0
6
5
4
0
7
8
9
1
3
5
2
4
6
6
1
7
8
9
2
4
3
5
0
5
0
2
7
8
9
3
4
6
1
4
6
1
3
7
8
9
5
0
2
9
5
0
2
4
7
8
6
1
3
7
1
0
6
8
9
6
1
3
5
7
0
2
4
8
7
2
1
7
8
9
0
2
4
6
1
3
5
9
8
7
3
1
2
3
4
5
6
0
7
8
9
1
9
8
7
2
3
4
5
6
0
1
8
9
7
3
2
9
8
3
4
5
6
0
1
2
9
7
8
5
4
3
9
8
7
6
0
1
2
2
3
4
5
6
0
1
7
9
8
4
5
6
0
1
2
3
8
7
9
6
0
1
2
3
4
5
9
8
7
COM BIN A TOR I A L
MAT HEM A TIC S
YEA R
February 1969 - June 1970
SOME PROPERTIES AND APPLICATIONS OF HERMITIAN VARIETIES IN A FINITE
PROJECTIVE SPACE PG(N~ q2) IN THE CONSTRUCTION OF
STRONGLY REGULAR GRAPHS (TWO-CLASS ASSOCIATION SCHEMES)
AND BLOCK DESIGNS
I. M. Chakravarti
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 600.23
MARCH 1970
This researah was partially supported by the Army Researah Offiae~
under Grant No. DA-ARO-D-31-124-G910 and the u.s. Air Forae
Offiae of Saientifia Researah under Contraat No. AFOSR-68-1406.
DurhaJ71~
SOME PROPERTIES AND APPLICATIONS OF
PROJECTIVE SPACE
PG(N,q2)
HERMITIfu~
VARIETIES IN A FINITE
IN THE CONSTRUCTION OF STRONGLY
REGULAR GRAPHS (TWO-CLASS ASSOCIATION SCHEMES) AND BLOCK DESIGNS
by
I.M. CHAKRAVARTI
Department of Statistics
University of North Carolina at Chapel Hill
1.
Introduction and Summary.
2
GF(q ),
where
q
H
=
«h
ij
is called Hermitian if
2
PG(N,q)
xTHx(q)
)),
h
ij
whose row vectors
is any element of a Galois field
Since
h.
i,j
h
q2
= h,
= O,1,2, ... ,N,
= hji for all i,j.
T
x
=
(xO,x1""'~)
h
x(q)
q q
q)
(xO,x 1 ""'XN
.
h
= h q is
is conjugate to
with elements from
h.
GF(q2)
satisfy the equation
if
H is
is the column vector whose transpose is
The variety
V_
N 1
is said to be non-degenerate if
H has
N+1.
The Hermitian form
to the canonical form
linear transformation
Hermitian variety
xTHx(q)
of order
o 0 + Y1 y- 1 +
~ = Ay
[4].
V_
N1
y y-
in
PG(N,q2)
N+1
-
+ YrYr
and rank
r
can be reduced
by a suitable non-singular
The equation of a non-degenerate
can then be taken in the canonical
form
(1.1)
A
The set of all points in
= 0 are said to form a Hermitian variety VN_1 ,
Hermitian and
rank
h
is a prime or a power of a prime, then
defined to be conjugate to
square matrix
If
X
q+1 +
o
q+1
xl
+ ... +
XNq+1 = O.
This research was partially supported by the Army Research Office,
Durham, under Grant No. DA-ARO-D-31-124-G910 and the U.S. Air Force
Office of Scientific Research under Contract No. AFOSR-68-l406.
2
The properties of the non-degenerate Hermitian variety
PG(2 ,q 2)
· ·ln canonlca
. 1 f orm lS
.
wh ose equa tlon
studied by R.C. Bose in [1].
in
2
o
2
in
= a,
were
The geometry of the Hermitian variety
V _
N l
was studied in some detail by R.C. Bose and by the author in
PG(N,q )
[4].
q+l+ x q+l
Xq+l+ xl
VI
In this last paper, the authors developed the theory of tangent and
polar hyperplanes of Hermitian varieties and characterized the sections of
these varieties by tangent hyperplanes and derived an algebraic expression
for the number of points on a Hermitian variety.
that if
N
=
2t+l
or 2t+2,
tains flats of dimension
a non-degenerate Hermitian variety
and no higher.
V _
N l
cpn-
The number of such subspaces
A study of the geometry of the surface
was derived.
= 0,
t
It was also shown there,
provided a new geometric interpretation of
X
q+l+xl
q+l+ x q+l+ x q+l
2
3
o
some statistical'"des1gns,
in particular, the balanced incomplete block design
v
= b = 45,
r
= k = 12,
A = 3.
