1.2.27 Symmetric Multilinear Map Definition

A symmetric multilinear map is a generalization of bilinear forms, preserving symmetry across multiple variables in tensor algebra.

Symmetric Multilinear Map Definition is the characterization of a multilinear map whose value is unchanged under any permutation of its arguments, so that reordering the inputs in any way leaves the output identical. Symmetric multilinear maps form the invariant counterpart to alternating multilinear maps within the general hierarchy of multilinear forms, and they underlie constructions such as quadratic forms, symmetric tensors, and polynomial functions on vector spaces.

Formal Definition

Let $V$ be a vector space over a field $F$ , and let

T : V \times V \times \dots \times V \to F

be a multilinear map taking $k$ arguments from $V$ . The map $T$ is called symmetric if, for every permutation $σ$ of the indices $1, \dots, k$ ,

T (v_{σ (1)}, \dots, v_{σ (k)}) = T (v_{1}, \dots, v_{k})

holds for all $v_{1}, \dots, v_{k} \in V$ . Because every permutation can be generated by successive transpositions of adjacent elements, it is sufficient to verify the condition for a single transposition of any two arguments:

T (\dots, v_{i}, \dots, v_{j}, \dots) = T (\dots, v_{j}, \dots, v_{i}, \dots)

for every pair of positions $i$ and $j$ , since any permutation is a composition of such transpositions and invariance under each transposition propagates to invariance under the full permutation.

Contrast with Alternating Maps

Symmetric multilinear maps stand at the opposite extreme from alternating multilinear maps within the classification of multilinear forms by their behavior under permutation of arguments. An alternating map changes sign under a transposition of two arguments and vanishes when two arguments coincide, whereas a symmetric map is unchanged under transposition and generally does not vanish on repeated arguments — indeed, the diagonal values $T (v, \dots, v)$ obtained by setting every argument equal to the same vector $v$ are typically nonzero and carry essential information about the map, forming the basis of its associated polynomial function.

A general multilinear form need not be symmetric or alternating. Over a field of characteristic other than two, however, any multilinear form can be decomposed into a sum of forms with definite symmetry type, and the fully symmetric part is obtained by averaging the map's value over every permutation of its arguments:

T^{sym} (v_{1}, \dots, v_{k}) = \frac{1}{k!} \sum_{σ} T (v_{σ (1)}, \dots, v_{σ (k)})

where the sum runs over all $k!$ permutations $σ$ of the indices.

Symmetric Bilinear Forms as the Simplest Case

Definition in Degree Two

The simplest nontrivial symmetric multilinear map occurs at degree $k = 2$ , where the condition reduces to the familiar symmetric bilinear form,

B (v, w) = B (w, v)

for all $v, w \in V$ . Every inner product on a real vector space is a symmetric bilinear form, in addition to satisfying positive-definiteness, and every symmetric bilinear form gives rise to an associated quadratic form through $Q (v) = B (v, v)$ .

Matrix Representation

On a finite-dimensional space with basis ${e_{1}, \dots, e_{n}}$ , a symmetric bilinear form corresponds to a symmetric matrix $M$ whose entries satisfy $M_{i j} = M_{j i}$ , since $M_{i j} = B (e_{i}, e_{j}) = B (e_{j}, e_{i}) = M_{j i}$ . This correspondence is exact: every symmetric matrix defines a symmetric bilinear form, and conversely, providing a complete finite-dimensional classification.

Higher-Degree Symmetric Maps and Polarization

A symmetric multilinear map of degree $k$ is closely tied to homogeneous polynomial functions of degree $k$ on $V$ . Setting all arguments equal produces the associated polynomial,

P (v) = T (v, \dots, v)

and, over a field of characteristic zero or greater than $k$ , this correspondence can be inverted by the polarization identity, which recovers the full symmetric multilinear map from its diagonal values alone. Polarization expresses $T (v_{1}, \dots, v_{k})$ as a finite linear combination of values of $P$ evaluated at signed sums of the $v_{i}$ , showing that a symmetric multilinear map and its associated homogeneous polynomial carry exactly the same information.

Relationship to Tensors and Symmetric Powers

The space of symmetric multilinear maps of degree $k$ on $V$ is naturally identified with the dual of the $k$ -th symmetric power ${Sym}^{k} V$ , playing a role directly analogous to the identification of alternating multilinear maps with the dual of the exterior power. Within the covariant tensor algebra on $V$ , the symmetric tensors of order $k$ and the alternating tensors of order $k$ occupy the two extremal subspaces of definite permutation symmetry, with the symmetrization and antisymmetrization operators projecting a general tensor onto each, and, in characteristic zero, every multilinear form of degree $k$ can be written as a sum of contributions of mixed symmetry type built from irreducible representations of the symmetric group acting on the $k$ argument slots.