4.21.2 Tensor Symmetric Argument Permutation

Tensor Symmetric Argument Permutation is the full group action of the symmetric group Sₙ on the arguments of an n-ary multilinear map, and the associated notion of a symmetric map as one whose output is invariant under every element of this action, not merely under individual transpositions. Viewing symmetry through the lens of the entire permutation group, rather than one exchange at a time, brings the machinery of group actions and representation theory to bear on symmetric multilinear structure.

The Group Action on Arguments

Sₙ Acting by Reindexing

The symmetric group Sₙ acts on the set of ordered n-tuples (v₁, ..., vₙ) of vectors from V by σ · (v₁,...,vₙ) = (v_{σ⁻¹(1)}, ..., v_{σ⁻¹(n)}), permuting which vector occupies which position. A multilinear map f: V × ... × V → W is symmetric precisely when it is constant on every orbit of this action, that is, f(σ · (v₁,...,vₙ)) = f(v₁,...,vₙ) for every σ ∈ Sₙ and every tuple.

Full Sₙ Invariance Versus Pairwise Exchange

Since Sₙ is generated by transpositions, checking invariance under every transposition (the slot exchange rule) is logically sufficient to guarantee invariance under the full group action; the argument-permutation perspective, however, treats this invariance as a single condition on the whole group rather than a family of pairwise conditions, which is the more natural framing when tools from group representation theory are to be applied.

Symmetrization as Group Averaging

The Symmetrization Operator

Given an arbitrary n-linear map g: V × ... × V → W, the symmetrization

averages g over the entire orbit generated by the group action, producing a genuinely symmetric multilinear map regardless of whether the original g had any symmetry at all.

Idempotence of Symmetrization

Applying the symmetrization operator to an already symmetric map leaves it unchanged, since every term in the averaging sum is then identical to the original value, and averaging n! copies of the same value returns that value; this idempotence, (g^{sym})^{sym} = g^{sym}, confirms symmetrization behaves as a projection onto the subspace of symmetric maps within the larger space of all multilinear maps.

Orbits, Stabilizers, and Repeated Arguments

Stabilizer Subgroups When Arguments Repeat

If some of the vectors v₁, ..., vₙ coincide, the stabilizer of the tuple (v₁,...,vₙ) under the Sₙ action, the subgroup of permutations fixing the tuple exactly, is nontrivial, isomorphic to a product of smaller symmetric groups permuting the repeated blocks among themselves; the orbit of such a tuple then has size n! / |stabilizer|, smaller than the generic orbit size n! obtained when all n vectors are distinct.

Multinomial Coefficients From Orbit Sizes

When symmetrizing a monomial-like expression with repeated variables, the orbit-size reduction due to a nontrivial stabilizer produces multinomial coefficients directly: symmetrizing the product v^{k₁}w^{k₂}⋯ (with k₁ copies of v, k₂ copies of w, and so on, totaling n factors) over all n! reorderings collapses to a sum with n!/(k₁!k₂!⋯) distinct terms, exactly the multinomial coefficient counting distinguishable rearrangements.

Representation-Theoretic Framing

The Trivial Representation

Symmetric multilinear maps correspond to the trivial one-dimensional representation of Sₙ, under which every group element acts as the identity; this is the representation-theoretic restatement of the argument-permutation invariance condition, situating symmetric maps as one specific isotypic component within the decomposition of the full space of multilinear maps under the Sₙ action.

Decomposition of General Multilinear Maps

The space of all n-ary multilinear maps on V decomposes, under the Sₙ action permuting arguments, into isotypic components corresponding to the irreducible representations of Sₙ; the symmetric maps form the component corresponding to the trivial representation, the alternating maps form the component corresponding to the sign representation, and for n ≥ 3 there are additional components corresponding to other irreducible representations of Sₙ, which are neither symmetric nor alternating.

Consequences for the Symmetric Power

Argument Permutation as the Quotienting Relation

The symmetric power Symⁿ(V) is built precisely by identifying elementary tensors related by the Sₙ action on their factors: v₁ ⊗ ... ⊗ vₙ and v_{σ(1)} ⊗ ... ⊗ v_{σ(n)} become the same element of Symⁿ(V) for every σ. The argument-permutation perspective on symmetric multilinear maps is thus mirrored exactly by an orbit-collapsing construction on the tensor power itself.

Basis Indexed by Multisets

Because argument permutation collapses each orbit to a single point, a basis for Symⁿ(V) is indexed by multisets of size n drawn from a basis of V, rather than by ordered tuples; the dimension formula dim(Symⁿ(V)) = C(d + n - 1, n) for V of dimension d counts exactly the number of such multisets, reflecting the orbit structure of the Sₙ action on tuples of basis vectors.

Practical Uses

Constructing Symmetric Invariants

Symmetrization over the argument-permutation action is the standard method for producing a symmetric multilinear map, or an associated symmetric tensor, from any given multilinear map when only the totally symmetric part of its behavior is of interest, such as extracting the symmetric part of a stress tensor in continuum mechanics or the symmetric part of a general bilinear coupling in a physical model.

Verifying Symmetry Efficiently

Rather than checking invariance for every one of the n! elements of Sₙ individually, verifying that a multilinear map is invariant under a generating set of transpositions, such as the adjacent transpositions (1 2), (2 3), ..., (n-1, n), suffices to establish full argument-permutation invariance, since these transpositions generate the entire symmetric group.