<s>
The	O
affine	B-Algorithm
symmetric	I-Algorithm
groups	I-Algorithm
are	O
a	O
family	O
of	O
mathematical	O
structures	O
that	O
describe	O
the	O
symmetries	O
of	O
the	O
number	O
line	O
and	O
the	O
regular	O
triangular	O
tiling	O
of	O
the	O
plane	O
,	O
as	O
well	O
as	O
related	O
higher-dimensional	O
objects	O
.	O
</s>
<s>
Each	O
one	O
is	O
an	O
infinite	O
extension	O
of	O
a	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
the	O
group	O
of	O
permutations	B-Algorithm
(	O
rearrangements	O
)	O
of	O
a	O
finite	O
set	O
.	O
</s>
<s>
In	O
addition	O
to	O
their	O
geometric	O
description	O
,	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
groups	I-Algorithm
may	O
be	O
defined	O
as	O
collections	O
of	O
permutations	B-Algorithm
of	O
the	O
integers	O
(	O
...	O
,	O
2	O
,	O
1	O
,	O
0	O
,	O
1	O
,	O
2	O
,	O
...	O
)	O
that	O
are	O
periodic	O
in	O
a	O
certain	O
sense	O
,	O
or	O
in	O
purely	O
algebraic	O
terms	O
as	O
a	O
group	O
with	O
certain	O
generators	O
and	O
relations	O
.	O
</s>
<s>
Many	O
important	O
combinatorial	O
properties	O
of	O
the	O
finite	O
symmetric	B-Algorithm
groups	I-Algorithm
can	O
be	O
extended	O
to	O
affine	B-Algorithm
symmetric	I-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Permutation	B-Algorithm
statistics	O
such	O
as	O
descents	O
and	O
inversions	B-Algorithm
can	O
be	O
defined	O
in	O
the	O
affine	O
case	O
.	O
</s>
<s>
The	O
affine	B-Algorithm
symmetric	I-Algorithm
groups	I-Algorithm
have	O
close	O
relationships	O
with	O
other	O
mathematical	O
objects	O
,	O
including	O
juggling	O
patterns	O
and	O
certain	O
complex	O
reflection	B-Algorithm
groups	O
.	O
</s>
<s>
Many	O
of	O
their	O
combinatorial	O
and	O
geometric	O
properties	O
extend	O
to	O
the	O
broader	O
family	O
of	O
affine	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
The	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
may	O
be	O
equivalently	O
defined	O
as	O
an	O
abstract	O
group	O
by	O
generators	O
and	O
relations	O
,	O
or	O
in	O
terms	O
of	O
concrete	O
geometric	O
and	O
combinatorial	O
models	O
.	O
</s>
<s>
(	O
the	O
generators	O
are	O
involutions	B-Algorithm
)	O
,	O
</s>
<s>
(	O
The	O
second	O
and	O
third	O
relation	O
are	O
sometimes	O
called	O
the	O
braid	B-Application
relations	I-Application
.	O
)	O
</s>
<s>
When	O
,	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
is	O
the	O
infinite	O
dihedral	O
group	O
generated	O
by	O
two	O
elements	O
subject	O
only	O
to	O
the	O
relations	O
.	O
</s>
<s>
These	O
relations	O
can	O
be	O
rewritten	O
in	O
the	O
special	O
form	O
that	O
defines	O
the	O
Coxeter	B-Algorithm
groups	I-Algorithm
,	O
so	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
groups	I-Algorithm
are	O
Coxeter	B-Algorithm
groups	I-Algorithm
,	O
with	O
the	O
as	O
their	O
Coxeter	O
generating	O
sets	O
.	O
</s>
<s>
In	O
these	O
diagrams	O
,	O
the	O
vertices	O
represent	O
the	O
generators	O
,	O
which	O
for	O
Coxeter	B-Algorithm
groups	I-Algorithm
must	O
be	O
involutions	B-Algorithm
.	O
</s>
<s>
For	O
every	O
pair	O
of	O
distinct	O
elements	O
and	O
of	O
and	O
every	O
integer	O
,	O
the	O
set	O
of	O
points	O
in	O
that	O
satisfy	O
forms	O
an	O
-dimensional	O
subspace	O
within	O
,	O
and	O
there	O
is	O
a	O
unique	O
reflection	B-Algorithm
of	O
that	O
fixes	O
this	O
subspace	O
.	O
</s>
<s>
Then	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
can	O
be	O
realized	O
geometrically	O
as	O
a	O
collection	O
of	O
maps	O
from	O
to	O
itself	O
,	O
the	O
compositions	O
