Generalized
polygons,
William
SCABs
and GABs
M. K a n t o r *
University
of O r e g o n
Contents
Introduction.
A.
B.
C.
Generalized
polygons.
A.I.
Definitions.
A. 2.
Numerical
A. 3.
Constructions:
n = 4.
A.4.
Constructions:
n = 6, 8.
Buildings
and coverings.
B.1.
Chamber
systems;
B.2.
Coxeter
groups.
B.3.
Buildings;
B.4.
Examples.
SCABs
SCABs.
coverings.
and GABs.
C.I.
Definitions;
C.2.
Covering
C.3.
Difference
The p r e p a r a t i o n
MCS
restrictions.
7903130-82.
criteria
for GABs.
GABs.
sets.
of this p a p e r w a s s u p p o r t e d
in p a r t by N S F G r a n t
80
C.4°
A table
of SCABs.
C.5.
E8
C.6.
Miscellaneous
C.7.
Connections
root
lattices.
problems
with
and examples.
finite
group
theory.
References.
Introduction
Recently
geometric
there has been
and group
theoretic
One of the d i r e c t i o n s
that
systems
are almost
are
"chamber
[Ti
7],
and
of type
are almost
systems
"Tits g e o m e t r i e s
of type
relationships
with
The study
his
few p r o o f s
On the other hand,
(especially
when
Proceedings),
devoted
difference
order
to c o n s t r u c t
there
GABs
("chamber
(Other names
for these
M" or
The m a i n
of type
"Tits
goal
M" in
chamber
systems
of this p a p e r
emphasis
paper
based
[Ti
that
our Chapter
are given.
this
survey.)
elementary
paper
in these
deal of space will
geometric
objects
can be easily u s e d
n e w or less
familiar
ones.
opportunities
is still
to enter
be
(e.g. hyper-
systems)
that the s u b j e c t
in
Consequently,
in its infancy
into
B
(In fact,
be somewhat
7] and Tits'
a great
familiar
While
of Tits,
sets or root
are m a n y
is
on their
the work
throughout
of v i e w will
w i t h [Ti
upon
7].
few proofs
be p r o v i d e d
our p o i n t
it will be clear b o t h
and that
will
compared
ovals,
concerned
"geometries
is p r i m a r i l y
results,
in w h i c h
geometries"
geometries.
in the sense
to ways
in the
and SCABs
w i t h special
on the seminal
describes
relatively
finite
of SCABs
and especially
briefly
Tim 1,2]. )
developments
has
buildings")
buildings").
of activity
"building-like
activity
M" and
these
deal
of
of type
M" in [AS;
to survey
study
of this
("geometries
that
a great
this
area.
81
Generalized polygons
SCABs
and GABs are obtained.
about these:
planes,
they
are rank
construct
2 buildings,
they
include projective
from a v a r i e t y
and c o m b i n a t o r i a l v i e w p o i n t s .
almost
from w h i c h
A g r e a t deal has b e e n w r i t t e n
and they are v e r y n a t u r a l
group theoretic
will
are the b u i l d i n g b r i c k s
all of the k n o w n
of g e o m e t r i c ,
In C h a p t e r A we
finite examples
that are
not p r o j e c t i v e planes, and i n d i c a t e t h e i r g r o u p t h e o r e t i c o r i g i n s
when
appropriate.
However,
we will not d i s c u s s
of c h a r a c t e r i z a t i o n s p r e s e n t l y k n o w n
the l a r g e n u m b e r
-- e s p e c i a l l y of f i n i t e
generalized quadrangles;
that
of this paper.
C h a p t e r A is i n t e n d e d to f u n c t i o n
Instead,
is o u t s i d e the s c o p e
the s o u r c e for g e n e r a l i z e d p o l y g o n s
Chapter A highlights
tions
arising
one b a s i c q u e s t i o n
in C h a p t e r C: w h a t n o n c l a s s i c a l
can a p p e a r in f i n i t e S C A B s ?
to be k n o w n
(In fact,
are s i m i l a r to t h o s e in
While Chapter
later.
the o n l y s u c h SCABs that seem
(C.6.11).)
ties of t h e i r m o r e g e o m e t r i c c o u n t e r p a r t s ,
the paper,
the final
t r a n s i t i v e SCABs,
While groups
section contains
and has
c o n c e r n i n g SCABs,
GABs.
to g i v e
results
list,
and the paper,
it s h o u l d be n o t e d that,
survey
cisely,
theoretic
1984.)
flavor.
of w h i c h results:
are a l r e a d y too long.
s i n c e this
open
a c o m p l e t e list of r e f e r e n c e s
Finally,,
area is g r o w i n g rapidly,
s h o u l d be out of d a t e even b e f o r e
this s u r v e y
Numerous
concerning chamber-
or a c o m p l e t e a c c o u n t of who p r o v e d w h i c h p a r t s
the p r e s e n t
and to p r o p e r -
appear frequently throughout
an e s p e c i a l l y g r o u p
No a t t e m p t has been m a d e
the c o n s t r u c -
finite generalized polygons
B contains generalities
are m e n t i o n e d .
as
Moreover,
concerning
C h a p t e r C is d e v o t e d b o t h to t h e i r c o n s t r u c t i o n
problems
(and size)
is b a s e d on m a t e r i a l
it appears.
this
(More p r e -
a v a i l a b l e to me
in April,
82
A.
Generalized
In this
n-gons
chapter
are c o m b i n a t o r i a l
we will
discuss
generalizations
generalized
of c l a s s i c a l
ing from forms on s m a l l - d i m e n s i o n a l
also
the rank
Chapters
cribe
2
objects
B and C.
some basic p r o p e r t i e s
However,
cially
relevant
many
embeddings,
to later
They
will
will
the k n o w n
aspects
be to des-
(cf. [PAT]).
that seem
espe-
we h a v e not
characterizations,
properties
in
finite models.
In particular,
topics:
are
be built
a book on this s u b j e c t
on those
aris-
of a u t o m o r p h i s m
projective
groups.
Definitions.
F
be a graph
except
iff
u
finite).
and
= ~u C V I u ~ V}o
V = VIO-O.UV r
v
An
for
r
that no two v e r t i c e s
joins v e r t i c e s
distance
of
Vi
If
V
r-partite
nonempty
of any
and
Vi
Vj
in the graph.
of edges
discussed
no m u l t i p l e
in B.I.4,
is its set of vertices,
by an edge.
graph
Also,
is a graph
pairwise
disjoint
are Joined;
for
some
two v e r t i c e s
in the obvious ways.
number
w i t h no loops,
circumstances
are Joined
d(u , v) b e t w e e n
are d e f i n e d
smallest
(undirected,
in r e s t r i c t e d
not n e c e s s a r i l y
u ~ v
spaces.
our o b j e c t i v e
chapters.
important
or general
Let
edges
focused
These
geometries
geometries
and to list
to w r i t e
we h a v e only
discussed
A.I.
from w h i c h
Consequently,
It w o u l d be tempting
vector
n-gons.
i # J.
u
and
The g i r t h
in a circuit
that
(i.e.,
write
set
r
sets
and
r(v)
such that
Vi
such
is, e a c h edge
Conneetedness,
v,
and
diameter
of a g r a p h
is the
polygon)
contained
the
83
Definition
bipartite
n ~ 2
A.l.1.
graph
(where
for each
~
In a rank
2
Let
are easy to r e c o v e r
be
3
from
those
of
V2
distance
between
that
of d i a m e t e r
[F(v)I
n,
any two v e r t i c e s
(Of course,
If
vi C Vi
I .
of
V1
then
Thus,
F
is J o i n e d
to e v e r y v e r t e x
of
"lines."
Then
two d i s t i n c t
points
pass
are J o i n e d
in
through
to a line,
V1
a n y two lines are J o i n e d
words,
is the i n c i d e n c e
Examp!e.
in w h i c h
plane
n = =.
all v e r t i c e s
which
of
V2 .
"points"
and
c a n n o t be at
2
(since a p a t h
and
V2).
is u n i q u e
Thus,
because
V2
Similarly,
every projective
VI
bipartite
F •
VI
F
V2
cannot
is a c o m p l e t e
the v e r t i c e s
4-gons
and
d(v I, v2)
Call
them must alternately
are no
path.
V1
n = 3 .
points
~ 3
"2-partite.")
~ 4 , and h e n c e m u s t be at d i s t a n c e
two d i s t i n c t
there
such
2n, for s o m e
F.)
n = 2 .
every vertex
Example.
and g i r t h
be as above.
and h e n c e m u s t be
graph:
is a c o n n e c t e d
are j o i n e d by a u n i q u e s h o r t e s t
V = V1 U V 2
Example.
n
bipartite means
building
n
building
is p e r m i t t e d ) ,
(Here,
n o t at d i s t a n c e
2
of d i a m e t e r
n = ~
v 6 V.
A rank
point.
g r a p h of a p r o j e c t i v e
produces
This
to a u n i q u e
a rank
time
have valence
F
2
plane,
building
and
in this manner.
is just an i n f i n i t e
~ 3.
In other
tree
84
Definition
building
A.I.2.
of d i a m e t e r
n
sets h a v e b e e n c a l l e d
have been called
ordered
more
in w h i c h
"points"
"lines."
the two sets
ing g e o m e t r i c
A generalized
satisfactory
seen from the case
n = 3 :
point
amounts
of dual p r o j e c t i v e
the labels
get
a new generalized
one.
A generalized
morphic
of o r d e r
Further
2
examples
w e will m e r e l y
2
"points"
n-gon
n-gon
to its dual,
morphism
a rank
and s e l f - p o l a r
is a s i n g l e
of d i a m e t e r
3
planes.
In general,
if
"lines,"
the dual
is c a l l e d
3-gon
as can b e
building
and
called
The r e s u l t -
but s l i g h t l y
of view,
a generalized
whereas
we interchange
we have merely
symmetric-looking,
p r o j e c t i v e plane,
to a p a i r
vertices
in one of two ways.
from a g e o m e t r i c
2
in one of the
and the r e m a i n i n g
V2
is less
is a rank
the v e r t i c e s
In o t h e r words,
V1,
object
n-gon
of the o r i g i n a l
self-dual
if there
we obviously
if it is iso-
is such
an iso-
(a p o l a r i t y ) .
will
consider
be g i v e n
in §§ A.3,
the d e f i n i t i o n
A.4.
from a m o r e
For n o w
geometric
v i e w point.
Let
distinct
~
be a g e n e r a l i z e d
l i n e s are j o i n e d
there are no 4 - g o n a l
its set
F(L)
as an edge
to at m o s t
circuits),
of a d j a c e n t
of the graph:
n-gon,
where
n > 2.
one c o m m o n point
and w e c a n i d e n t i f y
points.
T h e n two
A
an a d j a c e n t
flag o f
~
(since
each
line L w i t h
is the same
(or incident)
point-line
pair.
Example.
x
is c o l l i n e a r
n = 4 .
If a point
with a unique
x
y E L .
Y
L
/x
,,
is not
in a lime
L
then
85
For,
d ( x , L)
Example.
either
x
a unique
lines
is odd,
> i
and
~ 4 .
n = 6.
If a point
x
is not in a line
is c o l l i n e a r w i t h a ' unique point of ' L
sequenc e
Li.
(x , L I , Yl ' L2 ' Y2 ' L)
such that successive
I
L
then
or t h e r e is
of points
Yi
and
terms are incident.
L _
L
2
L
L
For,
fact
d ( x , L)
that
is
the
Similar
odd and
girth
assertions
further example,
Example°
there
is
~ 6 .
Uniqueness
is a unique
hold
whenever
n
A.2.
n
even.
Here
is
of
one
odd.
> 2
and
x 5 ;
8-gonal circuits
then
(and dually).
and u n i q u e n e s s
in the graph.
restrictions
In this section we w i l l consider
where
is
point c o l l i n e a r w i t h both of them
from the n o n e x i s t e n c e
Numerical
the
If two points are n0t ' c o l l i n e a r
For the points have e v e n distance
follows
from
12 .
this time w i t h
n = 5 .
follows
finite g e n e r a l i z e d
n-gons,
n > 2o
P r o p o s i t i o n A.2.1.
There are integers
e v e r y line is on e x a c t l y
exactly
t + I
lines.
s + I
If
n
s , t m 2
such that
points and e v e r y point is o n
is odd then
s = t .
88
Proof.
Then
If
u , v E V
d(u' , v) < n ,
distance
n - 2
= n - 2
or
for a g i v e n
IF(u) I ~ 3
and hence
from
I~(u) I = I~(v) I •
have distance
u';
let
there is a v e r t e x
moreover,
Moreover,
n,
if
v'
u' E r(u).
v' E r(v)
is unique.
~' E F(u')
~u}
at
Thus,
then
d(u" , v)
n , where
the first p o s s i b i l i t y
occurs only once
u' , and
Ir(u") I =
d(~' , v) = n .
for all
u,
IF(v) l
a connectedness
if
argument
completes
Since
the
proof.
The integers
n-gon.
s
O f course,
generalized
and
t
n-gon.
t
and
are the parameters
s
are the parameters
The m a i n results c o n c e r n i n g
of the g e n e r a l i z e d
of the dual
n ,
s
and
t
are as follows°
Theorem A.2.2
n = 3, 4, 6
or
(Feit-Higman T h e o r e m
planes
only focus on the cases
T h e o r e m A02.3o
[FH; Hi]
E Z,
(ii)
[Hi~
also [Bi; Hi; KS]).
8.
Since p r o j e c t i v e
(i)
[FH] ;
Let
(n = 3)
n = 4, 6
or
are so
f a m i l i a r we will
8.
n = 4 . Then
st(s + l)(t + l)(st + l ) / ~ s 2 t + t2s + s + t}
and
Ca]
t ~ s 2.
T h e o r e m A.2o4.
Let
n = 6.
Then
(i)
[FH; Hi]
~
E Z,
(ii)
[FH; Hi]
st(s + l)(t + l)(s2t 2 + st + 1)/
2[s2t + t2s - st + s + t -+ (s - l)(t - l)~s~} E Z
87
for both choices of signs, and
(iii)
[Hae;HR]
t ~ S3 .
Theorem A.2.5.
Let
n = 8 .
Then
(i)
[FH; Hi]
~
(ii)
[FH; Hi]
st(s + l)(t + l)(st + l)(s2t 2 + I)/
E ~.
2{s2t + st 2 + S + t} E 2 ,
st(s + l)(t + l)(st + l)(s2t 2 + i)/
4{s2t + st 2 - 2st + s + t -+ (s - l)(t - I ) 4 ~ }
E Z
for both c h o i c e s of si~Ds, ~D_d_
(iii)
[Hi]
t ~ s2 .
The most remarkable and important
first one:
n
is highly restricted~
of these results
is the
With the exception of
A.2.3(ii) , all proofs of the above theorems are algebraic.
Either a commutative
algebra
[FH; Bi]
is associated
are calculated.
r , and multiplicities
[Hi ]
matrix
of representations
The fact that these must be integers places
severe restriction on
of the theorems.
other than
to
or noncon~nutative
n,
s
and
t , as can be seen in all
A coherent exposition
A.2.4(iii)
A.2.3(i) , A.2.4(ii)
of all the results
can be found in [Hi]o
and
A.2.5(ii)
are,
(The integers
in
in fact~multiplicities--
as is shown in [FH; Hi]o)
The fact that
t ~ s2
in
A.2.3(ii)
is the only part of
these theorems presently having a short, elementary proof [Ca]o
In fact, that proof shows that equality can hold i ff the following
condition holds:
any
3
pairwise noncollinear
points are all
88
c011inear
with exactly
that a z e n e r a l i z e d
s + I
hexagon with
condition:
for a n y line
0 ~ i~
0~
2:
points.
j,
L
L , x
o n l y on
and
vertices
x, L
(ioeo,
y
or
and s u f f i c i e n t
t = s
3
agons
and
and
we will
x
k
and
see that
x
and
the f o l l o w i n g
y , and
2
the c o n f i g u r a t i o n
~Iowever,
seem
to h o l d
paths
no g e o m e t r i c
in the cases
from
L
necessary
for the e q u a l i t i e s
of g e n e r a l i z e d
In the next
can i n d e e d h o l d
speaking,
jective planes
hex-
two s e c t i o n s
in A.2.3(ii),
numbers
generalized
other words,
as
n
and v e r y
increases,
scarcer
avoid the f a m i l i a r
of the cases
of g e n e r a l i z e d
hexagons
to b e c o m e
generalized
there seem to be l a r g e n u m b e r s
(of p r i m e p o w e r o r d e r
n = 6
and
- and h a r d e r
(§ A.3),
8
in [ K a 4,
few g e n e r a l i z e d
case of p r o j e c t i v e
quadrangles
- e.g.
quadrangles,
finite
with
(§ A.4).
by
of the s u b g r a p h w h o s e
to be k n o w n
respectively.
equality
type
determined
and A . 2 . 5 ( i i i ) .
