1.2.43 Tensor Degree Definition

Tensor degree defines the rank of a tensor, indicating its dimensionality and the number of indices required to specify its components.

Tensor Degree Definition is the characterization of the total number of vector-space factors composing a tensor, used specifically in contexts where the tensor algebra of a vector space is treated as a graded algebra and its homogeneous components are indexed by this number in direct analogy with the degree of a monomial in a polynomial ring. The degree of a tensor is numerically identical to its order, but the term "degree" foregrounds the algebra-graded perspective, in which tensors of different degrees combine under the tensor product the way monomials of different degrees combine under multiplication.

Formal Definition

Let $V$ be a vector space over a field $F$ . The tensor algebra of $V$ is the direct sum

T (V) = \oplus_{n = 0}^{\infty} V^{\otimes} n

where $V^{\otimes} n$ denotes the $n$ -fold tensor product of $V$ with itself, with the convention $V^{\otimes} 0 = F$ . A tensor $T$ is said to have degree $n$ , or to be homogeneous of degree $n$ , if it lies entirely within the summand $V^{\otimes} n$ , meaning it is expressible as a finite sum of elementary tensors each built from exactly $n$ vectors of $V$ . An arbitrary element of $T (V)$ need not be homogeneous, but decomposes uniquely as a finite sum of homogeneous components, each of a distinct degree.

Grading and the Analogy with Polynomial Degree

The decomposition of $T (V)$ into summands indexed by degree is precisely what makes it a graded algebra: multiplication in $T (V)$ is given by the tensor product itself, and the tensor product of a degree- $m$ element with a degree- $n$ element produces a degree- $m + n$ element,

V^{\otimes} m \otimes V^{\otimes} n \subset V^{\otimes} (m + n)

This additivity of degree under multiplication is exactly the property that degree exhibits in a polynomial ring, where multiplying a degree- $m$ monomial by a degree- $n$ monomial produces a degree- $m + n$ monomial. The tensor algebra is, in fact, the free associative algebra generated by $V$ , and it specializes the familiar grading of a polynomial ring to the noncommutative setting, with the degree of a tensor playing the role that the degree of a monomial plays in that more elementary and commutative case.

Degree Zero and Degree One

A tensor of degree zero is a scalar in $F$ , playing the role of the multiplicative identity summand within the graded algebra. A tensor of degree one is simply an element of $V$ itself, so that the vector space $V$ embeds into its own tensor algebra as the degree-one homogeneous component, and every higher-degree summand is generated from this embedding by repeated application of the tensor product.

Homogeneous Versus General Elements

A general element of $T (V)$ is a finite sum of homogeneous pieces of possibly different degrees, analogous to a general polynomial being a sum of monomials of different degrees rather than a single monomial. Only the homogeneous components — the individual tensors of a single, well-defined degree — are what is typically meant by "a tensor" in the narrower sense used throughout tensor algebra and its applications; the notion of degree is not defined for a non-homogeneous sum unless each of its homogeneous parts is considered separately.

Degree Restricted to Covariant Tensors

When the tensor algebra is built purely from copies of the dual space $V^{*}$ , as in the algebra of covariant tensors used to house multilinear forms, the same grading by degree applies, with a degree- $n$ covariant tensor corresponding exactly to an $n$ -linear form on $V$ . This grading by degree is also inherited by the symmetric algebra and the exterior algebra, both of which are constructed as quotients of the full tensor algebra, with the degree of a symmetric or alternating tensor equal to the degree of any tensor representing it before passing to the quotient.

Relationship to Order and Type

For an ungraded tensor of mixed type $(r, s)$ , built from both $V$ and $V^{*}$ simultaneously, the total degree $r + s$ coincides with what is elsewhere called the order of the tensor; "degree" and "order" are used essentially interchangeably to denote this total factor count. The distinction in usage is primarily contextual: "degree" is favored when emphasizing the graded-algebra structure of $T (V)$ and its analogy to polynomial rings, while "order" is favored when emphasizing the number of indices required to write a tensor's components without reference to any surrounding algebra structure.

Role Within Tensor Algebra

Tensor degree is the grading invariant that organizes the tensor algebra into a direct sum of finite pieces, each spanned by tensors built from a fixed number of factors, and it governs how tensors combine: the tensor product respects and adds degrees, while contraction, when applied within a fixed covariant or contravariant algebra, reduces degree by two. This grading by degree is what allows the tensor algebra to be studied with the same structural tools — graded ideals, graded quotients, Hilbert series counting the dimension of each graded piece — used throughout the broader theory of graded algebras in abstract algebra.