4.12.1 Tensor Fixed Slot Evaluation

Tensor Fixed Slot Evaluation is the data of which arguments have been supplied to which slots of a tensor during partial evaluation, together with the specific vectors or covectors chosen for each fixed slot, considered as the input side of the partial evaluation operation rather than the reduced map it produces. It complements the tensor remaining slot map, which describes what is left over after fixing, by describing precisely what was fixed and where, and it is this data that must be specified before the remaining slot map can even be defined.

Specifying a Fixed Slot Evaluation

The Two Ingredients: Location and Value

A fixed slot evaluation for a type (p, q) tensor T on a vector space V consists of a subset S of the p + q slots, together with, for each slot r in S, a specific argument x_r of the appropriate kind, a covector if r is contravariant or a vector if r is covariant:

Both pieces of information, which slots are fixed and what values fill them, are required together; the subset S alone does not determine a fixed slot evaluation, since different choices of values x_r for the same subset S produce entirely different partial evaluations of T.

Validity of a Fixed Slot Evaluation

A fixed slot evaluation is valid only if every value x_r matches the variance required by its slot: supplying a vector to a contravariant slot, which expects a covector, or a covector to a covariant slot, which expects a vector, does not correspond to a well-defined fixed slot evaluation, since the pairing needed to contract that slot against T would not be defined.

The Fixed Slot Data as an Element of a Product Space

Membership in the Corresponding Product

The collection of values {x_r}_{r ∈ S} fixed for the slots in S is precisely an element of the Cartesian product of the argument spaces corresponding to those slots, matching the same kind of Cartesian product structure that describes a full tensor multilinear input tuple, restricted here to only the slots in S.

Equality of Fixed Slot Evaluations

Two fixed slot evaluations over the same subset S are considered equal exactly when their values agree in every fixed slot; if either the subset S differs or any individual value x_r differs, the two fixed slot evaluations are distinct, even if the resulting remaining slot maps happen to coincide as functions.

Effect of the Fixed Slot Data on the Resulting Map

Linear Dependence on Each Fixed Value

Holding the subset S constant, the remaining slot map produced by a fixed slot evaluation depends linearly on each individual value x_r, since T is multilinear in every slot, including those in S; scaling one fixed value by a constant scales the resulting remaining slot map by the same constant, and replacing one fixed value by a sum of two arguments replaces the resulting map by the sum of the two maps obtained from each argument separately.

Multilinear Dependence Across All Fixed Slots Jointly

More generally, the assignment sending an entire collection of fixed values {x_r}_{r ∈ S} to the resulting remaining slot map is itself multilinear in the fixed values taken together, mirroring the multilinearity of T on those slots before any of the remaining slots have been considered at all.

Fixed Slot Evaluation in Component Form

Coordinates of the Fixed Values

Once a basis is chosen for V and its dual, each fixed value x_r decomposes into coordinates relative to that basis, and the fixed slot evaluation as a whole is described by one coordinate array per fixed slot, together with the specification of which slot each array belongs to.

Contracting Only the Fixed Indices

In the component formula for the resulting remaining slot map, only the indices corresponding to slots in S are contracted against the coordinates of the fixed values, while the indices corresponding to the open slots are left free, exactly matching the general pattern used in tensor multilinear slot substitution applied simultaneously to every slot in S.

Choosing Fixed Slot Evaluations in Practice

Fixing Slots to Extract Specific Behavior

A fixed slot evaluation is often chosen deliberately to extract a particular kind of behavior from a tensor, such as fixing a direction to study how a physical quantity varies along that direction, or fixing a reference covector to study how a quantity responds to a particular kind of measurement, with the specific values x_r chosen according to the question being asked of the tensor.

Systematic Enumeration via Basis Values

When the fixed values are systematically taken to be basis vectors and basis covectors, ranging over every combination for the slots in S, the resulting collection of remaining slot maps enumerates exactly the components of T associated to those slots, which is the same mechanism used to build up the full component array of a tensor one fixed slot evaluation at a time.

Diagrammatic Summary

The diagram shows the fixed slot evaluation as a pairing between each slot in the chosen subset S and a specific value supplied for it, which together determine the remaining slot map obtained by partial evaluation.