6.22.4 Tensor Type Notation Relation

Tensor Type Notation Relation is the correspondence between the abstract type of a tensor, the pair (p, q) of contravariant and covariant slot counts, and the concrete index notation used to write its components, T^{i₁...i_p}_{j₁...j_q}, in which the number of superscript index positions equals p and the number of subscript index positions equals q. This relation is what allows a reader to determine a tensor's type by inspection of its written form alone, without any accompanying prose description, simply by counting raised and lowered indices.

Reading Type Directly from Notation

Superscripts Count Contravariant Slots

Each upper index attached to a tensor symbol corresponds to exactly one contravariant slot, that is, one factor of V in the tensor product space V ⊗ ... ⊗ V ⊗ V* ⊗ ... ⊗ V*. A tensor written T^{ij} has two upper indices and is therefore of type (2, 0); a tensor written T^{i} has one upper index and is of type (1, 0), matching the ordinary notation for vector components.

Subscripts Count Covariant Slots

Symmetrically, each lower index corresponds to one covariant slot, one factor of V*. A tensor written T_{ij} has type (0, 2); a tensor written T^{i}_{j} has one upper and one lower index and is of type (1, 1). The notation relation is exact and exhaustive: reading off the total count of superscripts gives p, the total count of subscripts gives q, and together they specify the type completely.

Why the Notation Encodes Transformation Behavior, Not Just Bookkeeping

Index Position Signals the Transformation Rule

The choice to place an index as a superscript or subscript is not a typographical preference; it signals precisely how that index behaves under a change of basis. An upper index transforms with the transition matrix A directly (contravariantly), while a lower index transforms with the inverse transition matrix (covariantly). The type notation relation therefore packs transformation information directly into the symbol, so that the correct change-of-basis formula for any tensor can be reconstructed purely from its index pattern.

Consistency Requirement in Every Valid Expression

Because index position carries transformation meaning, any well-formed tensor equation must have the same free upper indices and the same free lower indices on both sides — a requirement often called index balance. This consistency check is a direct application of the type notation relation: if the left side of an equation has type (1, 2) and the right side has type (2, 1), the equation cannot be correct regardless of what values the components take, because the two sides describe objects that transform differently under a change of basis.

Diagram of Index Position and Slot Type

Notational Variants and Their Limits

Mixed Ordering Conventions

Some texts interleave upper and lower indices to indicate a specific pairing with a fixed slot order, writing T^{i}_{~j}^{~k} rather than grouping all superscripts before all subscripts; the type notation relation is unaffected by this choice, since counting still yields the same p and q regardless of the order in which the indices are interleaved, though the interleaved form additionally records which slot position each index occupies.

Notation Alone Does Not Specify the Vector Space

While the index notation reliably encodes the type (p, q), it does not by itself specify which vector space V the slots refer to; two tensors with identical index patterns T^{i}_{j} but built over different vector spaces V and V′ share the same type but are elements of different, unrelated tensor product spaces. The type notation relation therefore determines type completely but must be supplemented by an explicit statement of the underlying vector space to fully specify the tensor's home space.