5.19.2 Tensor Product Vector Space Factor Order

Tensor Product Vector Space Factor Order is the question of what happens to the tensor product V ⊗ W when the two factor spaces are written in the opposite order, W ⊗ V, and more generally, the convention governing how the sequence of factors in an iterated tensor product V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ is tracked and interpreted. Although V ⊗ W and W ⊗ V turn out to be isomorphic, they are not identical as constructed objects, and factor order matters both for the precise bookkeeping of basis tensors and for keeping track of which slot of a multilinear map corresponds to which vector space.

The Swap Isomorphism

Constructing the Isomorphism

There is a canonical isomorphism τ: V ⊗ W → W ⊗ V, called the swap or twist map, defined on simple tensors by:

and extended linearly to all of V ⊗ W. This map is well-defined precisely because the assignment (v, w) ↦ w ⊗ v is bilinear in (v, w), and hence factors uniquely through V ⊗ W by the universal property.

Verifying τ Is an Isomorphism

The swap map τ: V ⊗ W → W ⊗ V has an evident inverse, the analogous swap map τ': W ⊗ V → V ⊗ W sending w ⊗ v ↦ v ⊗ w; composing the two in either order yields the identity on simple tensors, and hence on the whole space by linearity, confirming τ is indeed an isomorphism.

Why V ⊗ W and W ⊗ V Are Not Identical

Distinct as Constructed Objects

Although isomorphic, V ⊗ W and W ⊗ V are built from different free modules, F(V × W) versus F(W × V), and the canonical bilinear maps ⊗_{V,W} and ⊗_{W,V} have different domains, so the two spaces are equal only after explicitly applying the swap isomorphism, not automatically as sets.

Practical Consequence for Basis Ordering

Given bases {eᵢ} for V and {fⱼ} for W, the basis tensor eᵢ ⊗ fⱼ in V ⊗ W corresponds under τ to fⱼ ⊗ eᵢ in W ⊗ V; failing to track this correspondence correctly, for instance while converting between matrix representations of a tensor, can produce a transposed or otherwise incorrectly indexed result.

Factor Order in Iterated Tensor Products

Fixing a Convention for n Factors

For an iterated tensor product V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ, the order of the factors determines the order in which indices are listed on a basis tensor e_{i₁} ⊗ e_{i₂} ⊗ ... ⊗ e_{iₙ}, and this ordering convention must be fixed and maintained consistently throughout any computation involving multi-index tensors.

Permuting Factors via Generalized Swap Maps

Any permutation σ of the n factors induces a corresponding isomorphism between V₁ ⊗ ... ⊗ Vₙ and V_{σ(1)} ⊗ ... ⊗ V_{σ(n)}, built by composing elementary swap maps of adjacent factors, generalizing the two-factor swap isomorphism to the setting of arbitrarily many tensor factors.

Diagram of the Swap Isomorphism

Cases Where the Factor Order Distinction Disappears

Tensor Powers of a Single Space

When V = W, the swap map τ: V ⊗ V → V ⊗ V becomes an automorphism of the same space rather than a map between two different spaces; this automorphism is precisely the map used to define the symmetric and antisymmetric parts of V ⊗ V, since a tensor t satisfies τ(t) = t if and only if it lies in the symmetric part, and τ(t) = -t if and only if it lies in the antisymmetric part.

Symmetric Tensors as Order-Independent Elements

An element t ∈ V ⊗ V for which factor order genuinely does not matter, meaning τ(t) = t, is called a symmetric tensor; such elements form a proper subspace of V ⊗ V, illustrating that "not caring about factor order" is a special property of certain tensors rather than a general feature of the tensor product construction.

Broader Significance

Distinguishing Isomorphism from Identity

The relationship between V ⊗ W and W ⊗ V is a clear illustration of a broader principle in algebra: canonical isomorphism between two constructions does not mean the constructions are literally the same object, and careful bookkeeping of the specific isomorphism used, here the swap map, is necessary whenever results are transported between the two orderings.

Foundation for Tensor Index Conventions in Applications

In physics and multilinear algebra, where tensors are frequently written with explicit index notation, factor order corresponds directly to index order, and the swap isomorphism corresponds to the operation of permuting tensor indices, a routine but essential manipulation when comparing tensors expressed with different index conventions.