6.9.1 Tensor Contravariant Slot Count

Tensor Contravariant Slot Count is the number of distinct argument positions in a tensor's multilinear definition that must each be filled by a one-form in order to produce a scalar, this number coinciding with the contravariant order but emphasizing the concrete, positional nature of the count rather than its abstract classification role. Where contravariant 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 one-form, arranged in a definite sequence, with the value of the function depending in general on which one-form is placed in which position.

Slots as Positions Accepting One-Forms

The Slot as an Argument Position

A tensor with contravariant slot count p is most concretely pictured as a function of p one-form arguments, written with p empty positions waiting to be filled. Each position is a slot, and the slot count is simply the number of such positions before any one-form has been supplied. The tensor's upper indices are the bookkeeping device used to label these places once components are written down in a basis, but the slots themselves exist independently of any such labeling.

Filling Slots Produces a Scalar

Once every one of the p slots receives a one-form, the tensor returns a single scalar. Supplying one-forms one at a time into successive slots while holding the others fixed produces a linear function of the supplied one-form at each stage, this being precisely the multilinearity that defines a contravariant tensor. The slot count is exactly the number of one-forms that must be supplied before the process terminates in a scalar.

Order Sensitivity of Slots

The slots of a general contravariant tensor are ordered, meaning the position into which a one-form is placed matters: exchanging the one-forms occupying two different slots can change the resulting scalar unless the tensor possesses a symmetry making it insensitive to that exchange. The slot count therefore carries 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 Upper Index per Slot

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

Slot Count Under Change of Basis

Changing the basis, and hence the dual basis used to describe the one-forms 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 consequence of fixing a coordinate system.

Illustrating Slot Structure

Consequences of Fixing the Slot Count

Partial Evaluation and Slot Reduction

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

Slot Count and Tensor Product Combination

When two contravariant 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 one-forms to all of the first tensor's slots followed by all of the second tensor's slots, according to a fixed ordering convention, without any slot from either factor being removed or shared.

Distinguishing Slot Count From Total Argument Capacity

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