Roughly
reasonable
satisfies
lying on s h o r t e s t
y).
conditions
t = s
it is s h o w n
d ( z , x) = 2J , d ( z , y) = 2k] I
the i s o m o r p h i s m
to
octagons,
A.2.4(iii)
appear
J ,
are those v e r t i c e s
to
[Hae]
k ~ 3:
i ,
y
t = s3
and p o i n t s
I[z I d ( z , L) = 2i + i ,
depends
In
5]),
relatively
octagons.
generalized
to describe.
planes,
of pro-
few
In
n-gons
We will
and focus
on
a shorter discussion
to
89
A.3~
Constructions:
The classical
Let
V
n = 4 .
generalized
be a v e c t o r
singular
an orthogonal
be totally
isotropic
moreover,
a generalized
b e c a u s e girth
4
4 .
i-
in the b i p a r t i t e
to check that
quadrangle.
(The g i r t h is at
or
imply the existence
6
would
4
and line then
2-spaces,
It is s t r a i g h t f o r w a r d
diameter
if
totally
the case of
and
(i.e., a d j a c e n c e
isotropic or totally singular
because
contains
Then let points and lines
of a totally
is
V
exclude
or totally singular
g i v e n by inclusion.
8
that
symplectic,
2-spaces but no t o t a l l y isotropic
3-spaces;
w i t h incidence
this p r o d u c e s
least
Assume
space of d i m e n s i o n
respectively,
graph)
structure.
or totally singular
or totally
are o b t a i n e d as follows.
space e q u i p p e d w i t h a n o n d e g e n e r a t e
u n i t a r y or o r t h o g o n a l
isotropic
quadrangles
x
and
L
3-space.
The
are a n o n i n c i d e n t
x, (x , x ± n L>, x ± N L, L
point
is a p a t h of length
3.)
It is easy to construct other
rangles
(compare [Ti 4]).
we will b r i e f l y d e s c r i b e
there are the following
infinite g e n e r a l i z e d
In the remainder
of this section
the k n o w n finite ones.
classical
those of the c o r r e s p o n d i n g
examples,
classical
First of all,
w h o s e names
groups.
i n d i c a t e d the duals of the first two families.
any p r i m e p o w e r
quad-
are
We have also
(Here,
> i.)
Name
s
t
Name of dual
PSp(4 , q)
q
q
PO(5 , q)
P~-(6 , q)
q
q2
PSU(5 , q)
q2
q3
PSU(4 , q)
q
is
90
The
PSp(4 , q)
self-polar
iff
quadrangles
q
are self-dual
has the form
2 2e + 1
A l l cf the above examples,
iff
of that m e t h o d
itate the use of m a t r i c e s
Let
s
and
g r o u p and let
t
$
(i)
(ii)
(iii)
(Here
of
is dual to
be integers
[Ka2].
[Ka 2]
> i.
Let
be a family of subgroups of
The following
in order to facil-
A*
of
Q
Q
be a finite
Q.
Assume that
a s s o c i a t e d with each
3-element
subsets
3:
IQI = s 2 t ,
IJl = t + i ,
IAI = s ,
IA*I = st,
A < A*
Q = A*B , A * N B = I , and
AB N C = i .
A B = lab I a E A , b E B] ;
subgroup of
dence
e .
and m o s t of the k n o w n examples,
such that the following hold for all
{ A , B, C}
and
as in [CKS, § 3].
there is another s u b g r o u p
A E $
is even,
for some
can be obtained by a simple m e t h o d g i v e n in
description
q
Q.)
Given
structure
Q ,
Q(Q , $)
$
this u s u a l l y will not be a
and
as follows
A ~ A * , we define an inci(where
A E $
and
g E Q
are arbitrary).
Point.
Line.
3,
coset
Symbol
Incidence.
A g
; element
[ A] ~ coset
[A]
is on
~
g.
Ag.
and
A g ; all other incidences
are o b t a i n e d via inclusion.
Y
\g
91
Theorem A.3.1.
parameters
s
The proof
Q (Q,J)
and
q u a d r a n g l e with
t.
is a s t r a i g h t f o r w a r d
We will now p r e s e n t
A.3.1.
i_~s ~ q e n e r a l i z e d
counting
all of the k n o w n
argument.
examples o b t a i n e d via
Many of these examples will appear twice,
that both a q u a d r a n g l e
to list all k n o w n
and its dual will be d e s c r i b e d
instances
classes of examples
The first two
are easy to understand.
The remaining
but are included
the h o p e that their a v a i l a b i l i t y
(in this form)
s = t = q . Q
GF(q) , and
will aid in the d i s c o v e r y
$
is a
is an oval
3-dimensional vector
(i.eo, a set of
that is an oval of the c o r r e s p o n d i n g
projective
Also,
$
A*
is the tangent "line"
are all due to Tits
When
rangle.
~
generalized
is a conic,
inequivalent
s = q ~
space over
GF(q) ,
'~plane" to
$
p o 304].
If
quadrangle.
generalized
at
$
Q ( Q , 3)
inequivalent
quadrangles.
(A.3.3)
to
at
A.
q + i
plane
1-spaces
P G ( 2 , q)).
These e x a m p l e s
is Just the
ovals produce
For a d i s c u s s i o n
ovals,
t = ~2
see [Hit,
Q
P ~ ( 5 , q) quadnonisomorphic
of the number of pro-
pp. 176-182,
is a
416:
4-dimensional
is an ovoid,
and
A .
This example
is also due to Tits
is a quadric
Projectively
quadrangles.
there is an additional
then
A*
~ ( Q , $)
inequivalent
is the tangent
is the
[De,
P~-(6 , q)
ovoids produce n o n i s o m o r p h i c
In particular,
when
generalized quadrangle
52].
Pa 3].
vector
$
Suzuki o v o i d [De, p.
space
[De, p. 304].
ProJectively
Jectively
in
examples.
(A.3.2)
over
in order
of the theorem.
ones are fairly messy,
of further
in the sense
q = 22e + I > 2
o b t a i n e d from the
99
(A.3o4)
obtained
s = q ~ t = q.
as follows.
The
PSp(4 , q)
quadrangle
can be
Let
Q =I GF(q) 3
[a,~,~][~' ,~' ,y']
= [~ + ~' , ~ +
y'
~' , Y +
+ ~'
A(~)
= [ [ 0 , B, 0] I ~ E GF(q)} ,
A(t)
= [ [ - ~ t , ~ , 0] ] ~ E GF(q)} , t E GF(q) , and
= [A(t) I t E GF(q)
Then
Q
has order
of order
q.
If
U [®}}.
q3
and center
A E $
write
In order to see that
note that
Q
- ~'],
Z(Q) = ~ [ 0 , 0, y] I ¥ E GF(q)}
A* = AZ(Q) .
Q(Q , $)
is the
Then
A.3.1
PSp(4 , q)
applies.
quadrangle,
can be viewed as the set of all matrices
all of w h i c h preserve
1
0
0
1
1
the alternating
form
((x i) , (Yi)) = ElY 4 - x4Y I + x2Y 3 - x3y 2
and induce
the identity
Moreover,
A(~)
while
A(t)
these
1 + q
on
el/(el> , where
is the stabilizer
is the stabilizer
lines are totally
of
in
Q
of
<(0,0~0,i)
isotropic.
eI
(i, 0, 0 , 0)
((0,I,0,0) , (0,0,I,0)>,
, (011,t~0)~:
all of
g3
(A.3.5)
be obtained
s = 2
t = q3
as follows.
Q = [[¢t,~,~,H]
quadrangle
The PSU(5 , q)
can
Let
E GF(q2) 4 J t r y +
tr t ~ + C~ = 0}
[~,~,C,~][=',~',C',~'] = [:+:', ~+~' ,C+C' ,~+~'-E~' -~:' -~']
where
q7
~ = ~q
and
with center
consist
tr C = C + ~ .
Z(Q) = ~[0,0,0,~]
of the
I + q3
A(=)
A(b,c)
where
A*(m)
Then
of order
of order
q .
Let
J ~ C GF(q2)}
I ~ 6 GF(q2)}
let
= {[~,0,C,~}
I tr~ + (~T = 07
= [[-bB-cC~,~,C,D ] [ tr g + (~$ - C)(c~ - ~ ) = 0 } .
A.3.1 applies.
In order to see that
Q
is a group
groups
= [[~,~,~,0]
Also
Q
i tr ~ = 0}
= ~[g,0,0,0]
tr b + c~ = 0 .
A*(b,c)
Then
can be viewed
Q(Q , ~)
as the group
If°°°
001010io°
This group
preserves
is as asserted,
note that
of all matrices
with
the h e r m i t i a n
trn + tr=[ + C[ = 0.
form
5
((xi), (Yi)) =
while
inducing
E GF(q2) 5
the identity
More over,
on
A* (=)
~ xi~6_ i
i=l
el/<el) , where
is the stabilizer
e I = (I,0,0,0,0)
in
Q
of
94
(0,I,0,0,0) , A(~)
A*(b , c)
is the stabilizer
is the stabilizer
is the stabilizer
(A.3.6)
rangle
of
of
is obtained
whenever
q + I
(A.3.7)
rangle
= A(~),AI(~)
This produces
of order
q2
s = q3 ~ t = q2
can be obtained
Q : [[=,~,o,~,v]
A(b , c)
generalized
! tr(~ + ~ )
~.3.5):
= 0}
= A
(b,O) N QI
the required
of the group
Q1
The dual of the
as follows:
of
quad-
= a*(~) n Q:
= A ( b , 0) , Al(b)
tr b = 0.
subgroups
PSU(4 , q)
in the notation
Q1 = {[=,~,0,~]
Al(b)
, and
, (0,0,0,0,I)>
, (0,0,0,0,I)>.
The
as follows,
AI(=)
((0,i,0,0,0)
((0,l,c,b,0)>
<(0,l,c,b,0)
s = q2 ~ t = q.
of
set
~I
of
of order
q5.
P S U ( 5 , q)
quad-
use
E GF(q2) 5 I t r e + ~ - = 0 = tr B + v~'} ,
[~,~,q,~,v][~',~',o',~',v']
: [=+~'-Tu' , S + ~ ' - ~ '
A(=)
: {[~,0,0,~,0]
I tr
A (~) = {[t~,0,~,~,0]
,~+o'-~'
: + ~
= O}
] tr = + ~
= 0] ,
A(t) = {[tt~ , ~ , -tB , -tv ,v] I t r Y +
A*(t)
= {t7 - ~
,~+~' ,v+v'],
+ t~,~,o,-t~,v]
vv = 0} ,
I trY+
v~- = 0} , and
J = {A(t) It E GF(cl2) U ~ ] ]
The proof is similar
subgroup w i t h
to the preceding
~ = v = 0
ones.
Restricting
brings us back to the situation
to the
in
,
95
(A.3.3)
fines
w i t h a quadric
Z(Q).
=~ + ~ =
The quadric
is
(Note that
{[~,~,o,0,0]
U = v = 0
de-
I tr ~ = 0 = tr ~ ,
0}.)
Remark.
example
as ovoid.
Now that we have
arises via A.3.1,
explanation
decomposition
the points
Q ~ L
is a parabolic
p. 119].
at distance
- (A.3.6),
is presently
has been applied
In each classical
[Car,
(or lines)
As in (A.3.2)
There
(or line)
that every classical
it seems appropriate
for this fact.
izer of a point
indicated
case,
subgroup,
Here,
4
to provide an
Q
the stabilw i t h Levi
is regular
from the original
this produces
on
one.
the desired description°
onlx one other situation
in w h i c h
(in order to obtain a non-classical
A.3.1
generalized
quadrangle):
(A.3.8)
and
q ~ 2
s = q2 ~ t = q , w i t h
(mod 3).
q
a prime power,
q > 2
Let
Q = [(=,~,Y,~,~)
I ~,B,y,~,¢
E GF(q)}
(~,~,y,~,~)(=',~' ,y',~' ,~')
=
Then
(~. + ~.' , ~ +
Q
has order
has order
q o
13' , y +
q5
and
y'
+ ~'¢
3~'~
, ~ +
Z(Q) = { (0,0,y,O,0)
~'
, ~ +
~').
I Y E GF(q)}
Let
A(~) = [ (O,O,O,~,e)
A(t)
-
I ~ , ¢ E GF(q)}
= {(= , m r , -=2t3 , =t 2, ~ t 3 ) ( O , ~ , - 3 5 2 t , 25t , 3~t 2) I =,$ E GF(q)}
for
£ = [A(t) I t E GF(q)
A* = AZ(Q)
for
U {~]}
A E ~.
t E GF(q),
96
In
[Ka 2]
it is shown
generalized
quadrangle
(A.3.6)
to the dual
nor
For
further
that
A.3.1
applies
not i s o m o r p h i c
t_oany
of any of t h o s e
discussion
of all
and p r o d u c e s
in
of those
in
(A.3.3).
of the above
examples
see
[Ka 2].
It seems
However,
as if
conditions
In particular,
than
abelian,
ABC
in w h i c h
in
awkward.
(iii)
and
states
just
whenever
still m o r e
as
A ~ B.
(A. 3.3),
after
is
that
the m e m b e r s
of
~
are
A
and
(ii)
states
This
and is t a n t a l i z i n g l y
in our s i t u a t i o n
For
m u s t be solvable.
Conjecture.
p-group
example,
However,
for some p r i m e
In e a c h c l a s s i c a l
and
(A.3.8),
it is not
much more
The conditions
that
independence
However,
Q.
may be
Q
[De,
about
s3
if
in the c o n s t r u c t i o n
known
all,
does n o t
i__ss a group
involved
133].
to work with.
ABC
in the spreads
p.
easy
examples.
This c o n d i t i o n
s3 •
of course,
independent,"
(A. 3.2)
produce
are not very
has size
case
"independent"
visible
is very
s2t = IQI .
"triple-wis
are
( i ) - (iii)
(iii)
say that the set
larger
A. 3.1 s h o u l d
is e s p e c i a l l y
similar
of t r a n s l a t i o n
nothing
should
that
to that
planes
structural
even k n o w n
B
is
Q
be true:
in A. 3.1 force
Q
to be a
p.
case,
the h y p o t h e s e s
and m a n y
others
of one of the
in
(A.3.2),
following
(A.3.3)
conjectures
holds.
Conjecture.
one m e m b e r
Q
is a
of
2
p-group.
If
Aut Q
has a subgroup
and t r a n s i t i v e
of order
on the r e m a i n i n g
t
ones,
fixing
then
97
ConJectureo
group
of Lie
type
tion,
then
Sz(q)
induced)
t = q + 1
or
quadrangle,
a rank
2-transitive
classical
to the only other
found
These
or as in
1
representa-
(A.3.3)
(with
[Hall
one of its points
P G ( 3 , q)
to check
inclusion)
produce
lines
these
p
q
=
q
-
1
,
is a prime
Start w i t h a
Consider
p
but not
points
a generalized
s
[Pa i]o
not c o n t a i n i n g
containing
that
and
p o
generalized
parameters
t = q - 1 , where
in [ASz],
those
type of finite
have
s = q - I~ t = q + io
and
inducing
(A.3.8).
very easy
and
is either
at present.
pl , and
of
has a s u b g r o u p
in its usual
s = q + 1 ,
and w e r e
(A.3o9)
lines
J,
we turn
known
in
on
or
quadrangle
not
Aut Q
Q ( Q , ~)
Finally,
power,
If
PSp(4 , q)
those
as w e l l
inside
and lines
quadrangle
points
p
i
as those
o
It is
(and o r d i n a r y
with
s = q - 1
t = q + 1.
(A.3oI0)
a hyperoval
of
of
V = GF(q) 3
and
translates
produces
then we
obtain
All
as a set of
any
span
3
of w h i c h
of m e m b e r s
produce
of these
of
~
in
as in
If
(A.3.9).
nonisomorphic
generalized
V .
"lines."
quadrangle.
the example
exactly
t = q + i, q = 2 e > 2 .
PG(2 , q), v i e w e d
a generalized
hyperovals
(A.3o2)
s = q - 1 ~
D
N
Once
again,
arises
"points,"
inclusion
from a conic
inequivalent
quadrangles
arise
be
1-spaces
Call v e c t o r s
generalized
using
q + 2
ProJectively
quadrangles
~.3.9),
Let
from ones
the d i s t i n g u i s h e d
[Pa i].
in
point
p = ~ .
When
In this
q = 4
a subgroup
case the a u t o m o r p h i s m
of
P S L ( 3 , q)
group
induces
A6
of the g e n e r a l i z e d
on
quad-
~.
g8
tangle is flag-transitive.
(A. 3.10) h a v i n g
when
q = 16
There is exactly one other example
a flag-transitive
automorphism
group,
arising
[Hir, p. 177].
