<s>
In	O
mathematics	O
,	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
named	O
after	O
H	O
.	O
S	O
.	O
M	O
.	O
Coxeter	O
,	O
is	O
an	O
abstract	O
group	O
that	O
admits	O
a	O
formal	O
description	O
in	O
terms	O
of	O
reflections	B-Algorithm
(	O
or	O
kaleidoscopic	B-Application
mirrors	I-Application
)	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
finite	O
Coxeter	B-Algorithm
groups	I-Algorithm
are	O
precisely	O
the	O
finite	O
Euclidean	O
reflection	B-Algorithm
groups	I-Algorithm
;	O
the	O
symmetry	O
groups	O
of	O
regular	O
polyhedra	O
are	O
an	O
example	O
.	O
</s>
<s>
However	O
,	O
not	O
all	O
Coxeter	B-Algorithm
groups	I-Algorithm
are	O
finite	O
,	O
and	O
not	O
all	O
can	O
be	O
described	O
in	O
terms	O
of	O
symmetries	O
and	O
Euclidean	O
reflections	B-Algorithm
.	O
</s>
<s>
Coxeter	B-Algorithm
groups	I-Algorithm
were	O
introduced	O
in	O
1934	O
as	O
abstractions	O
of	O
reflection	B-Algorithm
groups	I-Algorithm
,	O
and	O
finite	O
Coxeter	B-Algorithm
groups	I-Algorithm
were	O
classified	O
in	O
1935	O
.	O
</s>
<s>
Coxeter	B-Algorithm
groups	I-Algorithm
find	O
applications	O
in	O
many	O
areas	O
of	O
mathematics	O
.	O
</s>
<s>
Examples	O
of	O
finite	O
Coxeter	B-Algorithm
groups	I-Algorithm
include	O
the	O
symmetry	O
groups	O
of	O
regular	O
polytopes	O
,	O
and	O
the	O
Weyl	O
groups	O
of	O
simple	O
Lie	O
algebras	O
.	O
</s>
<s>
Examples	O
of	O
infinite	O
Coxeter	B-Algorithm
groups	I-Algorithm
include	O
the	O
triangle	O
groups	O
corresponding	O
to	O
regular	O
tessellations	O
of	O
the	O
Euclidean	O
plane	O
and	O
the	O
hyperbolic	O
plane	O
,	O
and	O
the	O
Weyl	O
groups	O
of	O
infinite-dimensional	O
Kac	O
–	O
Moody	O
algebras	O
.	O
</s>
<s>
The	O
pair	O
where	O
is	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
with	O
generators	O
is	O
called	O
a	O
Coxeter	B-Algorithm
system	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Coxeter	B-Algorithm
groups	I-Algorithm
of	O
type	O
and	O
are	O
isomorphic	O
but	O
the	O
Coxeter	B-Algorithm
systems	I-Algorithm
are	O
not	O
equivalent	O
(	O
see	O
below	O
for	O
an	O
explanation	O
of	O
this	O
notation	O
)	O
.	O
</s>
<s>
The	O
relation	O
means	O
that	O
for	O
all	O
;	O
as	O
such	O
the	O
generators	O
are	O
involutions	B-Algorithm
.	O
</s>
<s>
Alternatively	O
,	O
since	O
the	O
generators	O
are	O
involutions	B-Algorithm
,	O
,	O
so	O
,	O
and	O
thus	O
is	O
equal	O
to	O
the	O
commutator	O
.	O
</s>
<s>
The	O
Coxeter	O
matrix	O
is	O
the	O
,	O
symmetric	B-Algorithm
matrix	I-Algorithm
with	O
entries	O
.	O
</s>
<s>
Indeed	O
,	O
every	O
symmetric	B-Algorithm
matrix	I-Algorithm
with	O
diagonal	O
entries	O
exclusively	O
1	O
and	O
nondiagonal	O
entries	O
in	O
the	O
set	O
is	O
a	O
Coxeter	O
matrix	O
.	O
</s>
<s>
Thus	O
the	O
disjoint	O
union	O
of	O
Coxeter	O
graphs	O
yields	O
a	O
direct	O
product	O
of	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
Schläfli	O
matrix	O
is	O
useful	O
because	O
its	O
eigenvalues	O
determine	O
whether	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
of	O
finite	O
type	O
(	O
