15.7 Tensor Symmetrization Operator Structure
The tensor symmetrization operator structure defines how symmetric tensors are formed through algebraic operations, key in multilinear algebra and physics.
Tensor Symmetrization Operator Structure is the complete algebraic description of the map that sends an arbitrary tensor to a totally symmetric tensor by averaging its components over every permutation of a chosen set of index positions, together with the properties that make this map a well-defined, idempotent projector onto the subspace of symmetric tensors. The operator, commonly denoted Sym, combines a permutation sum, which gathers every rearrangement of the relevant indices, with a normalization factor, which rescales that gathered sum to the correct magnitude, and the interaction of these two pieces is what gives the operator its defining behavior.
As a structure, Sym is best understood through the properties it must satisfy rather than through its formula alone: it must map every input into the subspace of tensors obeying the symmetric component equality constraint, it must leave any tensor already in that subspace unchanged, and it must be linear in its input. These three properties together characterize Sym as the orthogonal-style projector onto the symmetric subspace within the larger space of all tensors of a given rank.
Formula and Immediate Consequences
Combined Definition
The symmetrization operator applied to a rank-n tensor S is given by:
combining the permutation sum over the symmetric group S_n with the normalization factor 1 divided by n factorial.
Linearity of the Operator
Because the formula involves only a sum of components with a fixed scalar coefficient in front, Sym is linear: Sym(aS + bR) equals a Sym(S) plus b Sym(R) for any tensors S and R and scalars a and b, a property inherited directly from the linearity of ordinary summation.
The Two Defining Properties
Image Lies in the Symmetric Subspace
As established by the invariance of the permutation sum under reindexing, Sym(S) satisfies the equality constraint T_{i1...in} = T_{iσ(1)...iσ(n)} for every permutation σ and every input S, so the image of Sym consists entirely of totally symmetric tensors, regardless of the symmetry properties of the input.
Fixed Points Are Exactly the Symmetric Tensors
If T already satisfies the equality constraint, every term in the permutation sum defining Sym(T) equals T_{i1...in}, so the sum totals n factorial times that value, and the normalization factor divides this back down to exactly T_{i1...in}; thus Sym(T) equals T whenever T is already symmetric, and combined with the previous property this shows Sym is idempotent, Sym(Sym(S)) equals Sym(S) for every S.
Sym as a Projector
Projector Terminology
An operator satisfying idempotence, mapping every input into a fixed subspace and acting as the identity on that subspace, is called a projector onto that subspace; the symmetrization operator is precisely the projector onto the subspace of totally symmetric tensors within the full space of rank-n tensors.
Complementary Projector
The operator I minus Sym, where I is the identity, projects onto a complementary subspace containing tensors with no totally symmetric component; every tensor S decomposes uniquely as Sym(S) plus (S minus Sym(S)), splitting it into its symmetric part and a remainder that vanishes under further symmetrization.
Structural Variants of the Operator
Restriction to a Subset of Positions
The same operator structure applies when only a designated subset of index positions is to be symmetrized, replacing the full symmetric group S_n with the subgroup permuting only that subset, and replacing n factorial with the factorial of the subset's size; this restricted operator projects onto tensors symmetric in that subset alone, leaving other index positions untouched.
Composition With Other Symmetry Operators
Sym can be composed with other structural operators, such as an antisymmetrization operator built analogously but with alternating signs attached to each permutation term; applying Sym after antisymmetrization, or in the reverse order, on a generic tensor typically annihilates the result, reflecting the fact that the symmetric and antisymmetric subspaces intersect only in the zero tensor for rank two and above.