The Lasserre hierarchy for equiangular lines with a fixed angle

TL;DR

This paper advances the understanding of equiangular lines with a fixed angle by computing higher levels of the Lasserre hierarchy, providing new bounds through sophisticated algebraic and optimization techniques.

Contribution

It introduces a novel approach connecting group representation invariants to compute higher levels of the Lasserre hierarchy for the problem.

Findings

Computed second and third levels of the Lasserre hierarchy for the problem.

Derived new linear bounds on the maximum number of equiangular lines.

Performed asymptotic analysis to improve bounds in high dimensions.

Abstract

We compute the second and third levels of the Lasserre hierarchy for the spherical finite distance problem. A connection is used between invariants in representations of the orthogonal group and representations of the general linear group, which allows computations in high dimensions. We give new linear bounds on the maximum number of equiangular lines in dimension $n$ with common angle $\arccos \alpha$. These are obtained through asymptotic analysis in $n$ of the semidefinite programming bound given by the second level.