all	O
positive	O
)	O
,	O
affine	O
type	O
(	O
all	O
non-negative	O
,	O
at	O
least	O
one	O
zero	O
)	O
,	O
or	O
indefinite	O
type	O
(	O
otherwise	O
)	O
.	O
</s>
<s>
into	O
hyperbolic	O
and	O
other	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
However	O
,	O
there	O
are	O
multiple	O
non-equivalent	O
definitions	O
for	O
hyperbolic	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
graph	O
in	O
which	O
vertices	O
1	O
through	O
n	O
are	O
placed	O
in	O
a	O
row	O
with	O
each	O
vertex	O
connected	O
by	O
an	O
unlabelled	O
edge	O
to	O
its	O
immediate	O
neighbors	O
gives	O
rise	O
to	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn+1	O
;	O
the	O
generators	O
correspond	O
to	O
the	O
transpositions	B-Algorithm
(	O
1	O
2	O
)	O
,	O
(	O
2	O
3	O
)	O
,	O
...	O
,	O
(	O
n	O
n+1	O
)	O
.	O
</s>
<s>
Two	O
non-consecutive	O
transpositions	B-Algorithm
always	O
commute	O
,	O
while	O
(	O
k	O
k+1	O
)	O
(	O
k+1	O
k+2	O
)	O
gives	O
the	O
3-cycle	O
(	O
k	O
k+2	O
k+1	O
)	O
.	O
</s>
<s>
Of	O
course	O
,	O
this	O
only	O
shows	O
that	O
Sn+1	O
is	O
a	O
quotient	O
group	O
of	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
described	O
by	O
the	O
graph	O
,	O
but	O
it	O
is	O
not	O
too	O
difficult	O
to	O
check	O
that	O
equality	O
holds	O
.	O
</s>
<s>
Coxeter	B-Algorithm
groups	I-Algorithm
are	O
deeply	O
connected	O
with	O
reflection	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Simply	O
put	O
,	O
Coxeter	B-Algorithm
groups	I-Algorithm
are	O
abstract	O
groups	O
(	O
given	O
via	O
a	O
presentation	O
)	O
,	O
while	O
reflection	B-Algorithm
groups	I-Algorithm
are	O
concrete	O
groups	O
(	O
given	O
as	O
subgroups	O
of	O
linear	B-Algorithm
groups	I-Algorithm
or	O
various	O
generalizations	O
)	O
.	O
</s>
<s>
Coxeter	B-Algorithm
groups	I-Algorithm
grew	O
out	O
of	O
the	O
study	O
of	O
reflection	B-Algorithm
groups	I-Algorithm
—	O
they	O
are	O
an	O
abstraction	O
:	O
a	O
reflection	B-Algorithm
group	I-Algorithm
is	O
a	O
subgroup	O
of	O
a	O
linear	B-Algorithm
group	I-Algorithm
generated	O
by	O
reflections	B-Algorithm
(	O
which	O
have	O
order	O
2	O
)	O
,	O
while	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
an	O
abstract	O
group	O
generated	O
by	O
involutions	B-Algorithm
(	O
elements	O
of	O
order	O
2	O
,	O
abstracting	O
from	O
reflections	B-Algorithm
)	O
,	O
and	O
whose	O
relations	O
have	O
a	O
certain	O
form	O
(	O
,	O
corresponding	O
to	O
hyperplanes	O
meeting	O
at	O
an	O
angle	O
of	O
,	O
with	O
being	O
of	O
order	O
k	O
abstracting	O
from	O
a	O
rotation	O
by	O
)	O
.	O
</s>
<s>
The	O
abstract	O
group	O
of	O
a	O
reflection	B-Algorithm
group	I-Algorithm
is	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
while	O
conversely	O
a	O
reflection	B-Algorithm
group	I-Algorithm
can	O
be	O
seen	O
as	O
a	O
linear	O
representation	O
of	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
For	O
finite	O
reflection	B-Algorithm
groups	I-Algorithm
,	O
this	O
yields	O
an	O
exact	O
correspondence	O
:	O
every	O
finite	O
Coxeter	B-Algorithm
group	I-Algorithm
admits	O
a	O
faithful	O
