6.8.1 Tensor Covariant Slot Count

Tensor Covariant Slot Count is the number of distinct argument positions in a tensor's multilinear definition that must each be filled by a vector in order to produce a scalar, this number coinciding with the covariant order but emphasizing the concrete, positional nature of the count rather than its abstract classification role. Where covariant order names the grade of a tensor within a hierarchy of types, the slot count draws attention to the tensor viewed as a function: a machine with a fixed number of open positions, each open position accepting one vector, arranged in a definite left-to-right sequence, with the value of the function depending in general on which vector is placed in which position.

Slots as Positions in a Multilinear Map

The Slot as an Argument Position

A tensor with covariant slot count q is most concretely pictured as a function of q vector arguments, written with q empty positions waiting to be filled. Each position is a slot, and the slot count is simply the number of such positions before any vector has been supplied. A slot is not itself a number or a vector; it is a place where a vector may be inserted, and the tensor's lower indices are the bookkeeping device used to label these places once components are written down in a basis.

Filling Slots Produces a Scalar

Once every one of the q slots receives a vector, the tensor returns a single scalar. Supplying vectors one at a time into successive slots while holding the others fixed produces a linear function of the supplied vector at each stage, this being precisely the multilinearity that defines a covariant tensor. The slot count is exactly the number of vectors that must be supplied before this process terminates in a scalar; supplying fewer leaves a partially evaluated object of lower effective slot count, and supplying none returns the tensor itself unevaluated.

Order Sensitivity of Slots

The slots of a general covariant tensor are ordered, meaning the position into which a vector is placed matters: exchanging the vectors occupying two different slots can, and in general will, change the resulting scalar unless the tensor happens to possess a symmetry that makes it insensitive to that particular exchange. The slot count therefore carries with it an implicit sequence, and two tensors with the same slot count are only interchangeable term for term if their respective slots are matched up in the same order.

Slot Count in Relation to Components and Indices

One Lower Index per Slot

When a covariant tensor is expressed through components in a chosen basis, each slot corresponds to exactly one lower index on the component array, and the slot count equals the number of lower indices present. Filling a slot with a basis vector amounts to setting the corresponding index to the label of that basis vector, and summing over all basis choices for every slot, weighted by the components of the vectors actually supplied, recovers the scalar produced by the tensor.

Slot Count Under Change of Basis

Changing the basis used to describe the vectors supplied to each slot changes the numerical components of the tensor without changing the slot count itself: the number of open positions is a property of the tensor as an abstract multilinear map, entirely independent of any coordinate description, whereas the transformation law relating components in different bases is a separate, secondary consequence of fixing a coordinate system.

Illustrating Slot Structure

Consequences of Fixing the Slot Count

Partial Evaluation and Slot Reduction

Supplying a vector to only one slot of a tensor while leaving the remaining slots open produces a new object whose slot count is one less than the original, since one position has now been permanently occupied and only the remaining positions still await vectors. This partial evaluation is a common construction: it turns, for instance, a covariant slot count two tensor into a covariant slot count one object, namely a one-form, once a single vector has been inserted into one of its two slots.

Slot Count and Tensor Product Combination

When two covariant tensors are combined by tensor product, the slot count of the resulting tensor is the sum of the two original slot counts, since the product tensor is defined by supplying vectors to all of the first tensor's slots followed by all of the second tensor's slots, or any fixed ordering convention between the two groups, without any slot from either factor being removed or shared.

Distinguishing Slot Count From Total Argument Capacity

A tensor of mixed type also possesses upper slots awaiting one-forms in addition to its lower slots awaiting vectors, and the covariant slot count refers strictly to the lower, vector-accepting positions. The total number of arguments the tensor accepts before returning a scalar is the sum of its covariant slot count and its contravariant slot count, but the covariant slot count in isolation measures only the vector-facing half of that total capacity.