<s>
In	O
mathematics	O
,	O
a	O
generalized	B-Algorithm
permutation	I-Algorithm
matrix	I-Algorithm
(	O
or	O
monomial	B-Algorithm
matrix	I-Algorithm
)	O
is	O
a	O
matrix	B-Architecture
with	O
the	O
same	O
nonzero	O
pattern	O
as	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
,	O
i.e.	O
</s>
<s>
Unlike	O
a	O
permutation	B-Algorithm
matrix	I-Algorithm
,	O
where	O
the	O
nonzero	O
entry	O
must	O
be	O
1	O
,	O
in	O
a	O
generalized	B-Algorithm
permutation	I-Algorithm
matrix	I-Algorithm
the	O
nonzero	O
entry	O
can	O
be	O
any	O
nonzero	O
value	O
.	O
</s>
<s>
An	O
invertible	O
matrix	B-Architecture
A	O
is	O
a	O
generalized	B-Algorithm
permutation	I-Algorithm
matrix	I-Algorithm
if	O
and	O
only	O
if	O
it	O
can	O
be	O
written	O
as	O
a	O
product	O
of	O
an	O
invertible	O
diagonal	B-Algorithm
matrix	I-Algorithm
D	O
and	O
an	O
(	O
implicitly	O
invertible	O
)	O
permutation	B-Algorithm
matrix	I-Algorithm
P	O
:	O
i.e.	O
,	O
</s>
<s>
The	O
set	O
of	O
n×n	O
generalized	B-Algorithm
permutation	I-Algorithm
matrices	I-Algorithm
with	O
entries	O
in	O
a	O
field	O
F	O
forms	O
a	O
subgroup	O
of	O
the	O
general	O
linear	O
group	O
GL(n, F )	O
,	O
in	O
which	O
the	O
group	O
of	O
nonsingular	O
diagonal	B-Algorithm
matrices	I-Algorithm
Δ(n, F )	O
forms	O
a	O
normal	O
subgroup	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
generalized	B-Algorithm
permutation	I-Algorithm
matrices	I-Algorithm
are	O
the	O
normalizer	O
of	O
the	O
diagonal	B-Algorithm
matrices	I-Algorithm
,	O
meaning	O
that	O
the	O
generalized	B-Algorithm
permutation	I-Algorithm
matrices	I-Algorithm
are	O
the	O
largest	O
subgroup	O
of	O
GL(n, F )	O
in	O
which	O
diagonal	B-Algorithm
matrices	I-Algorithm
are	O
normal	O
.	O
</s>
<s>
The	O
abstract	O
group	O
of	O
generalized	B-Algorithm
permutation	I-Algorithm
matrices	I-Algorithm
is	O
the	O
wreath	O
product	O
of	O
F×	O
and	O
Sn	O
.	O
</s>
<s>
Concretely	O
,	O
this	O
means	O
that	O
it	O
is	O
the	O
semidirect	O
product	O
of	O
Δ(n, F )	O
by	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
Sn	O
:	O
</s>
<s>
where	O
Sn	O
acts	O
by	O
permuting	O
coordinates	O
and	O
the	O
diagonal	B-Algorithm
matrices	I-Algorithm
Δ(n, F )	O
are	O
isomorphic	O
to	O
the	O
n-fold	O
product	O
(	O
F×	O
)	O
n	O
.	O
</s>
<s>
To	O
be	O
precise	O
,	O
the	O
generalized	B-Algorithm
permutation	I-Algorithm
matrices	I-Algorithm
are	O
a	O
(	O
faithful	O
)	O
linear	O
representation	O
of	O
this	O
abstract	O
wreath	O
product	O
:	O
a	O
realization	O
of	O
the	O
abstract	O
group	O
as	O
a	O
subgroup	O
of	O
matrices	O
.	O
</s>
<s>
The	O
subgroup	O
where	O
all	O
entries	O
are	O
1	O
is	O
exactly	O
the	O
permutation	B-Algorithm
matrices	I-Algorithm
,	O
which	O
is	O
isomorphic	O
to	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
subgroup	O
where	O
all	O
entries	O
are	O
±1	O
is	O
the	O
signed	O
permutation	B-Algorithm
matrices	I-Algorithm
,	O
which	O
is	O
the	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
subgroup	O
where	O
the	O
entries	O
are	O
mth	O
roots	O
of	O
unity	O
is	O
isomorphic	O
to	O
a	O
generalized	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
subgroup	O
of	O
diagonal	B-Algorithm
matrices	I-Algorithm
is	O
abelian	O