representation	O
as	O
a	O
finite	O
reflection	B-Algorithm
group	I-Algorithm
of	O
some	O
Euclidean	O
space	O
.	O
</s>
<s>
For	O
infinite	O
Coxeter	B-Algorithm
groups	I-Algorithm
,	O
however	O
,	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
may	O
not	O
admit	O
a	O
representation	O
as	O
a	O
reflection	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Historically	O
,	O
proved	O
that	O
every	O
reflection	B-Algorithm
group	I-Algorithm
is	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
(	O
i.e.	O
,	O
has	O
a	O
presentation	O
where	O
all	O
relations	O
are	O
of	O
the	O
form	O
or	O
)	O
,	O
and	O
indeed	O
this	O
paper	O
introduced	O
the	O
notion	O
of	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
while	O
proved	O
that	O
every	O
finite	O
Coxeter	B-Algorithm
group	I-Algorithm
had	O
a	O
representation	O
as	O
a	O
reflection	B-Algorithm
group	I-Algorithm
,	O
and	O
classified	O
finite	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
finite	O
Coxeter	B-Algorithm
groups	I-Algorithm
were	O
classified	O
in	O
,	O
in	O
terms	O
of	O
Coxeter	O
–	O
Dynkin	O
diagrams	O
;	O
they	O
are	O
all	O
represented	O
by	O
reflection	B-Algorithm
groups	I-Algorithm
of	O
finite-dimensional	O
Euclidean	O
spaces	O
.	O
</s>
<s>
The	O
finite	O
Coxeter	B-Algorithm
groups	I-Algorithm
consist	O
of	O
three	O
one-parameter	O
families	O
of	O
increasing	O
rank	O
one	O
one-parameter	O
family	O
of	O
dimension	O
two	O
,	O
and	O
six	O
exceptional	O
groups	O
:	O
and	O
.	O
</s>
<s>
The	O
product	O
of	O
finitely	O
many	O
Coxeter	B-Algorithm
groups	I-Algorithm
in	O
this	O
list	O
is	O
again	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
and	O
all	O
finite	O
Coxeter	B-Algorithm
groups	I-Algorithm
arise	O
in	O
this	O
way	O
.	O
</s>
<s>
Many	O
,	O
but	O
not	O
all	O
of	O
these	O
,	O
are	O
Weyl	O
groups	O
,	O
and	O
every	O
Weyl	O
group	O
can	O
be	O
realized	O
as	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Also	O
note	O
that	O
every	O
finitely	O
generated	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
an	O
automatic	O
group	O
.	O
</s>
<s>
Note	O
further	O
that	O
the	O
(	O
directed	O
)	O
Dynkin	O
diagrams	O
Bn	O
and	O
Cn	O
give	O
rise	O
to	O
the	O
same	O
Weyl	O
group	O
(	O
hence	O
Coxeter	B-Algorithm
group	I-Algorithm
)	O
,	O
because	O
they	O
differ	O
as	O
directed	O
graphs	O
,	O
but	O
agree	O
as	O
undirected	O
graphs	O
–	O
direction	O
matters	O
for	O
root	O
systems	O
but	O
not	O
for	O
the	O
Weyl	O
group	O
;	O
this	O
corresponds	O
to	O
the	O
hypercube	B-Operating_System
and	O
cross-polytope	O
being	O
different	O
regular	O
polytopes	O
but	O
having	O
the	O
same	O
symmetry	O
group	O
.	O
</s>
<s>
Some	O
properties	O
of	O
the	O
finite	O
irreducible	O
Coxeter	B-Algorithm
groups	I-Algorithm
are	O
given	O
in	O
the	O
following	O
table	O
.	O
</s>
<s>
RanknGroupsymbol	O
Alternatesymbol	O
BracketnotationCoxetergraph	O
Reflectionsm	O
=	O
nhCoxeter	O
,	O
Regular	O
polytopes	O
,	O
§	O
12.6	O
The	O
number	O
of	O
reflections	B-Algorithm
,	O
equation	O
12.61Coxeter	O
numberh	O
