ADDITION.
On p. 294, at the end of the introduction to Chapter 10, add: `A recent
monograph on Coxeter groups is
.SC Humphreys
.[[
Humphreys 1990 reflection Coxeter groups
.]]'.
ADDITION.
On p. 297, Remark (iii), it is easy to verify that the
result of Hemmeter mentioned remains true without the
restriction `either $m <= 7$, or' (for noncomplete
half dual polar graphs).
ADDITION.
On p. 298, before Lemma 10.1.8, add `see also
.SC Shi Jianyi
.[[
Jianyi 1990 Bruhat order
.]]'.
CORRECTION.
On p. 306, in the proof of Theorem 10.2.10, change twice `10.2.1 (i)'
into `10.2.2 (i)' and in the one but last line of that proof,
change `S sub m-1' into `S sub m-2'.
CORRECTION.
On p. 309, line 20:
There are five, not four possibilities: also
$I sub { 2, "{"1,2"}" } sup m$, giving a $2m$-gon.
ADDITION.
On p. 313, after the description of the root system of type $E sub 7$,
add: `The 56 vectors of minimal norm $3o2$ in $E sub 7 star$ are
$1 over { sqrt 2 } "{" +- e sub i +- e sub j +- e sub k vb
ijk ~ roman "is a line" "}"$'.
CORRECTION.
On p. 313, line $-1$, the trace function should be defined
by $t( alpha + beta tau ) = 2 alpha$ $( alpha , beta mo Qq )$.
ADDITION.
On p. 314, line 4, add: `Eliminating the $tau$'s this yields
$PHI = 1 over { sqrt 2 } "{" +- 2 e sub i ,
+- e sub i +- e sub j +- e sub k +- e sub l vb
1 <= i,j,k,l <= 8 ,~ ijkl ~ roman "a block of" ~ S(3,4,8) "}"$'.
CORRECTION.
On p. 314, line 6, change `$v sub 1 mo PHI sub 7$'
into `$v sub 1 nm PHI sub 7$'.
ADDITION.
On p. 315 add at the end of Section 10.3.9:
.SC `M\*(:uhlherr
.[[
M\*(:uhlherr Coxeter 1993
.]]
describes quotients of Coxeter diagrams with respect to an
admissable partition; one particular example of his results is the
inclusion $W(H sub 4 ) ib W(E sub 8 )$.'
CORRECTION.
On p. 321, Theorem 10.4.11, delete `$!= H sub 4$'.
CORRECTION.
On p. 328, lines $-20$, $-18$: replace `$A sub n$' by `$A sub n,1$'.
CORRECTION.
On p. 332 in the subsection Incidence graphs:
Again there are five, not four cases, the fifth being
$I sub {2, "{"1,2"}" } sup m$ ($m = 6,8$), corresponding
to the incidence graphs of generalized $m$-gons.
ADDITION.
On p. 340, at the end of Section 10.7, add a subsection describing
the construction of the known generalized hexagons and of the known
generalized octagon of order (2,4). On p. 336, sentence below the Table,
add: `see below'.
.LP
.B
Construction of the known generalized hexagons
.LP
Set $F = Ff sub q$.
We define the \fICayley algebra\fP $O$ \fIover\fP $F$ as the set of matrices
.EQ
x = left ( matrix { ccol { alpha above b} ccol { a above beta }} right )
.EN
where $ alpha , beta mo F $ and
$a, b mo F sup {3}$.
We provide $O (F )$ with the usual (entrywise) vector space structure
over $F$ and with the following multiplication:
.EQ I
x y = left ( matrix {
ccol { alpha above b}
ccol { a above beta }} right )
left ( matrix {
ccol {alpha sup prime above b sup prime}
ccol {a sup prime above beta sup prime }
} right )
= left ( matrix {
ccol { alpha alpha sup prime - a cdot b sup prime
above
alpha sup prime b + beta b sup prime + a times a sup prime }
ccol { alpha a sup prime + beta sup prime a + b times b sup prime above
beta beta sup prime - b cdot a sup prime }
} right )
.EN
where $a cdot b$ and $a times b$ are the usual standard
inner and exterior product on $F sup 3$.
It is clear that $O $ is a non-associative algebra over $ F $ and
with unity $left ( matrix { ccol { 1 above 0} ccol { 0 above 1}} right )$.
The
quadratic form $Q : O -> F$ given by
.EQ
Q(x) = alpha beta + a cdot b
.EN
is obviously non-degenerate and of Witt index 4.
It turns $O$ into a composition algebra,
that is, it has the following property:
$Q(x y) = Q(x)Q(y)$ for all $x,y mo O $.
.LP
The algebra $O$ is uniquely determined up to isomorphism
by its dimension 8, the Witt index 4 of the quadratic form,
and the above identity, cf.
.SC Springer
.[[
Springer Oktaven lecture notes 1963
.]].
