1.2.47 Contravariant Order Definition

Contravariant Order Definition explores the structure and properties of order in tensor algebras, bridging algebraic frameworks with categorical mappings.

Contravariant Order Definition is the characterization of the number of contravariant, or upper, index slots possessed by a tensor, counted independently of any covariant slots the same tensor may also carry. For a tensor of type $(r, s)$ , the contravariant order is the first entry $r$ , and it records specifically how many copies of the vector space itself contribute to the tensor, independently of how many copies of the dual space are also present.

Formal Definition

Let $V$ be a vector space over a field $F$ , with dual space $V^{*}$ . For a tensor

T \in \underset{r}{⏨} V \otimes \dots \otimes V \otimes \underset{s}{⏨} V^{*} \otimes \dots \otimes V^{*}

the contravariant order of $T$ is the integer $r$ , the number of factors of $V$ appearing in the tensor product space in which $T$ lives. In component notation, the contravariant order equals the number of superscript indices carried by the tensor,

T_{j_{1} \dots j_{s}}^{i^{1} \dots i^{r}}

since, by the notational convention pairing superscripts with contravariant slots, exactly $r$ indices appear above the line regardless of how many appear below it.

Contravariant Order as an Independent Count

The defining feature of contravariant order is that it is tracked separately from covariant order and does not depend on it. A tensor of type $(1, 3)$ and a tensor of type $(1, 0)$ have the same contravariant order, one, despite differing substantially in total order and in covariant order; both accept exactly one covector argument when viewed as multilinear maps, even though only the second is an ordinary vector by itself in the narrowest sense. This independence is what makes contravariant order a useful invariant on its own, distinct from total order: it isolates precisely how many covector arguments — as opposed to vector arguments — a tensor requires, or equivalently, how many factors of $V$ it is built from.

Contravariant Order in Special Cases

Zero Contravariant Order

A tensor with contravariant order zero is purely covariant, of type $(0, s)$ , and accepts no covector arguments at all when viewed as a multilinear map — only vector arguments, or none, if $s$ is also zero.

Contravariant Order One

A tensor with contravariant order one and covariant order zero is an ordinary vector, an element of $V$ . More generally, any tensor with contravariant order one, regardless of its covariant order, accepts exactly one covector argument among its inputs.

Purely Contravariant Tensors

When covariant order is also zero, a tensor's contravariant order coincides with its total order, and the tensor is a multilinear form defined on covectors — that is, an element of the $r$ -fold tensor product of $V$ with itself, viewed as an $r$ -linear function of covector arguments through the natural pairing with $V^{*}$ .

Behavior Under Basic Operations

Tensor Product

Forming the tensor product of two tensors adds their contravariant orders together, independently of what happens to covariant order: a tensor of contravariant order $r_{1}$ combined with a tensor of contravariant order $r_{2}$ produces a tensor of contravariant order $r_{1} + r_{2}$ .

Contraction

Contracting a matched contravariant and covariant index pair reduces contravariant order by exactly one, together with a corresponding reduction of covariant order by one, since every valid contraction removes one slot of each kind simultaneously. As with covariant order, there is no operation within ordinary tensor algebra that reduces contravariant order alone while leaving covariant order untouched, since contraction inherently requires pairing an upper index with a lower one.

Relation to the Dimension of a Component Array

Once a basis is fixed for a finite-dimensional space $V$ of dimension $n$ , the contravariant order $r$ alone determines how many superscript positions, each ranging over $n$ values, appear in the tensor's component array, contributing a factor of $n^{r}$ to the total component count of $n^{r + s}$ . This makes contravariant order directly readable from the indexing structure of a tensor's components, independent of any information about its covariant part.

Role Within Tensor Algebra

Contravariant order provides the half of a tensor's type that specifically tracks its dependence on covector arguments, complementing covariant order, which tracks dependence on vector arguments. Isolating contravariant order is useful whenever attention is restricted to how a tensor behaves as a function of covectors alone — as when studying purely contravariant subalgebras such as the algebra of multivectors, symmetric contravariant tensors, or the higher exterior powers of $V$ itself, all of which are most naturally graded by contravariant order rather than by the finer type or the coarser total order.