of	O
these	O
reflections	O
.	O
</s>
<s>
Each	O
reflection	B-Algorithm
preserves	O
this	O
lattice	O
,	O
and	O
so	O
the	O
lattice	O
is	O
preserved	O
by	O
the	O
whole	O
group	O
.	O
</s>
<s>
To	O
translate	B-Algorithm
between	O
the	O
geometric	O
and	O
algebraic	O
definitions	O
,	O
one	O
fixes	O
an	O
alcove	O
and	O
consider	O
the	O
hyperplanes	O
that	O
form	O
its	O
boundary	O
.	O
</s>
<s>
For	O
,	O
one	O
may	O
identify	O
the	O
reflection	B-Algorithm
through	O
with	O
the	O
Coxeter	O
generator	O
,	O
and	O
also	O
identify	O
the	O
reflection	B-Algorithm
through	O
with	O
the	O
generator	O
.	O
</s>
<s>
The	O
elements	O
of	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
may	O
be	O
realized	O
as	O
a	O
group	O
of	O
periodic	O
permutations	B-Algorithm
of	O
the	O
integers	O
.	O
</s>
<s>
it	O
is	O
a	O
bijection	B-Algorithm
(	O
each	O
integer	O
appears	O
as	O
the	O
value	O
of	O
for	O
exactly	O
one	O
)	O
,	O
</s>
<s>
For	O
every	O
affine	B-Algorithm
permutation	I-Algorithm
,	O
and	O
more	O
generally	O
every	O
shift-equivariant	O
bijection	B-Algorithm
,	O
the	O
numbers	O
must	O
all	O
be	O
distinct	O
modulo	O
.	O
</s>
<s>
An	O
affine	B-Algorithm
permutation	I-Algorithm
is	O
uniquely	O
determined	O
by	O
its	O
window	O
notation	O
,	O
because	O
all	O
other	O
values	O
of	O
can	O
be	O
found	O
by	O
shifting	O
these	O
values	O
.	O
</s>
<s>
Thus	O
,	O
affine	B-Algorithm
permutations	I-Algorithm
may	O
also	O
be	O
identified	O
with	O
tuples	B-Application
of	O
integers	O
that	O
contain	O
one	O
element	O
from	O
each	O
congruence	O
class	O
modulo	O
and	O
sum	O
to	O
.	O
</s>
<s>
To	O
translate	B-Algorithm
between	O
the	O
combinatorial	O
and	O
algebraic	O
definitions	O
,	O
for	O
one	O
may	O
identify	O
the	O
Coxeter	O
generator	O
with	O
the	O
affine	B-Algorithm
permutation	I-Algorithm
that	O
has	O
window	O
notation	O
,	O
and	O
also	O
identify	O
the	O
generator	O
with	O
the	O
affine	B-Algorithm
permutation	I-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
every	O
reflection	B-Algorithm
(	O
that	O
is	O
,	O
a	O
conjugate	O
of	O
one	O
of	O
the	O
Coxeter	O
generators	O
)	O
can	O
be	O
described	O
uniquely	O
as	O
follows	O
:	O
for	O
distinct	O
integers	O
,	O
in	O
and	O
arbitrary	O
integer	O
,	O
it	O
maps	O
to	O
,	O
maps	O
to	O
,	O
and	O
fixes	O
all	O
inputs	O
not	O
congruent	O
to	O
or	O
modulo	O
.	O
</s>
<s>
Affine	B-Algorithm
permutations	I-Algorithm
can	O
be	O
represented	O
as	O
infinite	O
periodic	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
If	O
is	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
,	O
the	O
corresponding	O
matrix	O
has	O
entry	O
1	O
at	O
position	O
in	O
the	O
infinite	O
grid	O
for	O
each	O
integer	O
,	O
and	O
all	O
other	O
entries	O
are	O
equal	O
to	O
0	O
.	O
</s>
<s>
Since	O
is	O
a	O
bijection	B-Algorithm
,	O
the	O
resulting	O
matrix	O
contains	O
exactly	O
one	O
1	O
in	O
every	O
row	O
and	O
column	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
portion	O
of	O
the	O
matrix	O
for	O
the	O
affine	B-Algorithm
permutation	I-Algorithm
is	O
shown	O
in	O
the	O
figure	O
.	O
</s>
<s>
The	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
contains	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
permutations	B-Algorithm
on	O
elements	O
as	O
both	O
a	O
subgroup	O
and	O
a	O
quotient	O
group	O
.	O
</s>
<s>
There	O
is	O
a	O
canonical	O
way	O
to	O
choose	O
a	O
subgroup	O
of	O
that	O
is	O
isomorphic	O
to	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
