4.21 Tensor Symmetric Multilinear Pattern

Tensor Symmetric Multilinear Pattern is the structural regularity exhibited by multilinear maps whose output is unchanged under every permutation of their arguments, the direct counterpart to the alternating pattern, and the pattern underlying quadratic and higher polynomial forms, the symmetric algebra, and the classical relationship between multilinear forms and homogeneous polynomials.

The Pattern Stated

Invariance Under Permutation

A multilinear map f: V × ... × V → W (all n arguments from a single space V) is symmetric if

for every permutation σ of {1,...,n} and every choice of vectors. The pattern is entirely about reordering: no matter how the n inputs are rearranged among the argument slots, the output stays exactly the same.

Reduction to the Case of a Single Transposition

Since every permutation decomposes into transpositions, it suffices to check that f is invariant under swapping any two adjacent arguments; invariance under the full symmetric group then follows automatically, without needing to test every one of the n! possible reorderings directly.

Elementary Consequences

Determined by the Multiset of Arguments

Because the output depends only on which vectors appear among the n arguments, not on their order, a symmetric multilinear map is effectively a function of the multiset {v₁,...,vₙ} rather than of the ordered tuple (v₁,...,vₙ), which is the reason symmetric maps are naturally associated with unordered collections of inputs.

Coordinate Array Is Fully Symmetric

Relative to a basis, the component array T_{i₁...iₙ} = f(e_{i₁},...,e_{iₙ}) of a symmetric multilinear map satisfies T_{i_{σ(1)}...i_{σ(n)}} = T_{i₁...iₙ} for every permutation σ, so the array is unchanged by any reindexing of its indices, and is determined entirely by its values on non-decreasing index tuples i₁ ≤ i₂ ≤ ... ≤ iₙ.

Relation to Quadratic and Higher Polynomial Forms

Polarization

A symmetric bilinear form f on V gives rise to a quadratic form q(v) = f(v,v), and conversely, in characteristic not 2, f is recovered from q by the polarization identity f(v,w) = (q(v+w) - q(v) - q(w))/2. This bijection between symmetric bilinear forms and quadratic forms is the n = 2 instance of a general correspondence between symmetric n-linear forms and homogeneous degree-n polynomial functions on V.

Higher-Degree Homogeneous Polynomials

For general n, a symmetric n-linear form f produces a homogeneous degree-n polynomial function p(v) = f(v,...,v), and this correspondence is again invertible in characteristic zero (or characteristic greater than n) via a generalized polarization formula, so that symmetric multilinear forms of arity n and homogeneous polynomials of degree n on V carry equivalent information.

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 between elementary tensors related by a permutation of factors; this is the direct analogue, for the symmetric pattern, of the universal property connecting alternating maps to the exterior power.

Basis and Dimension

A basis for Symⁿ(V), for V of dimension d, is given by the symmetric products e_{i₁}⋯e_{iₙ} for non-decreasing index tuples i₁ ≤ ... ≤ iₙ, giving dimension C(d+n-1, n), the number of multisets of size n drawn from d elements, in contrast to the C(d,n) dimension of the exterior power, which counts only strictly increasing (repeat-free) tuples.

The Symmetric Algebra

Assembling All Degrees

The direct sum Sym(V) = ⊕_{n≥0} Symⁿ(V), with multiplication given by the symmetric product, forms the symmetric algebra of V, a commutative graded algebra: unlike the exterior algebra, which is only graded-commutative with sign factors, the symmetric algebra is commutative outright, since the underlying pattern imposes no sign changes at all.

Identification With Polynomial Functions

For V finite-dimensional with basis {e₁,...,e_d}, Sym(V) is naturally isomorphic to the polynomial ring F[x₁,...,x_d], with eᵢ corresponding to the variable xᵢ; this identification realizes the symmetric multilinear pattern concretely as ordinary polynomial algebra, connecting the abstract tensor-theoretic construction directly to elementary algebra.

Contrast With the Alternating Pattern

Opposite Sign Behavior

Where the alternating pattern demands f(...,v,...,w,...) = -f(...,w,...,v,...), the symmetric pattern demands equality with no sign change; a nonzero multilinear map cannot satisfy both patterns simultaneously for arity n ≥ 2 in a field where -1 ≠ 1, since that would force the shared value to equal its own negative.

Complementary but Incomplete Decomposition

For arity n = 2, every bilinear form decomposes as a sum of its symmetric and alternating parts, exhausting the space of bilinear forms; for arity n ≥ 3, the symmetric and alternating multilinear maps together span only part of the full space of multilinear forms, with the remainder corresponding to other irreducible representations of the symmetric group acting on the argument slots.

Occurrences of the Pattern

The Determinant's Complementary Pattern

Where the determinant is the prototypical alternating multilinear map, the permanent, defined by the same sum over permutations as the determinant but without the sign factor, is its symmetric counterpart, illustrating how the same combinatorial formula splits into two distinct multilinear patterns depending on whether a sign is included.

Symmetric Tensors in Physics

Stress tensors, strain tensors, and the metric tensor of Riemannian geometry are symmetric bilinear forms, reflecting a physical or geometric symmetry, such as the reciprocity of stress components or the symmetry of distance measurement, that is captured precisely by the symmetric multilinear pattern rather than by a general or alternating one.

Moments and Cumulants in Probability

Higher moments of a multivariate probability distribution are symmetric multilinear forms in the underlying random vector, since permuting the order in which expectations of products of coordinates are taken has no effect on the resulting moment, directly instantiating the symmetric pattern in a statistical context.