6.1.3 Tensor Valence Classification Scope

Tensor Valence Classification Scope is the delimitation of the subject matter that belongs to the classification of tensors by their valence, meaning the specific ordered sequence of upper (contravariant) and lower (covariant) argument slots a tensor exposes, as distinct from the bare count (p, q) of how many slots are upper and how many are lower. This scope statement fixes valence classification as concerned with slot arrangement and argument order, and separates it from type classification, which records only the totals p and q, and from order classification, which records only the sum p + q.

What Falls Inside This Scope

The Definition of Valence as Slot Arrangement

The valence of a tensor is the ordered list of variances attached to each argument slot in the order those slots are written, for example a tensor written T^i{}_{j}{}^{k} has valence "up, down, up," distinct from a tensor of the same type (2,1) written T^{i}{}^{k}{}_{j} with valence "up, up, down." Two tensors can share an identical type (p, q) while differing in valence whenever the upper and lower slots are interleaved differently.

Arrow-Diagram Notation for Valence

Within scope is the convention of drawing a tensor's valence as a horizontal sequence of up-arrows and down-arrows, one per slot, read left to right in the order the arguments are supplied.

Why Slot Order Matters Operationally

Within scope is the operational consequence that slot arrangement determines which argument a specific contraction, symmetrization, or basis substitution acts on: contracting "the first upper slot with the second lower slot" only has a well-defined meaning once the valence sequence, and not merely the type totals, is fixed.

Valence-Preserving Versus Valence-Permuting Operations

Within scope is the distinction between operations that preserve the exact valence sequence, such as addition of two tensors of identical valence, and operations that permute it, such as an explicit relabeling of argument order, which produces a tensor of the same type but a rearranged valence.

What Falls Outside This Scope

Type as Bare Upper/Lower Totals

The pair (p, q) alone, without regard to how the p upper and q lower slots are interleaved, belongs to tensor type classification and is referenced here only to note that a single type corresponds, in general, to several distinct valences (specifically to C(p+q, p) distinct interleavings).

Order as a Bare Slot Count

The total slot count p + q, without regard to upper/lower split or arrangement, belongs to tensor order classification and is outside this scope except as the length of the valence sequence.

Symmetry Classification of a Fixed Valence

Whether a tensor is symmetric or antisymmetric under exchange of two slots of identical variance is a structural property considered once a valence is already fixed, and is treated as a separate classification, not as part of valence classification itself.

Numerical Values of Components

The specific numbers stored in a tensor's components, and how they change under a basis substitution, belong to tensor transformation and representation, not to the purely combinatorial question of slot arrangement addressed here.

Boundary Cases Addressed Within This Scope

Valence of Scalars and Single-Slot Tensors

A scalar, having no slots at all, has the empty valence sequence, and a vector or covector, having exactly one slot, has a trivial valence sequence of length one, for which the interleaving question does not arise; both cases are within scope as the degenerate ends of the valence definition.

Valence Behavior Under Tensor Product

is within scope as the direct combinatorial consequence of forming a tensor product, since the argument slots of the second factor are simply appended, in order, after those of the first.

Valence Change Under Contraction

Within scope is the fact that contracting a specific upper slot at position i against a specific lower slot at position j removes exactly those two positions from the valence sequence, closing the gap and preserving the relative order of all remaining slots, which is the mechanism by which contraction is valence-specific rather than merely type-specific.

Purpose of Fixing This Scope

Refining the Classification Hierarchy Beyond Type

By isolating slot arrangement as its own classification axis, this scope allows the overall classification of tensors — order, then type, then valence — to proceed from the coarsest distinction to the finest, with each level adding exactly the information the previous level omitted.

Enabling Unambiguous Reference to Specific Argument Slots

Fixing valence as an explicit, ordered sequence gives every later discussion of contraction, symmetrization, or index substitution a precise way to refer to "the second slot" or "the first upper slot" of a tensor, a level of precision that type or order classification alone cannot supply.