What does type (r, s) mean?

I’d like to discuss the notation of the tensor type, commonly denoted $(r, s)$ as it relates to the tensor product. Specifically, the ordering of the vector spaces and dual vector spaces involved in the product. The order matters since tensors are typically categorized by the number of vectors and dual vectors they require as arguments. To avoid ambiguity, for a given tensor $T$, I will denote the number of vector arguments as $n_v$ and the number of dual vector arguments as $n_d$.

Preliminaries

Before we begin, recall that the dual vector space $V^\*$ is defined as the set of linear functionals from $V\rightarrow C$, where $C$ is the field over which $V$ is a vector space. Note, the dual space $V^\*$ is defined in terms of the vector space $V$. For similar reasons, the topic of dual spaces is introduced after the topic of vector spaces - in other words, epistemologically, the dual space follows the vector space. I only draw attention to this ordering between the vector space and the dual because it informs the aesthetic nature of the notation, as we’ll see.

Conventions

The tensor type $(r, s)$ is used to categories tensors based on the number of vectors and dual vectors they consume. Problem is, there is a choice to made between the $r=n_v$ or $r=n_d$ conventions - and this choice isn’t made consistently. As I mentioned above, the epistemological ordering of vectors and dual vectors is unambiguous; dual vectors follow vectors. It therefore seems natural to make the first number $r$ equal the number of vector arguments $n_v$. I’ll call this convention the vector first convention, or “VF” for short. Similarly, I’ll call the opposite convention, of using the number of dual vectors $n_d$ as the first number $r$, the dual first convention, or “DF” for short.

Usage of Conventions

Some sources use the VF convention. ^{1 2} More sources use the DF convention.^{3 4 5 6 7 8 9 10 11 12} To explicitly make clear the above conventions, the VF convention would define a type $(r, s)$ tensor $T$ as

$$T: V_1\times \cdot\cdot\cdot \times V_r \times V^\*_1 \times \cdot\cdot\cdot \times V^\*_s\rightarrow C$$

where $V_{i}=V$ and $V^\*_{i}=V^\*$ for all $i$. On the other hand, the DF convention would define a type $(r, s)$ tensor $T$ as

$$T: V^\*_1\times \cdot\cdot\cdot \times V^\*_r \times V_1 \times \cdot\cdot\cdot \times V_s\rightarrow C$$

where again $V_{I}=V$ and $V^\*_{i}=V^\*$ for all $i$.

Cartesian vs. Tensor Product Notations

The usage of Cartesian products to define the domain of $T$ is typically used before introducing the tensor product, as it is more familiar. In the Cartesian product notation, the VF convention places the vector spaces before the dual spaces, and in some sense is “aligned” with the way in which the subject is taught. When using the tensor product notation, however, this is no longer the case!

Recall the tensor product of two vector spaces $V$ and $W$ is denoted $V\otimes W$ and is the set of all multilinear functions from $V^\* \times W^\* \rightarrow C$. Notice how the usage of the tensor product $\otimes$ essentially replaces vector spaces with their duals in the Cartesian notation. This “replacement” effect combined with only using the spaces $V$ and $V^\*$ amounts to reversing the order of the input spaces the domain. For example, the VF convention would define a type $(r, s)$ tensor $T$ using the tensor product notation as

$$T: V^\*_1\otimes \cdot\cdot\cdot \otimes V^\*_r \otimes V_1 \otimes \cdot\cdot\cdot \otimes V_s\rightarrow C$$

For good measure, I will also note that the DF convention would define a type $(r, s)$ tensor $T$ using the tensor product notation as

$$T: V_1\otimes \cdot\cdot\cdot \otimes V_r \otimes V^\*_1 \otimes \cdot\cdot\cdot \otimes V^\*_s\rightarrow C$$

Notice that these definitions are equivalent to the previous definitions using Cartesian product notation, but that now the vector spaces are written first in what we called the dual first convention, not the vector first convention!

Why is DF convention preferred?

It seems that the $DF$ convention has wider usage and appeal; naturally I wonder why. Since it feels natural to align the notation with the epistemological order, in other words, to write $V$ before $V^\*$, then I am forced to conclude that the mathematical community, with malice of forethought, prefers to base the definition of a type $(r, s)$ tensor on the tensor product notation, rather than the Cartesian notation, since the former requires that $V$ be written before $V^\*$. I personally have no objection to the choice, as it seems sensible.

Parting note on terminology

In the DF convention, the number $r$ is often referred to as the covariant number and the number $s$ is called the contravariant number. These terms refer to the number of dual vectors and vectors respectively, since vectors are typically considered contravariant. Similar terms refer to lower and upper indices respectively.

I should note that my exploration of these conventions is limited to differential geometry, relativity, and linear algebra texts.