<s>
In	O
mathematics	O
,	O
a	O
Brauer	B-Application
algebra	I-Application
is	O
an	O
associative	O
algebra	O
introduced	O
by	O
Richard	O
Brauer	O
in	O
the	O
context	O
of	O
the	O
representation	O
theory	O
of	O
the	O
orthogonal	O
group	O
.	O
</s>
<s>
It	O
plays	O
the	O
same	O
role	O
that	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
does	O
for	O
the	O
representation	O
theory	O
of	O
the	O
general	O
linear	O
group	O
in	O
Schur	O
–	O
Weyl	O
duality	O
.	O
</s>
<s>
The	O
Brauer	B-Application
algebra	I-Application
is	O
a	O
-algebra	O
depending	O
on	O
the	O
choice	O
of	O
a	O
positive	O
integer	O
.	O
</s>
<s>
Relations	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
:	O
</s>
<s>
The	O
Brauer	B-Application
algebra	I-Application
is	O
a	O
subalgebra	O
of	O
the	O
partition	B-Application
algebra	I-Application
.	O
</s>
<s>
The	O
Brauer	B-Application
algebra	I-Application
is	O
semisimple	O
if	O
.	O
</s>
<s>
The	O
subalgebra	O
of	O
generated	O
by	O
the	O
generators	O
is	O
the	O
group	O
algebra	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
The	O
subalgebra	O
of	O
generated	O
by	O
the	O
generators	O
is	O
the	O
Temperley-Lieb	B-Application
algebra	I-Application
.	O
</s>
<s>
The	O
Brauer	B-Application
algebra	I-Application
is	O
a	O
cellular	O
algebra	O
.	O
</s>
<s>
Brauer-Specht	O
modules	O
are	O
finite-dimensional	O
modules	O
of	O
the	O
Brauer	B-Application
algebra	I-Application
.	O
</s>
<s>
These	O
modules	O
are	O
parametrized	O
by	O
partitions	O
,	O
because	O
they	O
are	O
built	O
from	O
the	O
Specht	O
modules	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
which	O
are	O
themselves	O
parametrized	O
by	O
partitions	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
Brauer	B-Application
algebra	I-Application
has	O
a	O
natural	O
action	O
on	O
the	O
space	O
of	O
polynomials	O
on	O
,	O
which	O
commutes	O
with	O
the	O
action	O
of	O
the	O
orthogonal	O
group	O
.	O
</s>
<s>
If	O
is	O
a	O
negative	O
even	O
integer	O
,	O
the	O
Brauer	B-Application
algebra	I-Application
is	O
related	O
by	O
Schur-Weyl	O
duality	O
to	O
the	O
symplectic	O
group	O
,	O
rather	O
than	O
the	O
orthogonal	O
group	O
.	O
</s>
<s>
The	O
walled	O
Brauer	B-Application
algebra	I-Application
is	O
a	O
subalgebra	O
of	O
.	O
</s>
<s>
The	O
walled	O
Brauer	B-Application
algebra	I-Application
is	O
generated	O
by	O
.	O
</s>
<s>
The	O
walled	O
Brauer	B-Application
algebra	I-Application
has	O
a	O
natural	O
action	O
on	O
,	O
which	O
is	O
related	O
by	O
Schur-Weyl	O
duality	O
to	O
the	O
action	O
of	O
.	O
</s>
