<s>
The	O
partition	B-Application
algebra	I-Application
is	O
an	O
associative	O
algebra	O
with	O
a	O
basis	O
of	O
set-partition	O
diagrams	O
and	O
multiplication	O
given	O
by	O
diagram	O
concatenation	O
.	O
</s>
<s>
Its	O
subalgebras	O
include	O
diagram	O
algebras	O
such	O
as	O
the	O
Brauer	B-Application
algebra	I-Application
,	O
the	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
,	O
or	O
the	O
group	O
algebra	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Representations	O
of	O
the	O
partition	B-Application
algebra	I-Application
are	O
built	O
from	O
sets	O
of	O
diagrams	O
and	O
from	O
representations	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
For	O
and	O
,	O
the	O
partition	B-Application
algebra	I-Application
is	O
defined	O
by	O
a	O
-basis	O
made	O
of	O
partitions	O
,	O
and	O
a	O
multiplication	O
given	O
by	O
diagram	O
concatenation	O
.	O
</s>
<s>
The	O
partition	B-Application
algebra	I-Application
is	O
an	O
associative	O
algebra	O
.	O
</s>
<s>
The	O
partition	B-Application
algebra	I-Application
is	O
semisimple	O
for	O
.	O
</s>
<s>
The	O
partition	B-Application
algebra	I-Application
is	O
finite-dimensional	O
,	O
with	O
(	O
a	O
Bell	O
number	O
)	O
.	O
</s>
<s>
Subalgebras	O
of	O
the	O
partition	B-Application
algebra	I-Application
can	O
be	O
defined	O
by	O
the	O
following	O
properties	O
:	O
</s>
<s>
Combining	O
these	O
properties	O
gives	O
rise	O
to	O
8	O
nontrivial	O
subalgebras	O
,	O
in	O
addition	O
to	O
the	O
partition	B-Application
algebra	I-Application
itself	O
:	O
</s>
<s>
The	O
symmetric	B-Algorithm
group	I-Algorithm
algebra	O
is	O
the	O
group	O
ring	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
over	O
.	O
</s>
<s>
The	O
Motzkin	O
algebra	O
is	O
sometimes	O
called	O
the	O
dilute	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
in	O
the	O
physics	O
literature	O
.	O
</s>
<s>
Partition	O
diagrams	O
act	O
on	O
from	O
the	O
bottom	O
,	O
while	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
acts	O
from	O
the	O
top	O
.	O
</s>
<s>
Representations	O
of	O
non-planar	O
subalgebras	O
have	O
similar	O
structures	O
as	O
representations	O
of	O
the	O
partition	B-Application
algebra	I-Application
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Brauer-Specht	O
modules	O
of	O
the	O
Brauer	B-Application
algebra	I-Application
are	O
built	O
from	O
Specht	O
modules	O
,	O
and	O
certain	O
sets	O
of	O
partitions	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
standard	O
module	O
of	O
the	O
Temperley	B-Application
–	I-Application
Lieb	I-Application
algebra	I-Application
is	O
parametrized	O
by	O
an	O
integer	O
with	O
,	O
and	O
a	O
basis	O
is	O
simply	O
given	O
by	O
a	O
set	O
of	O
partitions	O
.	O
</s>
<s>
The	O
following	O
table	O
lists	O
the	O
irreducible	O
representations	O
of	O
the	O
partition	B-Application
algebra	I-Application
and	O
eight	O
subalgebras	O
.	O
</s>
<s>
For	O
a	O
-dimensional	O
vector	O
space	O
with	O
basis	O
,	O
there	O
is	O
a	O
natural	O
action	O
of	O
the	O
partition	B-Application
algebra	I-Application
on	O
the	O
vector	O
space	O
.	O
</s>
<s>
Let	O
us	O
take	O
to	O
be	O
the	O
natural	O
permutation	O
representation	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Then	O
the	O
partition	B-Application
algebra	I-Application
is	O
the	O
centralizer	O
of	O
the	O
action	O
of	O
on	O
the	O
tensor	O
product	O
space	O
,	O
</s>
<s>
The	O
duality	O
between	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
and	O
the	O
partition	B-Application
algebra	I-Application
generalizes	O
the	O
original	O
Schur-Weyl	O
duality	O
between	O
the	O
general	O
linear	O
group	O
and	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Tensor	O
product	O
space	O
Group	O
or	O
algebra	O
Dual	O
algebra	O
or	O
group	O
Comments	O
The	O
duality	O
for	O
the	O
full	O
partition	B-Application
algebra	I-Application
Case	O
of	O
a	O
partition	B-Application
algebra	I-Application
with	O
a	O
half-integer	O
index	O
The	O
original	O
Schur-Weyl	O
duality	O
Duality	O
between	O
the	O
orthogonal	O
group	O
and	O
the	O
Brauer	B-Application
algebra	I-Application
Duality	O
between	O
the	O
orthogonal	O
group	O
and	O
the	O
rook	O
Brauer	B-Application
algebra	I-Application
Duality	O
between	O
the	O
rook	O
algebra	O
and	O
the	O
totally	O
propagating	O
partition	B-Application
algebra	I-Application
Duality	O
between	O
a	O
Lie	O
superalgebra	O
and	O
the	O
planar	O
rook	O
algebra	O
Duality	O
between	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
and	O
the	O
quasi-partition	O
algebra	O
Duality	O
involving	O
the	O
walled	O
Brauer	B-Application
algebra	I-Application
.	O
</s>
