16.10.1 Tensor Antisymmetrization Permutation Sum
Tensor antisymmetrization permutation sum calculates antisymmetric tensor components using permutation indices, key in differential geometry and physics.
Tensor Antisymmetrization Permutation Sum is the explicit summation formula, ranging over every element of the symmetric group S_k, that constructs the antisymmetrization operator by adding together sign-weighted copies of a tensor's arguments in every possible order, forming the computational core of the antisymmetrization process.
The Sum Formula Itself
Full Statement
For a general (not necessarily alternating) rank-k tensor S, the antisymmetrization permutation sum defining Alt(S) is:
Each of the k! terms in the sum corresponds to one specific reordering of the k arguments, weighted by +1 if the reordering is an even permutation and −1 if odd.
Role of Each Piece of the Formula
The sum Σ_σ ranges over the entire symmetric group, ensuring every possible ordering contributes; the sgn(σ) factor encodes the alternating sign pattern expected of the output; and the 1/k! prefactor normalizes the result so that antisymmetrizing an already-alternating tensor returns it unchanged.
Why the Sum Produces an Alternating Result
Verifying the Alternating Property Directly from the Sum
Applying any transposition τ to the arguments of Alt(S) permutes the summation index by τ, and reindexing the sum over σ' = στ shows the sum reproduces itself multiplied by sgn(τ) = −1:
confirming the output satisfies the alternating permutation rule regardless of what S itself looked like before summation.
The Normalization's Role in Idempotence
The 1/k! prefactor ensures the permutation sum is idempotent on tensors already alternating: applying it to a T that already satisfies σ · T = sgn(σ)T for all σ yields Alt(T) = (1/k!) · Σ_σ sgn(σ)² T = (1/k!)(k!)T = T, leaving T unchanged.
Worked Example of the Sum
Rank 2 Case in Full
For k = 2, S_2 has exactly two elements: the identity and the single transposition, giving:
directly matching the familiar formula for extracting the antisymmetric part of a bilinear form.
Rank 3 Case: Six Terms
For k = 3, the permutation sum expands into six terms, three with sign +1 (identity and the two 3-cycles) and three with sign −1 (the three transpositions):
with the shorthand S(a,b,c) denoting S(v_a, v_b, v_c).
Computational Cost of the Sum
Growth of Term Count
The permutation sum requires evaluating k! terms, a factorial growth rate that makes direct computation impractical for large k; in practice, the sum is rarely computed term-by-term for large rank, and instead the reconstruction rule or Levi-Civita expansion is used to achieve the same alternating result with less redundant computation.