In [3], R.C. Bose made a special study of the Hermitian variety
PG(3,3)
2
and derived a representation
with equation
of
V
in terms of external points, secant planes and self-conjugate
tetrahedra with respect to the Hermitian variety.
We shall assume the re-
suIts of the two papers [1] and [4] to which the reader may refer for detailed proofs.
In this paper, we derive the section of a non-degenerate Hermitian
in
variety
2
PG(N,q )
by a polar hyperplane (not necessarily a tan-
gent hyperplane) and the number of u-flats contained in a non-degenerate
in
2
PG(N,q )
integer less than or equal to
[N;1]-f1ats in
V _
N 1
where
for
x.
[x]
denotes the greatest
The number of points and the number of
were enumerated already in [4].
Next, considering the
geometry of external points, tangent lines and secant lines with respect to
3
a non-degenerate Hermitian variety
Z
in
VI
PG(Z,q )
we get a family of
two-class association schemes (or a strongly regular graph) with parameters,
= q Z(q Z-q+l),
v
n
1
= (q Z. 1) (q+l) ,
P
1
Z
P11
= q Z-Z+q Z and
11
Z
(q+l) .
Then, the configuration of external points, tangent lines and secant lines
with respect to a non-degenerate
Z
PG(3,q )
in
is used to derive a
family of two-class association schemes with parameters,
Z 2
2
3
Z
=
(q +l)(q -1), n = q (q -q+l)(q -q-l),
Z
l
Z
P
= (q Z-q)(q+l) Z .
ll
p
n
v
1
= q 2 -Z+q 4
11
= q3(qZ+l)(q_l),
and
Again, considering the geometry of generators in a non-degenerate
2
in
we get a family of two-class association schemes (equivalently,
PG(3,q ),
a strongly regular graph) with parameters
p
V
2
1
2
= q-l and
11
P11
=
Z
v
= (q 3+l)(q+l),
Taking the tangent planes as blocks
q +1.
we get a class of partially balanced incomplete block design with parameters
= (q3+ l )(qZ+l), v = (q 3+l)(q+l),
b
Z
P11
2.
2
q +1,
= q Z+1,
r
T
= O.
= (cO' c l ' ... ,~)
equivalently
V _
N l
= q(q Z+1),
A point
Hc (q)
= O.
V_
N l
2
PG(N,q),
V _
N l
in
2
PG(N,q),
V _
N l
if
cTH
points of
V _
N I
= 0 or
which is not singular is called
Thus a non-singular point may be a regular point
VN_ l ·
2
PG(N,q )
V N I
with
with row vector
V _ ,
N l
in which case we shall call it an
with respect to
I t is clear that a
VN_ l ·
non-degenerate Hermitian variety cannot possess a singular point.
other hand, if
= q-l,
= 1,
C in
A point of
or a point not lying on
external point of
1
P11
is called a singular point of
a regular point of
of
Al
1
Consider a Hermitian variety
Definitions.
equation
c
= q+l,
k
n
is degenerate and rank
H
r
<
N+I,
On the
the singular
will constitute a (N-r)-flat called the singular space of
4
Let
C
V_
N I
with respect to the Hermitian variety
with equation
PG(N,q)
C
=0
xTHx(q)
is
which satisfy the equation
o.
(2.1)
C is a singular point of
with the whole space
of
Then the polar space of
2
defined to be the set of points of
When
T
c .
be a point with row vector
2
PG(N,q ),
2
PG(N,q ).
V _ ,
N I
C is identical
the polar space of
When, however,
c
is a non-singular point
that is either a regular point of
or an external
point, the equation (2.1) is the equation of a hyperplane which will be
called the polar hyperplane of
be two points of
D,
2
PG(N,q).
C with respect to
V _ .
N I
If the polar hyperplane of
then the polar hyperplane of
D
passes through
defined to be conjugate to each other with respect to
points lying in the polar hyperplane of
conjugate to
plane of
C.
If
C,
C and
C passes through
Two such points are
VN- I .
Thus the
V_ ,
N I
the polar hyper-
C is self-conjugate and in this
that is,
case, the polar hyperplane is called the tangent hyperplane to
When
VN- I
D
C are all the points which are
C is a regular point of
C passes through
C.
Let
V_
N I
is non-degenerate, there is no singular point.
at
C.
To every
point there corresponds a unique polar hyperplane, and at every point of
V_
N I
3.
there is a unique tangent hyperplane.
Sections of Hermitian varieties with polar hyperplanes which are not
tangents.
PG(N,q2)
l.
Let the equation of a non-degenerate Hermitian variety
be
xTHx(q)
=0
and let
Then the polar hyperplane
VN_ I
in
D be an external point with row vector
of
D has the equation
Let
linearly independent and let
T
Then the row vector
F~l = D.
must be independent of the row vectors of
FO,Fl, ... ,F _ .
