V_{j₁} ⊗ V_{j₂} ≅ V_{|j₁−j₂|} ⊕ V_{|j₁−j₂|+1} ⊕ ··· ⊕ V_{j₁+j₂}. For example, spin-1 ⊗ spin-1 = spin-0 ⊕ spin-1 ⊕ spin-2, with dimensions 3·3 = 1+3+5 = 9. This is the mathematical basis for angular momentum addition in quantum mechanics.