12.9.1 Tensor Evaluation Input Slot Selection

Tensor Evaluation Input Slot Selection is the process of specifying exactly which of a tensor's multiple argument positions, whether covariant slots accepting vectors or contravariant slots accepting covectors, will receive a particular input when the tensor is evaluated, ensuring that each supplied argument is matched to the correct slot for the operation to be well defined.

The Nature of Slots

Slots Correspond to Indices

A tensor of type $$(p,q)$$, viewed as a multilinear map, has exactly $$p$$ slots designated to accept covectors and $$q$$ slots designated to accept vectors. Each such slot corresponds directly to one of the tensor's indices, with contravariant indices pairing with covector slots and covariant indices pairing with vector slots.

Distinguishing Slot Types

Because covector slots and vector slots accept fundamentally different kinds of objects, selecting an input for evaluation requires identifying not only which numbered slot is being filled but also whether that slot demands a vector or a covector, since supplying the wrong kind of object to a slot does not produce a valid evaluation.

Selecting a Slot for Evaluation

Explicit Slot Designation

When evaluating a tensor on fewer than all of its arguments, as occurs during partial evaluation, slot selection specifies precisely which position among the $$p+q$$ total slots receives the supplied vector or covector, while the remaining positions are left open:

Here the vector $$v$$ is explicitly placed into a particular slot, distinguishing this choice from placing $$v$$ into a different slot, which would generally produce a different resulting tensor.

Sensitivity to Which Slot Is Chosen

Because tensor components are generally not symmetric across their indices, filling one slot with a given vector typically produces a different result than filling a different slot with that same vector, so slot selection carries genuine significance for the outcome of the evaluation.

Componentwise Consequence of Slot Selection

Contraction Along the Chosen Index

Selecting a particular slot for evaluation corresponds, in component form, to contracting the tensor's array along the index associated with that specific slot, while leaving every other index free:

if the last covariant slot was the one selected. Selecting a different slot instead would contract against a different index of $$A$$, generally producing a distinct resulting tensor even when the same vector $$v$$ is used.

Practical Importance of Careful Slot Selection

Avoiding Ambiguity in Notation

When a tensor's evaluation is written with explicit placeholders or numbered slots, careful slot selection ensures that any reader or subsequent computation correctly identifies which vector or covector corresponds to which index of the original tensor, preventing ambiguity about which contraction was actually performed.

Consistency in Multi-Step Computations

In a longer computation involving several successive partial evaluations, keeping track of which slots have already been filled and which remain open is essential for correctly composing further evaluations, since each new input must be directed to one of the tensor's still-open slots.

Illustration