Page "Symmetric algebra" Paragraph 16

Conversely, symmetric tensors are defined as invariants: given the natural action of the symmetric group on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially.

Note that under the tensor product, symmetric tensors are not a subalgebra: given vectors v and w, they are trivially symmetric 1-tensors, but v ⊗ w is not a symmetric 2-tensor.