In	O
terms	O
of	O
the	O
algebraic	O
definition	O
,	O
this	O
is	O
the	O
subgroup	O
of	O
generated	O
by	O
(	O
excluding	O
the	O
simple	O
reflection	B-Algorithm
)	O
.	O
</s>
<s>
Geometrically	O
,	O
this	O
corresponds	O
to	O
the	O
subgroup	O
of	O
transformations	O
that	O
fix	O
the	O
origin	O
,	O
while	O
combinatorially	O
it	O
corresponds	O
to	O
the	O
window	O
notations	O
for	O
which	O
(	O
that	O
is	O
,	O
in	O
which	O
the	O
window	O
notation	O
is	O
the	O
one-line	O
notation	O
of	O
a	O
finite	O
permutation	B-Algorithm
)	O
.	O
</s>
<s>
If	O
is	O
the	O
window	O
notation	O
of	O
an	O
element	O
of	O
this	O
standard	O
copy	O
of	O
,	O
its	O
action	O
on	O
the	O
hyperplane	O
in	O
is	O
given	O
by	O
permutation	B-Algorithm
of	O
coordinates	O
:	O
.	O
</s>
<s>
(	O
In	O
this	O
article	O
,	O
the	O
geometric	O
action	O
of	O
permutations	B-Algorithm
and	O
affine	B-Algorithm
permutations	I-Algorithm
is	O
on	O
the	O
right	O
;	O
thus	O
,	O
if	O
and	O
are	O
two	O
affine	B-Algorithm
permutations	I-Algorithm
,	O
the	O
action	O
of	O
on	O
a	O
point	O
is	O
given	O
by	O
first	O
applying	O
,	O
then	O
applying	O
.	O
)	O
</s>
<s>
There	O
is	O
a	O
simple	O
map	O
(	O
technically	O
,	O
a	O
surjective	B-Algorithm
group	O
homomorphism	O
)	O
from	O
onto	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
In	O
terms	O
of	O
the	O
combinatorial	O
definition	O
,	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
can	O
be	O
mapped	O
to	O
a	O
permutation	B-Algorithm
by	O
reducing	O
the	O
window	O
entries	O
modulo	O
to	O
elements	O
of	O
,	O
leaving	O
the	O
one-line	O
notation	O
of	O
a	O
permutation	B-Algorithm
.	O
</s>
<s>
In	O
this	O
article	O
,	O
the	O
image	O
of	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
is	O
called	O
the	O
underlying	O
permutation	B-Algorithm
of	O
.	O
</s>
<s>
The	O
map	O
sends	O
the	O
Coxeter	O
generator	O
to	O
the	O
permutation	B-Algorithm
whose	O
one-line	O
notation	O
and	O
cycle	B-Algorithm
notation	I-Algorithm
are	O
and	O
,	O
respectively	O
.	O
</s>
<s>
The	O
kernel	O
of	O
is	O
by	O
definition	O
the	O
set	O
of	O
affine	B-Algorithm
permutations	I-Algorithm
whose	O
underlying	O
permutation	B-Algorithm
is	O
the	O
identity	O
.	O
</s>
<s>
The	O
window	O
notations	O
of	O
such	O
affine	B-Algorithm
permutations	I-Algorithm
are	O
of	O
the	O
form	O
,	O
where	O
is	O
an	O
integer	O
vector	O
such	O
that	O
,	O
that	O
is	O
,	O
where	O
.	O
</s>
<s>
Geometrically	O
,	O
this	O
kernel	O
consists	O
of	O
the	O
translations	B-Algorithm
,	O
the	O
isometries	O
that	O
shift	O
the	O
entire	O
space	O
without	O
rotating	O
or	O
reflecting	O
it	O
.	O
</s>
<s>
In	O
an	O
abuse	O
of	O
notation	O
,	O
the	O
symbol	O
is	O
used	O
in	O
this	O
article	O
for	O
all	O
three	O
of	O
these	O
sets	O
(	O
integer	O
vectors	O
in	O
,	O
affine	B-Algorithm
permutations	I-Algorithm
with	O
underlying	O
permutation	B-Algorithm
the	O
identity	O
,	O
and	O
translations	B-Algorithm
)	O
;	O
in	O
all	O
three	O
settings	O
,	O
the	O
natural	O
group	O
operation	O
turns	O
into	O
an	O
abelian	O
group	O
,	O
generated	O
freely	O
by	O
the	O
vectors	O
.	O
</s>
<s>
of	O
this	O
subgroup	O
with	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
where	O
the	O
action	O
of	O
on	O
is	O
by	O
permutation	B-Algorithm
of	O
coordinates	O
.	O
</s>
<s>
where	O
is	O
a	O
permutation	B-Algorithm
in	O
the	O
standard	O
copy	O
of	O
in	O
and	O
is	O
a	O
translation	O
in	O
.	O
</s>
<s>
Furthermore	O
,	O
as	O
with	O
every	O
