Equiangular lines and the Lemmens-Seidel conjecture

TL;DR

This paper proves longstanding claims about equiangular lines with a specific angle, generalizes a key theorem, and determines maximum sizes of such sets in dimensions 8 to 10.

Contribution

It confirms the Lemmens-Seidel conjecture for angle 1/5 and extends the Neumann Theorem to restrict angles when many lines are present.

Findings

Proof of Lemmens-Seidel claims for angle 1/5

Generalization of the Neumann Theorem

Exact maximum sizes of equiangular sets in dimensions 8, 9, 10

Abstract

In this paper, claims by Lemmens and Seidel in 1973 about equiangular sets of lines with angle $1/5$ are proved by carefully analyzing pillar decompositions, with the aid of the uniqueness of two-graphs on $276$ vertices. The Neumann Theorem is generalized in the sense that if there are more than $2r-2$ equiangular lines in $\mathbb{R}^r$, then the angle is quite restricted. Together with techniques on finding saturated equiangular sets, we determine the maximum size of equiangular sets "exactly" in an $r$-dimensional Euclidean space for $r = 8$, $9$, and $10$.