(A.3.11)
s = q + 1 2 t = q - I~ q = 2 e.
be as in (A.3.10).
Choose
A, B 6 ~ •
Let
Define
V
and
points and lines
as follows:
points:
containing
vectors;
A
lines:
A s usual,
or
translates
of those
2-spaces # <A , B>
B ;
translates
incidence
of m e m b e r s
of
~ - {A , B}.
is Just inclusion.
The v e r i f i c a t i o n
is
straightforward.
ProJectively
isomorphic
inequivalent
generalized
pairs
(n , [A , B]) produce non-
quadrangles with
The dual of one of these is isomorphic
there are
q
preserving
in [ P a i
elations
~.
This
PG(2 , q)
to one in (Ao3.10)
fixing
A
and
fact, and the construction,
B
iff
and
can be found
, 2].
Summ~o
W e have now listed all of the k n o w n types of
finite g e n e r a l i z e d
admitting
of
s = q + I , t = q - Io
quadrangles°
flag-transitive
cal ones and two in
The only k n o w n finite ones
automorphism
groups
(A.3.10) with p a r a m e t e r s
are the classi3, 5
and
15, 17.
99
A.4.
Constructions:
n = 6 ,8
Up to duality, only three classes of finite generalized
n-gons are known when
n = 6
groups of Lie type via their
or
8.
These arise from rank
2
BN-pairs [Ti 2, p. 40; Car, p. i07],
and these groups act flag-transitively.
Name
~
~
!
prime power q > 1
G2(q)
6
q
q
arbitrary
3D4(q)
6
q
q3
2F4(q)
8
q
q2
arbitrary
+
q = 2 2e I
The construction of these groups and their
complicated [Car],
BN-pairs is rather
especially in the case of octagons.
will outline a construction of the dual
G2(q)
We
and the
3D4(q)
hexagons analogous to A. 3.1 but significantly more complicated
(see [Ti 3]).
(N.B.--The
q = 3 e, and self-polar iff
G2(q) hexagon is self-dual iff
e
is odd.
In general,
the
G2(q)
hexagon is distinguished from its dual by the following property:
it can be realized by the set of all totally singular
points and certain totally singular lines of an
see [Ti i]
(A.4. i)
or
~(7, q)-space;
[CaK, pp. 409, 4 2 0 - 421].)
Th___ee 3D4 (q)
h_exagon o
Let
Q = [(a,~,c,8,e) I a,c,e E GF(q); ~,~ E GF(q3)}
(a, 8,c,8,e) (a' ,8' ,c' ,8' ,e' )
=
(a+a'
, ~+~
'
, c+c
'
+a
'e
-tr~
'4
, ~+8
'
, e+e
' ),
100
where
If
try
= y + yq + yq2.
i ~ i ~ 5
coordinate
is
of all such
x 6
x
x i.
G P ( q 3)
x6:
let
then
xl
Q
is a group of order
be the element
of
Q
and all others
0, and let
Then
= X3
Q'
= Z(Q)
x 6 E Aut
(a,~,c,~,e) ~
c
Then
Q,
where
has
x6
lt h
whose
X
i
De the set
order
is
q9.
q.
defined
If
by
(a , ~ + a x ,
a2x I + q + q 2
tr ~q + q 2
-
-
x
tr a~ q
-
+q2
+ a x q + q 2 + ~2xq 2 + ~q2x q ,
e+axl+q+q2tr
Now identify
sxq+q2+tr
(for typographical
~x).
reasons)
t 6 GF(q 3)
with
t6 ,
and define
AI(= ) = X 5 , A2(=)
Al(t ) = X~,
define
J C {1,2,3,4}
point:
line:
= X3X4X 5 , A4(=)
= X2X3X4X 5 ,
A2(t ) = (XIX2)t , A3(t ) = (XIX2X3)t ' A4(t ) = (XIX2X3X4)t.
As in A . 3 . 1 ,
where
= X4X 5 , A3(~)
, t E GF(q 3) U {~]
Symbol ~ ;
t;
incidence:
A4(t)g ;
A3(t)g ;
t
points and lines using cosets
and
A2(t)g ;
Aj(t)g,
g 6 Q :
g~
Al(t)g ~
is on
$
and
A4(t)g ;
all other incidences
are obtained via inclusion.
For more details,
There
see [Ti 3].
is a similar description
hexagon using
Q* = X2X3X4X5X 6
3 ~ J ~ 6 , by the elements
of
of the dual generalized
6
and conjugates of
~ Xi ,
i=J
XI .
101
(A.4o2)
and
t
to
of orders
Dual
G2(q)
GF(q).
hexagon.
In (A.4.1), restrict
This produces four families of
q , q2
q3
and
q4
q + I
~ ,
groups
The desired generalized hex-
agon is constructed exactly as in (A.4.1).
Remark.
q2
and
The group in (A.4.2), and the subgroups of order
q3 , are the same as those in (A.3.8).
This produces
the following strange construction of the generalized quadrangle
(A.3.8) from the generalized hexagon (A.4.2).
Fix a point
p
of the generalized hexagon, and define Points, Lines and Incidence
as follows°
Point:
p ;
distance
Line:
points at distance
3
from
lines on
Incidence:
from
lines at
p ;
p o
p ;
points at distance
a line on
Points at distance
p
6
p
2
4
is incident with
from it ;
from
p
p .
and all
a Line not incident with
is incident with all Points at distance
1
or
2
from it.
Point
Line/
fP
Line
~Point
This produces a generalized quadrangle with parameters
q2 , q
iff
Summary.
q -= 2
(mod 3).
Relatively few finite generalized hexagons or
octagons are known, all requiring fairly intricate constructions.
The
3D4(q)
and
and group-relatedo
2F4(q)
examples are especially complicated
102
An analogue of A.3ol can be proved, but involves
6
families of subgroups of a group,
4
or
satisfying complicated
generalizations of the conditions in Ao3olo
I have made many
attempts at finding new examples, using either these analogues
or other, very different methods°
My lack of success has led
to the following
Conjecture.
The only finite generalized hexagons or octa-
gons are those naturally associated with
However,
it is not at all clear how such a characterization
could ever be proved.
Consequently,
G2(q) , 3D4(q) or 2F4(q).
No automorphism group is hypothesized.
there is presently only one relevant characteriza-
tion known of all of the above generalized hexagons:
the beautiful
result in [Ro i], involving an additional but natural geometric
assumption.
No non-group theoretic characterization is known for
the
generalized octagons.
2F4(q)
103
Bo
Buildings and coverings
This chapter contains a summary of results of Tits [Ti 7].
These fundamental results form the theoretical
upon which the remaining chapters are built.
building-like
covering
B. I o
"geometries"
(called "SCABs") and their universal
SCABs.
Definitions B.Iol.
the finite set
with a family
I
A chamber system (~, {~i I i E 13) over
consists of a set
{~i I i E I}
member of
If
~i
are called
J ~ I
let
~j
of chambers,
~'.
~.
together
We will also
Two chambers in the same
i-adjacent.
be the Join of the partitions
~j ,
(i.e., the set of connected components of the graph
($,
, where
c
d =
c and
d
are
J-adjacent
for some
J).
Call
~
connected if
The rank of
If
J ~ I
G
and
is
of those members of
is any residue
resjX
llIo
(X,
nj
J ~ I
system
write
~(G/B,
then the residue
[~j I J E J}), where
lying in
with
construction Bol.2o
and a finite family
~I = {~]"
X E ~j
is the chamber system
If
~
of partitions of
call this "the chamber system
JE
We will define
SCABs, which usually are buildings.
Chamber systems ~
J E J
superstructure
X.
~
consists
A rank r residue of
IJI = r .
Consider a group
[Pi I i E I}
has
G,
a subgroup
of subgroups containing
Pj = (Pj I J E J) .
~Pi I i E I})
resjX = res X
B ,
Bo
The corresponding chamber
104
C~ = the set
Bg
If
J ~ I
iff
and
then
Bh
Bg
gh -I E Pj .
G/B
are
and
of cosets
of
B , and
i-adjacent = gh -I E P i "
Bh
lie in the same m e m b e r
In particular,
G
is c o n n e c t e d
of
~j
iff
G = <Pi I i E I> .
Definition
n-gon
(4 A.1)
B.I.3.
consists
The chamber
of the set of edges of the graph,
together w i t h the e q u i v a l e n c e
and "have a c o m m o n line."
diagram
~
n
system of a g e n e r a l i z e d
relations
"have a c o m m o n point"
This chamber
system is said to have
:.
Note that the e q u i v a l e n c e
classes here c o r r e s p o n d b i J e c t i v e l y
to the points and lines of the g e n e r a l i z e d
Definition
B.Io4.
A diagram
is the "graph" w i t h "vertex
i
and
J
D
set" I
are Joined by an "edge"
D(i , J) = D(J , i)
is either
~
D i a g r a m s are a b b r e v i a t e d
over
labeled
by r e p l a c i n g
.
:
by
=
: 6 :
by
-~----~-
3
-
4
and
III m i ,
D(i , J) , where
or an integer
by
=
4
I , where
such that distinct v e r t i c e s
2
o
n-gon.
~ 2 .
t05
(n = 2 , 3 , 4 , and
A. 2.2 and B.I.3,
relevant
6
are the most prevalent
these values of
to finite situations.)
n
and
instances.
n = 8
By
are the only ones
For example,
becomes
4
Moreover,
we now have the n o t i o n of a c o n n e c t e d diagram.
Definition
Just a
D-SCAB)
such that
ever
Bol.5o
A
X E ~ [ i , J} '
system
in
J c I ,
resjX
is a
J , with
I
(or
(~, [~i I i E I})
(Cfo B.I.4) w h e n -
IJI
~ 2, and
SCAB
whose diagram
the c o r r e s p o n d i n g
ch~bers
X E ~j,
are
Dj
consists
edges labeled as before.
i-adjacent
for at m o s t
i E I.
Whenever possible--and
the diagram
However,
Example°
.....
:
I
Tits [Ti 7] considers
I need not be finite,
required.
especially
instead of s p e c i f y i n g
Remark°
~
two flags b e i n g
their
over
i # J .
Also note that t~qo d i f f e r e n t
one
D
system
D(i,]).
has d i a g r a m
Note that, w h e n e v e r
of the v e r t i c e s
with diagram
is a c o n n e c t e d c h a m b e r
res[i ' j}X
the chamber
SCAB
and only
(B.I.5)
A projective
of rank
d
i-adjacent
(i - l)-spaces.
in Chapter C - - w e will draw
and the
D(i , j).
a m o r e general
situation:
"weak" g e n e r a l i z e d p o l y g o n s
seems to be a d e q u a t e
geometry
PG(d , K)
whose chambers
for our purposes.
produces
if
a
are the m a x i m a l
if they agree at all but
For example,
are
SCAB
flags,
(perhaps)
i ~ i < d - I
and
106
X 6 ~{i , i + i}
PG(2 , K)
and
then
of all
res X
i-l-
i+l-spaces
in
of
automorphism
~
of
[~i I i 6 I} ; if
automorphism
automorphism)
of
of
i-spaces
to the projective
lying between the
~
A type,preserving
automorphism
leaving each
invariant.
is a permutation
~
~i
~
of
is not type-preserving
$
i-2-
Sometimes
Aut G.
An
(or diagram-
a non-type-preserving
The group of all type-preserving
&
then it induces an
D , called a graph-automorphism
~ .
of
preserving
morphism is called a graph automorphism.
will be denoted by
plane
X .
Definition B.I.6.
is a permutation
and
corresponds
automorphisms
of
auto-
107
B.2.
Coxetergroups
In order to state the results
notation
is needed.
Definitions
is the group
relations
W = W(D)
if
wj
If
(i)
from
M
write
Definition
= i whenever
w .
M = M(D)
i ~ ri
The length
£(m)
of
is m i n {&(m) I m = w} , and
= £(~) o
i # J and D(i,J)
Call
Wj
J
<
spherical
are known.)
be the free m o n o i d on
extends
of
£(m)
presentation:
(All such
the n u m b e r
w E W
D
to a h o m o m o r p h i s m
W .
B.2.3.
i E I
group w i t h d i a g r a m
W j = <rj I J E J) .
Let
The map
onto
I.
i E I ;
B.2.2.
I o
over
h a v i n g the following
is a finite subgroup of
the a l p h a b e t
D
The C o x e t e r
(ri5)D~i,J~
J ~ I
Definition
m ~ ~
r i,
2
r i = i,
(ii)
Consider a diagram
B.2.1.
generators
in Tits [Ti 7] some a d d i t i o n a l
used to "spell"
(Compare
[Car,
p. 109].)
of a w o r d
m.
m E M
m E M
The length
is r e d u c e d
is
g(w)
if
108
B.3.
Buildings;
Let
D , W = W(D)
Let
($,
If
and
M = M(D)
[nil i E I})
be a
be as in § B.2.
SCAB
with diagram
m = ilo..i % E M , a gallery of type
sequence
are
coverings
C0Cl...c6
of chambers such that
ij-adJacent for each
J .
If
m
in
cj _ 1
% - I # cj
D.
~
is a
and
cj
for each
J
then the gallery will be called nonrepetitive.
Definition B.3.1.
~
is a building with diagram
D
if,
for any two nonrepetitive galleries starting at the same chamber,
ending at the same chamber, and having reduced types
m2,
we have
mI
and
ml = m 2 "
This is not the standard definition of buildings!
Tits
[Ti 7, Theorem 2] showed that this one is indeed equivalent to
the usual one [Ti 2].
We have used the present definition in
order to avoid discussing complexes and apartments
(see Tits'
paper in these Proceedings).
Definition B.3.2.
Consider two
and
(~',[ T~ii I i E I}).
A c0verin q
(C,
[~i I i E I})
(i)
(ii)
%0
If
D-SCABs
SCAB
is a map ~: ~' ~ ~
~:(~',[TTit I i E I])
such that
is surJective, and
J c_. I
and
IJI ~ 2 , then each member of
mapped biJectively onto a member of
IJI ~ 2
in (ii)).
a covering SCAB of
~.
!
nj
is
nj.
In [Ti 7] and [Ro 2] this is called a
(since
(~,[~i I i E I])
2-covering or 2-cover
Of course, we will say that
~'
is
Note that this is not the same as topological
covering spaces (but see [Ro 2]).
109
Definition
B.3.3.
~ : ~' ~ C
is a u n i v e r s a l
if it has the usual universal property:
covering
if
~ : ~" ~ ~
SCAB and c '~ = c ''~ for some c' 6 ~',
there is a c o v e r i n g
commutes.
Then
SCAB
covering
c" 6 ~",
then
k: ~' ~ C" such that
q
and h e n c e will be called
Proposition
is a
~' ~
\
is u n i q u e up to isomorphism, % ~ / ~
G'
"the" universal
B.3.4o
SCAB
~
"I
/
c o v e r i n g S C A B of
~.
D-SCAB ~ has a universal
Eyery
covering
SCAB ~ .
Proof.
Fix a c h a m b e r
s t a r t i n g at
if
A'
c0o
subgal!eries
from
sort:
Ai
C o n s i d e r all galleries
Call two such galleries
c a n be obtained
of the following
cO .
and
if
A
by a sequence
A = AIA2A 3
A~,
if
and
A2
and
first c h a m b e r and the same last chamber,
both lie in a m e m b e r
A
by
of
~j
for some
~
Define
be the set of all such
the p a r t i t i o n
~i
of
are in the same m e m b e r of
~i
iff
i-adjacent
Define
to the last term of
~:
~ ~ ~
We must verify
a
D-SCAB .
with
equivalent
of r e p l a c e m e n t s
A' = AIA~A 3
A~
for
have the same
and if
2-set
of
A2
and
A~
J ~ I , then replace
A'
Let
c
A ,A'
A
m e m b e r of
between
that
Statement
IJl ~ 2 ,
X
Ck-i
by
and let
and
~
Bo3.2
classes
as follows:
[A'] = [Ac]
[A] and [A']
A •
A o
(i) - (ii) hold and that
(i)
X ~ ~j .
[A].
for some chamber
[A] ~ = last t e r m of
B.3.2
has the form
equivalence
is obvious.
If
[A]E
X,
Consider
~
is
J ~ I
then e v e r y
[ACl.--c6] , w h e r e all a d J a c e n c i e s
c k , and
cI
and the last term
c
of
A ,
110
are
J-adJacencies
in a m e m b e r
of
X ~ E ~j.
Cl...c ~
galleries,
!
[ACl...c~]o
Thus,
If
together w i t h A.l.1,
be shrunk to the
in that m e m b e r
in the d e f i n i t i o n
in order
This proves B.3.2(ii).
