An upper bound for the cardinality of a set of equiangular lines in $R%5Ed,1$

An upper bound for the cardinality of a set of equiangular lines

in $R%5Ed,1$

Citation for published version (APA):

Blokhuis, A. (1981). An upper bound for the cardinality of a set of equiangular lines in $R%5Ed,1$. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8108). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1981

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics

Memorandum 1981- 08 i1une 1981

AN UPPER BOUND FOR THE CARDINALITY OF A SET OF EQUIANGULAR LINES IN Rd,l

by

A. Blokhuis

Eindhoven University of Technology Department of Mathematics

P.O. Box 513, Eindhoven The Netherlands

AN UPPER BOUND FOR THE CARDINALITY OF A SET OF EQUIANGULAR LINES IN Rd,l

by A. Blokhuis

o.

Abstract.

It is proved that the cardinality of a set S of equiangular lines in Rd,l satisfies card{S)

~ ~d(d+1).

Cases where equality holds are indicated.

1. Introduction.

2.

Let Rd,1 be the (d+l)-dimensional vectorspace over R provided with the inner product

(x,y)

=

-x

oYo +x1Y1+ ... +xdYd .

If two lines through the origin span a plane on which the inherited inner product becomes positive definite,we can define their angle to be arccos \(x,y)\ ' where x and yare unit vectors along the lines. A set of equiangular lines is a set of lines such that for each pair the angle is defined and equal. By methods similar to those of Koornwinder, Delsarte, Goethals and Seidel (1l it is possible to show that the cardinality of such a set is bounded by

~(d+1) (d+2). By use of techniques of Larman, Rogers and Seidel [2] which give bounds in terms of the particular angle, this bound can be improved.

Some lemmas.

Let S be a set of v equiangular lines in R d,1 at angle arccos 0(

Lemma 1. I f v

>

_2d+2 then 0< -1 is an odd integer. Lemma 2. I f d

>

3 and v

>

~d(d+1) then (d+1) 0< 2

~

1

.

-2-Proof of Lemma 1. (This is essentially Theorem 3.4 of Lemmens and Seidel [4] , due to P. M. Neumann).

Let A = 0<-1 (G - I) where G is the Gramian of S (each line

is represented by a unit~~ector). -1

eigenvalue - 0 ( of multiplicity

A is an integral matrix with -1 m

=

v-d-1. Therefore I - 0( is an algebraic integer, and every algebraic conjugate is an eigenvalue with multiplicity m. Since 2m

=

2v - (2d+2)

>

v , there is at most one eigenvalue of multiplicity m, therefore - 0(-1 is rational, hence an integer. The m-dimensional eigenspace of A

-1

corresponding to - a< , and the (v-i)-dimensional eigenspace of J , the all one matrix of size v x v , corresponding to the eigenvalue 0 have a non~trivial intersection since m> 1 . It's vectors are also eigenvectors of B

=

~(J - I - A) corresponding to the eigenvalue ~

=

~(-1- 0( -1) • _{Since B is an integral}

matrix and

_~

is rational, we infer that _{~ is an integer and -} 0(

an odd integer.

Proof of lemma 2. (The method is essentially that of Larman, Rogers and Seidel

(21).

-1

Fix a line 1 in S and a unit-vector w on l . For each line diffe-rent from 1 in S we choose a representing unit-vector x with

(x,w)

=

0(. Let U denote the set of unit-vectors you get in this manner (including w). For x,y & U define

d(x,y) = ~(x-y,x-y)

=

1 - (x,y) .

We may assume that

u,

considered as a set of points, is not con-tained in a lower-dimensional flat. Let D be the matrix

D

=

[d(X,y)] . Indexing the first row of D by w we have x,y€ U

where D

w

D

-3-Claim (Here

D2 has signature (+l2(_ld(Olv-d-2

(+) means that D has exactly two positive eigenvalues, etc.) Proof. We will show that D is the Gramian of a spanning set of vectors in R2,d. Define the map u ~ R2 ,d as follows

x -'"

x

=: (xii) where the inner product in R2,d reads

Then for any x,y ~ U we have

(x,y]

₌

-(x,y) + 1 = d(x,y)

~

R2,d _{u <:}

_6...1..

so 0 is the Gramian of U in Furthermore, let

in R2,d

,

then, with u

_""

(u',u

d+i ) we have

Lu,x

1

=

0 for all x I: U ,

which implies

(u' ,xl

=

u_d+₁

Since U is not contained in a lower dimensional flat we have u' "" 0 u

d+1 = 0, and U spans R 2

,d. This proves the claim. Now let D ' _w = 0 _w - J and

D' (1-o()jt)

