4.5.5 Tensor Argument Slot Variation

Tensor Argument Slot Variation is the behavior of a tensor's output as the input element occupying a single designated slot is allowed to range freely over its bound vector space while every other slot is held fixed. It is the complementary operation to slot fixing: where slot fixing removes a slot from consideration by nailing it to one value, slot variation isolates a slot and studies the resulting single-variable linear map that emerges once all other arguments are treated as constants.

Formal Definition

The Variation Map

Given a multilinear map of arity $k$,

and fixed elements $v_1, \ldots, v_{i-1}, v_{i+1}, \ldots, v_k$ for every slot except the $i$-th, the slot variation at position $i$ is the induced map

By the definition of slotwise linearity, $\phi_i$ is guaranteed to be an ordinary linear functional on $V_i$. Slot variation therefore converts the multilinear condition on $T$ into a family of ordinary linear-algebra statements, one for each choice of the fixed arguments.

Variation as the Definition of Multilinearity

Multilinearity itself can be restated entirely in terms of slot variation: $T$ is multilinear precisely when its slot variation at every position, for every choice of the other fixed arguments, is a linear functional. This reformulation makes explicit that multilinearity is a per-slot, not a joint, property.

Properties of Slot Variation

Dependence on the Fixed Context

The linear functional $\phi_i$ obtained by varying slot $i$ depends, in general, on the particular values chosen for the other $k-1$ fixed slots. Different choices of context typically produce different linear functionals, and the collection of all such functionals as the context ranges over every possible combination of fixed values encodes the full multilinear content of $T$.

Linearity of the Dependence on Context

The map sending a fixed context to the resulting linear functional $\phi_i$ is itself linear in each of the other slots, since those other slots are also subject to the original multilinearity condition. This nested linearity is what allows the entire tensor to be reconstructed from its component values via repeated slot variation and expansion in a basis.

Applications of Slot Variation

Verifying Multilinearity in Practice

To check whether a candidate map is genuinely multilinear, it suffices to fix all but one slot, arbitrarily, and confirm additivity and homogeneity in the remaining varying slot, and to repeat this test for each of the $k$ slots in turn. Slot variation is therefore the operational test used to verify the multilinearity condition slot by slot rather than attempting to reason about the map's behavior on the full domain product at once.

Connection to Partial Derivatives

For tensors arising in differential contexts, such as the derivative of a multilinear form along a smooth family of vector fields, varying one slot while holding others fixed is the algebraic analogue of taking a partial derivative with respect to one argument, and it underlies constructions such as the covariant derivative acting on tensor fields, where differentiation is applied one slot at a time.

Recovering Symmetry Properties

Slot variation also provides a direct way to test symmetry: fixing all slots except two specified ones, and comparing the resulting bilinear behavior under exchange of those two varying slots, isolates exactly the symmetry or antisymmetry condition relevant to that particular pair of positions, without needing to consider the remaining fixed slots at all.

Summary of Key Points

• Slot variation isolates one argument slot, holding all others fixed, to produce an ordinary linear functional on that slot's vector space.
• Multilinearity of a tensor is equivalent to the statement that every slot variation, for every fixed context, yields a linear functional.
• The linear functional obtained from varying one slot depends linearly on the values held fixed in the other slots.
• Slot variation is the standard technique for verifying multilinearity and for isolating symmetry properties between specific pairs of slots.
• It parallels the notion of a partial derivative, underlying differential operations that act on tensors one slot at a time.