We shall work with the projective space $fs P (O)$
of $O$. If $ x mo O $, $x != 0$ let
$<< x >> $ denote the point of $fs P ( O )$ determined by
$x$, i.e., the subspace $ F x$ of the vector space $O $.
.LP
Let $H$ be the set of points
$<< x >> $ of $fs P ( O )$ such that $x sup 2 = 0$.
Then $H$ is the intersection of the quadric $Q(x) = 0$ with
the hyperplane $trace (x) = 0$.
Let two points $<< x >> , << y >> $ of $H$ be adjacent
whenever $xy = 0$. Then the graph induced on $H$ is the collinearity
graph of a generalized hexagon of order $(q,q)$.
Given distinct adjacent $<< x >> , << y >> mo H$, the line on these two
points is the maximal clique
$"{" << z >> vb z mo << x , y >> , z != 0 "}"$ of $H$.
Details of this construction are to be found in
.SC Schellekens
.[[
Schellekens Indag hexagonic 1962
.]].
The same paper contains a twisted version of this construction
for the generalized hexagons of order $(q sup 3 ,q)$,
related to $"" sup {3} D sub {4} (q)$.
Suppose $F= GF(q sup 3 )$ and $sigma $ is a generator of the
Galois group of $F$ over $GF(q)$. Consider the algebra
$O sub {sigma}$ whose underlying vector space
coincides with that of $O$, and whose
multiplication $cs$ is given by
.EQ
x cs y = {x sup {sigma }} bar {y sup {{sigma } sup {2}}} bar =
{(( y sup {sigma } x) sup sigma )} bar ~~~ (x , y mo O sub {sigma } ) .
.EN
Here $sigma$ is an automorphism of $O $ obtained
by naturally extending the above Galois automorphism
indicated by the same symbol, and written in the exponent,
and $x bar$ is the usual antiautomorphism of $O$, i.e.,
.EQ
x bar =
left ( matrix { ccol {beta above - b } ccol { - a above alpha }} right ) .
.EN
.LP
Now let $H sub sigma$ be the set of all
$<< x >> mo P$ with $x cs x = 0$,
and call $<< x >>$ and $<< y >>$ adjacent if
$x cs y = 0$. Then adjacency is a symmetric relation and
the resulting graph is the collinearity graph
of a uniquely determined generalized hexagon
of order $(q sup 3 , q)$. Given
distinct adjacent $<< x >> , << y >> mo H sub sigma$, the line on these two
points is the maximal clique
$"{" << z >> vb z mo << x , y >> , z != 0 "}"$ of $H sub sigma$.
.B
The generalized octagon of order (2,4)
.LP
The Tits system of a Chevalley group gives rise to
relatively natural presentations (i.e., descriptions
in terms of generators and relations) of the group.
In the special instance of the group $F = "" sup 2 F sub 4 (2)$
of order 35942400 such a presentation (cf.
.SC Tits
.[[
Tits 1964 algebraic abstract
.]])
has the following shape.
Here, by $[x,y]$ we denote the commutator $x sup {-1} y sup {-1} x y$.
.EQ I
F:
.EN
.EQ I
{roman "generators:"} ~~~
u sub 1 , u sub 2 , u sub 3 , u sub 4 , u sub 5 ,
u sub 6 , u sub 7 , u sub 8 , v sub 1 , v sub 2 ,
v sub 3 , v sub 4 , v sub 5 , v sub 6 , v sub 7 ,
v sub 8 , r sub 1 , r sub 8 ;
.EN
.EQ I
{roman "relations:"} ~~~
u sub 1 supr(4) = u sub 3 supr(4) = u sub 5 supr(4) =
u sub 7 supr(4) = v sub 1 supr(4) = v sub 3 supr(4) =
v sub 5 supr(4) = v sub 7 supr(4) = 1 ,
.EN
.EQ I
u sub 2 supr(2) = u sub 4 supr(2) = u sub 6 supr(2) =
u sub 8 supr(2) = v sub 2 supr(2) = v sub 4 supr(2) =
v sub 6 supr(2) = v sub 8 supr(2) = 1 ,
.EN
.EQ I
[u sub 1 , u sub 2 ] = [u sub 1 , u sub 5 ] =
[u sub 2 , u sub 4 ] = [u sub 2 , u sub 6 ] = 1 ,
.EN
.EQ I
u sub 2 = [u sub 1 , u sub 3 sup {-1} ] , ~ ~
u sub 3 supr(2) = [u sub 1 , u sub 4 sup {-1} ] , ~ ~
u sub 4 u sub 6 = [u sub 2 , u sub 8 sup {-1} ] ,
.EN
.EQ I
