\[ \DeclareMathOperator{\Sym}{Sym} \]

The symmetrization operator \(\Sym\colon T^k(V^*)\to \Sigma^k(V^*)\) maps covariant \(k\)-tensors to symmetric ones. It is defined by

\begin{equation*} (\Sym \alpha)(v_1,\ldots,v_k)=\frac{1}{k!}\sum_{\sigma\in S_k} \alpha(v_{\sigma(1)},\ldots,v_{\sigma(k)}), \end{equation*}

where \(S_k\) is the group of permutations of the set \(\{1,\ldots,k\}\).

Remarks

A covariant \(k\)-tensor \(\alpha\) is symmetric if and only if \(\Sym \alpha = \alpha\) [1, Proposition 12.14].

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