) . D I

J w

Then D and D' have the same signature, because pO + (l-p)D' has the same 0-eigenvalues for all values of p

matrix of the quadratic form

t

u D'u =

Therefore D' is the

where the Li are linearly independent linear forms in u=(u

_{o' ..}

,u

-4-Now let L I

i be the form obtained from by putting u o =0 • Then

so either

t

L -2' , ••• , L d I

1

forms an independent system (0) , or

there is a dependence relation which, after a suitable linear trans-formation reads (1) L_1

=

0 or (2) L1

=

0 or (3) L₁ : ~ L_l • For the signature of the matrix D • _w there remain 4 possibilities :

( I ) ( II) (III) ( IV) (+)2 (_)d(0)v-d-3 (+)l(_)d(0)v-d-2 (+)2(_)d-l(0)v-d-2 (+)1(_)d-l(O)v-d-1 (case (0) ) (case (1) (case (2) (case (3)

Define the matrix E

=

and off-diagonal entries

0< -l(D I + I) w

!.

1, at least E v-d-1 (3) (3) \:>(

1=

1 ) .

has diagonal entries 0

-1

eigenvalues ) . « I and at least one of them is actually larger. Comparing eigenvalues and traces of E and ETE we get after arranging the eigenvalues in nondescending order : whence by

A

+ 2

<

<

a

(v-1) (v-2) (v-d-i) 2

L

(v-d-1)

J

-'---"':"l.~

<::

d (v-i) (v-2) - ~....:;;...;;..:... ~ ~~

and this implies

v-d-1

_<

-5-According to lemma 1. we may put we can rewrite (z) as

-1

CK =: 21<:-1 , an odd integer. Then

k<'l,.+l,.·~

\( ;:ttl

Now v

>

l,.d(d+1) yields 2k-1 <. '" d+2 I but this implies

(d+1)o<2~1

Remark. Relation (z) can be used to obtain better bounds when (d+l) ()(

2~

1 . This will be done after we proved the main theorem.

3. The main theorem.

Theorem. Let S be a set of equiangular lines in Rd,l then card (8) ~ l,.d (d+1) .

Proof. We first consider the case proof of lemma 2. For each u in

d

>

3 . Let U we define F (x) u 2 2 =: (u,x) - 0( (x,x)

and we define additional functions

f (x)

=

(x,x) o U f. (x) 1. = i=1, ... Id be as in the Rd,1~ R : F u

we

will show that the set F

=

5" F d d. \ u E U , i=1, ... ,d.}

- 1 u 0 1.

is independent. Since these functions are homogeneous of degree 2

the cardinality of F is bounded by l,.(d+l) (d+2). This will imply card{U) ~ l,.d(d+l) if the functions in F are independent. Suppose we have U€U a F (x) + u u d i=l

2.

a. f . (x) + a f (x) : 1. 1. 0 0

o •

-6-Inserting v E U in this relation and observing that F (v)

₌

(1 - 0< 2)

J

u uv we arrive at a (1- 0(2) +

L

a.u u, + a

₌

0 u i l. 0 l. 0 CO)

Comparing coefficEients of x 2

,

_xi 2 and x x, we get

0 o l. 2

L

2 2 0 x a (u + IX )

-

a

=

0 u 0 0 (1) u 2

L

a (u, 2 IX 2) ₀ x,

-

+ a

₌

l. U l. 0 (2) U X x.: o l. -2

Z

a u u. U 0 l. + ai = 0 (3) U

now add (1) and (2)

2:

2

_-L

2

a u

₌

a u

u 0 u i

u u

Summing both sides over i and using -u 2 2 o +u1 + 2 +u d

=

1 1 we obtain d

L

a U 2

=

-L

a (1+u 2) u 0 u 0 u U from (1) we obtain 2 1 L.a a

₌

0(

-d+l) 0 (4) U U

By lemma 2. (d+l) c>< 2

<

1 implies v

~ ~d

(d+l) . We may therefore assume (d+1) 0( 2

~

1 . Now, if (d+l) 0( 2

=

1 (4)

yields a =0 and i f (d+l) 0( 2

>

1 we can multiply (0) with

o

a and sum over u (using (3) and (4) to obtain u

L

_a 2 _{(1- 0( 2)} d 2 d+l 2 + ~

_Z

a. + (d+ 1) co( ... -1 a 0

.

U i=1 l. 0 u

"

-7-Now (d+l) 0( 2

>

1 implies that all a are

o ,

and the same

applies for (d+1) 0 ( 2 = 1 since then we can forget about the

which is

o .

This proves the theorem for d

>

3

.