N l
find a non-singular linear transformation
y = Ax.
51
of
Hence, we can
such that
FO,F1, ... ,F
N
become the fundamental points of the reference system with row vectors
(1,0, ... ,0). (0,1, ... ,0), ...
SN_l
becomes
YN
=
where
0
and
( 3.1)
VN l
and the equation of
i, j = 0,1, ... ,N
is conjugate to
FO,Fl, ... ,F _
= 0,
yTCy(q)
becomes
is Hermitian of rank
N+l.
Since
and is not self-conjugate, we have
Nl
~i
gNN 1= 0.
Thus the equation of
(0,0, ... ,1).
0,
i
0,1 •... ,N-l.
We can therefore write
gON-l
(3.2)
I
0
I
1---
G
°
gN-lN-l l
I
°
°
Now.
det G
=
~IG*I I O.
Regarding
section of
where
y*T
SN_l
Y
N
=
=
0
yTGy(q)
retaining only the first
o
N.
IG*I I 0
as a projective space of
and
=
and
[
0
J
~-t"NN
I gNN
gNN I 0,
(YO'Yl •...• yN-l)
shown above. is
1--
G* I
and thus rank
(N-l)
G*
N rows and
=
N.
dimensions the
can be expressed as
0
G*
y*TG*y*(q)
is the matrix obtained from
N
columns of
Hence the Hermitian variety
G.
V_
N 2
is a non-degenerate Hermitian variety of
But
0
G by
rank G*. as
with equation
rank N.
Thus we
have the
THEOREM
3.1:
Given a non-degenerate Hermitian variety
the polar hyperplane
SN-l
of an external point
in
D intersects
2
PG(N.q ).
in a
6
non-degenerate Hermitian variety
V _
Let
V _
N Z
of
rank N
contained in
SN_l.
be a non-degenerate Hermitian variety in
N l
and
Dl,DZ, •.. ,D _
be a set of N-m external points whose row vectors
Nm
T
T
~l' ... '~-m are linearly independent.
The intersection of the polar hyperplanes of these points is an m-flat
L
m
whose points satisfy the equations
(3.3)
L ,
m
row vectors are linearly independent and let
F +
m1
We can find a non-singular linear transformation
y
= Dl ,
..• , F
N
=~
whose
= DN_m•
such that
FO,Fl, •.• ,F , ..• FN become the fundamental points of the reference
m
The points of
system.
will then satisfy the equations
L
m
0,
(3.4)
and the equation of
V _
N 1
will be
(3.5)
where
G is Hermitian of rank N+1.
(3.6)
where
G*11-OJ
-
~°
G
G*
But
from zero.
is a
IGI
=
(n+1)
G
1
x
(m+1)
IG*IIGI :1 0,
G is a
matrix and
hence both
IG*I
Thus the equation of the section
be expressed as
and
V
m-l
(N-m)
IGI
of
x
(N-m)
matrix.
must be distinct
V _
N 1
with
L
m
can
7
(3.7)
where
~
y*
and
m
THEOREM
= (YO,Yl, ... ,Y m)
G*
is Hermitian of
3.2:
2
PG(N,q )
of
T
rank m+l.
Hence we have the
The section of a non-degenerate Hermitian variety
and an m-flat
(N-m)
is a point in m-dimensional projective space
~
m
VN~l
in
which is the intersection of the polar planes
linearly independent external points is a non-degenerate
Hermitian variety
V
m-l
of
rank m+l
contained in
~
m
.
We quote here, without proof, some theorems proved in [4] as we need
them for proving some new results.
THEOREM
3.3:
If the Hermitian variety
is degenerate with
r-l
~
rank r < N+l
V _
N l
and
~
disjoint with the singular space
~
=0
is a flat space of dimensions
r-l
of
N-r
V _ ,
N l
intersect in a non-degenerate Hermitian variety
r-l
xTHx(q)
with equation
V_
N l
then
V
and
contained in
r-2
~ r- 1·
This is the Theorem 7.1 in [4].
THEOREM
3.4:
The tangent spaces at two distinct regular points
of a Hermitian variety
A and
if
V N l
B
are identical if and only if the line joining
B meets the singular space of
V N l
A and
V _
N l
in a point.
is non-degenerate then the tangent spaces at
In particular,
A and
B must be
distinct.
This is Theorem 7.3 in [4].
THEOREM
3.5:
Given a non-degenerate Hermitian variety
space at a point
variety
V _
N 2
V _
N 2
of
C of
V _
N l
rank N-l
intersects
contained in
consists of the single point
C.
V _
N l
~N-l'
V _ ,
N l
the tangent
in a degenerate Hermitian
The singular space of
8
This is Theorem 7.4 in [4].
