5.16.4 Tensor Product Formal Class Representative

Tensor Product Formal Class Representative is any single element of the free module F(V × W) chosen to stand for an entire equivalence class under the tensor product's defining relation, serving as the concrete object manipulated in practice whenever an abstract element of V ⊗ W must be written down, computed with, or compared to another tensor. Because V ⊗ W is officially defined as a set of equivalence classes, every actual computation necessarily proceeds by selecting some representative from each relevant class, making the notion of a class representative the bridge between the abstract quotient definition and concrete symbolic manipulation.

What a Class Representative Is

Formal Definition

Given an element t ∈ V ⊗ W, formally the equivalence class [x] ∈ F(V × W)/R for some x ∈ F(V × W), a class representative for t is any element y ∈ F(V × W) satisfying y ∈ [x], equivalently y − x ∈ R.

Representatives Are Not Unique

Because equivalence classes generally contain infinitely many elements once F has more than a trivial number of elements, a given tensor t has infinitely many valid class representatives, all differing from one another only by elements of the relation submodule R.

Choosing a Convenient Representative

The Simple-Tensor Representative

For a rank-one tensor, the most natural representative is a single pair (v, w) corresponding to a chosen factorization v ⊗ w = t. This choice is convenient for computation but is itself non-unique, since (cv, c⁻¹w) represents the same tensor for any nonzero scalar c.

The Basis-Expansion Representative

For a general tensor t, the most standard representative is the formal sum:

in F(V × W) corresponding to t's unique expansion in a fixed pair of bases, which, although still just one representative among many, has the advantage of being computable in a canonical, algorithm-friendly way.

The Minimal-Rank Representative

For applications concerned with the complexity of a tensor, the preferred representative is one achieving the minimal number of terms, that is, a formal sum of length equal to the tensor's rank, which is generally distinct from the basis-expansion representative and often harder to compute.

Verifying Two Representatives Denote the Same Tensor

The Difference-in-R Criterion

Two class representatives x and y denote the same tensor if and only if x − y ∈ R, which in practice is checked by reducing both x and y to their canonical basis-expansion form and comparing coefficients directly, since the basis-expansion representative is unique.

Practical Algorithm for Comparison

Given two formal sums, the standard procedure expands each in a common basis, applies the additivity and homogeneity relations to collect coefficients on each basis tensor eᵢ ⊗ fⱼ, and then checks whether the resulting coefficient arrays match entry by entry.

Diagram of Multiple Representatives for One Class

Why the Notion of Representative Matters

Enabling Any Concrete Symbolic Manipulation

Since abstract elements of V ⊗ W cannot be manipulated directly except through some concrete expression, every algorithm, proof, or computation involving tensors necessarily works with a chosen class representative, making explicit awareness of representative-choice essential to avoid errors caused by assuming a property true of one representative extends automatically to the whole class.

Connection to Representative Independence

The correctness of any operation defined via a chosen representative depends entirely on representative independence, the property that the operation's result does not depend on which representative was picked; this dependency is why representative independence must always be verified whenever an operation on V ⊗ W is defined by referencing "a" representative from F(V × W).

Broader Context

Analogy with Other Quotient Constructions

The relationship between a tensor and its class representatives mirrors the relationship, in other areas of algebra, between an element of a quotient group and its coset representatives, or between a rational number and its representing pairs of integers, illustrating that the representative-versus-class distinction is a general feature of quotient-based constructions rather than something peculiar to tensor products.

Representatives in Iterated Tensor Constructions

The same representative concept extends to iterated tensor products V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ, where a class representative is an element of the corresponding free module on n-tuples, and the analogous notions of simple-tensor, basis-expansion, and minimal-rank representatives all generalize directly from the two-factor case described here.