15.6.1 Tensor Symmetric Multilinear Argument Set

Tensor Symmetric Multilinear Argument Set is the notion, generalized from the symmetric bilinear argument pair, of treating the n vector inputs supplied to a totally symmetric multilinear form as an unordered collection rather than an ordered sequence, since the permutation rule guarantees that the value returned by the form depends only on which vectors are supplied, never on the order in which they are listed. Where the argument pair concerns exactly two interchangeable inputs, the argument set concerns n interchangeable inputs, and the same underlying justification applies: the totally symmetric multilinear form is, by definition, invariant under every rearrangement of its arguments, so listing the arguments in one order versus another is a notational choice with no bearing on the result.

Framing the n inputs as a set rather than a tuple is a convenience that reflects the mathematics faithfully only because of the permutation rule; for a multilinear form lacking total symmetry, the inputs cannot be treated as a set, since different orderings of the same underlying vectors would generally produce different scalar outputs. The argument set formulation is therefore not an independent assumption but a direct consequence of, and a compact way of expressing, the tensor's total symmetry.

From Ordered Tuple to Unordered Collection

The Underlying Ordered Definition

A multilinear form B is formally defined on ordered n-tuples of vectors, written (v_1, ..., v_n), since multilinearity is a property that must be specified with respect to a fixed labeling of the argument positions in order to state linearity in each slot separately.

Collapsing Order Once Symmetry Holds

Once B is known to be totally symmetric, every one of the n factorial orderings of a given collection of n vectors produces the identical scalar output, so the value of B can be regarded as a function of the underlying multiset of vectors alone:

where the left-hand side denotes evaluation on the unordered argument set and the right-hand side denotes evaluation on any particular ordering of that same collection.

Repeated Elements Within the Argument Set

Multisets Rather Than Plain Sets

Because the same vector can be supplied to more than one argument slot, the argument set of a symmetric multilinear form is more precisely a multiset, allowing repetition, rather than a set in the strict sense that forbids repeated elements. The diagonal case, in which all n arguments coincide, is the extreme instance of a multiset with a single distinct element repeated n times, and it produces the associated homogeneous degree-n polynomial.

Partial Repetition

Intermediate cases, where some but not all of the n arguments coincide, correspond to multisets with more than one distinct element but fewer than n distinct elements, and the total symmetry of B guarantees that any arrangement of such a partially repeated multiset into an ordered tuple yields the same scalar value.

Consequences for Notation and Computation

Simplified Statement of Multilinearity Properties

Treating the arguments as an unordered multiset allows properties of the form, such as its behavior under scaling or addition of one argument, to be stated without reference to which position that argument occupies, since total symmetry guarantees the choice of position is immaterial; a statement proven for the first slot automatically holds for every other slot.

Avoiding Redundant Enumeration

When evaluating or reasoning about a symmetric multilinear form over a specific collection of vectors, the argument set framing avoids the need to consider all n factorial orderings of that collection separately, since every ordering yields an identical result and only one representative ordering needs to be examined.

Relation to the Rest of the Symmetric Multilinear Structure

Set Framing as a Restatement of the Permutation Rule

The argument set concept adds no new mathematical content beyond the permutation rule itself; it is a change in notation and viewpoint that becomes valid precisely because the permutation rule holds, mirroring how the symmetric bilinear argument pair is valid precisely because of the rank-2 special case of the same permutation rule.

Boundary of the Set Framing

The argument set framing applies only to the fully symmetrized argument positions of a tensor; if a tensor possesses only partial symmetry, restricted to a proper subset of its index positions, then only the arguments corresponding to that symmetric subset can be treated as an unordered collection, while arguments in non-symmetrized positions must retain their distinct, ordered roles.