4.7.5 Tensor Domain Slot Ordering

Tensor Domain Slot Ordering is the fixed sequencing convention that determines which position in a tensor's argument list corresponds to which factor of its multilinear domain structure, establishing a definite left-to-right (or otherwise indexed) arrangement of the domain's Cartesian product factors that must be respected whenever the tensor is evaluated, expressed in components, or combined with other tensors. It is the convention that turns an unordered collection of factor spaces into the specific, sequenced domain product on which a multilinear map is actually defined.

Formal Definition

Ordering as Part of the Domain Specification

A multilinear domain structure is not merely the set of factor spaces ${V_1, \ldots, V_k}$; it is the ordered tuple $(V_1, \ldots, V_k)$, together with a definite bijection between the integers $1, \ldots, k$ and the factors. The domain slot ordering is exactly this bijection: it specifies, for each position $i$, which factor space $V_i$ occupies that position, and it is this ordering that gives meaning to writing arguments as $(v_1, \ldots, v_k)$ rather than as an unordered multiset ${v_1, \ldots, v_k}$.

Effect of Reordering

Given a permutation $\sigma$ of ${1, \ldots, k}$, reordering the domain produces a new domain structure

and a multilinear map originally defined on the unreordered domain corresponds to a related map on this reordered domain via composition with $\sigma^{-1}$. Unless the original map is symmetric with respect to the permuted positions, this reordered map is a genuinely different multilinear map, even though it is built from the identical factor spaces and the identical original assignment rule.

Role of Ordering in the Tensor Formalism

Index Notation Depends on Ordering

Once a basis is chosen for each factor, the domain slot ordering translates directly into the order in which indices are written on the tensor's component array:

The specific position of each index in this array, first upper, second upper, first lower, and so on, is a direct readout of the domain slot ordering; changing the ordering convention changes which index is written first, even though it does not change the abstract tensor being described.

Ordering and the Tensor Product Construction

The tensor product $V_1 \otimes \cdots \otimes V_k$ is constructed with a specific factor order matching the domain slot ordering, and the associativity isomorphisms relating, for instance, $(V_1 \otimes V_2) \otimes V_3$ to $V_1 \otimes (V_2 \otimes V_3)$ preserve this ordering while only regrouping how the factors are parenthesized; permuting the order itself, by contrast, requires an explicit braiding or swap isomorphism, not merely a regrouping.

Ordering Under Tensor Operations

Concatenation in the Tensor Product

When two multilinear maps with slot orderings $(V_1, \ldots, V_m)$ and $(W_1, \ldots, W_n)$ are combined via the tensor product, the resulting arity $m+n$ map inherits a slot ordering formed by concatenation: the first map's slots retain their relative order and occupy positions $1$ through $m$, and the second map's slots follow in order at positions $m+1$ through $m+n$. This concatenation convention is what allows the original two tensors to be recovered from the combined one by fixing the appropriate block of slots.

Reordering as an Explicit Operation

Because reordering generally changes the tensor, any operation that requires a different slot arrangement, such as aligning two tensors' index conventions before contraction, must apply an explicit permutation to the domain slot ordering of one of them; this permutation is itself a linear isomorphism between the space of multilinear maps on the original ordering and the space of multilinear maps on the reordered domain.

Ordering and Symmetrization

Symmetrizing or antisymmetrizing a tensor with respect to a subset of its slots is defined by averaging (with appropriate signs) over all reorderings restricted to that subset, holding the ordering of the remaining slots fixed; the domain slot ordering of the untouched slots is therefore preserved exactly, while only the ordering within the specified subset is varied during the averaging process.

Summary of Key Points

• Domain slot ordering is the fixed bijection between argument positions and factor spaces that turns a set of vector spaces into a genuine ordered domain product.
• Reordering the slots generally produces a different multilinear map unless the original map has symmetry across the permuted positions.
• The ordering determines the left-to-right sequence of upper and lower indices once a basis is chosen for each factor.
• Tensor product concatenation preserves each original map's internal slot ordering while appending one ordered block after the other.
• Symmetrization and contraction operations depend explicitly on slot ordering, requiring reordering permutations to align tensors before combination.