4.1.3 Tensor Multilinear Map Argument Slot Structure

Tensor Multilinear Map Argument Slot Structure organizes inputs and outputs in algebra, enabling structured representation of multilinear operations through indexed slots.

Tensor Multilinear Map Argument Slot Structure is the fine-grained description of the individual positions, or slots, within a tensor's multilinear map presentation, covering how each slot is typed, indexed, filled, and permuted, and how the collection of slots as a whole governs contraction, symmetrization, and coordinate expression of the tensor.

Anatomy of a Slot

Definition of a Slot

Given a type (p, q) tensor T realized as a multilinear map on p copies of V* and q copies of V, each of the p + q argument positions is called a slot. A slot is characterized by two pieces of data: its type, either covector-accepting (V*) or vector-accepting (V), and its position index within the ordered list of arguments, conventionally written with upper indices for covector slots and lower indices for vector slots,

T (\underset{slot 1}{\dots}, \underset{slot 2}{\dots}, \dots, \underset{slot p+q}{\dots})

Slot Filling and Component Extraction

Filling every slot with a basis element, basis covectors e^i for the p covector slots and basis vectors e_j for the q vector slots, extracts a single scalar component of T. Systematically filling all slots with every combination of basis elements enumerates the full n^(p+q) component array, so the slot structure is the multilinear-map-level counterpart of the tensor's index structure in coordinates.

Slot Typing and Index Placement

Upper Versus Lower Slots

A slot of type V* is filled by a covector and, in components, produces an upper (contravariant) index, because the tensor evaluated on a basis covector e^i returns the coefficient attached to the corresponding basis vector direction. A slot of type V is filled by a vector and produces a lower (covariant) index, since the tensor evaluated on a basis vector e_j effectively measures the tensor's covariant response in that direction. This assignment is fixed at the time the tensor is defined and cannot be reassigned without producing a different multilinear map.

Slot Order and Non-Commutativity

Unless T possesses an explicit symmetry, the value produced by filling slot i with one covector and slot j with another is generally different from filling slot j with the first and slot i with the second, when i and j are slots of the same type. Formally, for a type (0, 2) tensor,

T (u, v) \neq T (v, u)

in general, and the slot structure is precisely what records which argument occupies which position so that this asymmetry is well-defined and reproducible.

Operations on Slots

Slot Permutation

Permuting the same-typed slots of a tensor's multilinear map produces a new tensor, related to the original by relabeling of indices. For a permutation σ of the q vector slots, the permuted tensor is defined by

(σ \cdot T) (v_{1}, \dots, v_{q}) = T (v_{σ (1)}, \dots, v_{σ (q)})

Symmetrizing or antisymmetrizing over a group of same-typed slots is done by averaging over all such permutations, with a sign factor in the antisymmetric case, and produces the symmetric or alternating part of the tensor restricted to those slots.

Slot Contraction

Contraction identifies one covector slot with one vector slot: fixing all other slots and summing the result of feeding in matched basis pairs across the chosen slots removes both slots from the argument list and lowers the valence by (1, 1). In index notation this is the familiar operation of setting an upper index equal to a lower index and summing,

T_{j \dots}^{i \dots} \to T_{i \dots}^{i \dots}

which at the level of slots corresponds to inserting the same basis element into both a covector slot and a vector slot and summing over the basis index.

Partial Slot Evaluation

Leaving some slots open while filling others with fixed vectors or covectors produces a new multilinear map on the remaining, unfilled slots. This partial evaluation is the slot-level description of what is sometimes called currying a tensor, and it is the mechanism by which a tensor of high valence is used to build a lower-valence tensor or a linear map between smaller spaces.

Slot Structure Under Linear Transformations

Consistent Slot Transformation

When a linear map f: V → W acts on a tensor, every slot of the same type transforms by the same rule: all vector slots transform together, either through f or its inverse depending on variance, and all covector slots transform together through f* or its inverse. The slot structure guarantees that this transformation is applied uniformly, position by position, so that the multilinear map obtained after transformation remains well-defined and consistent with the original tensor's contraction and symmetry relations.