6.24.1 Tensor Type Pair Notation

Tensor Type Pair Notation is the convention of writing a tensor's type as an ordered pair (p, q) enclosed in parentheses, with the first entry p always denoting the number of contravariant slots and the second entry q always denoting the number of covariant slots, so that the pair alone, independent of any index notation, communicates the complete variance classification of a tensor. This notation is the shorthand used in prose and in formal statements whenever a tensor's type needs to be referenced without writing out its full component expression.

The Ordering Convention

Contravariant First, Covariant Second

By nearly universal convention, the type pair (p, q) lists the contravariant count first and the covariant count second: a (2, 1) tensor has two contravariant slots and one covariant slot, never the reverse. This fixed ordering is what allows the bare pair of numbers to be read unambiguously, without needing to label which number refers to which kind of slot every time the notation is used.

Distinguishing the Pair From Other Numerical Data

The type pair notation (p, q) should not be confused with a tensor's shape tuple (d₁, ..., d_n), which records axis lengths rather than variance counts, nor with coordinates of a point such as (x, y). Context together with the fixed convention that both entries of a type pair are non-negative integers summing to the order is what disambiguates the notation in practice.

Reading and Writing the Notation

From Type Pair to Index Pattern

Given the pair (p, q), the corresponding index notation is immediate: write p superscript index slots followed by q subscript index slots, as in T^{i₁...i_p}_{j₁...j_q}. The type pair notation and the index notation are two views of the same information, related by the direct correspondence

so that either notation can be reconstructed from the other without loss of information.

From Index Pattern to Type Pair

Conversely, given any tensor expression, the type pair is recovered simply by counting: the number of superscripts gives p, and the number of subscripts gives q. This reading is purely mechanical and requires no knowledge of what the tensor represents physically or geometrically — only the written form of its indices.

Diagram of the Notation Correspondence

Common Named Types Expressed in the Notation

Familiar Special Cases

The type (0, 0) denotes scalars; (1, 0) denotes vectors; (0, 1) denotes covectors (linear functionals); (1, 1) denotes linear operators on a single space, equivalent to square matrices; (0, 2) denotes bilinear forms, including metric tensors when symmetric and non-degenerate; (2, 0) denotes objects such as certain bivectors or pairs of vectors combined multiplicatively. Referring to these objects by their type pair rather than by name is often preferred in formal contexts because it states their transformation behavior explicitly rather than relying on the reader's familiarity with the named object.

Extending to Higher Types

The same pair notation extends without modification to any order: a (3, 1) tensor, a (0, 4) tensor, or a (2, 2) tensor (such as the Riemann curvature tensor, typically written with one index raised as type (1, 3)) are all specified completely by their pair, with the notation making no distinction in form between low-order and high-order tensors beyond the values of p and q themselves.