Finally,
~
B.I.5
of
ct
lie
~j
arises
then use
of equivalent
to see that [ACl...c £] =
Moreover,
is c o n n e c t e d
1-element g a l l e r y
Now use
c , c I , o-. , and
[Ac I. .. c~] ~ = [Ac i. .. c~']~
' • .c6'
A 2' = c I.
and
r e s j X ~ r e s j X ~.
Jacencieso
J E J o
~j, and e v e r y c h a m b e r
in this manner:
A2 =
for
cO
if IJl=2 w e see that
since e v e r y g a l l e r y can
by a sequence
in order to see that
~
of adis a
D-SCABo
If
~:~' ~ ~
is a c o v e r i n g
e v e r y g a l l e r y b e g i n n i n g at
at
c$ , w i t h e q u i v a l e n t
Thus,
the m a p p i n g
the lift of
galleries
If
a deck t r a n s f o r m a t i o n
SCAB.
Let
regular
of
~'
SCAB then
such that
Let
~:~ ~ ~
be a u n i v e r s a ! c o v e r i n g
T h e n the group of deck t r a n s f o r m a t i o n s
is
o__n_n ~c E ~ I c~ = Co} .
Proof.
property
7; Ro 2].
is a c o v e r i n g
=
chamber
form a group.
B.3.6.
c o E ~.
galleries.
in B.3.3.
is an a u t o m o r p h i s m
These c l e a r l y
starting
[A] k = last term of
covering [Ti
~:~' ~ ~
Then
to a unique g a l l e r y
the above proof shows that any c o n n e c t e d
D e f i n i t i o n B.3.5.
Proposition
(c~) ~ = c O.
lifting to equivalent
as r e q u i r e d
system has a connected universal
~P = 9.
lifts
let
~: ~ ~ ~' , defined by
A , behaves
Of course,
cO
SCAB
This is a s t r a i g h t f o r w a r d
of galleries
consequence
of the lifting
a l r e a d y noted as the end of B.3.4o
111
It shouid be noted that the uniqueness of
~
(up to iso-
morphism) is also a consequence of the aforementioned lifting
property.
Corollary Bo3.7o
Every @utomorphism group
t_oo a__n_nautomorphism group
G
of
~
G
is chamber-transitive if
Proof°
If
g E G
Then
c~
~
and
= c ~ = c~g
Different choices of
G
Moreover,
g
G
~
and
lifts
T
of
G ~ G/T ,
iso
then the universal property of
that there are mappings
commute.
of
containing the ' group
deck transformations as a normal subgroup.
and
G
~
implies
making the diagrams
~
is an automorphism.
differ by a deck transformation.
This
proves the first assertion, and the remaining ones follow immediately. D
Finally, we come to one version of the main result in [Ti 7].
Theorem B.3.8
[Ti 7].
only if that is true for
set of size
3
and
~
resjX
(ii)
~
1 ~
J
is a spherical
Let
~
be a building°
is a universal covering SCAB°
I_~f ~ : ~ ~ @
the restriction of
whenever
X E ~j .
Proposition Bo3.9 [Ti 7]°
(i)
is covered by a buildin ~ if and
~
is a universal covering SCAB then so is
to each residue of
~.
The property B.3.9(i) is called simple connectedness
2-connectedness [Ti 7]) of a SCAB.
(or
There are SCABs having
spherical diagrams that are not buildings but that are simply
112
connected
([Ti 7] and C.2.4).
from c o n s i d e r a t i o n
B. 4.
partly eliminate
these
in § Co2.
Examples.
Examples
of b u i l d i n g s
Proceedings.
compare
building
see [Ti
later,
w e will
will be d i s c u s s e d
For s p h e r i c a l
B.2.1)
examples
now,
We will
when
examples
(i.e.,
2] and [Car].
for each p r i m e p o w e r
q
having
We will g i v e
they w i l l be n e e d e d
just n o t e that t h e r e
by Tits
(C.3.9,
in these
finite
W --
some
"affine"
§ C.5).
is at least one rank
For
r m 3
and each of the f o l l o w i n g
diagrams.
(B.4.1)
C
Connected
spherical
diagrams
of rank
r ~ 3.
r
F4
E6
E7
E8
:
--"
i
&
"
The c o r r e s p o n d i n g
group
as a normal
a precise
J
automorphism
subgroup
description
--
acting
group
a Chevalley
chamber-transitively.
of this situation,
For the case of a f f i n e b u i l d i n g s
in these P r o c e e d i n g s .
contains
For
see [Car].
we defer
to Tits'
paper
113
C.
SCABs
and GABs.
This
examples
chapter
is t h e h e a r t
of S C A B s
(B.I.5)
buildings) , many
of w h i c h
metric
The
objects.
infinite
in t h e
examples
construction
(B.3.4).)
another
C.l.
Definitions;
Definition
Recall
that
a member
~j
of
and t u r n
(for
or
C.I.I.
was
Let
Let
--- 0 V r
either
SCABs
of p r o b l e m s
of a g r o u p
theoretic
and
nature.
A vertex
be the
into
be
a rank
of t y p e
set of v e r t i c e s
a graph
r SCAB.
i
is
of t y p e
(V, ~) b y r e q u i r i n g
i,
that
u , v 6 V)
(Recall
that
u ~ v
v
say
to n o t e
V i.
In § C.3
the following
situation
Definition
= {~])
for
r - 1
that
u
each
u R v > ~.
and
is a set of p a i r w i s e
It is i m p o r t a n t
the sets
and
is a set of c h a m b e r s :
we will
(or c l i q u e )
any
ones,
GABs
U ~ V ~ u 6 V
If
geo-
(However,
covering
a section
in B.I.I.
Vi
almost
to f a m i l i a r
as u n i v e r s a l
with
are
examples.
(~, [ui I i E I ] )
defined
~I-[i]"
V = V1 0
for
that
of t h e f i n i t e
information
criteria
related
is on f i n i t e
concludes
It c o n t a i n s
(geometries
in t h e s t u d y
more
survey.
are c l o s e l y
implicitly
The chapter
containing
and G A B s
emphasis
arise
of t h i s
C.I.2.
i 6
v
are
incident
that
we will
a member
V
of s o m e
incident.
ni_[i].)
A flag
vertices.
is t h e d i s j o i n t
see m a n y
situations
union
of
in w h i c h
occurs:
C
I.
of t h e g r o u p s
is t i g h t
For
Pi
if
example,
V i = {~]
in B . I . 2
generate
G.
(i.e.,
this
ni_{i }
occurs
iff
114
However,
we will
diametrically
opposite
Definition
D-GAB)
be primarily
to
C.I.3.
is a t r i p l e
that
interested
=
a situation
in C . I . 2 :
A GAB with
F
in
(V,~,T)
diaqram
D
satisfying
(over
the
I)(or
following
condi-
tions:
(i)
(V, ~)
or
cliques
(ii)
~ : V ~
flags
uT ~ vT
If
= iv E V
I
Y);
D(i,j)
If
and
Y
- gon
chambers
c,
d E X
[Y
U {u,v]
Let
then
by
called
incidence,
u,
function:
where
all
is
r =
of
m
is
IIl
its
is c o n n e c t e d
v E V,
is t h e
IIl
size
y C Y]
any
flag
(ii)),
then
If
- 2
set
(using
(V,~,T)
c
and
j C
and
u ~ v ~
rank
and
of
I~);
F(Y)
of n e i g h b o r s
the
(or
restriction
lie
induction.
are
I,
i ~
in t h e
[]
that
F)
same
2,
the
and
graph
is a D - G A B ,
chambers
d
and
I If
is
j,
and
~
~
let
of
yT
a generalized
then
a D-SCAB
flags
of
V,
(c n d) ~ ~--- I - { i].
X E ~[i,j]"
i,j].
u ~ v] , so
if
of
are maximal
i-adjacent
Y = { v E c I vT
to p r o v e
of
I~(Y)
F =
D(i, j ) - g o n
connectedness
size
§ i.l).
follows:
i,
of
(cf.
as
d D
generalized
chambers
type
flag
I u , v 6 I~(Y)
In o r d e r
the
is a n y
~
in C . I . I ) "
the
,D(i,j)
is o b t a i n e d
two
as
graph,
C.l.4.
Proof.
the
is
F(Y)
(cf.
Lemma
~(I~)
I
(with
and
[i,j]
-
defined
for
then
(iv)
=
X
v ~ y
residue)
to
a graph
r-partite
(iii)
~
is
Thus,
that
If
X =
res{i,j]X
is
I" (Y) .
~(I~)
to p r o v e
member
of
is c o n n e c t e d
that
~I'
any
two
and
it s u f f i c e s
(by
intersecting
this
is
immediate
115
In v i e w of C.I.4,
by C.I.I
GABs
determine
cisely
then
we can v i e w
"are" p r e c i s e l y
those
all of the chambers
the m a x i m a l
F(X)
flags).
With
this
motivations
X
in w h i c h
in w h i c h
in mind,
of GABs
Conversely,
the v e r t i c e s
chambers
are pre-
is any flag of the GAB
from a r e s i d u e
correspondence
for the study
SCABs
(i.e.,
If
is a GAB arising
each GAB as a SCAB.
of
F,
C(~)-
we can state
(whose p r o o f
one of the
is, however,
elementary):
Proposition
arise
as
~(A)
C.I.5 [Ti
for a b u i l d i n g
In general,
explicitly,
it is not
and still
a GAB as in C.I.4.
with
2].
two c r i t e r i a
A diagram
graph
{G i I i 6 I]
of the
There
Gi
is also
If
J ~
Theorem
before,
Given
of subgroups
in
vertex
G, w i t h
a natural
I
let
the latter
D(i,j)
other
"nice"
GABs
a SCAB
arises
from
section
is c o n c e r n e d
problem.
if it has
> 2
the form
are erased.
Thus,
a group
generating
G
and a finite
G, let
set is the d i s j o i n t
Gig ~ Gjh
iff
typ~ function
F = F(G;
union
i ~ j
T:
[ Gi I i 6 I])
of the sets of cosets
and
Gig ~
collection
Gig ~ Gjh ~ 2.
i.
Gj = ~ { G j I J E J].
C.I.7 [ A
and s a t i s f y
GABS." (i.e.,
is just a path.
C.I.6.
be the g r a p h w h o s e
of this
(or a "string")
o---o..,o--o once all the n u m b e r s
Definition
to show that
The r e m a i n d e r
concerning
"are
4).
easy to c o n s t r u c t
harder
is linear
the u n d e r l y i n g
All b u i l d i n q s
I;MT].
Assume
the c o n d i t i o n s
that
G
and
Gi
are as
116
(i)
iE
If
i-J)
J c I
and.
I I - JI ~ 2
then
J c
and
I - J = [i,k]
Gj : <Gj U {i] I
;
(ii)
if
I
I~(Gj~ [Gj U [i] ' Gj U [k] ])
D(i,k)-gon
for some
(iii)
D(i,k)
The diagram . D
with
i ~ k, then
is the graph of a g e n e r a l i z e d
~ 2; and
determined
by the
D(i,k)
(as in B.I.4)
is linear.
Then the following
hold:
(1)
F(G;
(2)
G
(3)
The generalized
residue
{c i I i E I])
is chamber-transitive
r(Gj)
Another
sufficient
polygon
of the vertex
The preceding
C.3.4).
is a
result
fails
GAB criterion
amounts
on
in
I~;
(ii)
and
is isomorphic
to th@
Gj.
if
D
is not linear
can frequently
of transitivity
Theorem C.I.8 [A I]
D-GAB;
(e.g.,
be applied,
by
when
are available:
Assume that
G
and
[Gi I i E I ]
sat-
isfy the following conditions:
(i)
I_~f i E I
then F(Gi;
(ii)
I_~f i ~ j
then
(iii)
Any
that
G i N Gj
Gj/Gj
n ~.
3-element
[G i n Gj I J E I - [i])
is a GAB;
G ~ GiG j; and
subset of
I
has at most two orbits
can be ordered
on
Gi/G i ~ ~
i,j,k
and on
s__~o
117
Then the foil owing hold:
(i)
r(G:
transitivel~;
(2)
The GAB in (i) is isomorphic
Proof.
G
(iii).
for some
[Gi,Gj,~g],
Then
(Gj) n ( G / Q ) ,
also in
contradicts
By
for
= F(Gj)
and
3-set [i,j,k] ~ I.
i
and
j
are as in
G-orbit
as
has at least two orbits on
F(Gj)
n (G/Gk)-
~
and
~g,
each of w h i c h is
A (G/G k) ~ F(Gi)Then
(2)
For,
if
By symmetry,
GkG j = ~ G i ,
Fi
~:r i ~ F ( G i)
Moreover,
~
so that
G = <Gi,Gj>,
In fact,
this is clear,
3
and this
G
Consequently,
~(Gi) , and h e n c e
G
F ( G i)
Gi
is t r a n s i t i v e
(iii)
(i).
and maps
arises
edges to
in this man-
[ G i , G j , G k]
Gi, for some
j, k C I.
on the m a x i m a l
on the m a x i m a l
is n o w clear.
on the edges of type
This proves
(i),
(G i n Gj)g
can be sent to
is t r a n s i t i v e
c o n d i t i o n C.i.3
is t r a n s i t i v e
d e f i n e d by
and h e n c e of
Gi
the GAB in
since we h a v e seen that every
containing
by some element of
denotes
is b i j e c t i v e
We claim that every edge of
(i).
has the
we may assume that the
I~t is easy to see that
g E G i.
Moreover,
3
(ii).
clique of size
from
g E Gi
(iii),
there is a natural map
G
[i,j]-edges,
where
We can now p r o v e
ner.
and some
with representatives
= ~<Gi,Gj>.
edges.
on
G i n Gj
F(Gi).
F ( G i) n ( G / ~ )
Gig
h E G
Suppose that this is not in the same
[Gi,G j ,~].
~Gj
to the GAB d e t e r m i n e d
We first show that every flag of size
is t r a n s i t i v e
flag is
acts chamber-
F(Gi).
[Gi,Gj,~]h
Since
G
and
by the s u b g r a p h
form
is a GAB on w h i c h
[Gili E I ]
{i,j],
flags of
flags of
Finally,
C.I.3
F.
since
(iv) follows
118
Remarks.
The h y p o t h e s e s
for e s p e c i a l l y
they c o n c e r n
group
in C . 1 . 7
theoretic
structural
situations.
properties
along w i t h v a r i o u s
relations
C.I.8
to the fact that
(ii)
amounts
and Q . i . 8
are d e s i g n e d
For the m o s t part,
of the g r o u p s
Gj,
among t h e s e groups.
J ~
I,
N o t e that
the d i a g r a m
for
F
is
(3) and C.I.8
(2)
connected.
Isomorphisms
are c r u c i a l
such
ingredients
[ A 1;MT]
for g r a p h s
important
open p r o b l e m
(V,~,T)
Tits'
almost
in C.I.7
in m u c h m o r e g e n e r a l
that
results proved
are not n e c e s s a r i l y
to find f u r t h e r
GABS.
criteria
in
It is an
for a g r a p h
to b e a GAB.
Covering
C.2.
as those
GABs.
theorem
immediate
this w e will
Let
preserving
on u n i v e r s a l
consequence
first
~ =
B.3.8
define
(V,~,T)
for GABs.
"quotient
be a D-GAB.
automorphisms
of
covering
F,
SCABs h a s
In o r d e r
an
to d e s c r i b e
GAB
by
an a u t o m o r p h i s m
Let
H
be a g r o u p of type-
and let
A/H =
(V/H,~,T)
group.'
b e the
graph with
V/H = {vH [ V C V],
edges
[ u , v ] H = {u H,v HI
for edges
{u,v],
and
(vH) ~ = v T"
In general,
~/H
the f o l l o w i n g
will
hold:
not be a GAB.
However,
it is a D - G A B
if
119
Conditions
then
C.2.1.
(i)
If
is a f l a g
and
X
is a f l a g
of r a n k
Ill
- 2
[u E v H i {ui
U X
H x = i;
(ii)
If
a flag]
is e i t h e r
(iii)
forms
some
X
If
or an o r b i t
u, v E V, u ~ v,
a flag
together
element
{u,v} H.
~
v E V
of
(Here,
with
v H,
then
u
and
X
of
and
some
H~;
X
is
and
is a f l a g
element
forms
X
then
of
u
H
and
a flag with
"form a flag"
such
if
X
also with
some
{u~
that
element
U X
of
is a
flag.)
Each building
theorem
of [ T i
Theorem
Assume
that
buildinq,
Then
"is"
a GAB
(C.I.5).
7] is a p a r t i a l
C.2.2
[Tf
7].
and that
F ~ A/H
for
there
converse
Let
each residu e with
One version
F
of t h i s
be
a
diagram
~
is no r e s i d u e
a building
A
and
of t h e m a i n
fact.
