6.10.3 Tensor Mixed Index Arrangement

Tensor Mixed Index Arrangement is the convention governing how the upper and lower indices of a mixed type tensor are laid out relative to one another in written notation, including both the grouping of all upper indices together and all lower indices together, and the finer question of how the relative left-to-right order between a particular upper index and a particular lower index is recorded when that relative order carries meaning. Because a mixed type tensor generally lacks the symmetry that would make index order irrelevant, the arrangement chosen in notation is not a cosmetic matter but a load-bearing part of the tensor's definition, and misreading the arrangement can lead to contracting the wrong slot or misidentifying which vector or one-form argument a given index actually labels.

Grouped Versus Interleaved Arrangement

The Grouped Convention

The most common arrangement places every upper index together, immediately followed or preceded by every lower index together, as in a tensor written with two superscripts followed by one subscript. This grouped arrangement is compact and makes the contravariant and covariant orders easy to read at a glance, since counting the two groups separately immediately yields the type pair.

The Interleaved Convention

Some tensors, particularly those descended from an operation that raised or lowered specific indices out of an originally single-variance tensor, are written with upper and lower indices interleaved in an order that reflects the sequence of the original slots before any raising or lowering took place. In this arrangement the relative position of an upper index among the lower indices, rather than merely its presence in the upper group, is part of what identifies which original slot it corresponds to.

Recording Displaced Position Through Placeholders

The Need to Track Original Slot Position

When a lower index is raised to become an upper index, or an upper index is lowered to become a lower index, the resulting index moves from one group into the other, and if the tensor lacks symmetry, the position it vacated and the position it now occupies both carry information that must not be lost. Simply appending the newly raised or lowered index at the end of its new group would erase the record of which original slot it came from, producing ambiguity whenever more than one index of the same variance is present.

Placeholder Dots as an Arrangement Device

A common device for preserving this information is to leave an empty placeholder, often written as a dot, in the position an index vacated, so that the remaining indices in the old group keep their original relative spacing, while the new position of the raised or lowered index is marked explicitly rather than assumed to be at the end of its new group.

This notational discipline makes explicit that the index now in the upper group, in this arrangement, occupies the slot that used to sit between the two remaining lower indices, information that a purely grouped rewrite without placeholders would have discarded.

Consequences of Arrangement Choices

Arrangement Affects the Meaning of Contraction

When contracting a mixed tensor's index against another tensor, the arrangement determines unambiguously which slot is being contracted, since contraction always targets a specific named or positioned index rather than an unordered group. An interleaved arrangement with placeholders makes explicit exactly which original slot participates in a given contraction, whereas a careless regrouping that discards positional information can leave the intended contraction target ambiguous whenever the tensor is not symmetric across the affected indices.

Arrangement Does Not Affect Underlying Type

Regardless of whether a mixed tensor's indices are written grouped or interleaved, with or without placeholders, the total count of upper indices and the total count of lower indices, and hence the tensor's type, remain exactly the same; arrangement is a notational layer added on top of the type, not a modification of it. Two equivalent presentations of the same tensor, one grouped and one interleaved with placeholders, describe the identical multilinear object and the identical transformation law.

Consistency Requirements Across an Expression

Within a single tensorial expression involving several mixed tensors, the arrangement convention adopted must be applied consistently, since mixing grouped and interleaved conventions within the same calculation risks misidentifying which slot of one tensor is being contracted against which slot of another. Established conventions in a given context, once adopted, are maintained throughout to ensure that every contraction, symmetrization, and index relabeling refers unambiguously to the intended slot.