Enumeration of sets of equiangular lines with common angle $\arccos(1/3)$

Abstract

In 2018, Sz\"{o}ll\H{o}si and \"{O}sterg\r{a}rd used a computer to enumerate sets of equiangular lines with common angle $\arccos(1/3)$ in dimension $7$. They observed that the numbers $\omega(n)$ of sets of $n$ equiangular lines with common angle $\arccos(1/3)$ in dimension $7$ are almost symmetric around $n=14$. In this paper, we prove without a computer that the numbers $\omega(n)$ are indeed almost symmetric by considering isometries from root lattices of rank at most $8$ to the root lattice $\sE_8$ of rank $8$ and type $E$. Also, they determined the number $s(n)$ of sets of $n$ equiangular lines with common angle $\arccos(1/3)$ for $n \leq 13$. We construct all the sets of equiangular lines with common angle $\arccos(1/3)$ in dimension greater than $7$ from root lattices of type $A$ or $D$ with the aid of switching roots. As an application, we determine the number $s(n)$ for every positive integer $n$.