7.5.5 Tensor Component Index Ordering

Tensor Component Index Ordering is the convention fixing the sequence in which a tensor's upper indices are listed relative to one another, and its lower indices are listed relative to one another, within the abstract index notation used to write the tensor, independent of any later choice of memory layout for a stored array.

Definition and Scope

Ordering as Part of the Index Structure Itself

A tensor's index structure specifies not only how many upper and lower indices are present, but also the sequence in which they are named. Writing a tensor as (T^{ik}_{\ \ jl}) commits to a specific ordering, first upper index (i), second upper index (k), first lower index (j), second lower index (l), and this ordering is treated as part of the tensor's identity in any expression referencing it.

Ordering Without Symmetry Is Not Free

In the absence of a stated symmetry linking two indices of the same variance, their ordering cannot be altered without producing a generally different tensor. The expressions (T^{ik}{\ \ jl}) and (T^{ki}{\ \ jl}) name two distinct objects unless a symmetry between the first and second upper indices has been separately established.

Structural Properties

Ordering Conventions in Named Tensors

Well-known tensors adopt fixed index ordering conventions by long-standing use: the Riemann curvature tensor is conventionally written (R^{i}_{\ jkl}) with a specific assignment of roles to each position, and departing from that convention without comment risks being read as defining a different, related tensor, such as one obtained by a permutation of the original.

Ordering and Notational Equivalence

Some notational systems use named indices rather than position to convey meaning, in which case an index ordering can be permuted freely as long as the names travel with their original index, since the identity of each slot is carried by its name rather than by its position in the written sequence; this differs from positional systems, where the ordering itself is the only marker of which slot is which.

Ordering Under Contraction and Products

When a contraction or a tensor product is formed, the ordering of the surviving free indices must be explicitly stated, since the operation itself does not supply a canonical order for indices that were not involved in the pairing. Two equally valid conventions, keeping the free indices in their original relative order or grouping all upper indices before all lower indices, can both be adopted, provided the choice is fixed and applied consistently.

Role Within Tensor Algebra

Preventing Silent Misidentification

Fixing an explicit index ordering convention is what prevents two tensor expressions that look superficially similar from being silently conflated when they in fact refer to different orderings of the same underlying indices, a distinction that matters whenever a tensor lacks the symmetry that would make the ordering irrelevant.

Interface to the Array Level

Index ordering at the abstract, notational level is the layer from which a concrete array's axis ordering is ultimately derived: once an index ordering convention is fixed in notation, it can be mapped onto the axes of a stored array in a well-defined way, translating the abstract sequence of named or positioned indices into the concrete sequence of axes used for storage and computation.