Projective Equivalence for the Roots of Unity

Abstract

Let ${\mu }_{\infty }\subseteq \mathbb{C}$ be the collection of roots of unity and ${\mathcal{C}}_n:=\{(s_1,\cdots ,s_n)\in {\mu }_{\infty }^n:s_i\ne s_j\text{ for any }1\le i<j\le n\}$. 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 $1\le i\le n$. 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 $14$.

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