G A B o~f r a n k
-
~
with
~ 2.
is c o v e r e d
diagram
a qroup
H
~.
by a
: 5
satisfying
C.2.1.
Here,
One
which
(cf.
H
arises
important
says
that
as a g r o u p
consequence
H = 1
when
of C . 2 . 2
A
transformations
B.3.6.
is t h e f o l l o w i n g
result,
is f i n i t e .
Theorem
C.2.3
(i)
F
is a b u i l d i n q
if
(ii)
F
is a b u i l d i n g
if it is f i n i t e ,
B. 2.1)
buildings.
and all
[BCT ].
of d e c k
Let
residues
F
D
with
be
a
D-GAB.
i__ss An , D n
diag[~
o__~r E 6
D
C3
(cf.
B.4.1).
is s p h e r i c a l
are c o v e r e d
by
120
There
are several
adjacency
matrices
hold when
H = 1
that
H = 1
noticed
using
was
if
A
proof
There
seen
~
Example
acting
on
Fix a c o n j u g a c y
ponding
family
"planes".
class
Incidence
and each -
- is a
Conjecture.
with
a spherical
Some results
diagram
C3
hypothesis
in
7]
concerning
is
o-----L__j.
Let
"lines".
and call
structures
Each =
of
the corresPG(2,2)s
: is a PG(2,2),
quadrangle.
"A7-GAB"
that
concerning
in [Ti
is essential
of p o i n t s
PSL(3,2),
inclusion.
PSp(4,2)
are found
the same proof
given
GAB
triples
PSL(3,2) I = 15
diagram
in [CS],
GAB.
Call
is just
The
that
is an elegant
in C.2.3(ii)
this h y p o t h e s i s
of subgroups
IA7:
groups,
geometric.
The exceptional
7 points.
of
of C.2.3(i)
remarkable
C.2.4 [Ne].
proof
the fact
case)
spaces;
is p r i m a r i l y
That
in the f o l l o w i n g
G = A7,
F
~
uses
there
group
can only
and Cohen.
is an u n f o r t u n a t e
the d i a g r a m
Finally,
the p r o o f
for infinite
In[BCT~
A different
classical
and myself,
of finite p r o j e c t i v e
On the other hand,
and [ BCT]
from
in the classical
also found by BrouWer
(ii).
building.
arising
applies.
(again
correlations
A
by Surowski
immediately
of C.2.3
are u s e d to show that C.2.1
is a finite
for b u i l d i n g s
[ K a i, 2.1]
of parts
of graphs
independently
geometric
proofs
C.2.4
is not
this
lot
is the only
covered
conjecture
2], IRe]
finite
GAB
by a building.
in the case of the
and [ A 2].
121
C.3.
Difference
Some
sets.
Many
of
of
the
simplest
These
are
easy
them
are
tight
of
being
(by
Let
of
and
GABs
situation
that
(cf.
in
the
A
be
such
written
(C.I.2),
lines
If
elements
tuples
difference
d,
translates
A
and
F =
write
C.3.1.
A
E - F =
Let
such
E~
that
whenever
the
r
and
th
3
i et
component
(i)
complete
the
arise
(~,
graph
from
"
the
covered
of
order
n
D
2
by
buildings
by many
the
+ n
of
latter
of
A
+ 1
can
be
uniquely
D + a,
a 6
A.
ordered
Recall
elements
uniquely
determines
elements
are
+ i.
n
the
(e~),
1 ~ ~
elements
1 ~ ~
1
impression
n
of
a pro-
A
and
+ l-tuples
of
(e i, - fi ).
=
e i + a ..... e i + a)
profusion.
covered
are
(fi)
difference
section.
This
points
great
examples
a set
D.
be
from
the
all
to
element
E
the
of
set
d'
whose
(e i)
from
nonzero
of
in
give
are
give
this
is
n(A,D)
E =
Theorem
A
hence
out
group
in
every
of
arise
exist
they
We will
additive
for
are
and
turn
part
set
that
plane
will
SCABs
and
However,
second
an
of
construct
C.6.2).
d - d'
jective
whose
to
probably
a difference
A
examples
uninteresting.
B.3.8),
finite
sets.
< ~ ~
6
Ar
in
r.
I a E
~
r,
E~
be
ordered
- E~
form
n + la
Write
A,
equivalence
1 ~
relation
i ~ n
+ i],
"have
the
diagram
is
same
Then
{~j I 1 ~
on
r
(ii)
Each
rank
form
17"(A,
E~
j <
r] )
is
vertice_s_s,
2 residue
- E 8).
of
a SCAB
whose
a
and
~
is
a projective
plane
of
122
Proof.
we obtain
that
By r e s t r i c t i n g
the chamber
U{i,j]
= {~]
and
A
sets
in [Ro
3],
and let
n
due
at l e a s t
of
rank
2
Proof.
is
we
PG(2,q)
find
that
otherwise
of
elations
is easy
of
same
Aut ~
r SCABs
(mod q
2
is a unit.
are
let
r a 3,
that
C.3.1
r
the
produces
S C A B s oallf
PG(2,q).
Aut ~
~ PSL(3,q)
in C . 3 . 1
on
the
when
u (A, E l - E 2)
By r e s t r i c t i n g
to
We c l a i m
A ~ A u t ~.
in v i e w
By c o n s i d e r i n g
that
res[l,2]~
of the s u b g r o u p
action
res[ 2,3] ~
of a g r o u p
and on
For
structure
of
2
q
r e s [ l , 3 ] ~ , it
a contradiction.
E0 = 0
and
constructed
E1 = D
there
Then
any
in
using
C.3.1,
E 0, t h i s
E1
isomorphism
~ : ~ ~ ~'
must
must
lie
in
NpI~L(3,q)(A).
is i n d u c e d
of
there
are
by a m u l t i p l i e r
GF(q3)).
C.3.2.
and c o n s i d e r
this
~
in
p,
rank
that
$
r = 3
of d i f f e r e n c e
such
Then
assume
construct
E i - EJ).
r-tuples
A ~ A u t ~ n A u t ~' , w e m a y
Aut
use
+ q + i)
nonisomorphic
Aut ~' , and h e n c e
(forming
= ~(A,
3] w h e n
a prime
of u n o r d e r e d
Aut ~ ~ I~L(3,q).
A < PGL(3,q).
to
for
is cyclic.
res{l,2}~
let
q = pe
consider
A
to o b t a i n
Now
rank
residues
Aut ~
PSL(3,q).
Let
pairwise
and
[Ro
Ar
This proves
res[i,j]O
to R o n a n
of
to Ott. )
any t w o of t h e m
First
E 2 - El).
another
of i n t e g e r s
nr,q/3e
two c o m p o n e n t s
is a l s o
b e the n u m b e r
{ 0 , 1 , m 3 ..... mr]
u (A,
is d u e
(There
C.3.2.
r,q
difference
for
first
i ~ j, and t h a t
theorem
is a b e l i a n .
Corollary
whose
system
for
The p r e c e d i n g
to the
D;
Finally,
let
fixes
3e
E~ = m D
D
two
and the
send
Since
and h e n c e
such multipliers
in o r d e r
to
123
Remarks.
large
in C . 3 . 2
All
B.3.8,
For,
of t h e m
fields
those
with
cases
Definition
consists
groups
C.3.3.
of a g r o u p
of
(i)
(ii)
(iii)
whenever
G
such
IXil
and C . 3 . 2
prime
which,
are t i g h t
When
if any,
from
C.3.11).
For
q = 2
and
a family
or
on
8, a
~,
By
it w o u l d
buildings
of them
in C . 3 . 2
is t r a n s i t i v e
(cf.
r = 3
for C o m p l e t e
Some
appearing
q.
(C.I.2).
of t h e s e
PSL(3,K)
(C.3.9)).
powers
local
certainly
and
its proof.
and o t h e r
con-
PG(2,q)-family
[Xi I 1 ~ i ~ r]
of sub-
for
each
i,
is a F r o b e n i u s
and
group
of o r d e r
(q2 + q + l ) ( q
i ~ j.
a complete
C. 3.4.
~(G/I
qraph
on
Moreover,
Proof.
can
large
arbitrarily
that
= q + 1
<Xi,Xj}
then
PG(2,q).
G
can be m a d e
G = < X 1 . . . . . Xr>,
Proposition
i__nn G
(cf.
Aut ~
can be u s e d
r
a building.
obtained
q = 2, r = 3
in t h o s e
structions
in C . 3 . 1
[ BT]
that
sufficiently
to k n o w
ones
K
to see
produces
interesting
"classical"
(skew)
do:
by u s i n g
of the S C A B s
each
be v e r y
are
It is easy
I_~f { X i I 1 ~ i ~ r}
, { Xi I 1 ~ i ~ r~)
r
vertices
G
is t r a n s i t i v e
~(<Xi,Xj>/l
The preceding
result
PG(2,q)
a sharply
admit
needed
PG!2,q)-family
is a S C A B w h o s e
such
, [Xi,Xj])
is a
that
each
rank
diagram
is
2 residue
on C h a m b e r s .
is
q = 2
PG(2,q)
or
flag-transitive
8
for
since
group.
i ~ j. []
only
then
is
+ i)
124
Problem.
Generalize
Problem.
Construct
C.3.4 to all
finite
q.
SCABs ~
with nondesarguesian
residues.
The r e m a i n d e r
of C.3.4.
These
Example
of this
section
is c o n c e r n e d w i t h
are easy to find:
C.3.5.
Let
d
be any p o s i t i v e
G F ( q 2 + q + i) d ~
Zq+ 1
where
action
could
even be a F r o b e n i u s
(e.g.,
G
(q2 + q + i) (q + i)).
of o r d e r
equally
q + 1
Then
behaves
trivial
r ~ 4
pr~uce
finite
Problem.
C.3.4 when
extremely
the m o s t
Find
Study
r ~ 4.
C.3.11):
in C.3.4.
known
PG(2,q)-families
of o r d e r
This
in [ K a
of a
G =
to s c a l a r
of all s u b g r o u p s
can be f o u n d
those buildings
(In some cases
interesting
arise
group
Let
of
G
and m a n y
7].
PG(2,2)-family
a finite
of a r b i t r a r y
GAB [ K M W i].
s i z e that
GABs.
The c a s e of
that
as r e q u i r e d
integer.
corresponds
r = 4) that p r o d u c e s
well-behaved
the c a s e
Zq+ 1
is p r e s e n t l y
(in fact,
Problem.
the
the f a m i l y
constructions
Only one example
with
examples
r ~ 4.
buildings
cases
~
When
GABs
arising
all rank
(via B.3.8)
3 residues
- see C.3.11.
These
from
produce
are p r o b a b l y
of this p r o b l e m . )
in C . 3 . 4
q = 2
there
is m u c h b e t t e r b e h a v e d
are at m o s t
(and two of t h e m h a v e b e e n d e t e r m i n e d ]
4
than
buildings
cf. C . 3 . 1 0 ,
125
Theorem
produces
C.3.6
a SCAB
[Ro
3].
covered
by
C(G/I,{XI,X2,X3})
G
has
one
of
and
relations
and
one
I
-1
b
and
Type
XI =
following
(Type
I)
(ab)
(Type
II)
(ab)
(Type
III)
(ab)
(Type
IV)
(ab)
C.3.7.
),
Type
IV
and
2
2
2
c ~+ a
III
one
<a),
of
four
three
groups
buildings
X3 :
<c>,
generators
(ab) 7 =
I,
of
X 2 = <b>,
presentations:
(bc ~I ) 7 =
IGI
2
: ba,
(bc)
= ba,
(bc)
= cb,
(ca)
= cb,
(ac)
= ba,
(c-lb) 2 = bc-1 ,
2
and
a,b,c,
(ca) 7 :
I,
and
b
obvious
a ~ b
-1
~
c
-1
=
c
graph
-I
,
~
2
: ac,
2
(bc-I)2
= ba,
are
(a ~
a,
C.3.8
if
2
There
(a ~
Theorem
G
exactly
a3 = b3 = c3 :
(a ~+ b ~
b ~
in
the
PG(2,2)-family
of
Remark
Type
, where
Every
c ~
a
-I
: ca,
(ac) 2 : ca,
b,
(ac)
2
ca.
=
automorphisms
c),
and
Type
a ~ b
in
II
-1
,
or
(a ~ c
c ~
-I
c),
b ~c).
[KMW
I].
> 168
If
then
{ X I , X 2 , X 3}
the
is
a GF(2)-family
corresponding
SCAB
"is"
a
GAB.
This
relations
is p r o v e d
among
the
by
a careful
X
,
with
examination
some
help
of
from
the
possible
a computer.
1
We
now
to
identify
able
C.3.6,
we
turn
first
to
the
give
explicit
universal
a brief
constructions.
covers
in
description
Since
Types
of
we
I and
some
will
III
affine
be
of
buildings.
126
Definition
of f o r m a l
complete
formizer
be t h e
C.3.9.
Laurent
local
n
series
field.
(e.g.,
standard
R = PSL(3,K)
Let
the
basis
for
buildinq
rank
2 residues
tioned
each
Let
Let
is
PG(2,~/~O)
above.
p-adic
Let
R.
1
Li,
where
to the
generally
case).
Let
uni-
el,e2,e3
be the s t a b i l i z e r
in
1
1
L 2 = < ~ e l , ~ e 2 , e 3 > O.
B = R 1 n R 2 n R3.
~(R/B,[ RI,R2,R3])
- i.e.,
any
ring with
O-submodule
and t h a t
the field
~
generated
Then
(cf.
has
PG(2,p)
the
by
affine
B.I.2).
diagram
/X
in t h e c a s e s
and
men-
(See [ BT].)
Let
K = Q2'
k 2 + I + 2 = 0.
o:
refers
R i ~ SL(3,O),
Notation.
that
O
PSL(3,K)
that
or m o r e
field,
its v a l u ~ £ i o n
in t h e
K 3.
p-adic
1
L 1 = <~el,e2,e3,)O,
vectors.)
Note
be
O-module
subscript
for
be t h e
GF(p),
O
~ = p
of t h e
indicated
over
Let
L 0 = <el,e2,e3>o,
(Here t h e
K
and let
Define
X
a,T,b,c
be
a unit
6 GL(3,K)
0
0
0
1
-2%-2)/2
0
G = <a,b,c>.
Then
a 3 = b 3= c 3 = 1
and
in
O =
2
by
,b = a T , c = b T
T3
is a s c a l a r
matrix.
Theorem
affine
C.3.10
building
(ii)
If
m
rood
m",
then
rank
2 residues
for
[~JiW2].
(i)
~(G/l,[<a>,<b>,<c>])
is the
PSL(3,~2).
is any o d d
C(G~/I,[<a>~
PG(2,2)
integer
and
<b>~,<c>~] )
and T y p e
I (cf.
~
is t h e m a p
is a
such
~
C.3.6).
"pass aqe
SCAB with
127
(iii)
m m
The
above
(iv)
If
m
i,
4
2,
72SL(2,7)
and
for
21
It f o l l o w s
that
7),
G
Easy
a, b
Moreover,
is f l a g - t r a n s i t i v e
from
the
For
"is"
m
c.
from
Now
an e a s y
C.3.8.
For
view
of
of
~
for
(mod 7),
and
since
L I / 2 L I.
G ~ SL(3, Zg[~,$---~]),
the relations
proves
see [ K M W
C.3.10,
"residue"
of
~.
calculation
(iv)
group
= G ( R / B , [ R I , R 2 , R 3] )
(ii),
and
among
(iii)
2]. D
see C.5.9.
of o r d e r
the construction.
Note
that
3 in C . 3 . 7 ,
The same
T
and that
is t r u e
in
situation.
GF(2)
0
Let
K
in t h e
,
be the
G = <a,b,c>.
field
indeterminate
~ =
x + l + x -I
Let
R
of f o r m a l
x.
b e the
ring
x
Laurent
series
Write
x+l
1
Then
5, 6
is a F r o b e n i u s
and p r e s e r v e s
automorphism
simplified
Notation.
b =
is .........
S...L..( 3 , m )
a C R 0 n R I, b E R 1 N R 2
on t h e
on
Moreover,
is p o s s i b l e
just the graph
the next
over
connectedness
(i).
another
it g r e a t l y
3,
that
<a,b>
that
mod
G~
m =
show
is c h a m b e r - t r a n s i t i v e
and
follows
for
calculations
This proves
passage
SU(3,m)
then
m = 7.
c E R 3 N R I.
order
is a GAB.
is an od d p r i m e
(mod
Proof.
SCAB
, a =
(b -1) T
, c = b T and
x+
G F ( 2 ) [ x - I , (x 2 + x + i)-i].
G ~ SL(3,R).
Theorem
buildinq
for
C.3.11
[KMW
PSL(3,K).
3].
