v_k \mapsto \frac{1}{k!} \sum_{\sigma \in S_n} v_{\sigma(1)}\otimes \cdots \otimes v_{\sigma(k)}. For instance, vw \mapsto \frac{1}{2}(v\otimes w + w \otimes v). This is related to the representation theory of the symmetric group: in characteristic 0, over an algebraically closed field, the group algebra is semisimple , so every representation splits into a direct