4.4.4 Tensor Higher Arity Output Assignment

Tensor Higher Arity Output Assignment explores how multi-linear operations produce structured outputs in higher-dimensional algebraic frameworks.

Tensor Higher Arity Output Assignment is the rule by which a higher-arity multilinear map assigns a definite value, drawn from a fixed target space, to every tuple of inputs supplied across its argument slots. It is the "codomain side" of the multilinear map structure: once all $k$ slots of an arity-$k$ map are filled, the output assignment specifies exactly what element of the target space, whether a scalar field or a vector space, is produced, and it does so in a way that is uniquely determined by slotwise linearity.

Formal Definition

The Assignment Map

For a multilinear map of arity $k$,

T : V_{1} \times \dots \times V_{k} \to W

the output assignment is the correspondence

(v_{1}, \dots, v_{k}) \mapsto T (v_{1}, \dots, v_{k}) \in W

When $W = F$, the base field, $T$ is a scalar-valued multilinear map (the ordinary case of a covariant tensor evaluated on vectors, or a contravariant tensor evaluated on covectors). When $W$ is itself a vector space, $T$ is a vector-valued multilinear map, which is the setting for objects such as the torsion or curvature tensor viewed as producing a vector output rather than a number.

Well-Definedness of the Assignment

Because $T$ is required to be slotwise linear, the output assignment is completely determined once its value is known on all tuples of basis vectors drawn from each $V_i$. Any other tuple's output is fixed by the multilinear expansion of the domain product's basis decomposition; there is no freedom to assign outputs independently once the values on basis tuples, i.e. the components of the tensor, have been fixed.

Structural Aspects of the Assignment

Components as the Basic Output Data

Fixing a basis ${e_1, \ldots, e_n}$ for each domain factor, the output assignment restricted to basis tuples defines the components of the tensor:

T_{i_{1} \dots i_{k}} = T (e_{i_{1}}, \dots, e_{i_{k}})

The full output assignment on arbitrary tuples is then reconstructed as a finite weighted sum of these component values, with weights given by the coordinates of each argument in the chosen basis.

Codomain Choice and Tensor Type

The choice of codomain $W$ interacts with the type of the tensor being represented. A purely scalar output assignment, $W = F$, corresponds to viewing the tensor as a fully contracted object; allowing $W$ to retain one or more free vector factors corresponds to a tensor that has not been fully saturated by its arguments, i.e. a tensor of nonzero total valence remaining after some indices have been filled and others left open.

Output Assignment Under Tensor Operations

Effect of Contraction

Contraction changes the output assignment indirectly: by summing the original map's values over matched pairs of basis vectors in two slots, contraction produces a new, lower-arity map whose output assignment is a derived quantity, no longer requiring those two slots to be filled by explicit arguments.

Effect of Composition with Linear Maps

If $L : W \to W'$ is linear, composing gives a new higher-arity map $L \circ T$ whose output assignment on every input tuple is simply $L$ applied to the original assignment. This shows that the output assignment transforms covariantly and predictably under any linear post-processing of the codomain, which is the basis for how tensors change under change of frame or change of basis in the output space.

Consistency Under Symmetrization

When an output assignment is required to be symmetric or antisymmetric under permutation of input slots, the assignment rule itself must satisfy the corresponding symmetry equation for every tuple, not merely for basis tuples; slotwise linearity guarantees that verifying this on basis tuples is sufficient to guarantee it for all tuples.

Summary of Key Points

The output assignment is the rule sending each filled tuple of arguments to a definite element of the codomain.
It is fully determined by its values on basis tuples, i.e. by the components of the tensor.
The choice of scalar versus vector-valued codomain distinguishes fully contracted tensors from tensors with unsaturated valence.
Contraction and composition with linear maps act predictably on the output assignment, preserving its well-definedness.
Symmetry conditions on the output assignment need only be checked on basis tuples due to slotwise linearity.