##  [Symmetrization](/symmetrization-1) 

 Definition

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

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

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

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

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

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

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

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

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.