affine	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
acts	O
transitively	O
and	O
freely	O
on	O
the	O
set	O
of	O
alcoves	O
:	O
for	O
each	O
two	O
alcoves	O
,	O
a	O
unique	O
group	O
element	O
takes	O
one	O
alcove	O
to	O
the	O
other	O
.	O
</s>
<s>
Hence	O
,	O
making	O
an	O
arbitrary	O
choice	O
of	O
alcove	O
places	O
the	O
group	O
in	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
with	O
the	O
alcoves	O
:	O
the	O
identity	O
element	O
corresponds	O
to	O
,	O
and	O
every	O
other	O
group	O
element	O
corresponds	O
to	O
the	O
alcove	O
that	O
is	O
the	O
image	O
of	O
under	O
the	O
action	O
of	O
.	O
</s>
<s>
Combinatorially	O
,	O
the	O
affine	B-Algorithm
permutation	I-Algorithm
has	O
window	O
notation	O
,	O
corresponding	O
to	O
the	O
bijection	B-Algorithm
for	O
every	O
integer	O
.	O
</s>
<s>
The	O
affine	B-Algorithm
permutation	I-Algorithm
has	O
window	O
notation	O
,	O
corresponding	O
to	O
the	O
bijection	B-Algorithm
for	O
every	O
integer	O
.	O
</s>
<s>
It	O
is	O
natural	O
to	O
identify	O
the	O
line	O
with	O
the	O
real	O
line	O
,	O
with	O
reflection	B-Algorithm
around	O
the	O
point	O
,	O
and	O
with	O
reflection	B-Algorithm
around	O
the	O
point	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
reflection	B-Algorithm
reflects	O
across	O
the	O
point	O
for	O
any	O
integer	O
,	O
the	O
composition	O
translates	B-Algorithm
the	O
line	O
by	O
,	O
and	O
the	O
composition	O
translates	B-Algorithm
the	O
line	O
by	O
.	O
</s>
<s>
Many	O
permutation	B-Algorithm
statistics	O
and	O
other	O
features	O
of	O
the	O
combinatorics	O
of	O
finite	O
permutations	B-Algorithm
can	O
be	O
extended	O
to	O
the	O
affine	O
case	O
.	O
</s>
<s>
The	O
length	O
of	O
an	O
element	O
of	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
the	O
smallest	O
number	O
such	O
that	O
can	O
be	O
written	O
as	O
a	O
product	O
of	O
Coxeter	O
generators	O
of	O
.	O
</s>
<s>
Similarly	O
,	O
there	O
is	O
an	O
affine	O
analogue	O
of	O
descents	O
in	O
permutations	B-Algorithm
:	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
has	O
a	O
descent	O
in	O
position	O
if	O
.	O
</s>
<s>
Algebraically	O
,	O
the	O
descents	O
corresponds	O
to	O
the	O
right	O
descents	O
in	O
the	O
sense	O
of	O
Coxeter	B-Algorithm
groups	I-Algorithm
;	O
that	O
is	O
,	O
is	O
a	O
descent	O
of	O
if	O
and	O
only	O
if	O
.	O
</s>
<s>
The	O
left	O
descents	O
(	O
that	O
is	O
,	O
those	O
indices	O
such	O
that	O
are	O
the	O
descents	O
of	O
the	O
inverse	O
affine	B-Algorithm
permutation	I-Algorithm
;	O
equivalently	O
,	O
they	O
are	O
the	O
values	O
such	O
that	O
occurs	O
before	O
in	O
the	O
sequence	O
.	O
</s>
<s>
Because	O
there	O
are	O
only	O
finitely	O
many	O
possibilities	O
for	O
the	O
number	O
of	O
descents	O
of	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
,	O
but	O
infinitely	O
many	O
affine	B-Algorithm
permutations	I-Algorithm
,	O
it	O
is	O
not	O
possible	O
to	O
naively	O
form	O
a	O
generating	O
function	O
for	O
affine	B-Algorithm
permutations	I-Algorithm
by	O
number	O
of	O
descents	O
(	O
an	O
affine	O
analogue	O
of	O
Eulerian	O
polynomials	O
)	O
.	O
</s>
<s>
One	O
possible	O
resolution	O
is	O
to	O
consider	O
affine	O
descents	O
(	O
equivalently	O
,	O
cyclic	O
descents	O
)	O
in	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Another	O
is	O
to	O
consider	O
simultaneously	O
the	O
length	O
and	O
number	O
of	O
descents	O
of	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
number	O
of	O
descents	O
of	O
the	O
affine	B-Algorithm
