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Sets of equiangular lines in dimension $18$ constructed from $A_9 \oplus A_9 \oplus A_1$

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Title: Sets of equiangular lines in dimension $18$ constructed from $A_9 \oplus A_9 \oplus A_1$
Authors: Lin, Yen-chi Roger, Munemasa, Akihiro, Taniguchi, Tetsuji, Yoshino, Kiyoto
Publication Year: 2025
Collection: Mathematics
Subject Terms: Mathematics - Combinatorics, 05B40, 05B20, 05C50, 05C70
More Details: In 2023, Greaves et~al.\ constructed several sets of 57 equiangular lines in dimension 18. Using the concept of switching root introduced by Cao et~al.\ in 2021, these sets of equiangular lines are embedded in a lattice of rank 19 spanned by norm 3 vectors together with a switching root. We characterize this lattice as an overlattice of the root lattice $A_9\oplus A_9\oplus A_1$, and show that there are at least $246896$ sets of 57 equiangular lines in dimension $18$ arising in this way, up to isometry. Additionally, we prove that all of these sets of equiangular lines are strongly maximal. Here, a set of equiangular lines is said to be strongly maximal if there is no set of equiangular lines properly containing it even if the dimension of the underlying space is increased. Among these sets, there are ones with only six distinct Seidel eigenvalues.
Comment: 17 pages
Document Type: Working Paper
Access URL: http://arxiv.org/abs/2503.06377
Accession Number: edsarx.2503.06377
Database: arXiv
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  Data: Sets of equiangular lines in dimension $18$ constructed from $A_9 \oplus A_9 \oplus A_1$
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  Data: <searchLink fieldCode="AR" term="%22Lin%2C+Yen-chi+Roger%22">Lin, Yen-chi Roger</searchLink><br /><searchLink fieldCode="AR" term="%22Munemasa%2C+Akihiro%22">Munemasa, Akihiro</searchLink><br /><searchLink fieldCode="AR" term="%22Taniguchi%2C+Tetsuji%22">Taniguchi, Tetsuji</searchLink><br /><searchLink fieldCode="AR" term="%22Yoshino%2C+Kiyoto%22">Yoshino, Kiyoto</searchLink>
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  Data: 2025
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– Name: Abstract
  Label: Description
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  Data: In 2023, Greaves et~al.\ constructed several sets of 57 equiangular lines in dimension 18. Using the concept of switching root introduced by Cao et~al.\ in 2021, these sets of equiangular lines are embedded in a lattice of rank 19 spanned by norm 3 vectors together with a switching root. We characterize this lattice as an overlattice of the root lattice $A_9\oplus A_9\oplus A_1$, and show that there are at least $246896$ sets of 57 equiangular lines in dimension $18$ arising in this way, up to isometry. Additionally, we prove that all of these sets of equiangular lines are strongly maximal. Here, a set of equiangular lines is said to be strongly maximal if there is no set of equiangular lines properly containing it even if the dimension of the underlying space is increased. Among these sets, there are ones with only six distinct Seidel eigenvalues.<br />Comment: 17 pages
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RecordInfo BibRecord:
  BibEntity:
    Subjects:
      – SubjectFull: Mathematics - Combinatorics
        Type: general
      – SubjectFull: 05B40, 05B20, 05C50, 05C70
        Type: general
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      – TitleFull: Sets of equiangular lines in dimension $18$ constructed from $A_9 \oplus A_9 \oplus A_1$
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          Name:
            NameFull: Lin, Yen-chi Roger
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            NameFull: Munemasa, Akihiro
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            NameFull: Taniguchi, Tetsuji
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            NameFull: Yoshino, Kiyoto
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          Dates:
            – D: 08
              M: 03
              Type: published
              Y: 2025
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