8.2.1 Tensor Index Position Area

Tensor Index Position Area is the conceptual domain of tensor index notation concerned with the horizontal ordering of indices — which slot, first, second, third, and so on, each index letter occupies among the several superscript or subscript positions of a tensor — as distinct from the vertical placement (upper versus lower) that governs contravariant versus covariant transformation behavior. It addresses why, for a tensor without special symmetry, T^{ij} and T^{ji} generally denote different objects, and how index order is tracked once raising or lowering operations are applied to a mixed tensor.

Horizontal Order as an Independent Piece of Information

Position Distinguishes Argument Slots

A tensor of type (p, q) is, at heart, a multilinear map accepting p covector arguments and q vector arguments in a definite order; the horizontal position of each index in T^{i₁...iₚ}_{j₁...j_q} records which argument slot of that multilinear map the index refers to. Swapping the order of two indices in the same vertical row corresponds to swapping which argument slot each one labels, which generally changes the value being referred to unless the tensor happens to be symmetric in that pair.

in general, since the left side is the tensor evaluated with its first argument fed the index i and its second argument fed j, while the right side feeds them in the reverse order.

Symmetry and Antisymmetry as Statements About Position

Declaring a tensor symmetric in a pair of indices, T^{ij} = T^{ji}, or antisymmetric, T^{ij} = -T^{ji}, is precisely a statement that swapping horizontal position within that pair leaves the value unchanged or merely flips its sign; without such a stated symmetry, no assumption about the effect of reordering indices is justified.

Tracking Position Across Mixed Upper and Lower Indices

Staggered Notation for Mixed Tensors

When a tensor carries both superscript and subscript indices that originated, before any raising or lowering, from a single ordered list of argument slots, a staggered notation — leaving a small gap or using an empty placeholder slot, as in T^{i}{}_{j}{}^{k} — is used to record the original horizontal position of each index relative to the others, even though some have been moved to the upper row and some to the lower row by the operations of raising and lowering.

Why the Original Order Must Be Recorded

Raising or lowering a single index of a tensor that lacks full symmetry across all its slots can produce a different object depending on which original slot is being raised or lowered; retaining the horizontal position information through staggered notation ensures that T^{i}{}_{j}{}^{k}, obtained by raising the tensor's original third slot, is not confused with a different tensor obtained by raising its second slot, even though both results might otherwise be written with the same set of vertical index types.

with the position of j relative to i on the left preserved from the position of the corresponding raised index on the right.

Diagram of Index Position

Position in Common Operations

Matrix Representation and Row-Column Convention

For a rank-two tensor represented as a matrix, the standard convention reads the first index as the row and the second as the column; this is itself an index-position convention, and reversing it — reading the first index as the column instead — produces the transpose, which for a non-symmetric tensor is a genuinely different matrix of numbers.

Order Sensitivity in Contraction Chains

When contracting a chain of tensors, such as A^{i}_{j} B^{j}_{k} C^{k}_{l}, the horizontal position of the dummy indices j and k determines the order in which the corresponding matrices are multiplied; because matrix multiplication is not commutative in general, the position of each contracted index along the chain must be respected exactly as written, and reordering the factors without correspondingly adjusting the indices changes the result.

Practical Guidance

Never Assume Symmetry from Notation Alone

Because horizontal position and vertical position are independent pieces of information, seeing two indices of the same vertical type does not by itself justify treating their order as interchangeable; any claim that reordering a pair of indices leaves an expression unchanged must be justified by an explicitly stated or previously established symmetry property of the tensor in question.

Preserving Position Through Manipulation

Any manipulation of a tensor expression — raising or lowering an index, contracting with another tensor, taking a derivative — should be carried out while explicitly preserving the horizontal position of every remaining index relative to the others, using staggered notation where needed, so that the identity of each argument slot is never lost or silently reassigned during the manipulation.