Order	O
Group	O
structure	O
Related	O
polytopes1	O
A1	O
A1	O
[	O
]	O
1	O
2	O
2	O
{	O
}	O
2	O
A2	O
A2	O
 [ 3 ] 	O
3	O
3	O
6	O
 { 3 } 	O
3	O
A3	O
A3	O
 [ 3 , 3 ] 	O
6	O
4	O
24	O
 { 3 , 3 } 	O
4	O
A4	O
A4	O
 [ 3 , 3 , 3 ] 	O
10	O
5	O
120	O
 { 3 , 3 , 3 } 	O
5	O
A5	O
A5	O
 [ 3 , 3 , 3 , 3 ] 	O
15	O
6	O
720	O
 { 3 , 3 , 3 , 3 } 	O
n	O
An	O
An	O
 [ 3n−1 ] 	O
...	O
n( n	O
+	O
1	O
)	O
/2	O
n	O
+	O
1	O
(	O
n	O
+	O
1	O
)	O
!	O
</s>
<s>
n-simplex	O
2	O
B2	O
C2	O
 [ 4 ] 	O
4	O
4	O
8	O
{4}3	O
B3	O
C3	O
 [ 4 , 3 ] 	O
9	O
6	O
48	O
 { 4 , 3 } 	B-Application
/	O
 { 3 , 4 } 	O
4	O
B4	O
C4	O
 [ 4 , 3 , 3 ] 	O
16	O
8	O
384	O
 { 4 , 3 , 3 } 	B-Language
/	O
 { 3 , 3 , 4 } 	O
5	O
B5	O
C5	O
 [ 4 , 3 , 3 , 3 ] 	O
25	O
10	O
3840	O
 { 4 , 3 , 3 , 3 } 	O
/	O
 { 3 , 3 , 3 , 4 } 	O
n	O
Bn	O
Cn	O
 [ 4 , 3n−2 ] 	O
...	O
n2	O
2n	O
2n	O
n	O
!	O
</s>
<s>
n-cube	O
/	O
n-orthoplex	O
4	O
D4	O
B4	O
 [ 31 , 1 , 1 ] 	O
12	O
6	O
192	O
h{4,3,3}	O
/	O
{3,31,1}5	O
D5	O
B5	O
 [ 32 , 1 , 1 ] 	O
20	O
8	O
1920	O
h{4,3,3,3}	O
/	O
{3,3,31,1}n	O
Dn	O
Bn	O
 [ 3n−3 , 1 , 1 ] 	O
...	O
n( n	O
−	O
1	O
)	O
2( n	O
−	O
1	O
)	O
2n−1	O
n	O
!	O
</s>
<s>
All	O
symmetry	O
groups	O
of	O
regular	O
polytopes	O
are	O
finite	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
symmetry	O
group	O
of	O
a	O
regular	O
n-simplex	O
is	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn+1	O
,	O
also	O
known	O
as	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
of	O
type	O
An	O
.	O
</s>
<s>
The	O
symmetry	O
group	O
of	O
the	O
n-cube	O
and	O
its	O
dual	O
,	O
the	O
n-cross-polytope	O
,	O
is	O
Bn	O
,	O
and	O
is	O
known	O
as	O
the	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
exceptional	O
regular	O
polytopes	O
in	O
dimensions	O
two	O
,	O
three	O
,	O
and	O
four	O
,	O
correspond	O
to	O
other	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
the	O
dihedral	B-Algorithm
groups	I-Algorithm
,	O
which	O
are	O
the	O
symmetry	O
groups	O
of	O
regular	O
polygons	O
,	O
form	O
the	O
series	O
I2(p )	O
.	O
</s>
<s>
In	O
three	O
dimensions	O
,	O
the	O
symmetry	O
group	O
of	O
the	O
regular	O
dodecahedron	O
and	O
its	O
dual	O
,	O
the	O
regular	O
icosahedron	O
,	O
is	O
H3	O
,	O
known	O
as	O
the	O
full	B-Algorithm
icosahedral	I-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
Coxeter	B-Algorithm
groups	I-Algorithm
of	O
type	O
Dn	O
,	O
E6	O
,	O
E7	O
,	O
and	O
E8	O
are	O
the	O
symmetry	O
groups	O
of	O
certain	O
semiregular	O
polytopes	O
.	O
</s>
<s>
The	O
affine	O
Coxeter	B-Algorithm
groups	I-Algorithm
form	O
a	O
second	O
important	O
series	O
of	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
In	O
each	O
case	O
,	O
the	O
quotient	O
group	O
is	O
itself	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
and	O
the	O
Coxeter	O
graph	O
of	O
the	O
affine	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
obtained	O
from	O
the	O
Coxeter	O
graph	O
of	O
the	O
quotient	O
group	O
by	O
adding	O
another	O
vertex	O
and	O
one	O
or	O
two	O
additional	O
edges	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
n≥2	O
,	O
