4.5 Tensor Argument Slot Structure

Tensor Argument Slot Structure is the organizing framework that describes how a tensor, viewed as a multilinear map, exposes its inputs as a fixed collection of labeled positions, each with a determined variance (accepting either vectors or covectors), a determined order, and a determined dimension inherited from the underlying vector space. It is the general scaffold that governs how many arguments a tensor takes, in what sequence, and of what kind, before any particular values are substituted.

Formal Definition

Slots as the Primitive Units of Structure

For a tensor represented as a multilinear map

the argument slot structure is the ordered list of $r + s$ positions, each tagged with the space it draws from ($V^{*}$ for contravariant slots, $V$ for covariant slots). This structure is fixed independently of any particular vectors that will later be substituted into the slots; it is the "shape" of the tensor before it is evaluated.

Distinguishing Structure from Content

The slot structure specifies form, not content: it fixes how many arguments are needed and of what variance, while the actual multilinear map (the assignment of outputs to filled slots) is a separate piece of data layered on top of the slot structure. Two different tensors can share the identical slot structure, of the same type $(r,s)$, while having entirely different output values.

Components of the Slot Structure

Order

The slots are strictly ordered, since multilinear maps are, in general, not symmetric under permutation of their arguments. Swapping the order of two slots of the same variance produces a different map unless the tensor happens to be symmetric or antisymmetric in those positions.

Variance

Each slot is labeled as contravariant (accepting a covector from $V^{*}$) or covariant (accepting a vector from $V$). This labeling is what allows the slot structure to encode the type $(r, s)$ of the tensor: $r$ is the count of contravariant slots and $s$ is the count of covariant slots.

Dimension

Each slot inherits the dimension of the space it draws arguments from. If $V$ has dimension $n$, every slot, whether covariant or contravariant, effectively ranges over $n$ basis directions, and the total number of independent components determined by the slot structure is $n^{r+s}$.

Role Within the Tensor Formalism

Foundation for Index Notation

Once a basis is chosen, the slot structure translates directly into index notation: each contravariant slot becomes an upper index and each covariant slot becomes a lower index on the component array

The slot structure is thus the basis-independent description of what, after a basis is fixed, appears as the concrete index pattern of a tensor.

Constraint on Valid Operations

Many tensor operations are only valid when the slot structure is respected. Contraction requires selecting one contravariant slot and one covariant slot; the tensor product concatenates two slot structures end to end; raising or lowering an index (using a metric) converts one slot's variance from covariant to contravariant or vice versa without changing the total arity. Each of these operations is defined in terms of manipulating the slot structure directly.

Symmetrization Within a Slot Structure

Symmetry and antisymmetry are properties defined relative to a slot structure: they only make sense as constraints on how the output changes under permutations of slots that share the same variance. A slot structure with mixed variance (some upper, some lower indices) generally supports independent symmetrization of the upper block and of the lower block, but not, without further structure such as a metric, symmetrization mixing upper and lower slots together.

Summary of Key Points

• The argument slot structure fixes the number, order, variance, and dimension of a tensor's inputs, independent of the specific output values.
• Contravariant slots correspond to upper indices and covariant slots to lower indices once a basis is chosen.
• The slot structure determines the total component count of the tensor as $n^{r+s}$.
• Core tensor operations, including contraction, tensor product, and index raising or lowering, are defined as manipulations of the slot structure.
• Symmetry properties are always stated relative to a slot structure, applying to permutations within slots of matching variance.