15.3.2 Tensor Symmetric Argument Exchange Behavior

Tensor Symmetric Argument Exchange Behavior is the property, viewed from the multilinear-map perspective, that describes how a symmetric tensor's output value responds when its input arguments are exchanged with one another. Rather than describing symmetry through index components, this behavior describes the tensor as a multilinear function of several vector arguments and states that swapping any two of those arguments leaves the function's output completely unchanged.

Multilinear Function Viewpoint

Tensor as a Multilinear Map

A rank-( r ) tensor ( T ) can be regarded as a multilinear map taking ( r ) vector arguments ( v_1, \ldots, v_r ) from a vector space ( V ) and returning a scalar:

with ( r ) copies of ( V ) in the domain. The tensor is symmetric precisely when the value ( T(v_1, \ldots, v_r) ) does not depend on the order in which the arguments are listed.

The Exchange Condition

Argument exchange behavior is stated formally as: for every pair of positions ( k ) and ( l ), and for all vectors ( v_1, \ldots, v_r ),

where the two arguments occupying positions ( k ) and ( l ) have been swapped while every other argument remains in place. This must hold for arbitrary choices of the vectors, not merely for special cases such as repeated arguments.

Case of Two Arguments

Symmetric Bilinear Form Behavior

For rank 2, the exchange behavior reduces to the defining property of a symmetric bilinear form:

for every pair of vectors ( u, w ) in ( V ). This is the same exchange behavior familiar from the dot product and other symmetric inner-product-like forms, where reversing the order of the two arguments never changes the resulting scalar.

Diagram of the Exchange

Full Argument Exchange for Higher Rank

All Permutations, Not Only Transpositions

For rank ( r ) greater than 2, exchange behavior under a single transposition of two arguments is sufficient to guarantee invariance under exchanging the arguments in any order, since every permutation of the ( r ) arguments can be produced by successively applying transpositions, and each transposition preserves the output value.

Example with Three Arguments

For a rank-3 tensor, exchange behavior guarantees, among the six possible orderings of three vectors ( u, v, w ):

all producing an identical scalar output regardless of the argument ordering supplied.

Diagonal Recovery and Polarization

Diagonal Values Determine the Whole Function

Because of argument exchange behavior, a symmetric multilinear function is completely determined by its values along the diagonal, where every argument is set equal to the same vector ( v ):

This diagonal restriction is exactly a homogeneous polynomial of degree ( r ) in the components of ( v ), and the full multilinear function can be recovered from it through a polarization procedure that redistributes the diagonal values back across distinct arguments using exchange-consistent combinations.

Consequence for Verification

Because of this determinacy, argument exchange behavior can be checked entirely through the diagonal polynomial associated to ( T ): if the polarization of that polynomial reproduces ( T ) exactly, the exchange behavior holds; if the polarization requires an antisymmetric correction term, the tensor does not exhibit full argument exchange behavior.

Relation to Other Symmetric Structures

Equivalence with Slot Notation

Argument exchange behavior is the multilinear-map restatement of slot permutation invariance expressed on tensor components: swapping the vectors supplied to positions ( k ) and ( l ) has exactly the same effect as swapping the corresponding index labels in the component notation, so the two descriptions of symmetry agree completely.

Role in Defining the Symmetric Rank Area

Because argument exchange behavior forces the function's value to depend only on the multiset of arguments rather than their order, the space of such functions has the same dimension as the symmetric rank area, since both are counting the same set of order-independent configurations.