Tensor product of representations

For a group G, given representations ρ: G → GL(V) and σ: G → GL(W), the tensor product representation ρ⊗σ acts on V⊗W by (ρ⊗σ)(g)(v⊗w) = ρ(g)v ⊗ σ(g)w, extended linearly. This definition applies to Lie algebras, where the action is given by the Leibniz rule: X·(v⊗w) = (X·v)⊗w + v⊗(X·w). The dimension of the tensor product is the product of the dimensions, and the character of the tensor product is the pointwise product of characters: χ_ρ⊗σ(g) = χ_ρ(g)χ_σ(g).¹

For compact groups and semisimple Lie algebras, the tensor product of two irreducible representations is completely reducible, decomposing into a direct sum of irreducibles: V_λ⊗V_μ = ⊕_ν c_λμ^ν V_ν. The multiplicities c_λμ^ν are the Clebsch–Gordan coefficients, which appear in quantum mechanics when adding angular momenta. For SU(2), the decomposition follows the rule j₁⊗j₂ = ⊕_j=|j1−j2|^j₁+j₂ j, where each irreducible appears exactly once.²

In quantum mechanics, the tensor product of representations describes the combined state space of multiple particles, and the Clebsch–Gordan coefficients are used to compute the probabilities of coupled angular momentum states. In particle physics, the tensor product of representations of the Lorentz group or gauge groups (such as SU(3) color) determines the possible bound states and scattering amplitudes. In Lie theory, the tensor product of representations of a Lie algebra is used to construct higher-dimensional representations and to study the structure of the universal enveloping algebra. The tensor product also plays a role in the representation theory of quantum groups, where the braiding introduces non-trivial statistics.³

Beyond the standard group case, the tensor product of representations can be defined for Hopf algebras, where the comultiplication dictates the action on the tensor product. For super Lie algebras, the tensor product involves a sign factor: (X·(v⊗w)) = (X·v)⊗w + (−1)^|v||X| v⊗(X·w). The tensor product of representations of the symmetric group is related to the Kronecker product of symmetric functions, which remains a challenging computational problem. In the representation theory of finite groups, the tensor product of a representation with its dual contains the trivial representation exactly once, a fact used in the proof of Schur's lemma. The tensor product also appears in the theory of fusion categories, where the decomposition rules define the fusion rules of anyons in topological quantum computation.⁴

Irreducible representation

A representation with no non-trivial invariant subspaces.

Clebsch–Gordan coefficients

The multiplicities in the decomposition of a tensor product of irreducible representations.

Hopf algebra

An algebraic structure with a comultiplication that allows the definition of tensor products of representations.