6.17.4 Tensor p q Component Index Pattern
The Tensor p q Component Index Pattern describes how indices are arranged in a tensor with p contravariant and q covariant components.
Tensor p q Component Index Pattern is the general notational layout T^{i_1 ... i_p}_{j_1 ... j_q}, in which p upper indices are listed before q lower indices, that governs how the components of an arbitrary type (p, q) tensor are written, ordered, and manipulated in index calculations. This pattern is the master template underlying every specific index expression encountered in tensor algebra, and internalizing its conventions, index ordering, summation rules, and free-versus-dummy status, is what allows a practitioner to read and write tensor equations of any order without ambiguity.
The Structure of the Pattern
Ordering of Upper and Lower Indices
By convention, all p contravariant indices are written as superscripts, grouped together and typically listed in a fixed order i_1, i_2, ..., i_p, followed by all q covariant indices as subscripts, j_1, j_2, ..., j_q:
This ordering is a notational choice rather than a mathematical necessity, but consistent adherence to it throughout a calculation prevents ambiguity about which slot a given index occupies, particularly when the tensor is not fully symmetric and the order of the indices among themselves carries meaning.
Slot Identity Versus Numerical Value
Each position in the pattern, such as "the second upper index" or "the first lower index," is a distinct slot with its own transformation behavior when the tensor lacks symmetry, even though all upper slots individually transform the same way and all lower slots individually transform the same way. The component index pattern therefore encodes two layers of information simultaneously: which slots are contravariant or covariant, and the specific order in which multiple slots of the same variance appear.
Summation Convention Within the Pattern
Free Indices Define the Output Shape
In an expression such as T^{i_1 ... i_p}_{j_1 ... j_q} standing alone, every index is free, meaning the expression represents the entire array of components, with n^{p+q} distinct entries obtained by letting each index range independently over its n possible values.
Repeated Indices Trigger Summation
When an index appears exactly twice in a single term, once as an upper index and once as a lower index, the Einstein summation convention dictates an implicit sum over that index from 1 to n, and the repeated index is then called a dummy or contracted index rather than a free one. For example, in T^{i}_{j} S^{j}, the index j is summed, leaving only i as a free index in the resulting expression, which is itself the component pattern of a type (1,0) tensor.
Extending the Pattern to Multiple Tensors in One Expression
Consistent Index Naming Across Factors
When several tensors appear multiplied together in a single expression, the component index pattern requires that any index intended to be summed must appear exactly twice across the whole expression, once up and once down, while indices meant to remain free must appear only once and must match in name and position on both sides of an equation. A mismatch, such as an index appearing three times or appearing twice with the same variance, signals an invalid or ill-formed tensor expression.
Example of a Composite Pattern
The component index pattern for the product of a type (1,1) tensor and a type (0,1) covector, contracted to produce a type (1,0) result, is written T^i_j φ_j; if φ did not correctly carry a lower index matching the summed slot, the expression would violate the pattern and fail to represent a well-defined tensor operation.
Diagram of the Index Pattern Layout
Practical Value of Following the Pattern Rigorously
Preventing Sign and Contraction Errors
Careful adherence to the component index pattern prevents common computational errors, such as contracting two upper indices directly, which is not a legitimate operation without an intervening metric, or losing track of which free index in a long expression corresponds to which slot of the final result.
Compatibility with the General Transformation Law
The component index pattern is precisely what makes the general transformation law for type (p, q) tensors expressible in closed form, since the law simply states that every named upper index acquires a factor of the inverse transition matrix and every named lower index acquires a factor of the transition matrix, applied slot by slot according to the pattern's fixed ordering.