4.5.4 Tensor Argument Slot Fixing

Tensor Argument Slot Fixing is the operation of assigning a definite, unchanging input element to one or more of a tensor's argument slots while leaving the remaining slots open to vary, thereby producing a new multilinear map of lower arity from an original higher-arity tensor. It is the formal mechanism behind the intuitive idea of "plugging in" some but not all of a tensor's arguments, and it is the operation that makes the very definition of slotwise linearity meaningful in the first place.

Formal Definition

Fixing a Single Slot

Given a multilinear map of arity $k$,

fixing the $i$-th slot at a chosen element $a \in V_i$ produces the map

The resulting object $T_a$ is a multilinear map of arity $k - 1$, since slot $i$ no longer varies and has effectively been removed from the argument slot structure.

Fixing Multiple Slots

Fixing $m$ of the $k$ slots at chosen elements produces a multilinear map of arity $k - m$. Iterating this process down to $m = k$ leaves no free slots, and the result is a single scalar: the value of $T$ on one specific tuple.

Why Fixing Preserves Multilinearity

Inheritance from Slotwise Linearity

Because the original map $T$ is linear in each remaining slot independently of what values occupy the fixed slots, the reduced map $T_a$ automatically inherits linearity in every slot that was not fixed. No additional verification is needed: slot fixing never disturbs the linearity already guaranteed to hold in the untouched positions.

Dependence on the Fixed Value

The resulting lower-arity map $T_a$ depends linearly on the fixed element $a$ itself, precisely because slot $i$ was also subject to the original multilinearity condition before being fixed:

This shows slot fixing is itself a linear operation on the fixed argument, which is why the map sending $a$ to $T_a$ can itself be regarded as a linear map into the space of arity $(k-1)$ multilinear maps.

Applications Within Tensor Algebra

Currying the Tensor Structure

Iteratively fixing one slot at a time transforms an arity-$k$ tensor into a chain of linear maps: fixing the first slot yields a linear map from $V_1$ into the space of arity $(k-1)$ multilinear maps on the remaining slots, and repeating this process is the tensor-theoretic analogue of currying a multi-argument function into a sequence of single-argument functions.

Relation to Contraction

Slot fixing at a basis element, summed appropriately against a paired dual basis element in another slot, is the elementary building block from which the contraction operation is assembled: contraction can be understood as fixing pairs of matched slots simultaneously at corresponding dual basis elements and summing the results, rather than fixing at one arbitrary chosen value.

Partial Evaluation in Computation

In any computational or symbolic setting where a higher-arity tensor is represented, slot fixing corresponds to partial application: supplying some but not all of the tensor's arguments and obtaining, as output, a new tensor object of reduced arity rather than a final numeric result.

Summary of Key Points

• Slot fixing assigns a definite value to one or more argument slots, reducing the arity of the resulting multilinear map by the number of slots fixed.
• The reduced map automatically inherits linearity in every slot that was not fixed, without additional verification.
• The reduced map also depends linearly on the value used to fix each slot, since the original map was linear there too.
• Iteratively fixing every slot reduces a tensor to a single scalar, the value of the original map on one specific input tuple.
• Slot fixing underlies both the currying of multilinear maps into chains of linear maps and the elementary construction of contraction.