16.11.3 Tensor Exterior Product Degree Addition
Tensor Exterior Product Degree Addition combines degrees in exterior algebra, adding them to define the degree of the product of differential forms.
Tensor Exterior Product Degree Addition is the structural rule that combining a degree-p alternating form with a degree-q alternating form via the wedge product always produces a result of degree exactly p + q, never any other degree, making the exterior algebra a graded algebra in the precise technical sense.
Statement of the Degree Addition Rule
The Core Grading Law
For α ∈ Λᵖ(V*) and β ∈ Λᵍ(V*), the wedge product satisfies:
with no exceptions: the degree of the product is always the arithmetic sum of the input degrees, regardless of what specific forms α and β happen to be.
Why Addition, Not Some Other Combination
The degree adds rather than multiplies or otherwise combines because each wedge factor contributes its own independent set of argument slots to the result; a degree-p form consumes p vector arguments and a degree-q form consumes q vector arguments, so their combination naturally requires p + q total argument slots to be fully evaluated.
takes exactly p + q arguments, matching the degree addition rule directly.
Consequence for the Graded Algebra Structure
The Direct Sum Decomposition Respected
Because degree addition is exact and unconditional, the exterior algebra Λ•(V*) = ⊕ₖ Λᵏ(V*) genuinely deserves the label "graded algebra": multiplication respects the grading in the strict sense that the product of a degree-p and a degree-q element always lands in the degree-(p+q) graded piece, never spilling over into other degrees.
Termination Beyond Degree n
Because no graded piece exists above degree n = dim(V), degree addition forces any wedge product with p + q > n to land in the zero space:
directly linking the degree addition rule to the rank ceiling already established for alternating tensors: the product is not merely small, it is provably the zero element once the combined degree exceeds the ambient dimension.
Iterated Products and Total Degree
Multi-Factor Degree Addition
The degree addition rule extends by associativity to any number of factors: wedging together forms of degrees k₁, k₂, ..., kₘ produces a result of degree exactly k₁ + k₂ + ... + kₘ, with the same all-or-nothing behavior — either the total stays within [0, n] and the product can be nonzero, or it exceeds n and the product is forced to zero.
Special Case: Products of Covectors
For m covectors each of degree 1, the total degree after wedging all of them together is exactly m, so φ₁ ∧ ... ∧ φₘ always lands in Λᵐ(V*), providing the simplest possible illustration of the addition rule applied repeatedly.
Degree Addition and Commutativity Interaction
The Two Rules Act Independently
Degree addition determines which graded piece the product lands in, while the separate graded commutativity rule (α∧β = (−1)^{pq}β∧α) determines the sign relating the product to its reversed-order counterpart; these are two independent pieces of information about the same product, one fixing its degree and the other fixing how reordering the factors affects its sign, and neither rule can be derived from the other.