permutation	I-Algorithm
and	O
is	O
the	O
-exponential	O
function	O
.	O
</s>
<s>
Any	O
bijection	B-Algorithm
partitions	O
the	O
integers	O
into	O
a	O
(	O
possibly	O
infinite	O
)	O
list	O
of	O
(	O
possibly	O
infinite	O
)	O
cycles	O
:	O
for	O
each	O
integer	O
,	O
the	O
cycle	O
containing	O
is	O
the	O
sequence	O
where	O
exponentiation	O
represents	O
functional	O
composition	O
.	O
</s>
<s>
For	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
,	O
the	O
following	O
conditions	O
are	O
equivalent	O
:	O
all	O
cycles	O
of	O
are	O
finite	O
,	O
has	O
finite	O
order	O
,	O
and	O
the	O
geometric	O
action	O
of	O
on	O
the	O
space	O
has	O
at	O
least	O
one	O
fixed	O
point	O
.	O
</s>
<s>
The	O
reflection	B-Algorithm
length	O
of	O
an	O
element	O
of	O
is	O
the	O
smallest	O
number	O
such	O
that	O
there	O
exist	O
reflections	O
such	O
that	O
.	O
</s>
<s>
(	O
In	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
reflections	O
are	O
transpositions	O
,	O
and	O
the	O
reflection	B-Algorithm
length	O
of	O
a	O
permutation	B-Algorithm
is	O
,	O
where	O
is	O
the	O
number	O
of	O
cycles	O
of	O
.	O
)	O
</s>
<s>
In	O
,	O
the	O
following	O
formula	O
was	O
proved	O
for	O
the	O
reflection	B-Algorithm
length	O
of	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
:	O
for	O
each	O
cycle	O
of	O
,	O
define	O
the	O
weight	O
to	O
be	O
the	O
integer	O
k	O
such	O
that	O
consecutive	O
entries	O
congruent	O
modulo	O
differ	O
by	O
exactly	O
.	O
</s>
<s>
Form	O
a	O
tuple	B-Application
of	O
cycle	O
weights	O
of	O
(	O
counting	O
translates	B-Algorithm
of	O
the	O
same	O
cycle	O
by	O
multiples	O
of	O
only	O
once	O
)	O
,	O
and	O
define	O
the	O
nullity	O
to	O
be	O
the	O
size	O
of	O
the	O
smallest	O
set	O
partition	O
of	O
this	O
tuple	B-Application
so	O
that	O
each	O
part	O
sums	O
to	O
0	O
.	O
</s>
<s>
where	O
is	O
the	O
underlying	O
permutation	B-Algorithm
of	O
.	O
</s>
<s>
For	O
every	O
affine	B-Algorithm
permutation	I-Algorithm
,	O
there	O
is	O
a	O
choice	O
of	O
subgroup	O
of	O
such	O
that	O
,	O
,	O
and	O
for	O
the	O
standard	O
form	O
implied	O
by	O
this	O
semidirect	O
product	O
,	O
the	O
reflection	B-Algorithm
lengths	O
are	O
additive	O
,	O
that	O
is	O
,	O
.	O
</s>
<s>
A	O
reduced	O
word	O
for	O
an	O
element	O
of	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
a	O
tuple	B-Application
of	O
Coxeter	O
generators	O
of	O
minimum	O
possible	O
length	O
such	O
that	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
the	O
element	O
is	O
fully	O
commutative	O
,	O
since	O
its	O
two	O
reduced	O
words	O
and	O
can	O
be	O
connected	O
by	O
swapping	O
commuting	O
factors	O
,	O
but	O
is	O
not	O
fully	O
commutative	O
because	O
there	O
is	O
no	O
way	O
to	O
reach	O
the	O
reduced	O
word	O
starting	O
from	O
the	O
reduced	O
word	O
by	O
commutations	O
.	O
</s>
<s>
proved	O
that	O
in	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
a	O
permutation	B-Algorithm
is	O
fully	O
commutative	O
if	O
and	O
only	O
if	O
it	O
avoids	O
the	O
permutation	B-Algorithm
pattern	I-Algorithm
321	O
,	O
that	O
is	O
,	O
if	O
and	O
only	O
if	O
its	O
one-line	O
notation	O
contains	O
no	O
three-term	O
decreasing	O
subsequence	O
.	O
</s>
<s>
In	O
,	O
this	O
result	O
was	O
extended	O
to	O
affine	B-Algorithm
permutations	I-Algorithm
:	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
is	O
fully	O
commutative	O
if	O
and	O
only	O
if	O
there	O
do	O
not	O
exist	O
integers	O
such	O
that	O
.	O
</s>
<s>
The	O
number	O
of	O
affine	B-Algorithm
permutations	I-Algorithm
