Levi-Civita Symbol

$${ϵ}_{\sigma (n)}=\{\begin{array}{rl}1& \text{If σn is an even permutation}\\ -1& \text{If σn is an odd permutation}\end{array}$$

In the above we have

• $S_n$ is the permutation group of size $n$
• ${\sigma }_n\in S_n$ is an element of the permutation group
• The parity of a permutation $\sigma$ ( i.e. being even/odd) is defined via the parity of transpositions needed to decompose the permutation $\sigma$.