6.4 Tensor Degree Classification

Tensor Degree Classification is the classification of elements of the tensor algebra T(V) = ⊕_{n≥0} V^{⊗n} according to which graded summand, or combination of graded summands, they belong to, assigning the label "degree n" to any element lying entirely within V^{⊗n} and describing general elements as finite sums of such homogeneous parts across several degrees at once. It reinterprets the same integer that order classification computes for an individual tensor, this time as a coordinate locating that tensor within the larger algebraic structure that contains every order of tensor built from V simultaneously.

The Graded Algebra Underlying Degree Classification

Direct Sum Decomposition

The full tensor algebra decomposes as an infinite direct sum indexed by degree:

with V^{⊗0} = F, V^{⊗1} = V, V^{⊗2} = V ⊗ V, and so on; each summand is a full vector space in its own right, and degree classification is the assignment of elements to the summand, or combination of summands, they occupy.

Homogeneous Elements Defined by a Single Degree

An element x ∈ T(V) is homogeneous of degree n precisely when x ∈ V^{⊗n} and x ∉ V^{⊗m} for any m ≠ n; every simple tensor v₁ ⊗ ... ⊗ v_n built from n vectors is automatically homogeneous of degree n.

General, Inhomogeneous Elements

A general element of T(V) is a finite sum x = x₀ + x₁ + ... + x_m, with x_i ∈ V^{⊗i}; degree classification records the full set of degrees {i : x_i ≠ 0} appearing in this decomposition, rather than forcing a single degree label onto an element that may genuinely span several.

Diagram of Degree as Position in the Graded Sum

Degree Arithmetic Under the Algebra's Multiplication

Additivity Under Multiplication

The tensor algebra's multiplication, given by concatenating tensor factors, respects degree additively:

which is exactly the condition required for T(V) to be called a graded algebra rather than simply a direct sum of unrelated pieces glued together with no compatible multiplication.

Degree of Sums Within a Single Summand

Adding two homogeneous elements of the same degree n stays within V^{⊗n}, since each summand is itself closed under vector-space addition; degree is preserved by addition only when every term added shares that same degree.

Mixing Degrees Produces Inhomogeneous Elements

Adding a degree-1 element to a degree-2 element produces a genuinely inhomogeneous element with a nonzero part in both V^{⊗1} and V^{⊗2}, an outcome fully anticipated and accommodated by the direct-sum structure of T(V), since the algebra is explicitly built to hold such mixed-degree combinations.

Degree Classification Compared to Order Classification

Same Number, Different Frame of Reference

For a homogeneous simple tensor v₁ ⊗ ... ⊗ v_n, order classification and degree classification agree numerically, both reporting n; the distinction is that order is computed from the tensor considered in isolation as a multilinear object with n indices, while degree is computed from the same tensor's position inside the ambient algebra T(V).

Degree, Unlike Order, Extends Naturally to Sums

Order classification struggles to assign a single number to a sum of tensors of different orders, since the phrase "the order of a sum" is meaningful only when every summand shares the same order; degree classification, by contrast, is designed from the start to describe such sums, reporting the full set of degrees present rather than forcing an artificial single value.

Degree Requires the Ambient Algebra; Order Does Not

Order can be computed for a tensor considered entirely on its own, with no reference to any larger algebraic structure; degree is only meaningful once the tensor is regarded as living inside the specific graded algebra T(V), since it is that algebra's direct-sum decomposition that gives the word "degree" its meaning.

Why the Graded Perspective Matters

Extending Constructions Degree by Degree

Many important algebras built from T(V) — the symmetric algebra, the exterior algebra — are obtained by taking a quotient or a subspace of T(V) degree by degree, inheriting a grading of their own directly from the degree classification already established here, so that identities proved for one degree can often be generalized to all degrees by an argument that proceeds degree by degree.

Isolating Homogeneous Parts for Proofs

Many algebraic identities in T(V) are most easily proved first for homogeneous elements of a fixed degree and then extended by linearity to general inhomogeneous elements; degree classification is precisely what makes this reduction to the homogeneous case a well-defined first step.

A Natural Setting for Generating Functions

Because T(V) is graded by degree, quantities such as the dimension of each graded piece can be organized into a generating function indexed by degree, a bookkeeping device widely used to track how dimension or other invariants grow as degree increases across the whole graded algebra at once.