16.16.4 Tensor Alternating Basis Dimension Count

Tensor Alternating Basis Dimension Count examines the dimension of alternating tensor bases, revealing their role in spanning multilinear spaces through antisymmetry.

Tensor Alternating Basis Dimension Count is the combinatorial tally of how many linearly independent basis wedge products exist within the exterior power of a given degree, derived directly from counting the strictly increasing multi-indices available under the alternating basis structure, and expressed exactly by the binomial coefficient of the space's dimension and the chosen degree. It is the quantitative confirmation that the alternating basis construction produces exactly as many basis elements as the dimension formula predicts.

Counting the Basis Elements

Bijection With Subsets

The alternating basis dimension count relies on the direct correspondence between strictly increasing multi-indices of length k and k-element subsets of the index set {1, 2, ..., n}. Since specifying a subset of size k and specifying its unique increasing arrangement are equivalent, the number of basis wedge products at degree k equals the number of k-element subsets of an n-element set:

dim (Λ^{k} (V)) = (\binom{n}{k}) = \frac{n!}{k! (n - k)!}

Direct Verification via the Basis

Because the increasing index rule guarantees that the alternating basis contains no redundant or linearly dependent elements, and because every strictly increasing multi-index produces exactly one basis element, the size of the alternating basis is not merely an upper bound but the exact dimension of the exterior power, confirmed constructively rather than through an abstract counting argument alone.

Behavior of the Count Across Degrees

Values at the Extremes

At degree zero, the only multi-index is the empty sequence, giving a count of exactly one, corresponding to the scalar field itself. At degree n, the only strictly increasing multi-index of length n is the full sequence (1, 2, ..., n), giving a count of exactly one, corresponding to the one-dimensional top exterior power.

dim (Λ^{0} (V)) = dim (Λ^{n} (V)) = 1

Peak at the Midpoint

For fixed n, the dimension count C(n, k) increases from k = 0 up to the midpoint of the range and then decreases symmetrically back down to k = n, reaching its maximum value near k = n/2. This peak reflects the fact that the greatest number of distinct k-element subsets occurs when k is close to half of n.

Vanishing Beyond the Range

For any k greater than n, the count of strictly increasing multi-indices of length k drawn from only n available indices is exactly zero, since no such sequence can be formed, directly confirming the vanishing boundary of the exterior power at degrees beyond the ambient dimension.

Total Count Across All Degrees

Sum Over the Full Grading

Summing the alternating basis dimension count across every degree from 0 to n gives the total dimension of the graded exterior algebra:

\sum_{k = 0}^{n} (\binom{n}{k}) = 2^{n}

This total equals the number of subsets of the full n-element index set, since every basis element of the full exterior algebra corresponds to exactly one subset, be it the empty subset for degree zero, singletons for degree one, all the way up to the full set for degree n.

Symmetric Pairing of Counts

The dimension count satisfies C(n, k) = C(n, n − k), meaning the alternating bases at complementary degrees always have equal size. This symmetric pairing is the counting-level manifestation of the duality between Λᵏ(V) and Λⁿ⁻ᵏ(V) realized through operations such as the Hodge star.

Worked Example

Dimension Counts for n = 5

For a five-dimensional space, the alternating basis dimension counts across all degrees are 1, 5, 10, 10, 5, 1, corresponding to k = 0 through k = 5 respectively, summing to 32, which equals 2⁵.

Significance of the Count

The alternating basis dimension count is the direct combinatorial certification that the exterior power construction behaves exactly as its abstract dimension formula predicts. It confirms the vanishing boundary, the one-dimensionality of the extremal degrees, the symmetric pairing of complementary degrees, and the overall total dimension of 2ⁿ, tying every quantitative aspect of the exterior algebra back to simple subset-counting of the underlying basis index set.