Sets of equiangular lines in dimension 18 constructed from A9 A9 A1
Abstract
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 A9 A9 A1, 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.
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