4.4.5 Tensor Higher Arity Tensor Type Relation

Exploring how higher arity tensors relate to their type structures in algebraic contexts.

Tensor Higher Arity Tensor Type Relation is the correspondence that links the arity of a higher-arity multilinear map to the type $(r, s)$ classification of the tensor it represents, where $r$ counts the number of contravariant slots and $s$ counts the number of covariant slots, and the total arity is always their sum $k = r + s$. This relation is what allows every statement about "a multilinear map of arity $k$" to be translated into an equivalent statement about "a tensor of type $(r, s)$," and vice versa.

Formal Statement of the Relation

Arity as the Sum of Type Components

For a tensor represented as a multilinear map

T : \underset{r factors}{V^{*} \times \dots \times V^{*}} \times \underset{s factors}{V \times \dots \times V} \to F

the arity $k$ and the type $(r, s)$ satisfy

k = r + s

so that arity by itself is a strictly weaker invariant than type: many different type pairs can share the same total arity. For example, arity 3 admits the types $(3,0)$, $(2,1)$, $(1,2)$, and $(0,3)$, each describing a structurally distinct tensor even though all four consume exactly three arguments.

Recovering Type from Slot Variance

The relation is refined by recording, for each of the $k$ slots, whether it accepts a vector from $V$ or a covector from $V^{*}$. Given this variance labeling of the argument slot set, $r$ is the count of covector-accepting slots and $s$ is the count of vector-accepting slots. Arity alone forgets this labeling; the type relation restores it.

Consequences of the Relation

Type $(r, s)$ Determines Transformation Behavior

Under a change of basis given by a matrix $A$, the components of a type $(r, s)$ tensor transform with $r$ copies of $A^{-1}$ (or its transpose, depending on convention) and $s$ copies of $A$. Because this transformation law depends on the split between $r$ and $s$, not merely on their sum, two tensors of the same arity but different type transform differently under change of basis even though they have the same number of slots.

{\tilde{T}}_{b_{1} \dots b_{s}}^{a^{1} \dots a^{r}} = \sum_{indices} ({(A^{- 1})}_{⋮}^{⋮}) T_{⋮}^{⋮} (A_{⋮}^{⋮})

Type Governs Contraction Eligibility

Contraction, which pairs one covariant slot with one contravariant slot and sums over a shared index, is only meaningful between an $r$-slot and an $s$-slot. The tensor-type relation therefore also governs which pairs of slots in a higher-arity map can legally be contracted: contracting two slots of the same variance is not defined by the standard trace-type contraction, since there is no natural pairing between two vector slots or between two covector slots without an additional structure such as a metric.

The Relation Across Higher Arity

Growth in the Number of Compatible Types

As arity $k$ grows, the number of type pairs $(r, s)$ satisfying $r + s = k$ grows linearly, equal to $k + 1$ possibilities (including the extreme cases $(k, 0)$ and $(0, k)$). This means higher arity does not merely mean "more indices" in an undifferentiated sense; it means a richer combinatorial space of distinct tensor types sharing that arity, each with its own transformation law and contraction structure.

Special Cases

Type $(0, s)$: a purely covariant tensor of arity $s$, evaluated only on vectors; includes the metric tensor when $s = 2$.
Type $(r, 0)$: a purely contravariant tensor of arity $r$, evaluated only on covectors.
Type $(1, 1)$: arity 2, equivalent to a linear endomorphism of $V$ once one slot is used to represent the "input" and the other the coefficient dual to the "output."
Type $(1, 3)$: arity 4, the type of the Riemann curvature tensor with one raised index.

Summary of Key Points

Arity is the sum $r + s$ of a tensor's type, but the sum alone does not determine the type.
The full type relation requires knowing the variance (contravariant or covariant) of each individual slot in the arity-$k$ argument slot set.
Change-of-basis transformation laws depend on the specific split between $r$ and $s$, not merely on total arity.
Contraction is only naturally defined between a contravariant slot and a covariant slot, so the type relation constrains which slot pairs may be contracted.
The number of distinct tensor types compatible with a given arity $k$ grows as $k+1$, reflecting the combinatorial richness hidden inside a single arity value.