13.4.1 Tensor Single Contraction Slot Pair

Tensor Single Contraction Slot Pair is a tensor operation that reduces rank by summing over a single pair of indices.

Tensor Single Contraction Slot Pair is the specific pair of one contravariant slot and one covariant slot that has been selected and linked for the single index contraction operation, denoting the exact two index positions committed to summation within that particular application of the operation.

Identity of the Slot Pair

The Two Positions Comprising the Pair

A single contraction slot pair consists of exactly two index positions on the input tensor, one drawn from among its contravariant slots and one drawn from among its covariant slots, with these two positions being the only ones affected by the operation to which the pair belongs.

T_{j k}^{i l}

Here, a slot pair might consist of the contravariant slot holding $i$ together with the covariant slot holding $j$ , distinguishing this particular pairing from other pairings that could instead be formed from the same tensor's available slots.

Representation Through a Shared Symbol

Once identified, the slot pair is represented notationally by assigning both of its member positions the same index symbol, a convention that visually and formally marks these two slots as linked while leaving unpaired slots with their own distinct symbols.

T_{i k}^{i l}

Requirements Defining a Valid Slot Pair

Opposite Variance Between Members

A valid slot pair requires that its two member positions carry opposite variance, since a pairing of two contravariant slots or two covariant slots does not correspond to a defined contraction and therefore does not constitute a valid slot pair.

type (i) \neq type (j)

Equal Dimension Between Members

A valid slot pair requires that its two member positions range over vector spaces of the same dimension, since summing components across positions of differing dimension does not correspond to a well-defined operation.

Both Positions Genuinely Present on the Tensor

A valid slot pair requires that both of its member positions actually exist as slots on the input tensor, since a pairing referencing a position beyond the tensor's actual order fails to identify a genuine slot pair.

Role of the Slot Pair Within the Operation

Determining What Is Summed

The slot pair fixes exactly which two positions are set equal and summed during index summation, so identifying the slot pair is a necessary prerequisite for carrying out that summation, since the summation cannot proceed without first knowing which positions it concerns.

Determining What Remains Free

Every index position on the input tensor not belonging to the selected slot pair remains a free index of the output tensor, so the identity of the slot pair simultaneously determines both what is removed by the operation and what survives into its result.

Distinguishing a Slot Pair from Other Available Pairings

One Pairing Among Several Possibilities

When a tensor carries more than one contravariant or more than one covariant index, several different slot pairs may be available, each producing a generally different result, so identifying a specific slot pair distinguishes one legitimate application of the operation from another equally legitimate but distinct application on the same tensor.

T_{k}^{i} \neq T_{l}^{i}

Relationship to Tensor Operation Notation

The single contraction slot pair is made explicit in tensor operation notation through the deliberate reuse of one index symbol for exactly one upper and one lower position, with the specific choice of which two positions receive the shared symbol being precisely how notation identifies the slot pair intended for a given single index contraction operation.