N-dimensional Levi-Civita symbol (${ϵ}_{i_1\cdots i_n}$)

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