LN_ l
Let
be the tangent space at a point
Hermitian variety
equation of
V_ .
N l
L
_
N l
V _
N l
is
and let
c
T
xTHc(q) = 0,
V _
N Z
where
contained in
sists of the single point
Suppose
vector
SN-l
D,
of
xTHx(q)
N Z
L _
N l
L _
=0
is the equation of
V_
N l
and
Consider an external point
xTHd(q) = 0,
disjoint with
C.
is a degenerate
D.
D with row
Then the polar space
L_
will intersect
L
_ .
N Z
in a
N l
Hence by Theorem 3.4,
L _
and
V_
N 3
contained
N Z
V _
will intersect in a non-degenerate Hermitian variety
N Z
in
Then the
and its singular space con-
N l
C is not conjugate to
with equation
L _
(N-Z)-flat
C.
C.
be the row-vector of
By Theorem 3.5, the intersection of
Hermitian variety
C on a non-degenerate
Thus we have the
THEOREM 3.6:
VN_ ,
I
Given a non-degenerate Hermitian variety
a (N-Z)-flat
_
which is the intersection of the polar hyperplane of an external
N Z
L
point
D and the tangent hyperplane at a point
in a non-degenerate Hermitian variety
C and
V _
N 3
V_ ,
N I
C on
contained in
V_
meets
L _ '
N Z
N I
provided
D are not conjugates.
4.
Number of u-flats contained in a non-degenerate Hermitian variety
in
PG(N,q ).
Z
In this section, we derive an expression for the number of
u-flats contained in a non-degenerate Hermitian variety
O =:; u =:; [ N-zI] ,
to
x.
where
[x]
generate Hermitian variety
VN_ I
in
Z
PG(N,q ),
denotes the greatest integer less than or equal
It was shown in [4], that if
no higher.
VN_ I
V _
N I
N = Zt+l
or
Zt+Z,
then a non-de-
contains flat spaces of dimension
The expressions for the number of points
(u=O)
t
and
and the number
9
N-l
of [-2-]-flats
N-l
(u=[-])
2
contained in
V _
N l
were given in [4].
The same
method of reasoning as in [4], is used here to derive expressions for every
integer
u
in
0
u
$
$
[N;l].
We quote, without proof, a lemma and a corollary to it, proved in [4]
which we need here.
lEMMA 4.1:
V _
N l
is completely contained in
V_ .
jugate with respect to
CoROLLARY.
L
t
if and only if
V _
N l
C
and
D
The necessary and sufficient condition for any t-flat
jugate with respect to
of
D on a Hermitian variety
V_
Nl
VN_
at
l
is that any two points of
If
VN_ l ·
is a regular point of
tangent space to
are con-
This is Lemma 9.1 in [4].
N l
be completely contained in
C
C and
The line joining two points
L
V_
is contained in
t
V_ ,
N l
then
L
t
L
N l
t
L
t
to
are con-
and a point
is contained in the
C.
This is the corollary to Lemma 9.1 in [4].
Let
2
denote the number of u-flats contained in a non-de-
1jJ(N,u,q )
generate Hermitian variety
V_
N l
in
the number of points in a u-flat in
2
PG(N,q )
2
and let
2
f(u,q )
denote
Then it was shown in [4]
PG(N,q ).
that
2
2
(4.1)
1jJ(N,O,q )
(4.2)
1jJ(2t+l,u,q )
and
¢(N,q )
2
number of points in a non-degenerate
2
1jJ(2t+2,u,q )
o
for
u
>
t,
10
2
(4.3)
W(2 t+1 , t , q )
(4.4)
W(2t+2,t,q )
(q+l),
2
2
1jJ(N , u, q )
Here, we find the value of
Let
variety
V _
N 2
L N l
C
contained in
(Theorem 3.5).
N-l
o ::;
u ::; [-2-]'
and disjoint from
Now
V_
N 3
gether wi th
C
L _
N l
L _ ,
N l
cuts
V_
N l
for which
V _
N 3
contains
C,
C is the singular space
which intersects
L _
N 2
2
W(N-l,u-l,q)
L
is a u-flat contained in
C,
then it is contained in
u
L _
N l
in a (u-l)-flat contained in
in a non-degenerate
(u-l)-flats, anyone of which to-
determines a u-flat contained in
Conversely, i f
V_
N 2
contained
L _ •
N 2
contained in
C.
V _ .
N l
in a degenerate Hermitian
By Theorem 3.3, we can find a (N-2)-flat
Hermitian variety
L _
N 2
u,
C be any point on a non-degenerate Hermitian variety
Then the tangent space at
in
for all
V_
N l
and passing through
V_
N I
and passing th1Jl'ough
(Corollary to Lennna 4.1) and intersects
VN_ 3 ·
Hence the number of u-flats con-
tained in
V _
and passing through a fixed point C
N I
2
W(N-I,u-l,q). But the number of points on V _
is
N l
on
V _
N I
2
~(N,q).
is
We thus
~(N,q2) u-flats by considering all points of VN_ l .