For d=O and d=1 the question makes no sense, since there are no planes where the innerproduct becomes positive definite. Straight-forward carculations show that 3 lines is the maximum in R2,l • The three lines spanned by

(0,1,0) (0,0<' ,

~1-

0( 2 (c, 0< ,

where _{2 (0<'+1)(20<-1)}

c

=

1 - 0(

realize this maximum. It follows that ~::: 0( <. 1 case 0<

=~

corresponds to 3 lines in R2 at

and the special

600 •

a

0

For d=3 , 6 lines are realized by the diagonals of the icosahedron. Now suppose there is a set S

R3,1. We then necessarily have

containing 7 equiangular lines in 4

~

2

<

1 . An easy graphtheoretic argument (Ramsey) shows that it is possible to select 4 equiangular representing unit vectors. Let v

1, v2, v3' v4, be those vectors, Le.

(v"v,)=1

1. 1. (v"v,) 1. J = 0<

A first observation is that 0(

'f

-1/3 since otherwise their Gramian

would have the wrong signature. Relation (*) in lemma 2 : v-d-l

0(2

<.

d (v-2)

(which doesn't depend on the fact that and therefore let 0( f= -1/3 a

=

be a fifth vector in S so Let 2· d

>

3) implies c{

>

1/5,

v actually span R3,1 . Now

-8-Since (a,a) = 1 we have (1).

4

~

( ) 2=4ao<.2 yields (1- 0<) 2 A 2 1. Z +2 0( (1+ c( )AI =4 CJ( • (2) i=1 a,v. ~

Multiplying (1) with (1- 0<) and substracting (2) gives

2 2

1:1( (-1-3 0( ) A

=

1- CI( -4 "< 1

Since the left hand side is nonpositive, we infer a contradiction.

4. Final Remarks.

Apart from the case C( =1/3, equality seems to be a rare event.

For f.X =1/3 , d ~ 7 equality is realized by the following set of vectors in the orthoplement of w

=

(2~;

Id+1) in Rd+1,1;

The requirement d ~ 7 ensures that (w,w) ~ 0 where the case d=7 corresponds to the well known set of 28 equiangular lines in R7 . For 0( =1/5, equality occurs for d=23 realize. by the

276-two graph, (4] seen as a set of equiangular lines in R23 • Using indefinite metric they can be represented nicely as follows: 23 vectors (3

'fi;

_11, 122)

253 vectors (\f2; 17, 016) where the positions of the ones correspond to the blocks in the Steiner system 4-(23,7,1) •

23

These vectors are actually in R since they are in the orthoplement of the vector w

=

(3~~;

123) .

The lemmas suggest that something peculiar happens around (d+1) 0< 2 =1.

This is indeed the case. A slight modification of the proof of lemma 2. yields the following bound:

Theorem. If S, v and d are as before and (d+1) 0( 2

<

1 and v> 2d+2 then

v..::.

~(1-

tK

20

I'::

1-do< 2'

I.

-9-This is called the (indefinite) special bound cf. [4] Thro. 3.6. For 0( =1/5 the special bound gives interesting numbers, closely

related to open problems in the theory of strongly regular graphs: d=22 v ~ 176 d=21 v ~ 126; d=20, v ~ 96; d=19, v ~ 76. For d=22 equality is obtained by the vectors

where the positions of the ones correspond to the blocks of 4-(23,7,1) , that are orthonormal to the vector (0;1,0, .•. ,0) • For d=21 equality is obtained by the subset of vectors from the system above that are also orthogonal to ( { 2 ; 18,015) where the ones correspond to a particular block of 5-(24,8,1) containing the point corresponding to (0; 1,0, ••• ,0) •

Very interesting is the question concerning d=20 and d=19. The lifes of four unknown strongly regular graphs depend on the existence of 76 equiangular lines in R19,1, while three candidates relate to systems of 96 equiangular lines in R20,l .

For 0<

<

1/5 no case of equality for the absolute bound is known, so probably the bound may be improved here.

5. References.

[11 Ph. Delsarte, J.M. Goethals and J.J. Seidel; Spherical codes and designs. Geometriae Dedicata ~ (1977) 363-388.

(2] D.G. Larman, C.A. Rogers and J.J. Seidel i On two-distance sets

in Euclidean space. Bull. London Math. Soc. 9 (1977) 261-267 • [3] E. and E. Bannai; An upperbound for the cardinality of an

s distance subset in Euclidean space. (to appear). t41 P.W.H. Lemmens and J.J. Seidel Equiangular lines.

J. Algebra 24 (3) (1973) 494-512.

(5] D.E. Taylor; Regular two-graphs; Proc. London Math. Soc. (3) ~ (1977) 257-247.