4.4.2 Tensor Higher Arity Domain Product

The Tensor Higher Arity Domain Product generalizes multilinear operations across higher-dimensional spaces, enabling structured interactions in algebraic structures.

Tensor Higher Arity Domain Product is the Cartesian product of vector spaces that serves as the input domain of a higher-arity multilinear map. For a map of arity $k$ built from spaces $V_1, \ldots, V_k$, the domain product is the set of all ordered $k$-tuples of vectors, one drawn from each $V_i$, and it is precisely the structure that a multilinear map is defined on before any linearity condition is imposed.

Formal Definition

The Cartesian Product Structure

Given vector spaces $V_1, V_2, \ldots, V_k$ over a field $F$, the higher arity domain product is

V_{1} \times V_{2} \times \dots \times V_{k} = \{(v_{1}, \dots, v_{k}) : v_{i} \in V_{i}\}

This product is a set, not automatically a vector space in the way that matters for multilinearity: although it can be given a vector space structure componentwise, a multilinear map $T$ defined on it is emphatically not linear as a function on that vector space structure, only linear in each argument separately when the others are held fixed.

Product Versus Direct Sum

The domain product must be distinguished from the direct sum $V_1 \oplus \cdots \oplus V_k$. The direct sum is the natural domain for a single linear map that mixes all the $v_i$ together additively; the Cartesian product is the natural domain for a multilinear map that keeps the arguments in separate, independently-varying slots. Multilinearity is a statement about behavior on the product, not on the sum.

Role in Constructing the Tensor Space

Universal Property

The domain product is the object with respect to which the tensor product space is universal: for every multilinear map

T : V_{1} \times \dots \times V_{k} \to W

there exists a unique linear map

\tilde{T} : V_{1} \otimes \dots \otimes V_{k} \to W

such that $T$ factors through the canonical map from the domain product into the tensor product space. The domain product is thus the raw input structure, and the tensor product space is what results from forcing that input structure to behave linearly overall.

Basis Expansion Across the Product

If each $V_i$ has dimension $n_i$ with basis ${e^{(i)}1, \ldots, e^{(i)}{n_i}}$, then any element of the domain product can be expanded coordinatewise, and evaluating $T$ on a general tuple reduces, by repeated use of multilinearity, to a sum over all combinations of basis vectors, one chosen from each factor of the product:

T (v_{1}, \dots, v_{k}) = \sum_{i_{1}, \dots, i_{k}} v_{1}^{i_{1}} \dots v_{k}^{i_{k}} T (e_{i_{1}}^{(1)}, \dots, e_{i_{k}}^{(k)})

This is why the number of independent scalar components of a higher-arity tensor equals the product of the dimensions $n_1 n_2 \cdots n_k$ of the factors making up the domain product.

Higher Arity Consequences

Growth of the Domain

As arity $k$ increases, the domain product grows as an ordered $k$-fold structure, and the space of multilinear maps on it grows in dimension as the product $n_1 n_2 \cdots n_k$. This exponential-in-form growth is the direct reason higher-arity tensors carry rapidly increasing amounts of independent information as arity rises.

Restriction to a Single Space

When all factors coincide, $V_1 = V_2 = \cdots = V_k = V$, the domain product reduces to the $k$-fold self-product $V^{k} = V \times V \times \cdots \times V$, which is the standard domain for the higher-arity multilinear maps that define symmetric tensors, antisymmetric (alternating) tensors, and general covariant tensors of order $k$ on a single space.

Summary of Key Points

The domain product is the Cartesian product of the argument spaces, distinct from their direct sum.
Multilinearity is a condition imposed on functions defined over this product, not linearity with respect to the product's own vector space structure.
The domain product is the source object in the universal property defining the tensor product space.
Its total dimension count, as a product of factor dimensions, governs the number of independent components of the resulting tensor.
When all factors are equal, the domain product becomes the $k$-fold self-product used to define symmetric and antisymmetric tensors of order $k$.