avoiding	O
a	O
single	O
pattern	O
is	O
finite	O
if	O
and	O
only	O
if	O
avoids	O
the	O
pattern	O
321	O
,	O
so	O
in	O
particular	O
there	O
are	O
infinitely	O
many	O
fully	O
commutative	O
affine	B-Algorithm
permutations	I-Algorithm
.	O
</s>
<s>
Other	O
aspects	O
of	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
,	O
such	O
as	O
its	O
Bruhat	O
order	O
and	O
representation	O
theory	O
,	O
may	O
also	O
be	O
understood	O
via	O
combinatorial	O
models	O
.	O
</s>
<s>
A	O
standard	O
parabolic	O
subgroup	O
of	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
a	O
subgroup	O
generated	O
by	O
a	O
subset	O
of	O
its	O
Coxeter	O
generating	O
set	O
.	O
</s>
<s>
In	O
,	O
all	O
maximal	O
parabolic	O
subgroups	O
are	O
isomorphic	O
to	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
subgroup	O
generated	O
by	O
the	O
subset	O
consists	O
of	O
those	O
affine	B-Algorithm
permutations	I-Algorithm
that	O
stabilize	O
the	O
interval	O
,	O
that	O
is	O
,	O
that	O
map	O
every	O
element	O
of	O
this	O
interval	O
to	O
another	O
element	O
of	O
the	O
interval	O
.	O
</s>
<s>
The	O
collection	O
of	O
such	O
representatives	O
,	O
denoted	O
,	O
consists	O
of	O
the	O
following	O
affine	B-Algorithm
permutations	I-Algorithm
:	O
</s>
<s>
Then	O
for	O
two	O
affine	B-Algorithm
permutations	I-Algorithm
,	O
,	O
one	O
has	O
that	O
in	O
Bruhat	O
order	O
if	O
and	O
only	O
if	O
for	O
all	O
integers	O
,	O
.	O
</s>
<s>
In	O
the	O
finite	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
the	O
Robinson	B-Algorithm
–	I-Algorithm
Schensted	I-Algorithm
correspondence	I-Algorithm
gives	O
a	O
bijection	B-Algorithm
between	O
the	O
group	O
and	O
pairs	O
of	O
standard	O
Young	O
tableaux	O
of	O
the	O
same	O
shape	O
.	O
</s>
<s>
This	O
bijection	B-Algorithm
plays	O
a	O
central	O
role	O
in	O
the	O
combinatorics	O
and	O
the	O
representation	O
theory	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
language	O
of	O
Kazhdan	O
–	O
Lusztig	O
theory	O
,	O
two	O
permutations	B-Algorithm
lie	O
in	O
the	O
same	O
left	O
cell	O
if	O
and	O
only	O
if	O
their	O
images	O
under	O
Robinson	O
–	O
Schensted	O
have	O
the	O
same	O
tableau	O
,	O
and	O
in	O
the	O
same	O
right	O
cell	O
if	O
and	O
only	O
if	O
their	O
images	O
have	O
the	O
same	O
tableau	O
.	O
</s>
<s>
In	O
,	O
Jian-Yi	O
Shi	O
showed	O
that	O
left	O
cells	O
for	O
are	O
indexed	O
instead	O
by	O
tabloids	O
,	O
and	O
in	O
he	O
gave	O
an	O
algorithm	O
to	O
compute	O
the	O
tabloid	O
analogous	O
to	O
the	O
tableau	O
for	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
.	O
</s>
<s>
In	O
,	O
the	O
authors	O
extended	O
Shi	O
's	O
work	O
to	O
give	O
a	O
bijective	B-Algorithm
map	I-Algorithm
between	O
and	O
triples	O
consisting	O
of	O
two	O
tabloids	O
of	O
the	O
same	O
shape	O
and	O
an	O
integer	O
vector	O
whose	O
entries	O
satisfy	O
certain	O
inequalities	O
.	O
</s>
<s>
Their	O
procedure	O
uses	O
the	O
matrix	O
representation	O
of	O
affine	B-Algorithm
permutations	I-Algorithm
and	O
generalizes	O
the	O
shadow	O
construction	O
,	O
introduced	O
in	O
.	O
</s>
<s>
In	O
some	O
situations	O
,	O
one	O
may	O
wish	O
to	O
consider	O
the	O
action	O
of	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
on	O
or	O
on	O
alcoves	O
that	O
is	O
inverse	O
to	O
the	O
one	O
given	O
above	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
action	O
of	O
a	O
general	O
reflection	B-Algorithm
will	O
be	O
to	O
switch	O
the	O
entries	O
at	O
positions	O
and	O
for	O
each	O
,	O
