4.12.2 Tensor Remaining Slot Map

Tensor Remaining Slot Map is the multilinear map obtained as the output of a tensor multilinear partial evaluation operation, defined on precisely the slots that were left unfilled once a chosen subset of arguments has been substituted into a tensor. It is the object that remains available for further evaluation after some arguments have already been supplied, carrying forward exactly the multilinear dependence on whatever slots have not yet received a value.

Identifying the Remaining Slots

Complement of the Fixed Subset

If a type (p, q) tensor T on a vector space V has a subset S of its p + q slots filled by partial evaluation, the remaining slot map is defined on the complementary subset S̄, consisting of every slot not in S:

Each slot in S̄ retains the same variance, contravariant or covariant, that it had in the original tensor T, since partial evaluation only removes slots from consideration by filling them, without altering the type of any slot left open.

The Remaining Slot Map as a Function

The remaining slot map, written T_S, is the assignment sending any tuple of arguments matching the slots in S̄ to the scalar obtained by evaluating T on those arguments together with the arguments already fixed for the slots in S:

Multilinearity of the Remaining Slot Map

Linearity Inherited from the Original Tensor

Because T is multilinear in all of its p + q slots, restricting attention to the slots in S̄, with the slots in S held fixed, preserves linearity in each of the remaining arguments separately: the remaining slot map is itself a multilinear map, of type matching the count of contravariant and covariant slots that belong to S̄.

Independence of the Fixed Arguments' Internal Structure

The remaining slot map depends on the arguments fixed for S only through their specific values, not through any further decomposition of those values; once the fixed arguments are chosen, T_S is completely determined as a function of the open slots, and no additional information about how the fixed arguments were themselves constructed plays any further role.

The Remaining Slot Map as a Tensor

Correspondence with a Lower-Rank Tensor Product Element

Since T_S is multilinear on the slots in S̄, it corresponds to a unique element of the tensor product space built from copies of V and V* matching those slots, exactly as any multilinear map corresponds to an element of the appropriate tensor product space. This correspondence identifies the remaining slot map with a genuine tensor of type equal to the number of contravariant and covariant slots remaining in S̄.

Dependence on the Fixed Arguments

Varying the arguments fixed for the slots in S, while keeping S itself unchanged, produces a family of remaining slot maps, one for each choice of fixed arguments, and this family depends multilinearly on the fixed arguments, mirroring the multilinear dependence that T itself exhibits on all of its arguments jointly.

Composing Remaining Slot Maps

Further Partial Evaluation of the Remaining Map

The remaining slot map T_S can itself be partially evaluated further, by fixing some of the arguments in S̄ and leaving a smaller set of slots open; the resulting map coincides with the remaining slot map that would have been obtained by fixing the union of the original subset S and the newly fixed slots directly on T, confirming that partial evaluation and the formation of remaining slot maps compose consistently.

Full Reduction to a Scalar

If every slot in S̄ is eventually filled, the remaining slot map reduces to a single scalar, and this scalar agrees with the value produced by the full tensor multilinear evaluation operation applied to T on the complete input tuple, assembled from the arguments fixed for S together with the arguments subsequently supplied for S̄.

Uses of the Remaining Slot Map

Encoding Operators Derived from a Tensor

When S fixes all but one or two slots of T, the remaining slot map recovers, respectively, an associated linear operator or bilinear form derived from T, giving a systematic way to extract lower-arity operators from a single higher-rank tensor by choosing which slots to fix.

Describing Families of Constraints or Measurements

In settings where T encodes a physical or geometric quantity, the remaining slot map obtained by fixing certain arguments, such as a direction or a reference frame, describes how the quantity behaves as the remaining, unfixed arguments vary, which is the typical way a tensor is used to model a measurement that depends on some but not all of its natural inputs at any given moment.

Diagrammatic Summary

The diagram shows the remaining slot map T_S accepting arguments only for the slots not in the fixed subset S, having already absorbed the arguments supplied to the slots in S as constants.