16.5.4 Tensor Alternating Component Redundancy Reduction

Tensor Alternating Component Redundancy Reduction is the process, and the resulting count, of collapsing the raw n^k component slots of a general rank-k tensor down to the much smaller set of genuinely independent values once the alternating condition's redundancies — repeated-index vanishing and sign-linked reorderings — have been eliminated.

The Raw Redundancy Before Reduction

Starting Point: Full Component Count

A general rank-k tensor over an n-dimensional space has n^k component slots, one for every possible assignment of k index values, each ranging independently from 1 to n:

None of these n^k slots are assumed related to one another before any symmetry condition is imposed.

First Reduction Pass: Eliminating Repeated Indices

Removing Slots with Coincident Labels

The alternating condition forces every component with two or more equal index labels to vanish, removing all slots where the k chosen indices are not pairwise distinct. The count of surviving nonzero slot patterns (before accounting for sign linkage) is the number of ordered k-tuples of distinct values from n options:

This is already a substantial reduction from n^k, but it still counts each underlying value multiple times, once for every ordering of the same index set.

Second Reduction Pass: Eliminating Sign-Linked Reorderings

Collapsing Orderings of the Same Index Set

Every one of the k! orderings of a fixed set of k distinct index values produces a component equal, up to a known sign, to any other ordering of that same set. Since the value is fully determined by one representative ordering (conventionally the increasing one) plus the permutation sign, the redundancy reduction divides the distinct-ordered count by k!:

Interpretation of the Final Count

The resulting binomial coefficient C(n,k) counts unordered selections of k distinct index values from n, confirming that after redundancy reduction, an alternating tensor's information content is equivalent to a single scalar assigned to each unordered k-element subset of basis directions.

Worked Reduction Example

Rank 3 in Five Dimensions

For n = 5, k = 3, the raw slot count is 5³ = 125. After removing repeated-index slots, 5·4·3 = 60 ordered-distinct slots remain. After collapsing the 3! = 6 orderings per subset:

so the redundancy reduction takes a rank-3 tensor in 5 dimensions from 125 raw slots down to 10 truly independent scalar values, a reduction by more than a factor of 12.

Redundancy Reduction as a Storage Principle

Practical Consequence for Representation

Because of this reduction, an alternating tensor need only be stored as a list of C(n,k) numbers indexed by increasing index tuples, rather than as a full n^k array; every other component is recoverable by looking up the corresponding sorted tuple and applying the appropriate sign, with no additional data required.

Comparison to Symmetric Tensor Reduction

For contrast, a fully symmetric tensor of the same rank reduces its component count to C(n + k − 1, k) rather than C(n, k), since repeated indices are permitted (and not forced to zero) in the symmetric case; the alternating reduction is always the smaller of the two counts for equal n and k, reflecting the additional vanishing constraint unique to alternation.

Diagram of the Two-Stage Reduction