= N-dimensional Levi-Civita symbol
{title2=$\epsilon_{i_1\cdots i_n}$}
For indices in $\{1,\ldots,n\}$, this <Levi-Civita symbol> is zero when any indices repeat, and otherwise is the <sign of a permutation> taking $(1,\ldots,n)$ to $(i_1,\ldots,i_n)$. Thus $\epsilon_{1\cdots n}=1$ and exchanging any two index slots changes the sign. It expresses a <determinant> as $\det M=\epsilon_{i_1\cdots i_n}M_{i_1,1}\cdots M_{i_n,n}$ with repeated indices summed. Exactly $n!$ components are nonzero, so $\sum\epsilon_{i_1\cdots i_n}^2=n!$.
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