6.17 Tensor General p q Type Classification

Tensor General p q Type Classification is the complete labeling scheme that assigns to every tensor a pair of nonnegative integers (p, q), recording exactly p contravariant indices and q covariant indices, and thereby specifying, independently of any particular example, the full space of tensors T^{i_1 ... i_p}_{j_1 ... j_q} built from p copies of a vector space V and q copies of its dual V*. This classification is the organizing framework beneath every specific case studied in tensor algebra, including the purely contravariant type (2,0), the purely covariant type (0,2), and the mixed type (1,1), all of which are particular instances of the general (p, q) scheme.

The Abstract Definition of Type p q

Coordinate-Free Formulation

A tensor of type (p, q) on a vector space V is formally an element of the tensor product V ⊗ ... ⊗ V ⊗ V* ⊗ ... ⊗ V*, with p factors of V appearing first and q factors of V* appearing after, following the standard convention that upper indices are listed before lower indices. The order of the tensor is the total p + q, counting all indices regardless of variance.

Component Notation

Given a basis {e_i} of V and dual basis {e^j} of V*, a type (p, q) tensor is written with p superscripts and q subscripts:

with each repeated index summed over the dimension of V, following the summation convention.

Counting and Transformation Within the General Scheme

Number of Components

If V has dimension n, then a type (p, q) tensor has n^{p+q} components, since each of the p + q indices independently ranges over n values. This formula recovers the familiar special cases: n^1 for a vector, n^1 for a covector, n^2 for both type (2,0) and type (0,2) tensors, and n^2 for the mixed type (1,1) tensor.

The General Transformation Law

Under a change of basis with transition matrix A and inverse B, every upper index picks up a factor of B and every lower index picks up a factor of A, with all such factors multiplied together and summed over the corresponding old indices:

This single formula subsumes every transformation law discussed for smaller special cases, and it is the ultimate criterion for testing whether a given array of numbers, indexed with p upper and q lower slots, actually qualifies as a type (p, q) tensor.

The Grid of Types

Low-Order Examples Organized by p and q

The general (p, q) classification organizes low-order tensors into a grid, with p counting rows and q counting columns:

Interpretation of Extreme Cases

The type (0,0) case is the trivial one, consisting of scalars, which are unchanged under any transformation, and the types (1,0) and (0,1) correspond to vectors and covectors respectively, the smallest nontrivial cases within the general classification.

Operations Available Across the General Scheme

Tensor Product Increases Total Order

Given a type (p, q) tensor S and a type (r, s) tensor T, their tensor product S ⊗ T is a type (p+r, q+s) tensor, showing that the general classification is closed under tensor multiplication in a predictable, additive way on both the contravariant and covariant counts.

Contraction Reduces Total Order by Two

Contracting one upper index against one lower index of a type (p, q) tensor produces a type (p-1, q-1) tensor, reducing both counts by one and the total order by two; this operation is only available when both p ≥ 1 and q ≥ 1, which is why purely contravariant or purely covariant tensors, having q = 0 or p = 0 respectively, cannot be contracted internally without an auxiliary metric to convert one type of index into the other.