An operation on a multilinear tensor or function that projects it onto its symmetric part by averaging over all permutations of the input slots, yielding an invariant under permutation of arguments.

Principle

Apply the operator S = (1/n!) ∑_{σ∈S_n} σ, where σ permutes tensor factors (or arguments); in characteristic where n! is invertible this is an idempotent projection onto the symmetric subspace.

Demonstration

Given v⊗w in V⊗V, symmetrization gives (v⊗w + w⊗v)/2, which represents the corresponding element in the symmetric square Sym^2(V); applied to multilinear maps, it yields symmetric multilinear forms.

Misapplication

Averaging by 1/n! fails in characteristic p dividing n! because n! may be zero, so treating the naive average as a projection can be invalid; likewise, averaging over only a subset of permutations may not produce true symmetry.

Consequence

Produces symmetric tensors or forms used to define objects like symmetric powers, symmetric polynomials, and invariant theory constructions; makes permutation-invariant components explicit and computable.

Reversal

The complementing operation is antisymmetrization; reversing symmetry produces the alternating part by weighting permutations with their sign rather than uniformly, extracting skew-symmetric components instead.

Boundary

Applies to multilinear objects and tensor powers; requires care in fields or rings where factorial denominators are not invertible, and does not preserve nonsymmetric algebraic relations that depend on order-sensitive multiplication.

Semantic Tension

Close to 'symmetrize by summing' versus 'symmetrize by enforcing equal arguments' — one is an averaging projection, the other is taking the diagonal evaluation; they agree on multilinear tensors but differ in operational interpretation in some contexts.

Synthesis

Symmetrization is the canonical averaging projection that extracts the permutation-invariant (symmetric) component of a multilinear object, yielding elements of symmetric powers and enabling study of invariants under argument permutation.