1.3.3 Tensor Order Structure

Tensor Order Structure is the organizing framework by which tensors are classified and related according to their order, the total number of indices, p + q, they carry, establishing a hierarchy that ranges from scalars at order zero through vectors and covectors at order one, matrices and bilinear forms at order two, and onward to arbitrarily high orders, with algebraic operations such as the tensor product and contraction moving systematically between the levels of this hierarchy. The term "order" is used interchangeably with "rank" in most contexts to denote this quantity, and the order structure describes both how tensors are stratified by this number and how operations shift a tensor from one stratum to another.

The Order Hierarchy

Order Zero: Scalars

At order zero sit the scalars, elements of the field F with no indices at all, forming the base level of the hierarchy from which every higher order is built through repeated tensor products.

Order One: Vectors and Covectors

At order one sit two distinct families, contravariant vectors, type (1, 0), belonging to V, and covariant vectors, type (0, 1), belonging to V*. Both share the same order, one index, but occupy structurally distinct roles within the hierarchy, since their single index transforms in opposite ways under a change of basis.

Order Two and Beyond

At order two sit the type (2, 0), (1, 1), and (0, 2) tensors, generalizing matrices and bilinear forms, and at each successive order, the number of distinct types grows, since a tensor of order k can distribute its k indices between upper and lower positions in k + 1 distinct ways, corresponding to the type (p, q) with p + q = k for p ranging from 0 to k.

Operations That Move Within the Order Structure

The Tensor Product Increases Order

Taking the tensor product of a tensor of order k1 with a tensor of order k2 produces a tensor of order k1 + k2, so the tensor product operation always moves upward through the hierarchy, combining lower-order tensors into higher-order ones.

Contraction Decreases Order

Contracting a tensor over one matched upper and lower index reduces its order by exactly two, so contraction always moves downward through the hierarchy, and a tensor of order zero cannot be contracted further since it has no indices left to pair.

Raising and Lowering Preserves Order

Raising or lowering an index using a metric tensor changes the split between p and q but leaves the total order p + q unchanged, moving a tensor sideways within a single level of the hierarchy rather than up or down.

Order and the Component Count

Growth of Components with Order

The number of components of a tensor of order k in an n-dimensional space is n^k, so the order of a tensor directly governs how quickly its component count grows, with each additional unit of order multiplying the component count by a further factor of n.

Order Structure and the Graded Algebra

Order as a Grading

The collection of all tensors over V, assembled into the total tensor algebra, is naturally organized by order into a graded algebra, where the piece at grade k is the direct sum of all type (p, q) spaces with p + q = k. The tensor product respects this grading by adding grades, exactly mirroring how polynomial degree adds under multiplication of polynomials.

Analogy with Polynomial Degree

The order structure of tensors closely parallels the degree structure of polynomials: just as multiplying two polynomials adds their degrees, taking the tensor product of two tensors adds their orders, and just as a constant polynomial has degree zero, a scalar has tensor order zero.

Diagrammatic Summary

The diagram represents the tensor order structure as a vertical hierarchy, with scalars at the base, vectors and covectors at order one, and increasingly many types available at each successive order, connected by the tensor product moving upward and contraction moving downward.