Here each u-flat has been counted f(u,q2) = «q2(u+I)_1)/(q2_ l »
times.
obtain
W(N-2,u-l,q2)
Hence,
(4.5)
2
W(N,u,q )
2
2
W(N-2,u-l,q )p(N,q )
2
f(u,q )
Thus by successive reduction,
(4.6)
2
1jJ (N, u, q )
2
2
2
P(N,q )P(N-2,q ) ... P(N-2u,q )
222
f ( u, q ) f ( u-l, q ) ... f (0, q )
11
(qN+l_(_l)N+l) (qN_(_l)N) ••• (qN-2u+l_(_1)N-2u+l)(qN-2u_(_1)N-2u)
(q 2 (u+1) -1) ... (q 2-1)
2u+l
IT
(qN-2u+i_(_1)N-2u+i)
i=O
u
IT
i=O
5.
Association schemes, strongly regular graphs and block designs from
Following Bose and Shimamoto [6], we define a two-
Hermitian varieties.
class association scheme as relations between
(i)
(ii)
(iii)
v
objects as follows:
any two objects are either first associates or second associates,
each object has
n
i-th associates,
i
i = 1,2,
if two objects are i-th associates, then the number of objects which
are j-th associates of the first and k-th associates of the second is
i
and is independent of the pair of objects with which we start and
1
Bose and Clatworthy showed [5] that the constancy of
112
2
P2l' P22' P12' P2l
implies the constancy of
i
Pjk=Pkj'
n l ,n 2 ,Pll and
and
P
2
22
and the
Also it follows from their proof that
equalities
n
1
- P
1
n -
1
P
2
P 2l ,
A finite connected graph
G
1
2l ,
1
P 22
=
2
P 22
(without loops or multiple edges) is
called strongly regular [2], if it is regular of valence
of adjacent vertices is adjacent to exactly
n
l
and any pair
other vertices and any
pair of non-adjacent vertices is adjacent to exactly
vertices.
12
If
v
is the number of vertices,
parameters of the strongly regular graph
scheme defined on
v
G.
Given a two-class association
objects, the graph we obtain by taking the objects as
vertices and joining two vertices if the corresponding objects are first
associates and leaving them unjoined if they are second associates, is a
strongly regular graph and conversely [2].
1
2
I
2
(v,nl,Pll,Pll)
Hence a strongly regular graph
is equivalent to a two-class association scheme
(v,nl,Pll,Pll) .
A block design
B
defined on the
1
2
(v,nl,Pll,Pll)'
ation scheme
if it is an arrangement of
consists of
k
v
objects of a two-class associ-
is called a partially balanced block design
v
objects in
b
blocks such that every block
objects, every object occurs in exactly
objects which are first associates occur in exactly
r
A
l
blocks, any two
blocks and any two
objects which are second associates occur together in exactly
A
2
blocks.
The parameters of such a partially balanced incomplete block design are then
1
2
(b,v,r,k,A ,A ,n ,PII,PII)'
l 2 l
Al = A2 ,
If
the block design is called a
balanced incomplete block design.
Consider a non-degenerate Hermitian variety
5.1.
2
number of points in
is
(q3+1 )
PG(2,q )
is
4 2
q +q +1.
V
l
in
The number of points in
and hence the number of external points is
(q3+ l )
= q 2 (q 2-q+l).
meets
VI
either exactly at one point or intersects
erate
Va
which consists of
V
l
cannot contain any lines.
(q+l)
2
PG(2,q ).
v
points of the line.
V
l
(q4+q 2+ l )
A line of
V
l
The
2
PG(2,q )
in a non-degenIn the former
case, the line is called a tangent line at the point and in the latter it
is a secant line.
Thus any two external points will be either on a tangent
line or on a secant line.
Let us define two external points as first
13
associates if they are on a tangent line and as second associates if they
are on a secant line.
VI
at
q+l
through
P,
Consider an external point
points.
The
q+l
tangents at these
since these points are conjugate to
external point
P
pass
P.
(q+l)
q+l
P.
points must pass
Thus through every
2
tangent lines and
The number of first associates of a given point
Its polar line meets
q -q
P
secant lines.
is then,
2
(q -l)(q+l),
(5.1)
and the number of second associates
(5.2)
Let
P
2
2
(q -q)(q -q-l).
=
pass
P
q
tangent lines other than
tangent lines passing through
meets a tangent line through
l
q
(5.3)
2
Q other than
i.
