16.10.5 Tensor Antisymmetrization Result Space
Tensor Antisymmetrization Result Space captures antisymmetric tensor properties, forming a foundational space in algebraic structures and physical theories.
Tensor Antisymmetrization Result Space is Λᵏ(V*), the fixed target space into which every application of the antisymmetrization operator lands, examined here from the perspective of what this space looks like as a destination — its dimension, its relationship to the much larger space of inputs, and the structure of the fibers mapping onto it.
The Result Space Itself
Identifying the Target
Regardless of which rank-k multilinear tensor S is fed into Alt, the output always lands in the same fixed space:
for every S in the much larger space T^k(V*) of general rank-k multilinear forms.
Dimension Comparison
The result space has dimension C(n,k), while the domain T^k(V*) has the vastly larger dimension n^k; the antisymmetrization operator is a map from this large domain onto a comparatively small, fixed-size target, with the ratio C(n,k)/n^k shrinking rapidly as n grows for fixed k.
Surjectivity Onto the Result Space
Every Alternating Tensor Is Reachable
The map Alt is surjective onto Λᵏ(V*): for any alternating tensor T, choosing S = T as the input gives Alt(T) = T directly (by idempotence), so every element of the result space is achieved by some input, trivially including the alternating tensor itself.
Many Inputs Share the Same Output
Surjectivity combined with the large dimension gap means the fibers of Alt (the sets of inputs mapping to a single fixed output) are large: for a target T ∈ Λᵏ(V*), the fiber Alt⁻¹(T) includes T itself, any tensor differing from T by an element of the kernel, and every intermediate combination, since Alt is linear.
The Result Space's Internal Structure
Basis Inherited Regardless of Input
Since the result space is the same Λᵏ(V*) regardless of the input tensor S, it always admits the standard basis e*_{i₁} ∧ ... ∧ e*_{iₖ} for increasing index tuples, and any Alt(S) can be expanded in this fixed basis by computing its independent components directly.
Grading Consistency
Because Alt preserves rank (a rank-k input produces a rank-k output), the result space for rank-k antisymmetrization never mixes with result spaces of other ranks; applying Alt to inputs of different ranks lands in correspondingly different graded pieces Λ⁰, Λ¹, ..., Λⁿ of the full exterior algebra, never crossing between them.
Result Space Behavior at the Extremes
Rank 0 and Rank 1 Result Spaces
At k = 0, the result space is just the scalars, and Alt acts as the identity trivially, since there is nothing to antisymmetrize. At k = 1, the result space is V* itself, and again Alt acts as the identity, since a single argument has no partner to swap against.
Rank n Result Space
At k = n, the result space collapses to the one-dimensional top exterior power, and every rank-n multilinear input, no matter how complicated, antisymmetrizes down to a single scalar multiple of the Levi-Civita-associated volume form.