13.7.2 Tensor Partial Contraction Remaining Slot Set

Tensor Partial Contraction Remaining Slot Set is the collection of index positions that remain free and uncontracted after a partial contraction has been applied to a tensor, together with the variance type, ordering, and range associated with each of those positions. It characterizes what is left of a tensor's index structure once some, but not all, of its indices have been paired and summed, describing precisely which slots still require a value or further index assignment before the object can be fully evaluated.

Conceptual Basis

Slots Before and After Contraction

Every tensor index occupies a distinct slot with an associated variance, either contravariant or covariant, and an associated dimension. Partial contraction removes exactly two slots per contraction performed, one contravariant and one covariant, while the remaining slot set consists of every slot not involved in any of the pairings carried out.

Why the Remaining Set Matters

The remaining slot set determines the rank, variance signature, and transformation behavior of the resulting tensor. Unlike full contraction, which eliminates all slots and yields a scalar, partial contraction produces an object that still transforms under a change of basis according to the indices left in this remaining set.

Relationship to Rank Reduction

If a tensor of type $$(p,q)$$ undergoes $$k$$ contractions, the remaining slot set has cardinality:

with the contravariant and covariant portions of this set tracked separately.

Structure of the Remaining Slot Set

Variance Composition

The remaining slot set is partitioned into an upper subset and a lower subset. The upper subset contains all contravariant slots not consumed by contraction, and the lower subset contains all covariant slots not consumed. This partition determines the resulting tensor's type.

Positional Ordering

Slots in the remaining set retain a relative ordering inherited from the original tensor, since the position of an index within a tensor expression affects how it interacts with subsequent operations such as further contraction, multiplication, or symmetrization.

Dimensional Labeling

Each slot in the remaining set carries the dimension of the vector space it ranges over, which must be tracked explicitly when the tensor mixes indices associated with spaces of different dimension.

Example

A Rank-Four Tensor Under Partial Contraction

Consider a tensor $$T_{kl}^{ij}$$ contracted over the pair $$i,k$$:

Here the remaining slot set consists of the upper slot $$j$$ and the lower slot $$l$$, so the resulting object is a tensor of type $$(1,1)$$.

Consequences for Further Operations

Compatibility With Additional Contractions

Any further contraction applied to the resulting tensor must draw its index pairs exclusively from the remaining slot set, since the slots already consumed no longer exist as accessible positions.

Transformation Behavior

The remaining slot set dictates how the resulting tensor transforms under a change of basis: each upper slot contributes a factor from the Jacobian of the coordinate transformation, and each lower slot contributes a factor from its inverse, while contracted slots contribute no such factors since they no longer appear.

Interfacing With Tensor Products

When a partially contracted tensor is combined with another tensor through an outer product, the remaining slot set of the first tensor is simply appended to the full slot set of the second, producing a combined index structure that reflects both objects' uncontracted positions.

Distinguishing the Remaining Slot Set From the Contracted Slot Set

Contracted Slot Set

The contracted slot set records which slots were paired and summed away, and is relevant chiefly for verifying that dimensional compatibility and variance matching were respected during the contraction.

Remaining Slot Set

The remaining slot set records what is left to work with, and is the structure that governs the type, rank, and further usability of the tensor produced by the partial contraction.

Complementary Relationship

The contracted slot set and the remaining slot set are complementary and together account for every index slot present in the original tensor before any contraction was applied.