V 3 lines maximum size of equi lines in IR Question 23 41 42 6 7 - - PDF document

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Mar 17, 2023 409 likes •476 views

Equiangular lines with a fixed angle Joint with Alexandr Polyanskii, Jonathan Tidor, Yuan Y ao, Shengtong Zhang and Yufei Zhao arxiv arxiv 1708.02317 1907.12466 through 0 pairwise separated by same angle Lines in IR rt IRB blines V 3 lines

SLIDE 1

Equiangular lines with a fixed angle

Joint with Alexandr Polyanskii, Jonathan Tidor, Yuan Y ao, Shengtong Zhang and Yufei Zhao

arxiv 1708.02317

arxiv

1907.12466

Lines in IR

through 0 pairwiseseparatedby same angle

rt

IRB

3lines

V

blines

Question

maximumsizeof equi lines in IR

h

2 3 4

5 6

7 14

23 41 42

Max

3 6

10

16

28

276 276288

Cn

E

Max

E

NY

de

Caen 2000

Gerzon 1973

Question Whatif the angle is fixed

Each

Max size of equiangularlines

with anglearccos 2

in IR

1973

Lemmens Seidel

Eyz n

21h

for n

15 1989 Neumaier

Eys

Zz n

il

for not

1973 Neumann

Exch

E

Zn

unless ya is odd

2016

Bakh

Ex

E Ca

dependson d

2018 BallaDroixler Sudator

Edu

E 1.93n

i'f n

hold Keevash

and

at

13

SLIDE 2

Conjecture 1

Butch

Ey Cult In

EI

a n

Conjecture2

J

Polyanskii

ELCnl 2

n where K

Kal X

Spectral radius order

Kali

smallest k sit

I k vertexgraph Gat X

G

D

X CA

dock 3

a 31kcal

the eigenvalues ofadjacency matrix

h d

f

k

Eun

1 3

I

Too

2 2h

1 5

2 do 3

In

1 7

3 DX 4

9 n

ii a

THM

J

polyanskii

F

IR

conj 2 holds forall X E J

Spectralradii ofgraphs

4

is dense in a

TMM

JTYZZI

Barrier

Eun

L

il

when Kal

CA

uz noch

Es

n t

when KCN

Remark When 2 217

can show

Kat

k

hence Es

L

n h

hold

SLIDE 3

Equiangularlines in IR nin vertexgraph G

V

Set of

unit vectors

V

vertex

set

each vector represents a line

pgp XI

A EJ Z 0

Vi Vz

E L

J

Y

E

adj mat

allonesmatrix

Gram matrix

vi Vj7 i j to

CRANK

rank R2 A EJ

n

rank Gram mat

En

Think as if

rank

1dL A En Goal

Given n find largest m

Sx

an

m v.to graphG with PSD t CRAE

NK

Alternativegoal Given on find smallest

ns.t

an

m Vfx graph G Et

PSD

t

RANK

In other words given me

minimize rank 42

A

maximize

mult d

G

PSD

7 We need to deal with

2 cases

completely reducible

G

G U

G c where

each connected component Gi satisfies Xilai

X

Matt X G

I malt Cd Gi

C mkay

In this

case

it is optional to take

lait

KCA