9.4.4 Tensor Product Basis Ordered Tuple Role

Tensor Product Basis Ordered Tuple Role is the part played by fixing a definite sequence for the factors appearing in a tensor product basis element, so that e_i ⊗ f_j and f_j ⊗ e_i are treated as elements of different spaces, or as differently positioned within the same tensor, rather than as interchangeable; it is this ordering that gives meaning to statements about "the first slot" or "the second slot" of a tensor and that keeps the multi index address attached to each basis element unambiguous.

Why Order Cannot Be Discarded

Tensor Product Factors Are Not Automatically Interchangeable

The tensor product construction does not, by itself, identify V ⊗ W with W ⊗ V, even though the two spaces are isomorphic. The ordered tuple role is what fixes, once and for all, which factor is regarded as first and which as second, so that a basis element and its multi index address refer unambiguously to one specific arrangement rather than to an equivalence class of arrangements.

Slot Position Determines Which Argument a Tensor Accepts

A tensor built on an ordered tensor product basis accepts its arguments in a fixed sequence, matching the order fixed by the basis: the first slot accepts elements from the first factor, the second slot from the second factor, and so on. Swapping the order in which arguments are supplied to such a tensor, without also swapping the ordered tuple defining the basis, produces a different pairing and, in general, a different scalar result.

How the Ordered Tuple Role Is Carried Out

Fixing a Sequence Once, Using It Everywhere

The ordered tuple role is discharged by choosing, at the moment the tensor product basis is defined, a specific left-to-right sequence for the factors, and then maintaining that same sequence consistently in every basis element, every multi index address, and every coordinate expression built from that basis.

Every Basis Element Inherits the Same Sequence

Once the ordering of factors is fixed, every basis element of the tensor product, being formed as e_{i₁} ⊗ e_{i₂} ⊗ ⋯ ⊗ e_{iₖ}, automatically follows that same sequence, so no basis element requires its own separate ordering decision; the ordered tuple role is settled globally for the whole basis rather than locally for each element.

Relationship to Symmetric and Antisymmetric Tensors

Order Is the Basis for Defining Symmetry

Symmetry and antisymmetry of a tensor are properties defined relative to the exchange of ordered slots: a tensor is symmetric if exchanging the contents of two ordered slots leaves it unchanged, and antisymmetric if the exchange negates it. Neither property could be stated at all without the ordered tuple role first having fixed which slot is which.

Symmetrized Bases Are Built From an Ordered Starting Point

Even when a symmetric or antisymmetric combination of basis elements is formed by summing over permutations of the factors, the construction begins from an ordered tuple, and the permutations are explicitly defined as rearrangements of that fixed starting sequence.

Diagram of the Ordered Tuple Role

Consequences of the Ordered Tuple Role

Coordinate Expressions Remain Unambiguous

Because the ordered tuple role fixes which position in a multi index address belongs to which factor, any coordinate expression written down for a tensor can be interpreted without any further clarification of which index refers to which slot, since that assignment was settled once when the basis was defined.

Reordering Requires an Explicit, Separate Map

Whenever a tensor or its coordinate expression must be transferred from one ordering convention to another, this transfer requires an explicit permutation map applied on top of the original ordered tuple; the ordered tuple role guarantees that such a map is always well defined and always recoverable, since the starting order was fixed and known in the first place.