the	O
graph	O
consisting	O
of	O
n+1	O
vertices	O
in	O
a	O
circle	O
is	O
obtained	O
from	O
An	O
in	O
this	O
way	O
,	O
and	O
the	O
corresponding	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
the	O
affine	O
Weyl	O
group	O
of	O
An	O
(	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
)	O
.	O
</s>
<s>
The	O
affine	O
Coxeter	B-Algorithm
group	I-Algorithm
(	O
or	O
affine	O
Weyl	O
group	O
)	O
is	O
then	O
the	O
group	O
generated	O
by	O
the	O
(	O
affine	O
)	O
reflections	B-Algorithm
about	O
all	O
the	O
hyperplanes	O
in	O
the	O
diagram	O
.	O
</s>
<s>
The	O
Stiefel	O
diagram	O
divides	O
the	O
plane	O
into	O
infinitely	O
many	O
connected	O
components	O
called	O
alcoves	O
,	O
and	O
the	O
affine	O
Coxeter	B-Algorithm
group	I-Algorithm
acts	O
freely	O
and	O
transitively	O
on	O
the	O
alcoves	O
,	O
just	O
as	O
the	O
ordinary	O
Weyl	O
group	O
acts	O
freely	O
and	O
transitively	O
on	O
the	O
Weyl	O
chambers	O
.	O
</s>
<s>
Then	O
the	O
affine	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
generated	O
by	O
the	O
ordinary	O
(	O
linear	O
)	O
reflections	B-Algorithm
about	O
the	O
hyperplanes	O
perpendicular	O
to	O
,	O
together	O
with	O
an	O
affine	O
reflection	O
about	O
a	O
translate	O
of	O
the	O
hyperplane	O
perpendicular	O
to	O
.	O
</s>
<s>
A	O
list	O
of	O
the	O
affine	O
Coxeter	B-Algorithm
groups	I-Algorithm
follows	O
:	O
</s>
<s>
There	O
are	O
infinitely	O
many	O
hyperbolic	O
Coxeter	B-Algorithm
groups	I-Algorithm
describing	O
reflection	B-Algorithm
groups	I-Algorithm
in	O
hyperbolic	O
space	O
,	O
notably	O
including	O
the	O
hyperbolic	O
triangle	O
groups	O
.	O
</s>
<s>
A	O
choice	O
of	O
reflection	O
generators	O
gives	O
rise	O
to	O
a	O
length	O
function	O
ℓ	O
on	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
namely	O
the	O
minimum	O
number	O
of	O
uses	O
of	O
generators	O
required	O
to	O
express	O
a	O
group	O
element	O
;	O
this	O
is	O
precisely	O
the	O
length	O
in	O
the	O
word	O
metric	O
in	O
the	O
Cayley	O
graph	O
.	O
</s>
<s>
The	O
function	O
defines	O
a	O
map	O
generalizing	O
the	O
sign	O
map	O
for	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Using	O
reduced	O
words	O
one	O
may	O
define	O
three	O
partial	O
orders	O
on	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
the	O
(	O
right	O
)	O
weak	O
order	O
,	O
the	O
absolute	O
order	O
and	O
the	O
Bruhat	O
order	O
(	O
named	O
for	O
François	O
Bruhat	O
)	O
.	O
</s>
<s>
Since	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
generated	O
by	O
finitely	O
many	O
elements	O
of	O
order	O
2	O
,	O
its	O
abelianization	O
is	O
an	O
elementary	O
abelian	O
2-group	O
,	O
i.e.	O
,	O
it	O
is	O
isomorphic	O
to	O
the	O
direct	O
sum	O
of	O
several	O
copies	O
of	O
the	O
cyclic	O
group	O
.	O
</s>
<s>
The	O
Schur	O
multiplier	O
,	O
equal	O
to	O
the	O
second	O
homology	O
group	O
of	O
,	O
was	O
computed	O
in	O
for	O
finite	O
reflection	B-Algorithm
groups	I-Algorithm
and	O
in	O
for	O
affine	O
reflection	B-Algorithm
groups	I-Algorithm
,	O
with	O
a	O
more	O
unified	O
account	O
given	O
in	O
.	O
</s>