A tangent line through
Q exactly at one external point.
- 2 + q • q
Consider two external points
(q+l)
q
S
Rand
at
P
and
Q is
S
(q+l)
2
which are on a secant line
R intersect the
(q+l)
distinct external points.
2
(q+l) .
Hence we have the
THEOREM
5.1:
degenerate
The association scheme defined on points external to a nonin
PG(2,q2)
P
Hence num-
Thus we have
(5.4)
q
2
tangent lines passing through
tangent lines passing through
Through
and these are distinct from the
ber of external points which are first associates to both
Then the
i.
Q be two external points on a tangent line
and
by considering two external points as first
m.
14
associates if they are incident with a tangent line and as second associates
if they are incident with a secant line is a two-class association scheme
2 2
v = q (q -q+l),
with the parameters
2
2
Pll = (q+l) .
For
v
x
v
q=2,
the result was derived by Bose in [3].
B.
i = 1,2
1
scheme as follows.
on the
v
objects and the entry
column
S
of
B
l
v
p
1
2
2
= q -2+q
11
B.
is
1
B. 's
1
if
a.
Let us define a
B.
i = 1,2
1
at the intersection of row
and
S
a.
and
are i-th associates, zero other-
are called the association matrices.
1
correspond
On the other hand,
is the adjacency matrix of the strongly regular graph
and
B
2
is the adjacency matrix of the complementary graph.
If
1
2
Pu = PH'
plete block design
B
l
v
becomes the incidence matrix of a balanced incom-
= b,
In theorem 5.1, for
r
q
incomplete block design with
= k = nl ,
= 3,
v
B
l
1
A = Pll
2
= Pll.
is the incidence matrix of a balanced
= b = 63,
= k = 32,
r
A = 16
the incidence matrix of the balanced incomplete block design
r
and
objects of a two-class association
The rows and the columns of
to the
Then
= (q -1) (q+l),
1
Hence this is also a strongly regular graph.
matrix
wise.
2
n
= k = 31,
5.2.
2
PG(3,q )
was shown in [4] that the lines of
V ;
2
Hermitian variety
meets
V
2
entirely in
B
2
is
v = b = 63,
A = 15.
Consider a non-degenerate Hermitian variety
respect to
and
consisting of
at a single point and
V .
2
q+l
in
2
PG(3,q ).
It
fall into three classes with
(i) secant line which intersects
Va
V
2
points,
V
2
in a non-degenerate
(ii) tangent line which
(iii) generator which is contained
We quote here, without proof, some results proved in [4]
as lemmas which we need to derive some new results.
15
LEMMA 5.2.1:
Through any point
C of
V
2
there pass exactly
erators which constitute the intersection with
at
V
q+l
gen-
of the tangent plane
2
C.
LEMMA 5.2.2:
A line passing through a given point
tained in the tangent plane at
LEMMA 5.2.3:
C on
V '
Z
and not con-
C is a secant line.
Any plane through a generator is tangent to
point on the generator and intersects
V
in a set of
z
V
z
q+l
at some
generators
through the point of contact.
LEMMA 5.2.4:
i
Given a generator
there passes through
P
i
of
V
z and a point P of Vz not on
exactly one generator which meets
i
at one
point.
lEMMA. 5.2.5:
If two points
then there are exactly
the tangent planes at
q+l
and
points
C
l
1
DiC l
z
i
1
i
V
on
D.
i = 1,2 ••
D.
DiCO and
Thus the lines
Co
1,Z ..
i,
lie on a secant line
q+l,
on
V
such that
z
pass through the secant line
q+l
q+l
1, Z..
i.
are generators.
Next we prove
LEMMA 5.2.6:
Z
PG(3,q),
Through a tangent line
at a point
tangent plane at
PROOF.
Let
TI
O
Q on
VZ '
i
meeting a non-degenerate
qZ+l
pass
Q and the remaining
q
be the tangent plane at
Z
will contain
and
TI
QR,
R
on
a generator of
contain already
i.
V •
Z
V .
Z
in
planes of which one is a
planes are secant planes.
Q passing through i.
contrary to the hypothesis, that another plane
tangent plane at a point
V
2
TI
Suppose
passing through
Then it follows that both
This is impossible, because
i
is a
and
TI
O
TI
16
Now in
PG(3,q2)
(q3+l ) (q2+ l )
are
Let
P
there are
points.
(q4+ l ) (q2+ l )
Hence the number of points in
be an external point.
in a non-degenerate
VI
The polar plane of
consisting of
points are conjugate to
points and in
q3+ l
q+l
intersects
V
2
P.
this is also the number of tangent lines through
at
exter-
Each one of these
Hence the number of tangent planes passing through
V
2
there
PG(3,q2)
P
points.
V
2
P
meets
points.
