Equiangular lines and the Lemmens–Seidel conjecture

Yen Chi Roger Lin; Wei Hsuan Yu

doi:10.1016/j.disc.2019.111667

Equiangular lines and the Lemmens–Seidel conjecture

Yen Chi Roger Lin, Wei Hsuan Yu

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

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 decomposition, 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 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.

Original language	English
Article number	111667
Journal	Discrete Mathematics
Volume	343
Issue number	2
DOIs	https://doi.org/10.1016/j.disc.2019.111667
Publication status	Published - 2020 Feb

Keywords

Equiangular set
Lemmens–Seidel
Pillar methods

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

Access to Document

10.1016/j.disc.2019.111667

Fingerprint Dive into the research topics of 'Equiangular lines and the Lemmens–Seidel conjecture'. Together they form a unique fingerprint.

Cite this

@article{2278c607b89f4d8aa86dab49b5491843,

title = "Equiangular lines and the Lemmens–Seidel conjecture",

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 decomposition, 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 Rr, 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.",

keywords = "Equiangular set, Lemmens–Seidel, Pillar methods",

author = "Lin, {Yen Chi Roger} and Yu, {Wei Hsuan}",

note = "Funding Information: The authors thank Eiichi Bannai and Alexey Glazyrin for their helpful discussions on this work. This material is based upon work supported by the National Science Foundation, United States under Grant No. DMS-1439786 while the second author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Point configurations in Geometry, Physics and Computer Science Program. Part of this work was done when the second author visited National Center for Theoretical Sciences (NCTS), Taiwan, in the summer of 2018. The authors are grateful to the support of NCTS. Finally, the authors would like to thank anonymous referees who offer useful comments on improvement of Theorem 5.3 and give clues to the remark following Theorem 5.3 . ",

year = "2020",

month = feb,

doi = "10.1016/j.disc.2019.111667",

language = "English",

volume = "343",

journal = "Discrete Mathematics",

issn = "0012-365X",

publisher = "Elsevier",

number = "2",

}

TY - JOUR

T1 - Equiangular lines and the Lemmens–Seidel conjecture

AU - Lin, Yen Chi Roger

AU - Yu, Wei Hsuan

N1 - Funding Information: The authors thank Eiichi Bannai and Alexey Glazyrin for their helpful discussions on this work. This material is based upon work supported by the National Science Foundation, United States under Grant No. DMS-1439786 while the second author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Point configurations in Geometry, Physics and Computer Science Program. Part of this work was done when the second author visited National Center for Theoretical Sciences (NCTS), Taiwan, in the summer of 2018. The authors are grateful to the support of NCTS. Finally, the authors would like to thank anonymous referees who offer useful comments on improvement of Theorem 5.3 and give clues to the remark following Theorem 5.3 .

PY - 2020/2

Y1 - 2020/2

N2 - 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 decomposition, 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 Rr, 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.

AB - 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 decomposition, 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 Rr, 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.

KW - Equiangular set

KW - Lemmens–Seidel

KW - Pillar methods

UR - http://www.scopus.com/inward/record.url?scp=85072609684&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072609684&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/7ca03ffe-9d7c-3eab-9ab2-bd409cc2a7a1/

U2 - 10.1016/j.disc.2019.111667

DO - 10.1016/j.disc.2019.111667

M3 - Article

AN - SCOPUS:85072609684

VL - 343

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 2

M1 - 111667

ER -