5.21.4 Tensor Product Factor Reassociation

Tensor Product Factor Reassociation is the operational process of transforming one parenthesization of a multi-factor tensor product V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ into a different parenthesization by applying the triple-factor associativity isomorphism locally, at a chosen position within the chain, while leaving the surrounding factors untouched via tensoring with identity maps. Reassociation is the concrete mechanism by which the abstract fact of associativity is actually put to use in longer tensor products, converting a general regrouping problem into a sequence of elementary, local moves.

The Elementary Reassociation Move

Local Application of the Associative Isomorphism

Within a longer product, say V₁ ⊗ (V₂ ⊗ V₃) ⊗ V₄, an elementary reassociation move applies the associative isomorphism α_{V₂,V₃,·} only to the three consecutive factors V₂, V₃, and the fourth position, while the untouched factors are carried along via the identity, giving

converting V₁ ⊗ ((V₂ ⊗ V₃) ⊗ V₄)-style grouping at that position into the alternative local grouping, without affecting how V₁ or V₄ participate in the rest of the expression.

Locality of the Move

Because the elementary move is id ⊗ α ⊗ id (with identities on all factors outside the three being regrouped), it changes only the internal bracketing at the chosen position and acts as the identity everywhere else, which is what makes reassociation a genuinely local operation rather than one that requires reconstructing the whole tensor product from scratch.

Chaining Moves to Reach Any Parenthesization

Decomposing a Global Regrouping into Local Steps

Given any two parenthesizations of V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ, a finite sequence of elementary reassociation moves, each applied at some position in the chain, transforms one parenthesization into the other. This is because any full binary tree structure (representing a parenthesization) can be converted into any other full binary tree on the same leaves by a sequence of local rotations, and each such rotation corresponds exactly to one elementary reassociation move.

Composite Reassociation Isomorphism

The composite of the chosen sequence of elementary moves is itself a linear isomorphism between the two parenthesizations, and by the pentagon coherence condition, this composite isomorphism does not depend on which particular sequence of elementary moves was chosen to get from one parenthesization to the other — any valid sequence yields the same overall map.

Diagram of a Reassociation Move

Reassociation and Naturality

Compatibility with Maps on the Factors

Reassociation moves commute with tensor products of linear maps on the individual factors, by naturality of α: applying f₁ ⊗ f₂ ⊗ f₃ ⊗ f₄ before or after a reassociation move gives the same result, since the move itself is built from α and identities, both of which are natural. This ensures reassociation can be freely interleaved with applying transformations to the factors, in either order.

Reassociation Preserves Simple Tensors

A simple tensor v₁ ⊗ v₂ ⊗ v₃ ⊗ v₄, however it happens to be parenthesized, is sent by any reassociation move to the same simple tensor under the alternative parenthesization; reassociation never mixes simple tensors into sums, since each elementary move acts on simple tensors by relabeling their grouping only, matching the local formula α((u ⊗ v) ⊗ w) = u ⊗ (v ⊗ w).

Reassociation in Coordinates

Basis Elements Track the Same Index Pattern

If bases are chosen for each factor, an elementary reassociation move sends a basis simple tensor to the basis simple tensor with the same sequence of indices under the new grouping, so the coordinate array describing a general tensor is unaffected in content, only in the bracketing convention used to describe its indices.

Significance of Factor Reassociation

The Practical Engine Behind Associativity

While the associativity structure of the tensor product states, abstractly, that all parenthesizations of a product are canonically isomorphic, factor reassociation is the concrete procedure — an explicit, composable sequence of local moves — by which one actually converts an expression from one grouping to another in a calculation, making associativity usable rather than merely true in principle.

Foundation for Manipulating Long Tensor Expressions

Reassociation is used implicitly whenever a long tensor product expression is regrouped mid-calculation, for instance to apply a map to a particular pair of adjacent factors or to match the grouping conventions of two different expressions being compared, and its validity rests entirely on the coherence of the underlying associative isomorphism.