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| Format: | Preprint |
| Published: |
2023
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| Online Access: | https://arxiv.org/abs/2312.10384 |
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| _version_ | 1866915335569932288 |
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| author | Yoshino, Kiyoto |
| author_facet | Yoshino, Kiyoto |
| contents | 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 $\sE_8$ 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10384 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Enumeration of sets of equiangular lines with common angle $\arccos(1/3)$ Yoshino, Kiyoto Combinatorics 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 $\sE_8$ 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 $n$. |
| title | Enumeration of sets of equiangular lines with common angle $\arccos(1/3)$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.10384 |