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Projective equivalence for the roots of unity

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Abstract

Let \(\mu _{\infty }\subseteq \mathbb {C}\) be the collection of roots of unity and \(\mathcal {C}_{n}:=\{(s_{1},\ldots ,s_{n})\in \mu _{\infty }^{n}:s_{i}\ne s_{j}\text { for any }1\le i<j\le n\}\). Two elements \((s_{1},\ldots ,s_{n})\) and \((t_{1},\ldots ,t_{n})\) of \(\mathcal {C}_{n}\) are said to be projectively equivalent if there exists \(\gamma \in PGL (2,\mathbb {C})\) 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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Acknowledgements

The author is deeply grateful to Fedor Bogomolov for suggesting this problem and indicating the crucial Theorem 15.

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Correspondence to Hang Fu.

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Fu, H. Projective equivalence for the roots of unity. J Algebr Comb 56, 823–871 (2022). https://doi.org/10.1007/s10801-022-01134-1

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  • DOI: https://doi.org/10.1007/s10801-022-01134-1

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