fixing	O
all	O
inputs	O
at	O
positions	O
not	O
congruent	O
to	O
or	O
modulo	O
.	O
</s>
<s>
The	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
is	O
closely	O
related	O
to	O
a	O
variety	O
of	O
other	O
mathematical	O
objects	O
.	O
</s>
<s>
In	O
,	O
a	O
correspondence	O
is	O
given	O
between	O
affine	B-Algorithm
permutations	I-Algorithm
and	O
juggling	O
patterns	O
encoded	O
in	O
a	O
version	O
of	O
siteswap	O
notation	O
.	O
</s>
<s>
Then	O
is	O
an	O
affine	B-Algorithm
permutation	I-Algorithm
in	O
,	O
and	O
moreover	O
every	O
affine	B-Algorithm
permutation	I-Algorithm
arises	O
from	O
a	O
juggling	O
pattern	O
in	O
this	O
way	O
.	O
</s>
<s>
Under	O
this	O
bijection	B-Algorithm
,	O
the	O
length	O
of	O
the	O
affine	B-Algorithm
permutation	I-Algorithm
is	O
encoded	O
by	O
a	O
natural	O
statistic	O
in	O
the	O
juggling	O
pattern	O
:	O
</s>
<s>
This	O
allows	O
an	O
elementary	O
proof	O
of	O
the	O
generating	O
function	O
for	O
affine	B-Algorithm
permutations	I-Algorithm
by	O
length	O
.	O
</s>
<s>
Therefore	O
,	O
it	O
corresponds	O
to	O
the	O
affine	B-Algorithm
permutation	I-Algorithm
.	O
</s>
<s>
The	O
juggling	O
pattern	O
has	O
four	O
crossings	O
,	O
and	O
the	O
affine	B-Algorithm
permutation	I-Algorithm
has	O
length	O
.	O
</s>
<s>
In	O
a	O
finite-dimensional	O
real	O
inner	O
product	O
space	O
,	O
a	O
reflection	B-Algorithm
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
that	O
fixes	O
a	O
linear	O
hyperplane	O
pointwise	O
and	O
negates	O
the	O
vector	O
orthogonal	O
to	O
the	O
plane	O
.	O
</s>
<s>
In	O
particular	O
,	O
in	O
a	O
complex	O
inner	O
product	O
space	O
,	O
a	O
reflection	B-Algorithm
is	O
a	O
unitary	B-Algorithm
transformation	I-Algorithm
of	O
finite	O
order	O
that	O
fixes	O
a	O
hyperplane	O
.	O
</s>
<s>
A	O
complex	O
reflection	B-Algorithm
group	O
is	O
a	O
finite	O
group	O
of	O
linear	B-Architecture
transformations	I-Architecture
on	O
a	O
complex	O
vector	O
space	O
generated	O
by	O
reflections	O
.	O
</s>
<s>
The	O
complex	O
reflection	B-Algorithm
groups	O
were	O
fully	O
classified	O
by	O
:	O
each	O
complex	O
reflection	B-Algorithm
group	O
is	O
isomorphic	O
to	O
a	O
product	O
of	O
irreducible	O
complex	O
reflection	B-Algorithm
groups	O
,	O
and	O
every	O
irreducible	O
either	O
belongs	O
to	O
an	O
infinite	O
family	O
(	O
where	O
,	O
,	O
and	O
are	O
positive	O
integers	O
such	O
that	O
divides	O
)	O
or	O
is	O
one	O
of	O
34	O
other	O
(	O
so-called	O
"	O
exceptional	O
"	O
)	O
examples	O
.	O
</s>
<s>
The	O
group	O
is	O
the	O
generalized	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
:	O
algebraically	O
,	O
it	O
is	O
the	O
wreath	O
product	O
of	O
the	O
cyclic	O
group	O
with	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Concretely	O
,	O
the	O
elements	O
of	O
the	O
group	O
may	O
be	O
represented	O
by	O
monomial	B-Algorithm
matrices	I-Algorithm
(	O
matrices	O
having	O
one	O
nonzero	O
entry	O
in	O
every	O
row	O
and	O
column	O
)	O
whose	O
nonzero	O
entries	O
are	O
all	O
th	O
roots	O
of	O
unity	O
.	O
</s>
<s>
In	O
,	O
Shi	O
showed	O
that	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
is	O
a	O
generic	O
cover	O
of	O
the	O
family	O
,	O
in	O
the	O
following	O
sense	O
:	O
for	O
every	O
positive	O
integer	O
,	O
there	O
is	O
a	O
surjection	B-Algorithm
from	O
to	O
,	O
and	O
these	O
maps	O
are	O
compatible	O
with	O
the	O
natural	O
surjections	B-Algorithm
when	O
that	O
come	O
from	O
raising	O
each	O
entry	O
to	O
the	O
th	O
power	O
.	O
</s>
<s>