P
is equal to the number of points in
non-tangent line divided by
P.
P
is
q3+ l
and
A secant line through
Hence the number of secant lines through
q+l.
V
2
which are joined to
This number is
2
P
by a
2
q (q -q+l).
Let us define two external points as first associates if they are on a
tangent line and as second associates if they are on a secant line.
the number of first associates of a given external point
3
(5.5)
is
2
(q +l)(q -1),
and the number of second associates of
P
222
(5.6)
q (q -q+l)(q -q-l).
Suppose
meeting
V
2
PI
and
P
2
at the point
are two external points on a tangent line
Q.
There are
of which only one is a tangent plane at
are secant planes.
line
P
Then
l
Now there are
2
q -2
q2+l
in the plane.
l
intersects
V
2
planes passing through
Q and the remaining
PI
and
PI
P .
2
2
planes
Again a secant
in a non-degenerate
From Section 5.1, we have that there are
plane, passing through
q
l,
external points on the tangent
which are common first associates of
plane passing through
l
which are tangent lines to
q+l
VI
VI
contained
lines in this
and similarly
17
q+l
lines through
P
to both the sets of
PI
which are tangent lines to
2
q+l
will intersect the
2 2
q .q
= q4
associates of both
Ql
Now through
q+l
q
Hence the
q
q
2
and
P .
2
is common
P
2
at
q
2
distinct
secant planes passing through
q
2
f.
Thus
secant planes, which are first
Hence
2
are two external points on a secant line
Q2
and
q2+ 1
m pass
are tangent planes.
2
q -q
planes of which
m.
are secant planes and
Thus the number of external points which are at
the intersection of tangent lines passing through
the number of secant planes passing through
sections of tangent lines passing through
plane.
f
But
tangents passing through
tangents passing through
external points in
PI
=
(5.7)
Suppose
q
But there are
external points.
there are
tangents.
VI.
Ql
and
Q2
is equal to
m times the number of interP
1
and
P
2
in a given secant
Hence
(5.8)
=
2
(q -q)(q+1)
2
.
Hence we have the
THEOREM
5.2.
degenerate
The association scheme defined on points external to a nonV
2
in
2
PG(3,q )
by considering two external points as first
associates if they are incident with the same tangent line and as second
associates i f they are incident with the same secant line is a two-class
association scheme with the parameters
(5.9)
3 2
q (q +1) (q-1)
v
=
3
2
(q +1) (q -1)
18
=
222
q (q -q+l)(q -q-l)
124
Pll
q
- 2 + q
Hence this is also a strongly regular graph.
For
q=2,
we have
Hence in this case,
block design
v = 40,
B
l
v = 40,
n
l
1
= 27,
Pll
=
18
is the incidence matrix of a balanced incomplete
r = k = 27,
A = 18.
PG(3,22)
external to
scheme on 40 points of
The two-class association
V
2
was earlier obtained by
R.C. Bose in [3].
A non-degenerate
5.3.
(q3+l ) (q+l)
in
generators.
2
PG(3,q )
h~
Through each point of
and two tangent planes at two distinct points on
V
passes a tangent plane
2
V
2
are distinct.
Consider the association scheme where two generators are first associates if they are incident with the same tangent plane, otherwise they are
second associates.
Through a generator
at
q2+l
on
l
Hence
(5.10)
l
pass
q2+ 1
points of the generator.
will contain besides
l,
q
planes which are tangent planes
The tangent plane at a given point
other generators meeting
the number of first associates of
l
is
2
q(q +1),
and the number of second associates
(5.11)
4
q •
l
at
Q.
Q
19
II
Consider two generators
plane
II
TI.
also contains
TI
q-1
l2 which lie in the same tangent
and
generators other than
l2 which are first associates have q-1
and
first associates.
II
Suppose
and
and
l2'
Hence
generators as common
Thus
=
(5.12)
II
q - 1.
l3
are two generators which do not lie on the same
tangent plane.
By Lemma 5.2.4, there is a unique generator passing through a given
point
P
on
l3'
which meets
at a point.
II
erators which are first associates of both
(5.13)
q
2
II
Hence there are
and
l3'
q2+ 1
gen-
Hence
+ 1.
We thus have the
THEOREM
5.3:
The association scheme defined on
V
of a non-degenerate
2
in
2
3
v = (q +1) (q+1)
generators
by considering two generators as
PG(3,q )
first associates if they are incident with the same tangent plane and as
second associates if they are skew, define a two-class association scheme
with the parameters
(5.14)
v
n
n
1
2
1
PH
(q3+ 1 ) (q+1)
2
q(q +1)
q
4
q - 1,
2
PH
q
2
+1.
This is also a strongly regular graph.
