15.6.2 Tensor Symmetric Multilinear Permutation Rule

The Tensor Symmetric Multilinear Permutation Rule governs how tensor components rearrange under index permutations while preserving symmetry and multilinearity.

Tensor Symmetric Multilinear Permutation Rule is the precise statement governing how a totally symmetric multilinear form responds when the order of its n arguments is rearranged by an arbitrary permutation: the value of the form is completely unaffected, regardless of which of the n factorial possible permutations is applied. This rule is the operational core of the symmetric multilinear form structure, expressing in a single, checkable statement what it means for a rank-n tensor to be totally symmetric when viewed as a map from n-tuples of vectors to scalars.

The rule applies uniformly to every permutation, not merely to adjacent transpositions or to a generating subset chosen for convenience, though such subsets can be used to verify the rule efficiently. Understanding the permutation rule requires distinguishing between the permutation acting on the argument positions of the multilinear form and the permutation acting on the index positions of the underlying tensor components, since these two actions correspond exactly and the rule can be stated equivalently in either language.

Statement of the Rule

Permutation of Arguments

For a totally symmetric multilinear form B built from a rank-n tensor T, and for any permutation σ of the set {1, ..., n}, the permutation rule states:

B (v_{1}, \dots, v_{n}) = B (v_{σ (1)}, \dots, v_{σ (n)})

holding for every choice of vectors v_1 through v_n and every permutation σ drawn from the full symmetric group on n elements.

Equivalent Statement on Index Components

The same rule stated at the component level requires that the index tuple (i_1, ..., i_n) and its permuted form (i_σ(1), ..., i_σ(n)) label components of equal value:

T_{i_{1} \dots i_{n}} = T_{i_{σ (1)} \dots i_{σ (n)}}

for every permutation σ, which is exactly the symmetric component equality constraint restated in permutation notation.

Reduction to Generators

Adjacent Transpositions Generate the Group

The symmetric group on n elements is generated by the n minus one adjacent transpositions that swap positions k and k+1 for k from 1 to n-1, meaning any permutation can be written as a composition of these adjacent swaps. Consequently, verifying the permutation rule only for adjacent transpositions is logically sufficient to establish it for every permutation in the group.

Composition Argument

If B is invariant under each adjacent transposition individually, then applying a sequence of such transpositions to reach an arbitrary permutation σ preserves invariance at every step, since each step is itself a rule-preserving operation; composing rule-preserving operations yields a rule-preserving operation, so invariance under the generating set extends to invariance under the whole group.

Distinguishing the Rule From Partial Symmetry

Full Symmetric Group Versus a Subgroup

The permutation rule as stated applies to the entire symmetric group on all n argument positions. A weaker condition, in which invariance holds only under permutations belonging to a proper subgroup, describes a tensor with partial or mixed symmetry rather than total symmetry, and such a tensor does not satisfy the full multilinear permutation rule even though it may satisfy a restricted version of it on a subset of index positions.

Sign-Twisted Permutation Rules

The permutation rule for total symmetry should be distinguished from the analogous rule governing totally antisymmetric tensors, where applying a permutation σ multiplies the component value by the sign of σ rather than leaving it unchanged; the symmetric permutation rule corresponds to the trivial representation of the symmetric group, assigning every permutation a multiplicative factor of one.

Practical Use of the Rule

Verifying Total Symmetry

To confirm that a given rank-n tensor is totally symmetric, it suffices to check the permutation rule against the n minus one adjacent transpositions rather than against all n factorial permutations individually, substantially reducing the number of equality checks required, particularly as n grows large.

Constructing Symmetric Tensors From Arbitrary Ones

Given an arbitrary rank-n tensor that does not satisfy the permutation rule, a totally symmetric tensor can be produced by averaging its components over every permutation of the index positions, and the resulting averaged tensor satisfies the permutation rule exactly, since re-permuting an already-averaged sum over the full group merely reorders the terms of the sum without changing its value.