Moreover	O
,	O
these	O
projections	O
respect	O
the	O
reflection	B-Algorithm
group	O
structure	O
,	O
in	O
that	O
the	O
image	O
of	O
every	O
reflection	B-Algorithm
in	O
under	O
is	O
a	O
reflection	B-Algorithm
in	O
;	O
and	O
similarly	O
when	O
the	O
image	O
of	O
the	O
standard	O
Coxeter	O
element	O
in	O
is	O
a	O
Coxeter	O
element	O
in	O
.	O
</s>
<s>
Each	O
affine	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
associated	O
to	O
an	O
affine	O
Lie	O
algebra	O
,	O
a	O
certain	O
infinite-dimensional	O
non-associative	O
algebra	O
with	O
unusually	O
nice	O
representation-theoretic	O
properties	O
.	O
</s>
<s>
In	O
this	O
association	O
,	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
arises	O
as	O
a	O
group	O
of	O
symmetries	O
of	O
the	O
root	O
space	O
of	O
the	O
Lie	O
algebra	O
(	O
the	O
dual	O
of	O
the	O
Cartan	O
subalgebra	O
)	O
.	O
</s>
<s>
In	O
particular	O
,	O
for	O
the	O
affine	O
Lie	O
algebra	O
of	O
type	O
,	O
associated	O
to	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
,	O
the	O
corresponding	O
Macdonald	O
identity	O
is	O
equivalent	O
to	O
the	O
Jacobi	O
triple	O
product	O
.	O
</s>
<s>
The	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
is	O
a	O
subgroup	O
of	O
the	O
extended	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Its	O
elements	O
are	O
extended	O
affine	B-Algorithm
permutations	I-Algorithm
:	O
bijections	B-Algorithm
such	O
that	O
for	O
all	O
integers	O
.	O
</s>
<s>
Unlike	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
,	O
the	O
extended	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
is	O
not	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
geometric	O
action	O
of	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
places	O
it	O
naturally	O
in	O
the	O
family	O
of	O
affine	O
Coxeter	B-Algorithm
groups	I-Algorithm
,	O
each	O
of	O
which	O
has	O
a	O
similar	O
geometric	O
action	O
on	O
an	O
affine	O
space	O
.	O
</s>
<s>
The	O
combinatorial	O
description	O
of	O
the	O
may	O
also	O
be	O
extended	O
to	O
many	O
of	O
these	O
groups	O
:	O
in	O
,	O
an	O
axiomatic	O
description	O
is	O
given	O
of	O
certain	O
permutation	B-Algorithm
groups	O
acting	O
on	O
(	O
the	O
"	O
George	O
groups	O
"	O
,	O
in	O
honor	O
of	O
George	O
Lusztig	O
)	O
,	O
and	O
it	O
is	O
shown	O
that	O
they	O
are	O
exactly	O
the	O
"	O
classical	O
"	O
Coxeter	B-Algorithm
groups	I-Algorithm
of	O
finite	O
and	O
affine	O
types	O
A	O
,	O
B	O
,	O
C	O
,	O
and	O
D	O
.	O
(	O
In	O
the	O
classification	O
of	O
affine	O
Coxeter	B-Algorithm
groups	I-Algorithm
,	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
is	O
type	O
A	O
.	O
)	O
</s>
<s>
Thus	O
,	O
the	O
combinatorial	O
interpretations	O
of	O
descents	O
,	O
inversions	B-Algorithm
,	O
etc.	O
,	O
carry	O
over	O
in	O
these	O
cases	O
.	O
</s>
<s>
The	O
study	O
of	O
Coxeter	B-Algorithm
groups	I-Algorithm
in	O
general	O
could	O
be	O
said	O
to	O
first	O
arise	O
in	O
the	O
classification	O
of	O
regular	O
polyhedra	O
(	O
the	O
Platonic	O
solids	O
)	O
in	O
ancient	O
Greece	O
.	O
</s>
<s>
The	O
modern	O
systematic	O
study	O
(	O
connecting	O
the	O
algebraic	O
and	O
geometric	O
definitions	O
of	O
finite	O
and	O
affine	O
Coxeter	B-Algorithm
groups	I-Algorithm
)	O
began	O
in	O
work	O
of	O
Coxeter	O
in	O
the	O
1930s	O
.	O
</s>
<s>
The	O
combinatorial	O
description	O
of	O
the	O
affine	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
first	O
appears	O
in	O
work	O
of	O
,	O
and	O
was	O
expanded	O
upon	O
by	O
.	O
</s>