,	O
normal	O
,	O
and	O
a	O
maximal	O
abelian	O
subgroup	O
.	O
</s>
<s>
The	O
quotient	O
group	O
is	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
and	O
this	O
construction	O
is	O
in	O
fact	O
the	O
Weyl	O
group	O
of	O
the	O
general	O
linear	O
group	O
:	O
the	O
diagonal	B-Algorithm
matrices	I-Algorithm
are	O
a	O
maximal	O
torus	O
in	O
the	O
general	O
linear	O
group	O
(	O
and	O
are	O
their	O
own	O
centralizer	O
)	O
,	O
the	O
generalized	B-Algorithm
permutation	I-Algorithm
matrices	I-Algorithm
are	O
the	O
normalizer	O
of	O
this	O
torus	O
,	O
and	O
the	O
quotient	O
,	O
is	O
the	O
Weyl	O
group	O
.	O
</s>
<s>
If	O
a	O
nonsingular	O
matrix	B-Architecture
and	O
its	O
inverse	O
are	O
both	O
nonnegative	B-Algorithm
matrices	I-Algorithm
(	O
i.e.	O
</s>
<s>
matrices	O
with	O
nonnegative	O
entries	O
)	O
,	O
then	O
the	O
matrix	B-Architecture
is	O
a	O
generalized	B-Algorithm
permutation	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
generalized	B-Algorithm
permutation	I-Algorithm
matrix	I-Algorithm
is	O
given	O
by	O
where	O
is	O
the	O
sign	O
of	O
the	O
permutation	B-Algorithm
associated	O
with	O
and	O
are	O
the	O
diagonal	O
elements	O
of	O
.	O
</s>
<s>
One	O
may	O
also	O
schematically	O
allow	O
the	O
non-zero	O
entries	O
to	O
lie	O
in	O
a	O
group	O
G	O
,	O
with	O
the	O
understanding	O
that	O
matrix	B-Architecture
multiplication	O
will	O
only	O
involve	O
multiplying	O
a	O
single	O
pair	O
of	O
group	O
elements	O
,	O
not	O
"	O
adding	O
"	O
group	O
elements	O
.	O
</s>
<s>
This	O
is	O
an	O
abuse	O
of	O
notation	O
,	O
since	O
element	O
of	O
matrices	O
being	O
multiplied	O
must	O
allow	O
multiplication	O
and	O
addition	O
,	O
but	O
is	O
suggestive	O
notion	O
for	O
the	O
(	O
formally	O
correct	O
)	O
abstract	O
group	O
(	O
the	O
wreath	O
product	O
of	O
the	O
group	O
G	O
by	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
)	O
.	O
</s>
<s>
A	O
signed	O
permutation	B-Algorithm
matrix	I-Algorithm
is	O
a	O
generalized	B-Algorithm
permutation	I-Algorithm
matrix	I-Algorithm
whose	O
nonzero	O
entries	O
are	O
±1	O
,	O
and	O
are	O
the	O
integer	O
generalized	B-Algorithm
permutation	I-Algorithm
matrices	I-Algorithm
with	O
integer	O
inverse	O
.	O
</s>
<s>
It	O
is	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
and	O
has	O
order	O
.	O
</s>
<s>
It	O
is	O
the	O
symmetry	O
group	O
of	O
the	O
hypercube	B-Operating_System
and	O
(	O
dually	O
)	O
of	O
the	O
cross-polytope	O
.	O
</s>
<s>
Its	O
index	O
2	O
subgroup	O
of	O
matrices	O
with	O
determinant	O
equal	O
to	O
their	O
underlying	O
(	O
unsigned	O
)	O
permutation	B-Algorithm
is	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
and	O
is	O
the	O
symmetry	O
group	O
of	O
the	O
demihypercube	O
.	O
</s>
<s>
Monomial	B-Algorithm
matrices	I-Algorithm
occur	O
in	O
representation	O
theory	O
in	O
the	O
context	O
of	O
monomial	O
representations	O
.	O
</s>
<s>
A	O
monomial	O
representation	O
of	O
a	O
group	O
G	O
is	O
a	O
linear	O
representation	O
of	O
G	O
(	O
here	O
F	O
is	O
the	O
defining	O
field	O
of	O
the	O
representation	O
)	O
such	O
that	O
the	O
image	O
ρ(G )	O
is	O
a	O
subgroup	O
of	O
the	O
group	O
of	O
monomial	B-Algorithm
matrices	I-Algorithm
.	O
</s>
