8.4.1 Tensor Upper Index Position Role

Tensor Upper Index Position Role is the specific function played by the horizontal ordering among a tensor's several superscript indices: the left-to-right sequence in which upper indices are written, which determines which covector-accepting argument slot, among the several such slots a tensor may have, each particular upper index refers to. It narrows the general upper index role down to the question of order among indices that already share the upper, contravariant position.

Order Among Multiple Upper Indices

Each Position Names a Distinct Argument Slot

A tensor of type (p, q) with p ≥ 2 has more than one contravariant argument slot, and the horizontal order of its superscript indices in an expression such as T^{ijk} records which slot each of i, j, and k is understood to fill: i names the first covector-accepting slot, j the second, k the third, and reordering them generally refers to a different assignment of arguments to slots.

in general, since the left side feeds its first slot the value assigned to i and its second slot the value assigned to j, while the right side does the reverse.

Position Is Independent of Which Letters Are Used

The role played by position among upper indices is entirely about order, not about which specific letters are chosen to fill each position; renaming the indices in T^{ijk} to T^{abc}, preserving their relative order, refers to exactly the same slot assignment, whereas permuting the letters while keeping the same names, as in T^{jik}, changes which slot each named index refers to.

Symmetry as a Statement About Upper Index Position

Symmetric Pairs of Upper Indices

When a tensor happens to be symmetric in a pair of its upper indices, T^{ij} = T^{ji}, this is a substantive claim specifically about the upper index position role: it asserts that, for this particular tensor, the assignment of arguments to its first and second contravariant slots may be swapped without changing the value, a property that does not hold for a generic tensor and must be established or assumed explicitly.

Antisymmetric Pairs

Conversely, T^{ij} = -T^{ji} asserts that swapping the position of a pair of upper indices flips the sign of the result; both symmetric and antisymmetric statements are only meaningful once the upper index position role is understood, since they are precisely claims about what happens under reordering.

Diagram of Upper Index Position

Upper Index Position in Staggered Notation

Recording Original Position After Raising or Lowering

When one index of a mixed tensor is raised from a lower to an upper position, staggered notation, such as T^{i}{}_{j}{}^{k}, is used to preserve a record of the original horizontal position that the newly raised index occupied among all the tensor's slots before the raising operation, ensuring that the raised index is not mistaken for occupying a different original slot than it actually did.

Why This Matters Specifically for Upper Positions

Because raising an index moves it into upper position without necessarily placing it at the leftmost or rightmost upper slot, the position role for upper indices must account for indices that arrived at their upper status through raising, not only those that were upper from the tensor's original definition; staggered notation is the standard device for keeping this positional history unambiguous.

Practical Consequences

Contraction Order in a Chain

When several upper indices participate in different contractions within a longer expression, their relative horizontal position, together with the corresponding lower indices they are paired against, determines the order in which the associated operations combine, mirroring the non-commutativity of the operations being represented, such as matrix or operator composition.

Verifying Claimed Symmetries

Any claimed symmetry or antisymmetry among a tensor's upper indices should be checked specifically against the positional role: verifying that swapping the horizontal position of the claimed pair, while leaving all other indices fixed, produces the stated relationship, rather than assuming the relationship holds merely because the indices in question happen to share the same vertical, upper placement.