4.4.1 Tensor Higher Arity Argument Slot Set

Tensor Higher Arity Argument Slot Set organizes multiple input slots for tensor operations, enabling advanced algebraic manipulations in higher dimensions.

Tensor Higher Arity Argument Slot Set is the ordered collection of input positions, or "slots," that a higher-arity multilinear map exposes for its vector and covector arguments. Each slot is a designated position in the map's signature that accepts an element from a specific vector space or its dual, and the set of all such slots, taken together with the type (covariant or contravariant) assigned to each, fully determines how many arguments the tensor consumes and in what order they must be supplied before a scalar output is produced.

Formal Definition

Slots as Ordered Positions

For a multilinear map of arity $k$,

T : V_{1} \times V_{2} \times \dots \times V_{k} \to F

the argument slot set is the indexed set

S = \{1, 2, \dots, k\}

together with the assignment of each element $i \in S$ to a vector space $V_i$. No arithmetic is performed on the slot set itself; it exists purely to specify arity, argument order, and argument type before any evaluation of $T$ takes place.

Covariant and Contravariant Slot Partition

For a tensor of type $(r, s)$, the slot set $S$ is partitioned into two disjoint subsets:

S = S^{*} \cup S_{⊥}

where the first subset, of size $r$, contains slots that accept covectors from $V^{*}$ (contravariant indices of the tensor), and the second subset, of size $s$, contains slots that accept vectors from $V$ (covariant indices of the tensor). The total arity is $k = r + s$, and the slot set records not just the count but the precise ordered interleaving of the two kinds of slots.

Structural Role of the Slot Set

Determining Arity

The cardinality of the slot set is, by definition, the arity of the map. A slot set of size 1 gives an ordinary linear functional; a slot set of size 2 gives a bilinear form; a slot set of size $k$ gives a genuinely higher-arity multilinear map requiring $k$ simultaneous, independently-linear inputs.

Fixing Evaluation Order

Because multilinearity is only guaranteed slot-by-slot, the slot set also fixes the order in which arguments must be presented. Reordering the slot set (permuting which vector space is associated with which position) produces, in general, a genuinely different multilinear map, unless the underlying tensor happens to possess symmetry across those slots.

Slot Labeling in Index Notation

In component form with a fixed basis, each element of the slot set corresponds to exactly one tensor index. A type $(r,s)$ tensor's components are written as

T_{j_{1} \dots j_{s}}^{i^{1} \dots i^{r}}

where each upper index $i^{m}$ labels one contravariant slot and each lower index $j_{n}$ labels one covariant slot. The slot set is thus the abstract, basis-free counterpart of the concrete list of upper and lower indices seen in coordinate expressions.

Operations That Act on the Slot Set

Permutation

A permutation $\sigma$ of the slot set produces a new multilinear map by re-routing which argument enters which original position:

T_{σ} (v_{1}, \dots, v_{k}) = T (v_{σ (1)}, \dots, v_{σ (k)})

Permutations restricted to slots of the same variance type (all covariant, or all contravariant) are the ones relevant to defining symmetric and antisymmetric tensors.

Slot Removal via Contraction

Contraction selects one contravariant slot and one covariant slot from the set, identifies them through a chosen basis, and sums over that shared index, removing both slots from the set and reducing the arity by two. This is the primary operation by which the slot set shrinks without altering the values already assigned to the remaining slots.

Slot Insertion via Tensor Product

Given two multilinear maps with slot sets $S_1$ and $S_2$, their tensor product has a slot set of size $|S_1| + |S_2|$, formed by concatenating the two original slot sets in order. This is the operation by which higher-arity maps are built from lower-arity ones.

Summary of Key Points

The argument slot set fixes the arity, order, and variance type of every input to a multilinear map.
Partitioning the slot set into contravariant and covariant subsets recovers the type $(r,s)$ classification of a tensor.
Permuting the slot set underlies the definitions of symmetric and antisymmetric tensors.
Contraction removes a matched covariant-contravariant pair of slots, lowering arity by two.
Tensor product concatenates slot sets, raising arity by combining two lower-arity maps into one higher-arity map.