(i)
O(G/I,
{(a>,<b},<c~)
is the
affine
128
(ii)
f
Passage modulo
GABs
in a r b i t r a r i l y
(iii)
of d e q r e e
nonisomorphic
~
The p r o o f
(iii)
(but
n ~ i0
in [ K M W
exponentially
same g r o u p
in the same manner.
C.3.12.
(i 3 7)(2 6 4) p r o d u c e
The
2-,
3-,
duce covers
from
of this
group
SL(3,2n).
elements
one [Ro
3].
and p a s s a g e
(ii)
and
but m e s s i e r
exist
for all
n ~ 3
It seems u n e x p e c t e d
examples
central
pairwise
can arise from
(2 3 5)(4
in
7 6),(1
A7
extensions
All
and
the
2 6)(4
of T y p e
of
of t h e s e
A7
II.
pro-
examples
arise
Are there
infinitely
many
Are there
any e x a m p l e s
finite
examples
of
II?
Problem.
In [ K M W
were
2 n/4
3].
Problem.
Type
A7
than
a better
GF(2)-family
and 6 - f o l d p e r f e c t
GABs [Ro
C.3. 6
a
polynomials
Parts
examples
many
The
and
more
GABs.
i]).
finite
qroups.
irreducible
finite
occur [ K M W
that
produces
to that of C.3.10(i),
3], w h e r e
remarkable
Construction
R
chamber-transitive
in p a r t i c u l a r ,
cannot
of
produces
ideals p r o d u c e s
is given;
PSL(3,4)
for the
(i) is s i m i l a r
are d i s c u s s e d
ideals
solvable
Rf
GABs w i t h
of
suitable
estimate
large
Passaqe modulo
f E GF(2)[x]
modulo
suitable
i] all s i m p l e
examined
groups
to see w h i c h p r o d u c e
of Type
IV?
of order
less t h a n
a million
PG(2,2
-families
of size
3.
We h a v e
just d i s c u s s e d
No a n a l o g u e
of C.2.6
an a n a l o g u e
of C.3.11
is k n o w n
PG(2,2)-families
for
is k n o w n for
in some detail.
PG(2,8)-families.
GF(8)
[ K M W 4].
However,
7 5),
129
C.4.
A table
In t h i s
(of r a n k
other
section
than
spherical
first
example
here
values
of
m
names
all
alized
column
associated
in
residue
whether
out
that
concerns
2-adic
group
the
SCAB
the universal
(in
(1)-(6))
The
In
the
is t i g h t
cover,
or,
? = unknown,
but
is a b u i l d i n g
?? = u n k n o w n ,
but
is n o t
the quadratic
k
2
fk = E x i
i
k
fl~ =
2E
1
2
xi
forms
more
a building
a r e as f o l l o w s :
k-i
-
E
i=l
xixi+
1.
column
(other than
gener-
polygon
the commutator
SCAB
B.3.8).
(1)-(5)
fourth
column.
group.
follows:
(cf.
in t h e f i f t h
contains
or n o t
general
is a c l a s s i c a l
Chevalley
gives
is an i n t e g e r
but more
polygons
this
group,
where
column
If t h e r e
(1)-(3).)
(26),
corresponding
points
The second
generalized
on t h e
6 indicates
occasionally
last
Except
in t h i s p a p e r
in t h e t a b l e ,
in
SCABs
in § C . 3 .
groups.
odd
finite
automorphism
- see the reference
is m e r e l y
induced
of the
Column
the
(if it is).
of t h e r e s i d u a l
and the group
a location
it is a p r i m e
m
the known
and t h o s e
automorphism
2-gons).
subgroup
gives
also occur
(For e x a m p l e ,
all
a chamber-transitive
is m e n t i o n e d
then
list
buildings
column
chamber-transitive
"m"
we will
a 3) a d m i t t i n g
The
the
of SCABs.
is a G A B
(C.I.2).
giving
- and
Finally,
the
frequently,
as
130
The order
follows:
in w h i c h
first
Numbers
The
due
of r a n k
does
(14),
(15).
The universal
rank
(compare
is as
covers;
then
and t h e s i z e of t h e f i e l d s
are
exceptional,
in t h a t
the rank
3.
not
indicate
rank
of space.
3
(or h i g h e r )
S o m e of t h o s e
residues,
residues
merit
(18)
-
is No.
One
3 residues
(23),
(24).
O n e of t h e t y p e s
(7).
t y p e of
covering
(21),
rank
SCAB
3 residue
~: ~ ~ ~
is No.
(3).
does not have
are a l w a y s m a p p e d
the property
isomorphically
by
B.3.2(ii)).
No.
(19).
m = 3,
One
of t h e t y p e s
and h e n c e
building.
However,
to a r i s e
as a r e s i d u e
in fact,
(19) m a y
situation
in fact,
apply,
Problem.
or n o t
is a b u i l d i n g
~ = ~.
relatively
guarantee
has
a known,
there
of the u n i v e r s a l
B.3.8 does
B.3.9
states
coverings
Given
it is s i m p l y
covering
apply.
that
~,
(3)
covering
that building
S C A B of
(N.B.
Whenever
residues
find
(19);
- This
B.3.8
of u n i v e r s a l
a simple way
connected.
, find
(In o t h e r w o r d s ,
~
to e x p e c t
connected!
not
is No.
is a
does,
coverings
of r e s i d u e s . )
a SCAB
(by B.3.8)
3 residues
2-adic universal
is no r e a s o n
few assumptions
that
of r a n k
e v e n by s i m p l y
in w h i c h
are u n i v e r s a l
or n o t
2-adic universal
of t h e r a n k
(26)
(15),
3 residues
No.
whether
listed
comments:
Nos.
with
and
at l e a s t
table
have been
polygons.
to c o n s i d e r a t i o n s
further
that
in t e r m s
(25)
is any n u m b e r
examples
those with known
lexicographically
of t h e r e s i d u a l
the
In p a r t i c u l a r ,
a simple way
criteria
and o n l y
is a b u i l d i n g . )
to d e t e c t
simple
to d e c i d e
are n e e d e d ,
tests,
if
whether
involving
that will
,
~
(9)
+
~==mm:=m
_-
(8)
(7)C.2. 4
(6)
.
P~+ (8,2)
Suz
A7
G 2 (m) ,m> 2
n(5,7)
PG(2,2)
P~+ (6,m),
m>2 ,m~7
~-+(6 ,m) ,
m>2 ,m~7
75~(5,7)
(5)C.5.7
PSp(4,2)
~
lhq (5,m) ,
m> 3
34A6 ,m=3
-
(4)C.5.7
PSp(4,3),
[~(5,3)
PSp(4,2),
PSp(4,2)
PG(2,2),
PSp(4,2)
G2(2)
PC(2,2),
PSp(4,2)
P~-+ (6 ,m ) ,
m> 2
~-+ (6,m),
m> 2
(3)C.5.7
PG(2,2)
P~+(8,m),
m>2
PG(2,2),
PSp(4,2)
Polyqons
Group
Pn(7,m),
m>2
X
Diagram
(2)C.5.7
(I)C.5.6
No.
SCABs
Y
Y
Y
Y
Y
GAB?
[Ka 3]
[RSm]
[Ne; AS; Ka 6]
for m=3
Y
y
[Ka 6]; [co ; AS] Y
[Ka 7]; [AS]
for m=3
[KMW 4]
[ Ka 6] ;
[RS; Ka 2; AS]
for m=3
[Ka 6] ; [AS]
for m=3
for m=3
[Ka 6]: ;As]
References
Chamber-transitive
cover
?
Self [ Ro 4]
G2(~ 2 )
PQ (Q2' f5
l~q (@2, f 6 )
B3(@ 2 )
=P~ (@2 'f7 )
=PQ (~2,f8)
P~+(8,~ 2 )
Universal
(19)
Fq
Mc
PG(2,2),
PSp(4,2)
Ps~(4,2)
PG(2,2),
Psu(3,5)
v
PSp(4,3)
Pn+(8,2)
(17)
(18)
PSp(4,3)
PSU(6,m) ,
m~3
(16)
<
PG(2,2),
PSp(4,2)
~(7,m)
(15)
m
PSp(4,2),
PG(2,2)
PG(2,5),
G2(5)
G2[2) and
its dual
A7
G2L3J
Polygons
(14)
_
Group
LyS
_
Diagr~
(13)
(12;
No °
[ RSt]
[Ne; Ka 6]
Compare [ Ka 3]
and No. (9)
Compare [Ka 73
and No. (i0)
[Ks 6], [AS]
for m=3
[KMW 4; RO 43
[ Ka 3]
L AS]
References
Y
Y
Tight
Tight
Y
Tight
Y
Y
GAB?
7?
77
?
?
Known [ Li]
77
?
?
Universal
cover
k/",,./
4]
(s,t) =
(3,5) o r
(15,17)
[KMW 4]
Tight
??
Tight
[ KMW 4]
PG(2,2)
PSp(4,2)
(6,3)
P~
PSp (4,2)
??
Tight
??
Known [ Li]
Universal cover
[ ~Mw 4]
[KMW 4]
[~w
Y
GAB ?
PG(2,2)
PSp(4,2)
PSp(4,2)
PG(2,2)
PSp(4,2)
[ KMW 4]
References
(6,3)
(6,3)
PG(2,2)
PSp(4,2)
Polygons
P~
P~
P~t(6,m)
m>2
ii
ii
Group
Diaqram
(26)C.6.10 <
(251
(24)
(23)
(22)
(20
~o.
CO
134
C.5.
E8
root
In [ K a
constructed
this
6] s e v e r a l
families
using
root
section
paper
and of l a t e r
--
~, w h e r e
equipped
with
and let
Write
E8
we w i l l
Following
and
lattices.
~i
C.3.9
lattices
provide
work
we
[Ka
7, K M W
introduced
k - X = -~-~.
2-adic
the
dot p r o d u c t
2-adic
(u,v)
e2 =
(3,l,k,X,l,k,k,0)
e3 =
(~', 3 , X ,~ ,k-,k , ~ - , 0 )
e0 =
(I,i,i,i,i,i,1,
and
buildings.
In
view
of that
integers
the v e c t o r
(~,k,k,~,~,l,3,0)
ei
were
4].
Consider
from
GABs
different
eI =
be o b t a i n e d
f0 = ~ 0 / 1 4
and
a slightly
--
the u s u a l
of c h a m b e r - t r a n s i t i v e
k
8
space
~2'
8
for u , v E ~2"
Let
-~)
by
interchanging
~
and
k
~.
1 7T 7!~/~-i
fi = ~ a i j e j , (ai3)
J
where
1 ~ i,
j ~ 3.
Finally,
=
71
21
7~
7~
~
for
all
i, j, and
~ ~2f°r
all
i,j,
let
8
A
= { (a i) E Q8 I a i + a 3
(C.5.1)
8
[ = {(%) ~ Q 2 1 % + a j
SO that
and
A
is the u s u a l
A ~-A ®ZZZ~ 2.
E8
E 2Z
root
lattice
and
(compare
Za
E 2Z~
1 z
8
Z a i E 2ZZ2~,
1
[Car,
§3.6])
135
As in C.3.9,
8
~2
generated
basis
by
Sl,...,s8,
if
8
S ~ ~2
S.
(For example,
write
let
(S~
be the
~2-submodule
A = (A>~)
If
d e t < S > ~ 2 = det((si,sj)).
S
of
is a
This is deterW
mined only
up to
multiplication
by
an
element
of
the
group
2~2
of 2-adic units.
Lemma C.5.2.
for
(i)
(ei,e j) = 0 =
(fi,fj)
and
(ei,f i) = 8ij
0 ~ i, j < 3.
(ii)
3
® (ei, fi>2Z 2
i=O
A =
Proof.
(i)
(ii)
This is just a straightforward
calculation.
Let
L 0 = (e 0,e I ,e 2,e 3,e0/14,e I ,e 2,e3>
ZZ 2
Since
det(aij)
another
= 7-4 E ~ ,
simple calculation,
SO that
L 0 ~ A.
integral
lattice.
Let
Let
Then
det A E ~
Thus,
since
det L 0 E ~ .
of
L0
A
By
is in
A,
is a unimodular
L 0 = A. []
be the reflection
interchanges
(i) we have
each generator
S = {ui I 1 { i m 85
r = r8
r
Here,
by
e0
and
be the standard
basis of
v ~ v - 2(v,u8)u 8
in
e0, and hence sends
8
Q2"
u~.
L0
to
r
L 1 = L 0 = (½e0,el,e2,e3,2f0,fl,f2,f3>Tz 2 .
+
The group
~
(8,~ 2)
group of all isometries
building
A8
is the commutator
of
8
@2"
has as its chambers
under this group,
where
subgroup
The corresponding
all images
of
of the
(affine)
{L0,LI,L2,L3,L4]
136
L 2 = <½e0,½e],,e2,e3,f0,fl,f2,f3)vz 2
L3
< 2e 0, 2e I, 2e 2, 2 e 3 , f 0 , f l , f 2 , f 3 > z Z 2
L 4 = <ge0,½el,~e2,½f3,f0,fl,f2,e3>2Z
The vertices
Let
of
W0
A8
are
consist
of t h o s e
determinant
1 which
S)
to itself.
send
A
of the Weyl
group
Q(X
when
acting
in the n a t u r a l
+ 2A) = ½(k,k)
Set
C.5.3.
(i)
G
(iii)
W0
Proof.
taining
L I.
(ii).
is the
(i)
L 0.
Thus,
(In fact,
GF(2)-space
8
of t h e
W0
matrices
o__n_n A 8.
GL0
indicated
acts
W0
W1
form
orthogonal
r = diag(l,l,l,l,l,l,l,-l),
Thus,
Also,
8 X
(using
subgroup
the q u a d r a t i c
of
of
~ [½].
stabilizer
For,
By C.5.2,
L0/2L 0 ~ A/2A.
consists
is c h a m b e r - t r a n s i t i v e
of m a t r i c e s
(ii)
preserving
G = <W0,WI>.
G
matrices
transformations
8-dimensional
and
in the r i n g
orthogonal
W 0 m 2~+(8,2).
2).)
(ii)
consists
and
(mod
entries
L i-
is the c o m m u t a t o r
on the
manner,
of t h e s e
as l i n e a r
W0
E8,
8
r
W 1 = W0
Lemma
with
Then
images
8 ×
viewed
of t y p e
W0/<-I > ~ Q+(8,2),
A/2A
just the
2
on
L0
is t r a n s i t i v e
of
A
W0
type.
and
is t r a n s i t i v e
the c o n n e c t e d n e s s
and
induces
Q+(8,2)
on the c h a m b e r s
on the
chambers
(as a c h a m b e r
on
con-
containing
system)
implies
137
(±ii)
of
A
in
@. []
T h e Weyl g r o u p of t y p e
in the g r o u p of all
Now define
8 X
W4~ = G L
E8
8
for
is p r e c i s e l y
orthogonal
the s t a b i l i z e r
matrices
with
entries
0 ~ i ~ 4.
1
L e m m a C.5.4.
Proof.
in
W0
W0 ~ W 1 ~ W 3 ~ W4 ~
If
i = 3
of a t o t a l l y
W0 n W i
induces
totally
singular
or
4
singular
on
Li/2L i
subspace;
difficult
that only
and this
readily
In [ K a 6] a l i n e a r
in GL(8,~)
proof
of C.5.4,
transitive
G (m)
W0
That
showing
C.5.5.
Let
in
in
Wi
Also,
~+(8,2)
of a
If f o l l o w s
that
It is not
can i n d u c e
1
on
the lemma.
and
~
W3
is d e f i n e d
as well
transformation
that
G
has
as
that lies
W1
and
provides
another
an a u t o m o r p h i s m
group
m
be any odd
integer > l,
i n d u c e d by p a s s a g e m o d m
and let
(cf.
Let
(m)
(cf. C.I.6),
and let
system obtained
Theorem
(m)
w(m)
~ 8(m) = ~ (A 8(m))
be the c o r r e s p o n d i n g
chamber
as in C.I.4.
C.5.6.
PG(2,2)
universal
L0/2L0[
L i / 2 L i.
transformation
be the h o m o m o r p h i s m
C.5.3(i)).
X ,
on
is the s t a b i l i z e r
on { W 0 , W 1 , W 3 , W 4 ] .
Definition
G ~
~i
implies
W 2.
while
of
W 1 N W i.
~+(8,2)
and i n t e r c h a n g e s
W4, w h i l e n o r m a l i z i n g
1-space
so does
induces
Li/2Li,
W0 n Wi
the s t a b i l i z e r
<W 0 Q W i , W 1 Q Wi>
to check
then
2~+(8,2).
[ K a 61 .
residues,
covering
SCAB
A 8(m)
is a f i n i t e
chamber-transitive
A 8.
GAB with diagram
group
G (m)
I
and
138
Proof.
where
The map
O8 = ~(A 8)
G ~ G (m)
induces
(cf. C.I.4).
is a GAB by C.I.8.
