4.11.2 Tensor Multilinear Slot Substitution

Tensor Multilinear Slot Substitution is a method to replace slots in tensors using multilinear mappings, preserving structural properties across algebraic operations.

Tensor Multilinear Slot Substitution is the operation of inserting a specific vector or covector into a single designated slot of a multilinear map while leaving every other slot open, producing a new multilinear map of lower rank that still awaits the remaining arguments. It isolates the effect of one argument at a time, and repeated slot substitution across all slots of a tensor is precisely how the tensor multilinear evaluation operation can be decomposed into a sequence of simpler steps.

The Substitution Operation

Fixing a Single Argument

Given a type (p, q) tensor T on a vector space V, slot substitution designates one of its p + q slots and fills it with a specific vector or covector, depending on whether the slot is contravariant or covariant, while the remaining p + q - 1 slots are left as open placeholders. If the substitution fills the k-th covariant slot with a fixed vector v, the result is written as

(⊛_{v} T) (α^{1}, \dots, α^{p}, v_{1}, \dots, v_{k - 1}, v_{k + 1}, \dots, v_{q}) = T (α^{1}, \dots, α^{p}, v_{1}, \dots, v, \dots, v_{q})

where v has replaced the k-th slot and every other argument remains a free variable of the resulting map.

Rank Reduction Produced by Substitution

The map obtained after substituting into one slot, denoted ⊛_v T above, is itself multilinear in its remaining p + q - 1 arguments, and its type has dropped from (p, q) to (p, q - 1) if the filled slot was covariant, or to (p - 1, q) if the filled slot was contravariant. This rank reduction is the defining outcome of the substitution operation, distinguishing it from full evaluation, which fills every slot at once and reduces the type all the way to (0, 0), a scalar.

Substitution as a Building Block of Evaluation

Composing Substitutions

Full evaluation of T on an input tuple can be reconstructed by performing slot substitution once for each argument in turn, feeding the result of one substitution into the next:

T (α^{1}, \dots, v_{q}) = (⊛_{v_{q}} \dots ⊛_{v_{1}} ⊛_{α^{p}} \dots ⊛_{α^{1}} T)

so that repeated substitution, one argument at a time until no slots remain, agrees with directly applying the full evaluation operation to the entire input tuple at once. Multilinearity guarantees that this decomposition into individual substitutions produces the same scalar regardless of the order in which the slots are filled.

Substitution Independence from Slot Order

Because each substitution acts on a slot independent of the others, filling the slots in a different order produces the same intermediate structure up to a relabeling of which slots remain open; the final scalar obtained after all substitutions are performed does not depend on the order chosen, which is a direct consequence of multilinearity holding separately and simultaneously across every slot.

Component View of Substitution

Contracting a Single Index

In component form, relative to a basis e_1, ..., e_n of V and its dual basis e^1, ..., e^n, substituting a vector v = v^j e_j into the k-th covariant slot of T corresponds to contracting the k-th lower index of T against the coordinates v^j of v, leaving the remaining p + q - 1 indices free:

{(⊛_{v} T)}_{j_{1} \dots \hat{j_{k}} \dots j_{q}}^{i_{1} \dots i_{p}} = T_{j_{1} \dots j_{k} \dots j_{q}}^{i_{1} \dots i_{p}} v^{j_{k}}

where the hat over j_k on the left-hand side marks that this index has been removed from the free indices of the resulting lower-rank tensor, having been summed away against v^{j_k} on the right-hand side.

Substitution into Contravariant Slots

Substituting a covector α = α_i e^i into a contravariant slot proceeds identically, contracting the corresponding upper index of T against the coordinates α_i of α, and reducing the count of free upper indices by one.

Uses of Slot Substitution

Defining Associated Linear Maps

Slot substitution into all but one slot of a type (1, 1) tensor recovers the linear map associated to it, since fixing the single vector argument and leaving the covector slot open, or vice versa, produces a linear function of the one remaining argument, matching the standard identification of (1, 1) tensors with linear operators.

Constructing Interior Products and Contractions

Substituting a single vector into one slot of a higher type tensor, while leaving all others open, is the operation underlying the interior product used in exterior algebra and the general notion of contracting a tensor against a fixed vector, both of which reduce the rank of the tensor by exactly one.

Diagrammatic Summary

The diagram shows one slot of the tensor T filled by a fixed vector v, while the surrounding slots remain open, illustrating how slot substitution produces a lower-rank multilinear map that still depends on the unfilled arguments.