4.4 Tensor Higher Arity Multilinear Map Structure

Tensor Higher Arity Multilinear Map Structure is the mathematical framework describing a tensor as a function of several vector (and covector) arguments that is separately linear in each argument. Instead of viewing a tensor purely as an element of a tensor product space, this structure treats it as a multilinear map of arity $k$, meaning it accepts $k$ input arguments drawn from vector spaces and their duals, and produces a scalar (or, more generally, a vector) output while respecting linearity in every slot independently.

Formal Definition

Multilinear Maps of Arity k

Let $V_1, V_2, \ldots, V_k$ be vector spaces over a field $F$. A map

is called multilinear (or $k$-linear) if, for every index $i$ and every fixed choice of the remaining arguments, the map is linear in the $i$-th argument. That is, for scalars $a, b \in F$ and vectors $u, v \in V_i$:

The arity $k$ is the number of independent slots the map consumes. A tensor of type $(r, s)$ over a vector space $V$ is realized as a multilinear map taking $r$ covectors from the dual space $V^{*}$ and $s$ vectors from $V$, producing a scalar.

The Multilinear Map Perspective on Tensors

From Tensor Product to Multilinear Map

The tensor product space $V_1 \otimes V_2 \otimes \cdots \otimes V_k$ and the space of $k$-linear maps are linked by a universal property: every multilinear map out of the product $V_1 \times \cdots \times V_k$ factors uniquely through the tensor product. Concretely, a tensor $T \in V_1^{} \otimes \cdots \otimes V_k^{}$ can be identified with the multilinear functional

that it induces on $V_1 \times \cdots \times V_k$. This identification is the operational bridge between the algebraic definition of a tensor (as an abstract element of a tensor product) and its analytic or computational definition (as a rule that eats vectors and returns a number).

Arity and Tensor Rank

The arity of the multilinear map equals the total number of indices carried by the tensor, i.e. the sum $r + s$ for a tensor of type $(r, s)$. Increasing arity increases the "higher-order" character of the tensor: an arity-2 tensor is a bilinear form (matrix-like), an arity-3 tensor is a trilinear map, and in general an arity-$k$ tensor requires $k$ independent slots to be saturated before a scalar output is produced.

Component Representation

Index Notation

Fixing a basis ${e_1, \ldots, e_n}$ for $V$ and its dual basis ${e^1, \ldots, e^n}$ for $V^{*}$, a type $(r,s)$ tensor with arity $k = r+s$ is represented by a component array with $r$ upper (contravariant) indices and $s$ lower (covariant) indices:

Applying the tensor as a multilinear map to $r$ covector arguments and $s$ vector arguments recovers a scalar by full contraction of every index against the corresponding basis coefficient of each argument.

Slot-by-Slot Evaluation

Each of the $k$ slots of the multilinear map corresponds to exactly one index of the component array. Substituting a basis vector into slot $m$ "reads off" the $m$-th index of the tensor; this is why higher arity directly corresponds to higher-rank component arrays, and why the multilinear-map structure and the multi-index component structure are two faces of the same object.

Structural Properties

Symmetry and Antisymmetry Under Arity

Because a higher-arity multilinear map has multiple slots of the same type (all covariant or all contravariant), it can exhibit symmetry properties under permutation of those slots:

• Totally symmetric: $T(v_1, \ldots, v_k) = T(v_{\sigma(1)}, \ldots, v_{\sigma(k)})$ for every permutation $\sigma$.
• Totally antisymmetric: $T(v_1, \ldots, v_k) = \operatorname{sgn}(\sigma), T(v_{\sigma(1)}, \ldots, v_{\sigma(k)})$.
• Mixed symmetry: symmetric or antisymmetric only on a subset of slots, described by Young symmetrizers.

These symmetry classes only become meaningful once arity is at least 2, since a single-slot (arity 1) map has no pair of slots to permute.

Composition and Contraction

Higher-arity multilinear maps can be reduced in arity through contraction: pairing one covariant slot with one contravariant slot and summing over the shared index lowers the arity by two. This operation is the multilinear-map-level description of the trace and of tensor contraction more generally, and it is what allows higher-arity tensors to be built up from, or broken down into, lower-arity pieces.

Why Higher Arity Matters

Generalizing Bilinear Forms

Bilinear forms such as inner products and the metric tensor are the arity-2 case of this structure. Extending arity to 3, 4, or higher yields objects such as the Riemann curvature tensor (arity 4, encoding sectional curvature information) and higher structure constants in algebraic systems, where a single scalar output must depend jointly and linearly on more than two independent directions.

Connection to Multilinear Algebra

The higher-arity multilinear map structure underlies the entire framework of multilinear algebra: exterior algebra (antisymmetric arity-$k$ maps), symmetric algebra (symmetric arity-$k$ maps), and general tensor algebra (unconstrained arity-$k$ maps) are all obtained by imposing or relaxing symmetry constraints on multilinear maps of a given arity.

Summary of Key Correspondences

• Arity 1 multilinear map: a linear functional, equivalent to a vector or covector.
• Arity 2 multilinear map: a bilinear form, equivalent to a matrix.
• Arity $k$ multilinear map: a tensor of total order $k$, equivalent to a $k$-dimensional array of components once a basis is fixed.
• Contraction: lowers arity by 2 by pairing and summing over one covariant and one contravariant slot.
• Symmetrization or antisymmetrization: imposes structure on the permutation behavior of the $k$ slots without changing the arity itself.