By B.3.9,
Thus,
~8
(m)
a covering
~(m)
O8 ~ ~8
is a SCAB.
is its universal
'
It
covering
SCAB. []
In [Ka 6] it is noted that
prime
p, and that
G (p)
~[w(P
i ) I i = 0,1,3,4}
on
§C.4.
is.
has an automorphism
Keeping
G (m)
in mind
p.)
S Cl)
=
further
transitive
(This group
The above
GABs are No. (i) in
(m)
(m)/<
Aut A 8
: only
G
-i>
is not in
the fact that faithfulness
sential part of the construction
now define
group
can consist of inner automorphisms,
upon the prime
Of course,
for each odd
W 2(P)
and n o r m a l i z i n g
of graph automorphisms
depending
G (p) = Q+(8,p)
"subGABs"
is not an es-
of GABs via C.I.6,
of the above ones.
we can
(Notation:
G. )
Theorem
C.5.7
[Ka 6; Ka 7; KMW 4!.
is a GAB w i t h ~ the i n d i c a t e d ' diagram
(i)
F((G
Is); ~ ( W ) ( m )
[
u8
universal
covering
(ii)
via
and universal
I i = 0,2,3,4]),
coverinq
diaqram
•
SCAB:
<
,
u8
m = 1
(cf. ~c.4,
No.
(2)) ;
~ ( ( G u 7 , u 8 ) (m), { ((Wi)u 7 ,Us) (m) I i = 0,2,3} ), diagram
universal
(iii)
ii
i
Each of the followinq
covering
via
m = 1
(cf. ~C.4,
No.
V ( ( G e 0 ,f0) (m). { ((Wi)e 0 ,f0)(m) I i = 0,i,3,4~),
• Universal
coverinq via
m = 1
(cf. ~C.4,
No.
(3));
diagram
(5));
and
(iv)
diagram
No.
(4)).
~((G u
,
)(m)
7,Us,e0
; [ ((Wi)u7,US,e0
-, universal
coverinq
via
)(m) I i = 0 # 2 t 3 !~ ) s
m = 1 (cf. ~C.4,
139
In [ K a 6] there
•
is also
~ GABs c o v e r e d
These
are o b t a i n e d
automorphism
of
by the
SCAB
from
48
GABs
related
family,
whose
J
GAB w h o s e
m = 1
C.5.8
in [Li~
intrigling.
share
three
C.5.7(i)
GAB.
by
a triality
2-adic
universal
buil dinqs.
covering
(Wi)(m)li. = 0,2,33)
u8
covering
SCAB
of this
Let
C.5.9.
H
e0,f0,<el,e2,e3},
on the chambers
(el,e2,e3>.
GAB induces
idea
lemma
the u n i v e r s a l
that
f o u r types
SCAB of §C.4,
a general
Remark
concerning
The same
covering
situations
manner.
universal
of their
of a C.5.8
Moreover,
•
It is n o t e d
covering
versal
(6)).
is o b t a i n e d
but is not a buildinq.
The proof
is very
No.
C.5.8 [ K a 6; Li].
u7> ,u s
via
by using
are c o v e r e d
(m)
:
(§C.4,
of
a building:
Theorem
is a
building
(m)
G
and
of a family
P~+(8,@2).
is a closely
is not
G2(~ 2)
(m)
All of the p r e c e d i n g
There
a construction
the GABs
No.
in [Li~
SCAB
in C.5.7(i)
of vertices,
a covering
is u s e d
covering
and
and that
any
of the c o r r e s p o n d i n g
in [Li]
to obtain
(20) by m e a n s
may h a v e
the uni-
of C.5.7(iii).
applications
to other
sort.
The SCABs
in C°3.10
be the s t a b i l i z e r
and (fl,f2,f3>.
of the affine
The element
T
in
Then
SL(3,~ 2)
used
arise
G
in the f o l l o w i n g
of each of the following:
H
acts t r a n s i t i v e l y
building
in C.3.10
defined
on
can be v i e w e d
as
140
e I ~ e 2 ~ e 3 ~ el,
ment
of
H.
A
-8(m) "
inside
formation
further
Nos.
There
automorhism
group
the
Moreover,
(20)
and G A B s
accounts
manner
in C . 3 . 1 0
centralizes
and this
are
GABs
-
to an e l e -
are v i s i b l e
the o r t h o g o n a l
for
the u n i t a r y
transgroups
been
the
transitivity
.
found
"inside"
A~ m)
[KMW 4]:
(24).
groups
using
have
3-adic versions
<Wo,W~
PQ+(8,3).
H
in an o b v i o u s
in C . 3 . 1 0 ( i v ) .
Many
obtained
of the S C A B s
e. + ~ e . ,
i
1
appearing
§C.4,
All
and extends
having
same
The
3
W0,
of the G A B s
in C . 5 . 6 ,
orbits
on chambers.
a new
r,
a n d the
admitting
These
are
resulting
small
number
of c h a m b e r - o r b i t s
properties
of the
subgroup
~+(8,2)
is due
of
new
to
141
C.6.
Miscellaneous
problems
T h e r e are m a n y
or GABs w e h a v e
and GABs.
open p r o b l e m s
constructed
C.6.1.
All that
are s e v e r a l
3 sporadic
universal
is k n o w n
reasons
covering
fields [BT].
in m o r e
(As a n o t h e r
instance
affine buildings,
7] b e t w e e n
(somewhat
(i0) and
c o v e r e d by
to h a v e
C.5).
as i n d i c a t e d
a relationship
over
covering
related
a
the SCABs
defined
(~§C.3,
somehow
the
into t h e s e groups.
(ii) and c e r t a i n
SCABs h a v i n g
or C.6.3
on
local
SCABs
Others
after C.5.8.
to c l a s s i c a l
was noted
in
5- and 6 - d i m e n -
covering
SCABs
chamber-transitive
below).
There
should
groups
is also the f o l l o w -
(compare C.6.3):
Under w h a t
a building
wise nonisomorphic
that
There
in ~C.4,
between
affine buildings
of the u n i v e r s a l
finite
C.6.2.
covering
groups.)
knowledge
ing r e l a t e d q u e s t i o n
Problem
appearing
it w o u l d be d e s i r a b l e
of e x a m p l e s
as in § C . 5
in §C.4.
a few of the u n i v e r s a l
w e remark
~os.
produce many more
the u n i v e r s a l
In the cases of the
MC,LyS)
i n d i r e c t manners,
3-adfc u n i t a r y
Finally,
C.6.1.
to be such b u i l d i n g s
are r e l a t e d
SCABs
some of t h e s e p r o b l e m s .
is l i s t e d
of the r e l a t i o n s h i p s
Of course,
the SCABs
of o t h e r
SCABs m a y g i v e n e w i n s i g h t
"classical"
are a l r e a d y k n o w n
sional
(Suz,
of the cases
and the
either
in §C.4.
for s t u d y i n g
understanding
the list
discuss
determine
about C.6.1
s~mple groups
In the m a j o r i t y
[Ka
or the e x i s t e n c e
Explicitly
of m o r e of the SCABs
better
concerning
In this s e c t i o n w e will
Problem
SCABs
and examples.
circumstances
also be c o v e r e d by
finite
SCABs
will
infinitely
(or even G A B s ) ?
a finite
SCAB
many pair-
142
This
is r e a l l y
of a SCAB
mations
~
of the c o v e r i n g
groups).
many
C.6.2
group defined
quotient
C.6.3 [Ti
over
are m a n y
&/H
covers many
~/H
discrete
(§C.2)
GABs.
Aut
(£/H)
only r a r e l y w i l l
n = 3, 4 or 6, a r i s e s
Examples
restriction
of C . 6 . 3
assumed
For an e x a m p l e
We n e x t
in order
Buildings
F =
structural
especially
(V,~,~):
index
G
If
in
for
and subg r o u p has
of GABs :
algebraic
such that the
H
is such
Property
small;
finite
(C.6.4
below).
in p a r t i c u l a r ,
of any type.
in i n f i n i t e l y m a n y
fields
a
"small"
generalized
in C.3.10
4.
H, so that
on v e r t i c e s
to g u a r a n t e e
have various
only s t a t e o n e
of a D - G A B
0
in c h a r a c t e r i s t i c
consider
of
is s u f f i c i e n t l y
classical
to c h a r a c t e r i s t i c
H
D-GAB.
very
can be seen
22
and a f f i n e b u i l d i n g
Intersection
as r e s i d u e s
transfor-
f i e l d of c h a r a c t e r i s t i c
D
H
is g e n e r a l l y
every
be a simple
subgroups
the
coverings
large numbers
G
compact
it b e t r a n s i t i v e
In p a r t i c u l a r ,
will
Let
if
cf. [Ro
the f u n d a m e n t a l
cover
of f i n i t e
Moreover,
of deck
group
images.
is a f i n i t e
even s a t i s f i e s
However,
§C.4.
6].
a locally
so is any s u b g r o u p
w a s only
between
rank m 3, affine d i a g r a m
Then t h e r e
(B.3.6);
asks w h e t h e r
affine buildings
Construction
then
G ~ C
finite homomorphic
"Classical"
0, h a v i n g
SCAB
the f u n d a m e n t a l
to the g r o u p
type of c o r r e s p o n d e n c e
Namely,
infinitely
concerning
(which is i s o m o r p h i c
the s t a n d a r d
group,
a question
GABs.
and C.5.6.
is not
n-gon,
The
essential:
that a group
H
it
exists.
2, see C.3.11.
properties
"intersection
geometric
of t h e GABs
properties."
property
in
We
of this sort
143
Intersection
Property
[v E V I v T = i] .
C.6.4.
if
v E V - Vi
if
v E V i.
Let
i E I
let
and
Vi =
I'i(v) = l~(v) ~ Vi,
and
t
let
l~i(v) = Iv]
~'i (u) N l~i(v)
is e i t h e r
for s o m e
F
flag
are flags,
just
Problem
Property?
C.6.5.
some
special
(3)
for
m = 3,
t i o n of
GAB
For
[Bu:
Ti
being
of t h e
(6)
for
certain
U F
is a flag]
U F
and
Iv]
U F
u ~ v.
satisfy
have
C.6.4
the
[Ti
2,
7],
Intersection
of s p e c i a l
cases
conditions[).
of § C . 4
is it k n o w n
Intersection
m = 3,
(7),
(8),
nice
(9),
or
holds
(i0)
in w h i c h
vector
of
whether
Property
situations
in the p r o o f
for
any
space
(Nos.
m = 2,
c a s e of
realiza-
this p r o p e r t y ,
with
subspaces.
discussion
are u s u a l l y
is i s o m o r p h i c
2,
to
7].
Various
apartments
in GABs
as
(e.g.,
3; Ro 3] ).
[Ka
be defined
can b e
u , v 6 V,
of the
Intersection
Property
see
7].
of w h i c h
cover
for
in t e r m s
is an e s p e c i a l l y
involved
further
Buildings
see [Ti
geometric
instances
[u]
in § C . 4
in m o s t k n o w n
there
the
GABs
case
Moreover,
C.6.4 holds
both
buildings
Which
any
[x E V I Ix]
characterizations
in a few
not
for
l~i(u) ~ F i ( v )
( or any o t h e r
Only
or
for w h i c h
indicated,
Obtain
of C . 6 . 4
vertices
l~
l~ioreover,
As w e
(13)).
of
~
Then,
defined
@(D)
found
only
However,
for r a r e
have
embedded"
It is n o t
-- and u n d o u b t e d l y
all of §C.4.
= @(W(D)/I,
attempts
"nicely
in t e r m s
clear
there
it m a y
classes
of a p a r t m e n t s ,
{<ri)
been made
morphic
how
I i E I] ) -to d e f i n e
images
"apartments"
is no d e f i n i t i o n
be that usable
of GABs,
each
of
~(D)
should
that
can
definitions
and t h a t
characteri-
144
zations
than
can be o b t a i n e d
buildings)
Another
either
of
property
defined
graph
Both of t h e s e n o t i o n s
IWI).
few additional
is known.
examples
(e.g.,
The examples
Problem
C.6.6.
if
of d i a m e t e r
has been
in [ K a
in C.6.3
3]).
suggest
Characterize
(other
is the d i a m e t e r
~
those
--
happens
are v e r y
(where the c h a m b e r
The s e c o n d n o t i o n
GABs
"apartments."
~
in the case of b u i l d i n g s
diameter
interesting
of a S C A B
or of a s u i t a b l y
to be a GAB.
priate
in terms of s u i t a b l e
natural
~,
for e s p e c i a l l y
appro-
s y s t e m has
considered
However,
for a
very
little
the f o l l o w i n g
SCABs of very
small
diameter.
Finally
most basic
we r e t u r n
one is s u g g e s t e d
generalized
polygons
Problem
polygon
C.6.7.
occurs
desarguesian
arose
projective
likely
residue
the
eyery f i n i t e
GAB.
Prove this
cases of this i n v o l v i n g
planes
generalized
--
either non-
or some of the g e n e r a l i z e d
quad-
in §A.3.
For example,
in a
Another
ence sets.
Probably
that
in some f i n i t e
The case of n o n d e s a r g u e s i a n
esting.
Probably
by the fact that o n l y c l a s s i c a l
It s e e m s
special
questions.
in C.6.3:
as a r e s i d u e
or at least p r o v e
rangles
to e x i s t e n c e
/ ~
seems
can each t r a n s l a t i o n
especially
plane occur
inter-
as a
GAB?
intriguing
By C.3.1,
these lift to
representation
planes
special
case of C.6.7
each d i f f e r e n c e
finite
theoretic
GABs
methods
involves
set p r o d u c e s
(cf. C.6.2).
[Lie;
Ot 17 m a y
//~
differSCABs.
At this point,
apply:
145
Problem
C.6.8.
Study
planes
are d i f f e r e n c e
planes
of
admits
a sharply
The
each
p = n
n + 1
p.
rank
prove
C.6.8
results
finite
this
SCABs with
let
V
as a set
of
residual
or
8)
q + 2
in the
group
latter
of
case,
GF(p).
conclusions
that
are:
In o t h e r w o r d s ,
asserts
by c o n s t r u c t i n g
that planes
Let
O
subspaces
the only known
residues.
q >
3-dimensional
2
be a p o w e r
vector
space
be a h y p e r o v a l
of d i m e n s i o n
in
of
2,
over
V
1 I cf.
£ V
GF(q)
(A.3.10)).
= [T~ I i ~ j ~ q + 2~
G = [ (Vl,.--,Vm,J) I v
let
(viewed
Write
O~
desar-
of o r d e r
group
be classical."
section
let
that
are d e s a r g u e s i a n .
C.6.10.
and
1~
general
whose
of t h e o n e
8]
is e i t h e r
GF(p)*
additive
theorems
must
of
if
Frobenius
that,
residual
group.
fact [Ka
plane
the m o r e
nonclassical
be a
1 ~ ~ ~ m,
the
to t h e
projective
in the
Prove
3-space
Construction
m ~ 2,
refers
suggests
reminiscent
We c o n c l u d e
that
2
whose
automorphism
the subgroup
set
residues
of a p r o j e c t i v e
for
then
C.6.9.
2
1~
(of o r d e r
It is w e l l - k n o w n
is a d i f f e r e n c e
Problem
GABs
a sharply flag-transitive
+ n + 1
Of c o u r s e ,
"all
of C . 6 . 8
flag-transitive
degree
2
Show
chamber-transitive
or a d m i t s
of p r i m e
A _
set p l a n e s .
are desarguesian
last part
finite
guesian
if
F
finite
, 1 ~ j ~ q + 2]
146
Call
iff
c =
and
(Vl,...,Vm,J)
c' =
'
(Vl,..
.
"~ - a d j a c e n t "
'
,v m,j)
either
1 ~ ~ ~ m
th
and
component,
= m + 1
Theorem
Proof.
v I E Vl,
easy
j ~ q + 2,
at m o s t
for
th
1
the
~
residues
subscripts
~
arbitrarily
ordered
that
at m o s t
then
in t h e
the same
Corollary
is a r a n k
are
q - i, q + i.
of
Z[~,$]
~(~jT~/I,[T~,T~]
identified
with
the pairs
i-adjacence
components
).
the generalized
(Vl,T~ + v I)
3
corresponding
differ.
~
is a G A B by
the q u a d r a n g l e s
the quadrangles
in
to
Finally,
it is
C.I.7
Note
residues
C.6.12.
m + 1
and
o__rr 16
chamber-transitive
and quadrangl e residues
O~,
numbers
with
(A.3.10). C7
and
are
all
since we have
the c o n s t r u c -
of n o n i s o m o r p h i c
and the same
.
for d i f f e r e n t
Moreover,
numbers
I_~f q = 4
~ [ ~ , m + i]
occurring
e a c h of the h y p e r o v a l s
enormous
C.6.11
SCAB with diagram
each member
2-gon
chambers
with
m + 1
c a n be n o n i s o m o r p h i c .
tion will produce
Proof.
or
have parameters
can be
m = 2,
1 ~ ~ ~ m,
but
i
in t h e
is c o n n e c t e d .
