For $\sigma$ a permutation in $S_n$, the sign of $\sigma$ denoted $\text{sgn}(\sigma)$ is the determinant of the linear transformation $P_\sigma:\mathbb R^n\to\mathbb R^n$ given by \(P_\sigma(x_1,\dots,x_n) =(x_{\sigma(1)},\dots, x_{\sigma(n)}).\)
Wikidata ID: Q1064405