20
Now if we define the tangent planes as blocks and the generators as
treatments (or objects) we get a block design with parameters
(5.15)
b
3
2
(q +l)(q +1),
=
r
1
PH
q
=
2
+ 1,
n
2
q(q +1),
l
2
PH
q - 1,
3
(q +1) (q+l),
v
q
2
n
+ 1,
q + 1,
4
q ,
2
=
Al
k
1,
O.
A
2
This is a partially balanced incomplete block design based on a twoclass association scheme defined in (5.14)
For
q=2,
we get the
3,
=
design:
(5.16)
45,
b
27,
v
16,
The design (5.16)
k
s,
r
5,
1,
10,
o.
1,
can be obtained from the famous Steiner configu-
ration of 45 triangles formed by 27 straight lines contained in a general
cubic surface.
We quote from Jordan [7]:
"Steiner a fait connai tre
(Journal de M. Borchardt, t. Llll) les theoremes suivants:
Toute surface du troisieme degre contient vingt-sept droites;
L'une quelconque d'entre elles, a, en rencontre dix autres, se coupant
elles-memes deux
a
deux, et formant ainsi avec a cinq triangles.
Le nombre
total des triangles ainsi formes sur la surface par les vingt-sept droites
est de quarante-cinq.
Si deux triangles
abc,
cote commun, on peut leur en associer un troisieme
al b' cl
a" b" c"
nlont aucun
tel, que les
cotes correspondants de ces trois triangles se coupent, et forment trois
nouveaux triangles
associes
a b c,
a a l a" ,
al b' cl,
b b I b" ,
a" b" c"
D'apres cela, designons par les lett res
C
c l c".
Les trois triangles
s'appellent Ie triedre de Steiner.
a, b, c, d, e, f, g, h, i, k,
m, n, p, q, r, s, t, u, ml , n l , pI, q I, r l , s I , t I , u l
les vingt-sept
.t,
Zl
droites:
on formera sans peine le tableau suivant des quarante-cinq triangles,
ou la designation des droites reste seule arbitraire:
abc,
ade,
afg,
ahi,
bIlU1,
bpq,
brs,
btu,
cm'n' ,
(5.17)
cp 'q' ,
ak.t,
cr's' ,
ct 'u' ,
dmm' ,
dpp' ,
drr' ,
dtt' ,
enn' ,
eqq' ,
ess' ,
euu' ,
fmq' ,
fpn' ,
fst' ,
fur' ,
gnp' ,
gqm' ,
gru' ,
gts' ,
hms' ,
hrn' ,
hqt' ,
hup' ,
inr' ,
ism' ,
itq' ,
ipu' ,
kmu' ,
ktn' ,
kqr' ,
ksp' ,
.tnt' ,
.tum' ,
.trq' ,
.tps ' •"
Consider the block design (5.15).
external to
q3+l
V '
Z
Then the points on
Let
V
z
P
be a point in
which are conjugate to
in number and they constitute a non-degenerate
section of the polar plane of
P
with
Z
PG(3,q )
V .
2
V
l
Pare
which is the inter-
No two of the
q3+l
planes at these
q3+l
points will contain the same generator.
Hence these
q3+ l
tangent planes regarded as blocks and
tangent
q+l
generators
in each tangent plane as objects, provide us a parallel class of
v
=
3
(q +1) (q+l)
objects in
q3+l
blocks or equivalently, a set of
maximal cliques in the strongly regular graph.
q3+l
22
REFERENCES
[1]
R.C. Bose, On the application of finite geometry for deriving certain
series of balanced Kirkman arrangements, The Golden Jubilee
Commemorative Volume~ Calcutta Mathematical Society (1958 1959), 341-354.
[2]
R.C. Bose, Strongly regular graphs, partial geometries and partially
balanced designs, Pacific J. Math.~ 13(1963), 389-419.
[3 ]
R.C. Bose, Se1f-conju~ate tetrahedra with respect to the Hermitian
variety x5+xr+x2+x§=0 in PG(3,2 2 ) and a representation of
PG(3,3), (to be published in: Proc. of the Symposia in Pure
Mathematics~ Combinatorics~ Vol. 19, Amer. Math. Soc.
[4]
R.C. Bose and I.M. Chakravarti, Hermitian varieties in a finite projective space P~N,q2), Canadian J. Math.~ 18(1966), 1161 1182.
[5]
R.C. Bose and W.H. Clatworthy, Some classes of partially balanced
designs, Annals Math. Statist.~ 26(1955), 212-232.
[6]
R.C. Bose and T. Shimamoto, Classification and analysis of partially
balanced designs with two associates classes, Journal of the
American Statist. Assoc.~ 47(1952),151-184.
[7]
C. Jordan, Traite des Substitutions et des Equations Algebriques.
Gauthier-Villars, Paris (1870).