u sub 3 supr(2) u sub 4 u sub 5 supr(2) =
[u sub 1 , u sub 6 sup {-1} ] , ~ ~
u sub 2 u sub 3 supr(3) u sub 5 =
[u sub 1 , u sub 7 sup {-1} ] ,
.EN
.EQ I
u sub 2 u sub 3 supr(2) u sub 4 u sub 5 supr(3) u sub 6 u sub 7 =
[u sub 1 , u sub 8 sup {-1} ] ,
.EN
.EQ I
r sub 1 = u sub 1 v sub 1 supr(2) u sub 1 sup {-1} , ~ ~
r sub 8 = u sub 8 v sub 8 u sub 8 sup {-1} , ~ ~
(r sub 1 r sub 8 ) sup 8 = 1 ,
.EN
.EQ I
r sub 1 u sub 1 r sub 1 = v sub 1 , ~ ~ r sub 1 u sub 2 r sub 1 =
u sub 8 , ~ ~ r sub 1 u sub 3 r sub 1 = u sub 7 ,
.EN
.EQ I
r sub 1 u sub 4 r sub 1 = u sub 6 , ~ ~
r sub 1 u sub 5 r sub 1 = u sub 5 , ~ ~
r sub 1 v sub 2 r sub 1 = v sub 8 ,
.EN
.EQ I
r sub 1 v sub 3 r sub 1 = v sub 7 , ~ ~
r sub 1 v sub 4 r sub 1 = v sub 6 , ~ ~
r sub 1 v sub 5 r sub 1 = v sub 5 ,
.EN
.EQ I
r sub 8 u sub 1 r sub 8 = u sub 7 , ~ ~
r sub 8 u sub 2 r sub 8 = u sub 6 , ~ ~
r sub 8 u sub 3 r sub 8 = u sub 5 ,
.EN
.EQ I
r sub 8 u sub 4 r sub 8 = u sub 4 , ~ ~
r sub 8 u sub 8 r sub 8 = v sub 8 , ~ ~
r sub 8 v sub 1 r sub 8 = v sub 7 ,
.EN
.EQ I
r sub 8 v sub 2 r sub 8 = v sub 6 , ~ ~
r sub 8 v sub 3 r sub 8 = v sub 5 , ~ ~
r sub 8 v sub 4 r sub 8 = v sub 4 .
.EN
.LP
Here the subgroups $B$ and $N$ of the Tits system of
$F$ used to construct this presentation are
$B = << u sub 1 , ... , u sub 8 >>$ and $N = << r sub 1 , r sub 8 >>$.
Thus the collinearity graph $GAM$ of the generalized
octagon of order $(2,4)$ associated with $F$ arises as the
graph $GAMMA (F, << B , r sub 1 >> , r sub 2 )$. Using
that $u sub 5 supr(2)$ lies in the center of
$<< B , r sub 1 >> $ and that the latter group is a
maximal subgroup of $F$ (so that it coincides with the
centralizer in $F$ of $u sub 5 supr(2)$), we obtain the
following more direct (but equivalent) description
of the generalized octagon.
Its points and lines are the involutions
$F$-conjugate to $u sub {2i+1} supr(2)$ and $u sub {2i}$, respectively.
A point $x$ is incident to a line $y$ if the pair
$x,y$ is $F$-conjugate to $ u sub 1 supr(2) , u sub 2$;
this is equivalent to $y$ lying in $C sub F (x) sup {prime prime prime}$,
the third commutator subgroup of the centralizer of $x$
in $F$.
The chain $u sub 1 supr(2) , u sub 2 , u sub 3 supr(2) , ... ,
u sub 8 , v sub 1 supr(2) , v sub 2 , ... , v sub 8 , u sub 1 supr(2)$
forms a 16-circuit in the incidence graph - an apartment in the
generalized octagon.
.LP
The Rudvalis graph is a strongly regular graph with parameters
$(v,k, lam , mu ) = (4060,1755,730,780)$ which is locally $GAM$.
Its automorphism group is the simple Rudvalis group. For a construction,
see
.SC Conway & Wales
.[[
Conway Wales 1973 Rudvalis
.]]
and the $Atlas$
.[[
Conway Parker Norton Atlas
.]].
CORRECTION.
On p. 343, line $-13$, replace `\fI(ii)\fP' by `\fI(iii)\fP'.
ADDITION.
On p. 344, add at the end of the page:
.LP
$Re$. In the case of infinite fields more examples may be expected
of distance-transitive graphs from algebraic groups than those
suggested by the above theorem. For example, if $G$ is an algebraic
$F$-group of type $E sub 7$ for some field $F$ possessing an anisotropic
kernel of type $D sub 4$ (cf.
.SC Tits
.[[
Tits buildings of spherical type finite pairs
.]]),
then the maximal parabolic subgroups of $G$ lead to a polar space
of rank 3 (over the Cayley division algebra with center $F$)
on which $G$ acts highly transitively. Just as in the finite case,
the collinearity graph of the polar space and that of the dual polar space
are distance transitive.