When
isomorphic,
there
is a r a n k
~[l,m, + i]
Remarks.
having
~
A.3.10 whose
1 ~
at m o s t
disagree
to t h e g e n e r a l i z e d
to s e e t h a t
that,
c'
1 ~ ~ < 8 ~ m
of
quadrangle
disagree
- v ' ~ E T ~,
j
v
and
quadranqle
is i s o m o r p h i c
c'
component.
If
Each member
c
C.6.11.
whose
and
while
and
(m + I) st
having
c
SCABs
of c h a m b e r s .
m ~ 2
SCAB having
parameters
then
diagram
q - i, q + I.
147
There
For
are o t h e r
example,
GF(q),
q
let
even
a hyperoval
be
and
in
V,
mutations
of
free
i ~ j.
for
V
variations
a
q >
3-dimensional
2,
let
and let
adjacent"
Then
~
iff
they
is a g a i n
and q u a d r a n g l e
Of
course,
residues
m ~ q + 2
the
~i'
for
each
i.
As
another
the s i m p l e s t
p,
let
x
a point
q
the family
write
O~
be
of
6
(T~)v I.
j ~
~2
let
to
subgroups
define
with
of o r d e r
of an o d d p r i m e
space.
q
This produces
a SCAB
iff
and let
j = j'
as b e f o r e .
and
Let
appearing
(v I .... ~Vm,J)
m + 1-adjacent
for
i-i
~i = ~2
G ~ PSp(4,q)
symplectic
c =
<
choices
and
)] = {~j I 1 ~ j ~ q + 2],
time
to be
are m a n y
be a p o w e r
O p ( G x)
"j-
q - I, q + i.
q + 2-cycle
q
4-dimensional
q + 1
This
a
per-
component.
diagram
parameters
There
~
th
3
the
SCAB with
time.
has
isomorphic
= ~ U [Z(V
(v i .... ,v~,j')
this
m ~ 2
be
is f i x e d - p o i n t -
ele~ments of
in at m o s t
having
over
+ v ..... Ti~ m + v,v) I
two
m + 1
variation,
of the
as in C.6.10.
v
a rank
be
u i o j-I
that
~ = { (Tlol
disagree
space
O = [Tj I i ~ j < q + 23
such
1 ~ i s q + 2, v 6 V] , and call
construction.
vector
~I = i'~2 .... '~m
{1 ..... q + 2]
Write
on t h e p r e c e d i n g
and
and
~
be
in A.3.4,
~
be
c' =
148
C.7.
Connections with
T h e r e has been
towards
studying
f i n i t e g r o u p theory.
an i n c r e a s i n g
amount
and c h a r a c t e r i z i n g
finite groups
transitively
(and f a i t h f u l l y )
effect,
a n e w v i e w of the t h e o r y
been
more precisely,
Ti 2, ~13].
The goal has g e n e r a l l y
v a l u e to
improving
the a p p r o a c h e s
groups.
additional
"revisionism":
hypotheses
In o r d e r
to h a v e
to o b t a i n
the p r o c e s s
theoretic
theoretic.
a brief
summary
incomplete~
in p a r t
a firm h o l d
result
or,
[Cu;
results
of r e w o r k i n g
of
and
of f i n i t e
results
nature,
in
involve
and m o s t
Therefore
in this
of this research.
groups,
it is
on the case of buildings.
is due to
Seitz.
the f o l l o w i n g
from [KL],
[Carl
relations
to deal w i t h c h a m b e r - t r a n s i t i v e
the f u n d a m e n t a l
quoted
been
chamber-
This has,
of B N - p a i r s
some of t h e s e
group
only give
first n e c e s s a r y
was
or GABs.
directed
acting
in the c l a s s i f i c a t i o n
of a g r o u p
are h i g h l y
section w e will
of [Se]
used
Unfortunately,
of the p r o o f s
Here,
on SCABs
of the C u r t i s - S t e i n b e r g - T i t s
potential
simple
of r e s e a r c h
(The
statement
where numerous
original
of his r e s u l t
variations
version
is
can also be
found.)
Theorem
rank
Let
~ 2
K
C.7.1 [Se].
having
be a g r o u p
Let
a connected
(ii)
(iii)
be a
diagram
of t y p e - p r e s e r v i n g
c h a m b e r l t r a n s i t i v e l [ o n_n ~.
(i)
G
IKI = 73-9,
g r o u p of L i e t[pe o_f_f
and c o r r e s p o n d i n g
automorphisms
G = PSL(3,2);
G = PSL(3,8);
building
acting
Then one of the f o l l o w i n g
K ~ G;
IKI = 7-3,
finite
holds:
~.
~49
(iv)
K
is
A7,
inside
PSL(4,2)
(v)
K
is
A6,
inside
PSp (4,2) ;
(vi)
K
is a s e m i d i r e c t
group
o__ffo r d e r
16
o_~_r 32
of o r d e r
20,
inside
(vii)
K m
PSL(3,4)'2,
(viii)
However,
note
that
with
(ii)
the
C.7.2.
(other
than generalized
that
following
(ii or
member
ably
diagram
in [ T i m
section.)
case
of g r e a t e s t
in ~§C.3,
of this
section
G
be a f i n i t e
D,
all of w h o s e
2-gons)
~
may
[Tim
iii)
n
occur
2] for
are
1
"be"
D
(6,3);
interest.
C.5.
we will
be con-
a GAB,
D
2 residues
of L i e
r = 3, t h e n
~.
is
_/~
o n e of
and g r o u p s
planes,
4.
possibility
(The l a t t e r
diagrams.
be
type.
of
on the r e s i d u a l
that will
r a 3
by C.I.4.
is linear,
for n o n l i n e a r
reasons
rank
of a u t o m o r p h i s m s
if
size
S C A B of rank
from groups
that,
induced
has
arise
group
2~ s h o w e d
occurs:
of s o m e
cannot
~_ A u t P ~
o_r_r
(v) o c c u r r e d
be a c h a m b e r - t r a n s i t i v e
Timmesfeld
C.7.1
g rgup
2F4(2).
is the
Let
a connected
the
or a F r o b e n i u s
AutPSU(4,3)
G2(2) ;
remainder
having
Recall
$5,
abelian
the f o l l o w i n g
~ypothesis
G
an e l e m e n t a r y
~ AutP~(5,3);
inside
-
of
A5,
AutPSp(4,3)
C.7.1(i)
Throughout
Let
with
K = 2F4(2)' , i n s i d e
Of course,
cerned
product
K = G2(2)' , i n s i d e
(ix)
~ P~+(6,2);
It w a s
explained
not
at the
or s o m e
prob-
eliminated
end of this
150
Timmesfeld
also c l a s s i f i e d
tiple edges [Tim i].
all d i a g r ~ s
He showed that
~
he determined
all of the groups
it was n o t e d that
cases
~.
~
we saw e x ~ p l e s
G
an~
and X
~
are u n i q u e l y
~
in these
G
form
f
C.5.)
Moreover,
in [ Ka 6, 7~
in the
Fo~
approach.
and h e n c e has been d e t e r m i n e d
Proceedings).
Some m a n i p u l a -
G ~ Aut ~
(up to conjugacy).
is a finite h o m o m o r p h i c
a suitable quadratic
or X
J ~ I.
and the lifted g r o u p
determined
, ~
could also be c l a s s i f i e d
cides w i t h the a p p r o p r i a t e group
Now
for
, using the f o l l o w i n g
(see his p a p e r
tion shows that
_/~k_
in ~§C.3,
Pj
This is a b u i l d i n g by B.3.8,
by Tits
h a v i n g no mul-
must be a building,
except in the case of diagrams of the form
(In the latter cases,
D
Moreover,
in C.5.3(i)
image of
(cf. B.3.7)
G
coin-
or C.5.7(iii).
G = P~(~[½],f)
for
(compare this w i t h the last
column
in the table in §C.4).
known,
in v i e w of the recent work of Prasad on the C o n g r u e n c e
Subgroup
Probl~
above a p p r o a c h
[Pr~.
All such h o m o m o r p h i s m s
(N.B. - It is also p o s s i b l e
to p r o v e the a f o r ~ e n t i o n e d
are now
to use the
result in [Tim i].
One first reduces to the case of affine d i a g r ~ s
of r ~ k
Various
Aut ~,
groups
Pj
lift to finite subgroups
~aS] can then be applied to greatly
Pj
and b u i l d i n g s
Ex~ple
paragraphs)
present
of
~.)
C.2.4
(which we will call
is clearly both b e a u t i f u l
context.
Several
and a n u i s a n c e
in § C . 4 show that
rank
o c c u r r e n c e in a GAB is p r e s e n t l y known).
showed that,
4
listings
'~7" in the next few
in rank
Recall
SCABs and even rank
that we are assuming C.7.2.
if
and
restrict the p o s s i b l e groups
reappear
5
> 3.
~
is a SCAB h a v i n g
4
GABs
in the
~7
can
(although no
Aschbacher
a linear d i a g r ~
[ A 2~
and an
151
~Z7
residue,
then
~
that o n l y b u i l d i n g s
Along
then
D
can o c c u r
Stroth [St
if
D
2-gons,
!] s h o w e d
or
PG(2,2) 's
it f o l l o w s
is spherical.
that,
2
residue, and all rank
can be f o u n d in § C . 4
first w h e n
r = 3
also o b t a i n e d
there
d7
In v i e w of C.2.3,
if
residues
Sp(4,2)
quadrangles,
is one of the f o l l o w i n g :
Examples
has
CZ7
lines,
an
are g e n e r a l i z e d
~Z7.
and
the s a m e
is a G A B h a v i n g
is
is an
or
4,
further
Sp(6,2)
for s o m e of t h e s e diagrams:
and the last three.
results when
residue,
Stroth [St
there
and all rank
the
is no
2
~7
residues
2]
residue,
are as
before.
The p r i n c i p a l
ations
in w h i c h
focus
in [ T i m
all rank
2
One r e a s o n
for this
techniques
can b e a p p l i e d
the i n t e n d e d
to g r o u p s
provided
1,2]
residues
is that a v a r i e t y
have
of c h a r a c t e r i s t i c
type.
is on situ-
characteristic
Another
applications
2
1,2]
of r e p r e s e n t a t i o n
in that case.
group theoretic
by a b e a u t i f u l
and [St
reason
theoretic
is that
of t h e s e r e s u l t s
An a d d i t i o n a l
r e s u l t of N i l e s [Ni]
2.
is
reason
w h i c h w e will
is
now
describe.
Definition
C.7.3.
in a f i n i t e g r o u p
groups
G
of
distinct
(i)
i
G
A parabolic
is a f a m i l y
s y s t e m of c h a r a c t e r i s t i c
{PI,...,Pr}
such that the f o l l o w i n g
and
j:
G = <PI ..... Pr >;
conditions
of
r ~ 2
p
sub-
h o l d for all
152
r
~P
1 1
(ii)
contains
a Sylow
p-subgroup
of
each group
P{ i, j] = {Pi'Pj >;
(iii)
1
0 p' ( P i / O p ( P i ) )
group
of Lie
(iv)
of
Li
2
and
(N.B.
normal
t y p e of
group
Lj
- If
T
subgroup
be c l e a r
7
that
where
In fact,
t y p e or
is a g r o u p
which
T
is n o t
(i) -
(iv)
are v e r y
a given
chamber
is a S C A B
group
obvious
system
from
of the p r o j e c t i o n s
in
in the
G
It s h o u l d
context
of
of t h a t m e m b e r
in C.7.3.
of
Namely,
let
and w r i t e
P* = P B.
1
1
Then
is a SCAB.
has
is the s m a l l e s t
c E ~.
in C . 7 . 3 ( i i ) ,
(S) I i ~ i < r>
and
l
= G ( G / B , [ P* I 1 ~ i ~ r] )
course,
extension
to be d i r e c t .
natural
implicit
B = {Np
each p a r a b o l i c
a rank
p'-group.)
assumed
is the s t a b i l i z e r
there
is a c e n t r a l
oP' (T)
is a
(iv)
Pi
of
and
the p r o d u c t
then
in
be the S y l o w
that
p;
extension
Lij.
modulo
containing
1
characteristic
of L i e
into
The product
C.7.2,
is a c e n t r a l
O p' ( P [ i , j ] / O p ( P { i , j ] )) = Lij
a rank
of
= Li
In p a r t i c u l a r ,
a diagram
D
(which
the d e f i n i t i o n )
and produces
of N i l e s
following
is,
we
see
of
an i n s t a n c e
of C.7.2.
The main
Theorem
system
in
(a)
Z(Lij),
G
result
C.7.4
such
[Nil.
is t h e
Assume
that
[PI,...,Pr]
is a p~arabolic
that
Each product
appearing
in
C.7.2(iv)
is d i r e c t
modulo
153
(b)
~__oo Li/Z(Li)
PSU(3,2),
Sz(2)
(c)
Then
N0_o Lij/Z(Lij)
G
contains
The p r o o f
has a
is isomorphic
is a b u i l d i n g
the c o r r e s p o n d i n g
in [Ni]
BN-pair.
unfortunate
to
PSL(2,2),
PSL(2,3),
o__~r 2G2(3) , and
~ ( G , [ P 1 ..... Pr])
by
G
is isomorphic
is elegant
However,
to
PSL(3,4).
and the group induced on
Chevalley
and short:
assumptions
qroup.
it is shown that
(a) - (c)
are both
and at least somewhat necessary:
Examples where
C.7.4
fails w i t h o u t
one of
(a) - (c).
We refer to the table in §C.4.
Li/Z(L i)
is
Li/Z(L i)
is
In No.
However,
PSL(2,2)
in m u c h of the table.
PSL(2,3)
(13), with diagram
LI3 = SL(2,9)
in Nos.
~
= SL(2,5)-SL(2,5)
Thus,
case
b e h a v e properly).
and
examples w i t h
Problems.
No.
L23
p = 2
the theorem
in § C.4
Eliminate
occurs
"barely"
Eliminate
in w h i c h both
the r e s t r i c t i o n
the r e s t r i c t i o n s
(17).
(c) hold.
as a p r o d u c t
fails in this
There are many other
(a) and
C.7.4(c).
(13) in terms of the failure of condition
p ~ 5.
(16),
, (b) and
that is not direct.
(LI2
(9) - (ii),
(b) fail.
Characterize
C.7.4(a)
with
Li/Z(L i) ~ PSU(3,2),
Sz(2),
in C.7.4(b).
Note that, when the diagram of C.7.3
is connected,
cannot occur in C.7.4 since it never appears
in any
2G2(q)
Lij.
154
The r e s t r i c t i o n s
avoided
They
in the r e s u l t
also p r o v i d e
in C.7.4(b)
of T i m m e s f e l d
of C.7.2,
1.,2; St 2]).
bility
GF(3)
was
2] m e n t i o n e d
for f o c u s i n g
earlier.
on the c a s e
2.
It is not clear that C . 7 . 3
[Tim
[Tim
further motivation
of c h a r a c t e r i s t i c
situation
explain why
although
this
seems
likely
holds
in the
(compare
we n o t e
that t h e r e
is also a p o s s i -
that m o r e of the e x c e p t i o n a l
situations
in C.7.1
i n d u c e d on r e s i d u e s
is c l e a r l y
Finally,
automatically
in c h a m b e r - t r a n s i t i v e
of interest,
at least
from
SCABs.
a geometric
can be
This s i t u a t i o n
p o i n t of view.
155
References
[A I]
M. Aschbacher, Flag
14 (1983), 21-32.
[A 2]
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[AS]
_ _
and S.D. Smith, Tits g e o m e t r i e s over
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GF(3).
Comm. in Alg. 11 (1983), 1675-1684.
[ASz]
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EBCT]
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N.L. Biggs, A l g e b r a i c
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F. Bruhat and J. Tits,
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[Bu]
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P.J. Cameron,
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_ _
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R.W.
[CS]
J. C h i m a and E. Shult,
[Co]
B.N. Cooperstein, A finite
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[Cu]
C.W. Curtis, Central e x t e n s i o n s of groups
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,
W.M. Kantor and G.M. Seitz,
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[De]
P. Dembowski, Finite
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[FH]
W. Felt and G. Higman, The n o n e x i s t e n c e
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W. Haemers, E i g e n v a l u e techniques
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[HaR]
_ _
Geom.
Carter,
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graph
Partial
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Ded.
flag t r a n s i t i v e
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on Tits
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Groupes r@ductifs sur un corps local.
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Quart.
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Wiley,
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