Let $V$ be a vector space. The $m$-fold symmetric product of $V$ is the quotient vector space of the $m$-fold tensor product $V^{\otimes m}$ by the subspace generated by the tensors \(v_1\otimes\cdots\otimes v_m - v_{\sigma(1)}\otimes \cdots\otimes v_{\sigma(m)}\) for all $v_i \in V$ and all $\sigma \in S_m$ (the set of permutations of ${1,\dots,m}$). It is denoted $\text{Sym}^m(V)$.