2023/12/16 by Kiyoto Yoshino, Yoshino, Kiyoto
Computer Science · Materials Science · Mathematics · #Advanced Combinatorial Mathematics #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2312.10384
openalex publication_date 2023/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2018, Szöllősi and Östergård used a computer to enumerate sets of equiangular lines with common angle \arccos(1/3) in dimension 7. They observed that the numbers ω(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 ω(n) are indeed almost symmetric by considering isometries from root lattices of rank at most 8 to the root lattice \sE8 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 ≤ 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.