6.24 Tensor Order and Type Notation

Tensor Order and Type Notation is the collection of written conventions used to communicate a tensor's order and type — its total number of index slots and the split of those slots into contravariant and covariant categories — encompassing index notation with raised and lowered letters, the parenthesized type pair (p, q), and the descriptive order-only terminology used when the variance split is not the point under discussion. This notation family exists because order and type are abstract classificatory data that must be communicated in writing, and different notational choices make different aspects of that data easiest to read at a glance.

The Three Principal Notational Devices

Index Notation

Writing a tensor with explicit upper and lower indices, T^{i₁...i_p}_{j₁...j_q}, encodes both the order (total index count) and the type (split between superscripts and subscripts) simultaneously, and additionally names the indices so that contractions and other operations can be written out explicitly. This is the most information-dense notation, but it requires committing to actual index letters even when the specific letters are not important to the point being made.

Type Pair Notation

Writing the type alone as (p, q), without index letters, communicates the order and type split without committing to any particular indices; this is the preferred notation when discussing a tensor's classification abstractly, independent of any specific expression it might appear in.

Order-Only Terminology

Referring to a tensor simply as "an order-n tensor" or "a rank-n tensor" communicates only the total index count, suppressing the contravariant-covariant split entirely; this is used when the split is either irrelevant to the point being made or has already been fixed by context (for example, when discussing purely covariant tensors, so that stating "order 2" already implies type (0,2)).

Relations Between the Three Devices

All Three Encode a Common Underlying Classification

Each arrow in this chain is a one-way simplification: index notation determines the type pair by counting, and the type pair determines the order by addition, but neither step is reversible without additional information, since recovering index notation from a type pair requires choosing index letters, and recovering a type pair from an order requires knowing the split.

Choosing the Right Notation for the Task

Because each device discards some information relative to the ones above it in the chain, selecting a notation is a matter of choosing the coarsest device that still carries the information relevant to the discussion at hand: a proof involving explicit contraction requires index notation, a classification statement requires the type pair, and a remark about computational cost or storage often requires only the order.

Diagram of the Notation Hierarchy

Historical and Contextual Notes on Usage

Variation by Mathematical Subfield

Differential geometry and general relativity favor explicit index notation because contraction and covariant differentiation are ubiquitous and must be tracked precisely; abstract algebra and functional analysis more often favor the type pair or order-only terminology, since the objects of interest are frequently discussed structurally rather than through explicit component manipulation. Recognizing which notational convention a given source uses is often the first step to correctly interpreting its statements about tensors.

Rank as a Sometimes-Ambiguous Synonym

The word "rank" is sometimes used as a synonym for order, but in matrix theory "rank" already has an unrelated, specific meaning — the dimension of the image of a linear map — so the order and type notation described here favors "order" specifically to avoid this collision, reserving "rank" for its matrix-theoretic sense whenever both concepts are in play.