5.21.1 Tensor Product Triple Factor Grouping

Tensor Product Triple Factor Grouping is the specific instance of tensor product associativity concerned with three vector spaces U, V, W, namely the choice of which two of the three factors are combined first — (U ⊗ V) ⊗ W or U ⊗ (V ⊗ W) — and the canonical isomorphism identifying these two groupings as the same underlying space. As the smallest nontrivial case of iterated tensor products, triple factor grouping is the foundational example from which the general associativity structure, and its coherence for even more factors, is built and understood.

The Two Groupings of Three Factors

Left Grouping

The left grouping first forms U ⊗ V and then tensors the result with W, producing (U ⊗ V) ⊗ W, whose simple tensors have the nested form (u ⊗ v) ⊗ w.

Right Grouping

The right grouping first forms V ⊗ W and then tensors U with the result, producing U ⊗ (V ⊗ W), whose simple tensors have the nested form u ⊗ (v ⊗ w).

No A Priori Reason the Two Coincide

Before any argument is given, (U ⊗ V) ⊗ W and U ⊗ (V ⊗ W) are, syntactically, two different spaces built by two different sequences of pairwise tensor product constructions; triple factor grouping is precisely the study of why and how these two different-looking constructions turn out to be canonically the same object.

The Canonical Identification

The Grouping Isomorphism

There is a unique linear isomorphism α : (U ⊗ V) ⊗ W → U ⊗ (V ⊗ W) determined by its action on simple tensors,

matching each nested simple tensor on the left grouping with the corresponding nested simple tensor on the right grouping, factor by factor.

Two-Step Universal Property Argument

The isomorphism is constructed in two applications of the universal property: for fixed w, the assignment (u, v) ↦ u ⊗ (v ⊗ w) is bilinear and induces a linear map on U ⊗ V; letting w vary produces a bilinear map ((U ⊗ V), W) → U ⊗ (V ⊗ W), which the universal property of (U ⊗ V) ⊗ W then converts into the single linear map α. An entirely symmetric construction produces the inverse map, confirming α is bijective.

Diagram of Triple Factor Grouping

Consequences for Simple and General Tensors

Simple Tensors Correspond Termwise

A triple simple tensor u ⊗ v ⊗ w, understood via either grouping, corresponds under α to itself in the other grouping, so simple tensors are unambiguous regardless of grouping; only intermediate bookkeeping, not the final content, differs between the two constructions.

Bases Match Under Grouping

If {eᵢ}, {fⱼ}, {gₖ} are bases of U, V, W, the basis {(eᵢ ⊗ fⱼ) ⊗ gₖ} of the left grouping corresponds exactly, index by index, to the basis {eᵢ ⊗ (fⱼ ⊗ gₖ)} of the right grouping under α, so dimension counting and coordinate assignment are identical no matter which grouping is used to construct the space.

Extending Grouping to More Factors

The Pentagon for Four Factors

With a fourth factor X, there are five distinct groupings of U ⊗ V ⊗ W ⊗ X, and the coherence theorem for tensor products (Mac Lane's pentagon condition) guarantees that any two chains of grouping isomorphisms connecting two groupings agree, so triple factor grouping, iterated, extends consistently without hidden ambiguity to any number of factors.

Triple Grouping as the Base Case

Because every higher-order regrouping can be decomposed into a sequence of individual triple-factor regroupings (reassociating three factors at a time within a longer product), the triple factor grouping isomorphism α is the atomic building block from which the associativity of tensor products with any number of factors is assembled.

Significance of Triple Factor Grouping

Licensing Unparenthesized Triple Products

Triple factor grouping is what justifies writing U ⊗ V ⊗ W without parentheses at all, since the two possible parenthesizations are canonically identified; this notational simplification, taken for granted throughout multilinear algebra, rests entirely on the existence and coherence of the grouping isomorphism α.

Minimal Case for Understanding Coherence

Because it is the simplest case involving more than two factors, triple factor grouping serves as the essential worked example through which the more abstract coherence conditions governing tensor products of arbitrarily many factors are first understood and verified.