1.2.44 Tensor Valence Definition

Tensor valence defines the number of indices required to express a tensor, indicating its rank and how it transforms under coordinate changes.

Tensor Valence Definition is the characterization of the total number of index slots a tensor possesses, together with the classification of each slot as either an upper, contravariant slot or a lower, covariant slot, using terminology borrowed from chemistry by analogy with the number of bonds an atom can form. The valence of a tensor is numerically the same information carried by its type $(r, s)$ , but the term is used specifically to emphasize the tensor as an object with a fixed number of "slots" waiting to be filled by vector or covector arguments, a framing especially common in physics and in abstract index notation.

Formal Definition

Let $V$ be a vector space over a field $F$ , with dual space $V^{*}$ . A tensor of type $(r, s)$ , an element of

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

is said to have valence $(r, s)$ , meaning it presents $r$ upper (contravariant) index slots and $s$ lower (covariant) index slots. Equivalently, viewing the tensor as a multilinear map

T : \underset{r}{⏨} V^{*} \times \dots \times V^{*} \times \underset{s}{⏨} V \times \dots \times V \to F

the valence records the number and kind of arguments the tensor is prepared to accept before it evaluates to a scalar: $r$ covector arguments and $s$ vector arguments.

The Chemical Analogy

Slots as Bonds

The term valence is borrowed directly from chemistry, where the valence of an atom denotes the number of bonds it can form with other atoms. In the tensor setting, each index slot of a tensor functions analogously to a bond site: an upper slot can be "bonded" to, or contracted with, a lower slot of a compatible tensor, and the total number and type of available slots governs which combinations are algebraically meaningful. This analogy is most vivid in tensor network diagrams, where a tensor is drawn as a node with a number of legs equal to its valence, and a contraction between two tensors is represented by joining a leg from each.

Valence as a Predictor of Combinability

Just as an atom's valence predicts how many bonds it can participate in, a tensor's valence predicts how many contractions it can undergo before every index is exhausted: a tensor of valence $(r, s)$ can be fully contracted against compatible tensors at most $\min (r, s)$ times using its own indices alone, since each contraction consumes exactly one upper and one lower slot.

Valence in Abstract Index Notation

Abstract index notation, widely used in physics, writes a tensor's valence directly and explicitly as part of its symbol, with each upper index letter marking a contravariant slot and each lower index letter marking a covariant slot, as in $T_{b c}^{a}$ for a valence $(1, 2)$ tensor. Unlike ordinary component notation relative to a chosen basis, the abstract indices here are labels for the slots themselves rather than numerical values ranging over a basis, so the valence is read off immediately from the number of distinct abstract index letters attached to the tensor symbol, split by their position above or below the line.

Valence Versus Order, Type, and Degree

Valence conveys the identical information as type: a tensor of type $(r, s)$ has valence $(r, s)$ , with no distinction between the two beyond terminology and emphasis. Order and degree, by contrast, retain only the total $r + s$ , discarding the split between upper and lower slots; valence, like type, preserves this finer information. The choice between saying "valence" and saying "type" is a matter of disciplinary convention: valence is more common in physics literature and in discussions emphasizing tensors as operations with a fixed number of input slots, while type is more common in purely algebraic treatments of the tensor product construction.

Total Number of Components Determined by Valence

Once a basis of an $n$ -dimensional vector space $V$ is fixed, a tensor of valence $(r, s)$ has

n^{r + s}

independent components, one for every assignment of a basis index to each of its $r$ upper and $s$ lower slots, since the dimension of the tensor product space equals the product of the dimensions of its factors, here all equal to $n$ or to the dimension of the dual space, which coincides with $n$ in finite dimensions.

Role Within Tensor Algebra

Valence functions as the practical, slot-counting description of a tensor's type throughout applied and physical presentations of tensor algebra: it fixes how many vector or covector arguments a tensor accepts, determines which contractions between tensors are legal, and, through abstract index notation, provides the bookkeeping device by which tensor equations are written and verified for consistency without ever needing to refer to a specific choice of basis.