Projective Equivalence for the Roots of Unity

Let $\mu_{\infty}\subseteq\mathbb{C}$ be the collection of roots of unity and $\mathcal{C}_{n}:=\{(s_{1},\cdots,s_{n})\in\mu_{\infty}^{n}:s_{i}\neq s_{j}\text{ for any }1\leq i<j\leq n\}$. Two elements $(s_{1},\cdots,s_{n})$ and $(t_{1},\cdots,t_{n})$ of $\mathcal{C}_{n}$ are said to be projectively equivalent if there exists $\gamma\in\text{PGL}(2,\mathbb{C})$ such that $\gamma(s_{i})=t_{i}$ for any $1\leq i\leq n$. In this article, we will give a complete classification for the projectively equivalent pairs. As a consequence, we will show that the maximal length for the nontrivial projectively equivalent pairs is $14$.