4.7.3 Tensor Repeated Vector Space Domain

The Tensor Repeated Vector Space Domain explores how repeated vectors interact within algebraic structures, forming foundational frameworks for advanced tensor operations.

Tensor Repeated Vector Space Domain is the special case of a tensor's multilinear domain structure in which the same vector space, either $V$ or its dual $V^{*}$, is used as the factor for more than one argument slot, so that several positions in the domain product draw from an identical underlying space rather than from mutually distinct ones. It is the structural condition that makes symmetry, antisymmetry, and permutation of arguments meaningful, since exchanging two arguments only makes sense when both are drawn from the same space.

Formal Definition

Repetition Within the Domain Product

For a type $(r, s)$ tensor over a single space $V$, the domain structure is

\underset{r copies}{V^{*} \times \dots \times V^{*}} \times \underset{s copies}{V \times \dots \times V}

Here $V^{*}$ is repeated $r$ times and $V$ is repeated $s$ times: the domain is not built from $r + s$ mutually unrelated spaces, but from only two distinct spaces, each appearing multiple times. This repetition is what distinguishes an ordinary tensor over a single vector space from the more general case of a multiple vector space domain built from genuinely different factors.

Full Repetition: The k-Fold Self-Product

When all factors are equal, either $V^{k} = V \times \cdots \times V$ ($s = k$, $r = 0$) or $(V^{*})^{k}$ ($r = k$, $s = 0$), the domain is said to be fully repeated. This is the setting for purely covariant or purely contravariant tensors of order $k$, and it is the domain on which the symmetric and exterior (antisymmetric) algebras are constructed.

Consequences of Repetition

Well-Defined Permutation of Slots

Because repeated slots draw from the identical space, a permutation $\sigma$ of those slots is a well-typed operation: swapping the vectors occupying two slots bound to the same $V$ produces another valid tuple in the domain. This is precisely what fails in a domain built from genuinely distinct spaces, and it is why symmetry and antisymmetry are properties that can only be stated relative to a repeated (or partially repeated) vector space domain.

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

Partial Repetition

A domain may repeat one factor space across some slots while other slots draw from different spaces, producing a mixed structure in which permutation symmetry can only be meaningfully imposed within each repeated block separately. For a type $(r, s)$ tensor with $r, s \geq 2$, this means symmetrization can be applied independently among the $r$ contravariant slots and among the $s$ covariant slots, but not across the two blocks without additional structure such as a metric.

Repetition and the Component Structure

Equal Dimension Across Repeated Slots

Since repeated slots draw from the identical space $V$, each has the same dimension $n = \dim V$, and the total number of independent components of the tensor is $n^{r+s}$, a uniform power rather than a product of possibly unequal factor dimensions as would occur in a domain built from genuinely distinct spaces.

Enabling Symmetric and Antisymmetric Subspaces

A repeated vector space domain is the prerequisite for decomposing the space of multilinear maps on that domain into invariant subspaces under the permutation action of the symmetric group on the repeated slots, of which the fully symmetric subspace and the fully antisymmetric (alternating) subspace are the two most important examples, each spanned by tensors satisfying the corresponding sign rule under slot exchange.

Summary of Key Points

A repeated vector space domain occurs when the same underlying space, $V$ or $V^{*}$, serves as the factor for more than one argument slot.
Repetition is the structural precondition for permutation of arguments to be well-typed, and hence for symmetry or antisymmetry to be meaningfully defined.
Full repetition across all $k$ slots gives the $k$-fold self-product domain used to define purely covariant or purely contravariant tensors of order $k$.
Partial repetition allows symmetrization within a repeated block of slots while keeping distinct blocks (such as upper versus lower indices) independent.
Repeated slots share a common dimension, giving a uniform component count of $n^{r+s}$ rather than a product of differing factor dimensions.