5.22.1 Tensor Product Factor Swap Map

Tensor Product Factor Swap Map is the special case of the symmetry isomorphism obtained when both tensor factors come from the same vector space V, namely the linear operator τ : V ⊗ V → V ⊗ V defined by τ(v ⊗ v′) = v′ ⊗ v, which — unlike the general symmetry isomorphism between distinct spaces V ⊗ W and W ⊗ V — is an endomorphism of a single space and can therefore be studied through its eigenvalues, invariant subspaces, and role in decomposing V ⊗ V into symmetric and antisymmetric parts.

Definition of the Swap Map on V ⊗ V

The Operator on Simple Tensors

For a single vector space V, the factor swap map is the linear operator

extended by linearity to all of V ⊗ V, existing by the same universal-property argument used for the general symmetry isomorphism, now specialized to the case W = V.

Involution Property

Since τ(τ(v ⊗ v′)) = τ(v′ ⊗ v) = v ⊗ v′, the swap map satisfies τ² = id, making it an involution on V ⊗ V: applying it twice returns every element unchanged.

Eigenvalues and Eigenspace Decomposition

Only Two Possible Eigenvalues

Because τ² = id, the minimal polynomial of τ divides x² − 1 = (x − 1)(x + 1), so the only possible eigenvalues of τ are +1 and −1; when the underlying field does not have characteristic 2, V ⊗ V decomposes as the direct sum of the +1-eigenspace and the −1-eigenspace of τ.

The Symmetric and Antisymmetric Projections

Explicit projections onto these eigenspaces are given by

so that P₊(v ⊗ v′) = ½(v ⊗ v′ + v′ ⊗ v) lands in the +1-eigenspace (the symmetric tensors) and P₋(v ⊗ v′) = ½(v ⊗ v′ − v′ ⊗ v) lands in the −1-eigenspace (the antisymmetric tensors), with P₊ + P₋ = id and P₊P₋ = P₋P₊ = 0 confirming the direct sum decomposition.

Diagram of the Eigenspace Decomposition

Connection to Symmetric and Exterior Squares

Symmetric Square as the +1 Eigenspace

The image of P₊, consisting of all tensors fixed by τ, is precisely the symmetric square Sym²(V), spanned by symmetrized simple tensors v ⊗ v′ + v′ ⊗ v, and coincides with the usual quotient or subspace construction of the second symmetric power of V.

Exterior Square as the −1 Eigenspace

The image of P₋, consisting of all tensors negated by τ, is precisely the exterior square Λ²(V), spanned by antisymmetrized simple tensors v ⊗ v′ − v′ ⊗ v, matching the usual construction of the second exterior power of V as the alternating part of V ⊗ V.

Basis Description of the Swap Map

Action on a Basis

Given a basis {eᵢ} of V, the swap map acts on the basis of V ⊗ V by τ(eᵢ ⊗ eⱼ) = eⱼ ⊗ eᵢ, fixing the diagonal elements eᵢ ⊗ eᵢ (which are eigenvectors with eigenvalue +1) and exchanging each off-diagonal pair eᵢ ⊗ eⱼ with eⱼ ⊗ eᵢ for i ≠ j.

Diagonalization in a Basis-Adapted Form

Relative to the basis obtained by pairing each off-diagonal pair {eᵢ ⊗ eⱼ, eⱼ ⊗ eᵢ} into their symmetric and antisymmetric combinations, together with the fixed diagonal elements eᵢ ⊗ eᵢ, the swap map is diagonal, with eigenvalue +1 on the diagonal elements and each symmetric combination, and eigenvalue −1 on each antisymmetric combination, exhibiting the eigenspace decomposition explicitly in coordinates.

Significance of the Factor Swap Map

Foundation for Symmetric and Alternating Tensor Theory

The factor swap map on V ⊗ V, and its natural generalization to the action of the full symmetric group on V^{⊗n} for higher tensor powers, is the algebraic mechanism by which symmetric and exterior powers are extracted from ordinary tensor powers, underlying the construction of symmetric and alternating multilinear forms used throughout algebra, geometry, and physics.

Distinguishing the Endomorphism Case from the General Symmetry Isomorphism

Because the swap map is an operator on a single space V ⊗ V rather than an isomorphism between two different spaces V ⊗ W and W ⊗ V, it supports a richer structural analysis — eigenvalues, invariant subspaces, projections — that has no direct analogue for the general symmetry isomorphism τ_{V,W} when V and W are distinct.