6.5.3 Tensor Valence Ordered Pair

Tensor Valence Ordered Pair is the pair (p, q) obtained by combining a tensor's contravariant count and covariant count, recovered together from a fixed valence sequence, into a single object in which the order of the two entries is fixed by convention — contravariant count first, covariant count second — so that (2, 1) and (1, 2) are treated as genuinely distinct type labels rather than as two ways of writing the same unordered pair of numbers. This distinguishes the combined pair, as an object with a fixed internal order, from the two counts considered separately, and is what allows the pair to serve as type classification's defining label rather than merely a convenient shorthand for two independently useful numbers.

Why the Pair Must Be Ordered

(p, q) and (q, p) Describe Different Tensors in General

A tensor of type (2, 1) has two contravariant slots and one covariant slot, while a tensor of type (1, 2) has one contravariant slot and two covariant slots; these are structurally different objects — a bilinear-form-like object with an extra vector argument in the first case, and one with an extra covector argument in the second — so treating the pair as unordered, an ordinary set {p, q}, would erase a distinction that matters.

The Fixed Convention: Contravariant First, Covariant Second

By convention, the first entry of the pair is always the contravariant count p and the second is always the covariant count q, matching the standard placement of upper indices before lower indices in the component expression T_{j₁...j_q}^{i₁...i_p}; adopting the opposite convention consistently would not be mathematically wrong, but mixing the two conventions within a single discussion would make every pair ambiguous.

Diagram Contrasting the Two Types

Recovering the Ordered Pair From Valence

Reading Off Both Counts at Once

Given a valence sequence, scanning once and tallying up-arrows and down-arrows in a single pass yields both p and q simultaneously, and placing them into the fixed order gives the ordered pair directly:

Many Valences, One Ordered Pair

Every one of the (p+q)! / (p! q!) distinct valences compatible with a given type collapses, under this extraction, to the exact same ordered pair; the ordered pair is therefore a many-to-one summary of valence, discarding positional information while retaining the two counts that positional information was built from.

The Ordered Pair as a Coordinate

Types as Points on a Lattice

Treating (p, q) as an ordered pair of non-negative integers naturally places every type at a distinct point on the two-dimensional lattice N × N, with p as one axis and q as the other; this coordinate-like structure is only available because the pair is ordered, since an unordered pair would fold the lattice's off-diagonal points together.

Componentwise Addition Under the Tensor Product

is the type-arithmetic rule for the tensor product, and it is precisely componentwise vector addition on the lattice, an operation that only makes sense because each entry of the pair is fixed to a specific, unambiguous role (first entry always tracks contravariant totals, second always tracks covariant totals).

Componentwise Subtraction Under Contraction

Contraction subtracts (1, 1) from the ordered pair, (p, q) → (p - 1, q - 1), another componentwise operation that depends on the ordering convention remaining fixed throughout a calculation; swapping the convention partway through would silently turn this rule into an incorrect one.

Why the Ordered Pair Matters

The Precise Object Type Classification Is Built On

Type classification's defining label is not "two numbers describing a tensor" but specifically the ordered pair (p, q), and every property attributed to type — its arithmetic under tensor product and contraction, its role in transformation laws, its lattice structure — depends on the ordering convention being respected consistently.

A Compact Summary That Remains Faithful to Valence's Counts

By packaging the contravariant and covariant counts into a single ordered object rather than leaving them as two separately floating numbers, the ordered pair provides type classification with a compact, unambiguous label that is always recoverable from, and always consistent with, whatever more detailed valence sequence a specific tensor happens to have.