Alternation

\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \DeclareMathOperator{\Alt}{Alt} \DeclareMathOperator{\sgn}{sgn} \]

The alternation operator \(\Alt\colon T^k(V^*)\to \Lambda^k(V^*)\) maps covariant \(k\)-tensors to alternatig ones. It is defined by

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

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