4.21.1 Tensor Symmetric Slot Exchange Rule

Tensor Symmetric Slot Exchange Rule is the rule that exchanging the contents of any two argument slots of a symmetric multilinear map leaves its output completely unchanged: f(..., v, ..., w, ...) = f(..., w, ..., v, ...), where the two indicated positions have traded contents and every other slot is untouched. This is the direct counterpart to the alternating slot exchange rule, replacing a sign flip with exact invariance, and it is the operational core of what it means for a multilinear map to be symmetric.

Statement of the Rule

Exchange Without Consequence

For a symmetric multilinear map f: V × ... × V → W and any two slot positions i ≠ j,

with the vectors at positions i and j swapped and every other argument held fixed. Unlike the alternating case, this rule places no restriction on repeated values: it holds regardless of whether vᵢ and vⱼ are equal, distinct, or linearly related in any other way.

Deriving the Rule for Higher Permutations

From One Transposition to All Permutations

Since any permutation σ ∈ Sₙ can be written as a composite of transpositions, applying the basic two-slot exchange rule repeatedly, once for each transposition in a chosen decomposition of σ, shows that

for every permutation σ, with no sign correction of any kind, unlike the alternating case where each transposition contributes a factor of -1. Because the identity element of {+1} under multiplication never changes, the outcome after any number of exchanges is simply the original value.

No Dependence on Parity

Since the exchange rule produces no sign change at all, it makes no difference whether the permutation used to reorder the arguments is even or odd; both even and odd permutations leave the output of a symmetric map exactly as it was, in sharp contrast to the alternating case where the parity of the permutation determines the sign of the result.

Consequences for Computation

Values Depend Only on the Multiset of Arguments

Because every reordering of the arguments produces the same output, the value of a symmetric multilinear map on a tuple (v₁, ..., vₙ) depends only on the multiset {v₁, ..., vₙ}, not on any particular sequence in which the vectors are listed; this is why symmetric forms can be evaluated on an unordered collection of inputs without ambiguity.

Simplifying Sums Over Permutations

When a formula for some quantity is expressed as a sum over all n! permutations of a set of vectors fed into a symmetric multilinear map, the slot exchange rule shows every term in the sum is identical, so the sum collapses to n! times a single term; this simplification is used routinely when symmetrizing an otherwise asymmetric expression, since applying the symmetrization operator to an already-symmetric map has no effect beyond this multiplicative factor.

Examples

The Standard Inner Product

The dot product f(v,w) = ∑ᵢ vᵢwᵢ on Rⁿ obeys the exchange rule directly: f(v,w) = f(w,v) by commutativity of multiplication of real numbers, matching the general pattern for n = 2.

Elementary Symmetric Polynomials

The elementary symmetric polynomial eₖ(x₁,...,xₙ) = ∑_{i₁<...<iₖ} x_{i₁}⋯x_{iₖ}, viewed as a symmetric multilinear map when polarized appropriately, is invariant under any exchange of its variables by construction, since the sum ranges over all subsets rather than ordered tuples, automatically satisfying the exchange rule for the associated multilinear form.

The Symmetric Part of a General Bilinear Form

Given any bilinear form f, its symmetric part f_sym(v,w) = (f(v,w) + f(w,v))/2 obeys the exchange rule by construction, since swapping v and w in the defining formula simply exchanges the two terms being averaged, leaving the average unchanged.

Relation to the Symmetric Power

Factorization Through Symⁿ(V)

Every symmetric multilinear map factors uniquely through the symmetric power Symⁿ(V), the quotient of V ⊗ ... ⊗ V by the subspace generated by all differences v₁ ⊗ ... ⊗ vₙ - v_{σ(1)} ⊗ ... ⊗ v_{σ(n)} for permutations σ. The exchange rule is the multilinear-map-level statement of exactly the relation being quotiented by in this construction: two elementary tensors related by a permutation of factors become identified in Symⁿ(V).

Symmetric Product Notation

The image of v₁ ⊗ ... ⊗ vₙ in Symⁿ(V) is written v₁ ⋯ vₙ (or sometimes v₁ ⊙ ... ⊙ vₙ), and the exchange rule is encoded directly into this notation: v w = w v for the symmetric product, exactly paralleling how the alternating exchange rule is encoded into the anticommutativity of the wedge product.

Contrast With the Alternating Slot Exchange Rule

Same Structural Position, Opposite Content

The symmetric and alternating slot exchange rules occupy exactly the same structural position, both govern what happens to a multilinear map's value when two arguments are swapped, but they specify opposite outcomes: invariance for the symmetric rule, sign reversal for the alternating rule. A multilinear map obeying one rule for a specific pair of arguments generally cannot obey the other for the same pair unless the corresponding value is zero, since a quantity equal to both itself and its negative must vanish (outside characteristic 2).

Both Are Instances of a General Group-Representation Pattern

Both rules are special cases of the general fact that permuting the argument slots of any n-ary multilinear map gives an action of the symmetric group Sₙ on the space of such maps; the symmetric exchange rule identifies the maps transforming according to the trivial representation, while the alternating exchange rule identifies those transforming according to the sign representation, with general multilinear maps of arity three or more admitting further representations beyond these two extremes.