14.21.4 Tensor Map Product Representation Role

The Tensor Map Product Representation Role describes how tensor maps interact through product structures, enabling algebraic operations in multilinear algebra.

Tensor Map Product Representation Role is the use of the tensor product of maps to combine two representations of the same group or algebra into a single new representation on the tensor product of the underlying spaces, by applying the tensor product construction to the pair of operators $ρ (g)$ and $σ (g)$ for one and the same group element $g$ , rather than to independent elements of two separate groups.

Constructing the Tensor Product Representation

The Defining Assignment

Given representations $ρ : G \to GL (V)$ and $σ : G \to GL (W)$ of the same group $G$ , the tensor product representation is the assignment

(ρ \otimes σ) (g) = ρ (g) \otimes σ (g)

using the same group element $g$ to build both operators being tensored, in contrast with the operator construction role, where two independent operators from unrelated algebras are combined.

Verifying the Homomorphism Property

For this to be a representation of $G$ on $V \otimes W$ , it must satisfy $(ρ \otimes σ) (g h) = (ρ \otimes σ) (g) (ρ \otimes σ) (h)$ , which follows directly from the interchange law,

ρ (g h) \otimes σ (g h) = (ρ (g) ρ (h)) \otimes (σ (g) σ (h)) = (ρ (g) \otimes σ (g)) (ρ (h) \otimes σ (h))

using the fact that $ρ (g h) = ρ (g) ρ (h)$ and $σ (g h) = σ (g) σ (h)$ , each a consequence of $ρ$ and $σ$ being representations in their own right.

Character of the Tensor Product Representation

Multiplicativity of the Character

Recalling the trace behavior established for the tensor product of operators, the character of the tensor product representation satisfies

χ_{ρ \otimes σ} (g) = tr (ρ (g) \otimes σ (g)) = tr (ρ (g)) \cdot tr (σ (g)) = χ_{ρ} (g) \cdot χ_{σ} (g)

for every $g \in G$ , giving a purely numerical relation between the characters of the two original representations and the character of their tensor product, without any need to compute the tensor product matrices themselves.

Practical Use of the Character Formula

Since characters determine representations up to isomorphism in many settings, the character formula allows the isomorphism type of $ρ \otimes σ$ to be identified from a table of known character values, entirely bypassing direct manipulation of the underlying matrices.

Decomposition Into Irreducible Constituents

The Tensor Product Representation Need Not Be Irreducible

Even when $ρ$ and $σ$ are both irreducible representations, their tensor product representation is typically reducible, decomposing as a direct sum

ρ \otimes σ ≅ ⨁_{k} N_{k} τ_{k}

for some irreducible representations $τ_{k}$ occurring with multiplicities $N_{k}$ , a decomposition whose coefficients can be computed directly from the character formula above using orthogonality relations among irreducible characters.

Role in Combining Independent Systems

This decomposition is the standard mathematical description of what happens when two independently transforming systems are combined into one, since the tensor product representation governs how the combined system transforms under the shared symmetry group $G$ , while the decomposition into irreducibles identifies the distinct types of behavior present in that combined system.

Contrast With the Operator Construction Role

Same Group Versus Product of Groups

The operator construction role produces a representation of the product group $G \times H$ on $V \otimes W$ from representations of $G$ and $H$ separately, evaluated at independent elements $(g, h)$ ; the representation role instead restricts attention to a single group $G$ and evaluates both factors at the same element $g$ . The representation role can be recovered from the product-group construction by restricting along the diagonal embedding $G \to G \times G$ , $g \mapsto (g, g)$ , showing the two roles are closely related but not identical.