As examples we will treat the 
-group algebras 
of symmetric groups.
Here the standard modules 
are known to be the
Specht modules ([GL], 5.7,
see [JK], chapter 7 for details on Specht-modules). The reader
not familiar with cellular algebras may check imediately that the
conditions stated in the begining of section 5
can be realised using this modules.
They are labelled by the set
of partitions of n and are defined as
submodules of permutation modules 
on the cosets of parabolic
subgroups corresponding to the partition 
.
As first example, let us look at 
.
Then
is the permutation representation on the cosets of the copy of 
in
fixing the last element n. As a -module this is free
of rank n with basis 
, say. The operation of
from the right is given by 
.
The Specht module is the free -submodule with
basis 
for 
.
The symmetric bilinear form 
is
given by restriction of the canonical bilinear form on
(given by the unit matrix) to . Thus the Gram matrix
of is nothing but the Cartan matrix
corresponding to the Dynkin diagram of type 
.
This in turn is just the coefficient matrix of
with respect to the basis 
on and the corresponding dual basis 
on
. The image of 
is consequently
the span of the columns of this matrix and is easily calculated to be
the span of 
and 
for 
. Therefore 
- the cokernel of - is generated
by 
and is obviously isomorphic to 
as a -module.
Let us show that the operation of 
on is
trivial. We only need to check this for the long cycle 
or its inverse and 
.
On the operation
is given by
and 
for i>1
in the case of 
whereas we have 
and 
for all i>2. Transposing the
corresponding coefficient matrices we get 
. Since 
is congruent to modulo
the image of we are done.
The Specht module of the partition 
is just the one-dimensional
trivial representation with bilinear form 
being
given by the trivial Gram matrix. Therefore
holds in
. Moreover, we have shown the following equation in
:
In the case of the
symmetric group 
there are three partitions
and 
. The
corresponding Specht modules are free -modules
of rank 1, 2 and 1, respectively. The cases 
and
have been treated in general above. The Specht module
is the sign representation and 
. Therefore we have 
. From the ordinary decomposition matrices
of for the primes 2 and 3 we get
and
leading to the
global decomposition matrix of :
Note that the corresponding equations according to condition (c)
of Proposition 5.6 even hold in . This will not
be true in the next example 
as well as the fact that the
global decomposition matrix can entirely be constructed from
the knowledge of the ordinary decomposition matrices for the primes
2 and 3. On the other hand, the
ordinary decomposition numbers are detected from the
above by multiplying the whole table by [2] resp. [3] and then
setting [2]=1 resp. [3]=1.
Before proceeding to the next example let us write down all the
relations between the 
with the help of
Proposition 5.8.
We have to consider two pairs 
such that 
. These are 
and 
. We calculate
and
. This gives
and
leading to the following complete set of relations for 
and
:
Turning to we have 5 partitions
and 
. Let us first picture
the global decomposition matrix and then comment it:
First note that the entries [8] and [4] cannot be reconstructed from
the knowledge of the ordinary decomposition matrices of .
As in the preceeding example the first row follows easily
from 
and the knowledge
of 
and
. Note that
does not hold in
but surely in 
.
In contrast, the equations corresponding to
the rows 3,4,5 hold in the last two of which have
been treated in the beginning of this chapter.
We leave the calculation of 
as an exercise and turn to the
most interesting case 
.
Here the Specht module is a free -module of rank three with basis
, say. The long cycle 
and
operate from right by 
and 
. The
Gram matrix of the bilinear form is given by
and therefore the image of is spanned by 
and 
. Clearly, is isomorphic
to [8]+[8]+[2] as a -module. But anyway, as an A-module
it is irreducible by -sums as defined in chapter 2. Therefore it cannot be splitted in anymore.
But we can calculate in as in the proof of Lemma
5.1. From the ordinary decomposition matrix for the
prime 2, we get
and by dimension arguments the 2-residual series must be
which completes the verification. In order to determine the relations
among the one calculates
As in the case of a complete set of relations can be
obtained from these formulas with help of Proposition 5.8.
Problems:
and etc. or is it independed up to
permutations on rows and columns?
if you take additional structure on the algebra A into
acount, for example informations on the open subset in 
corresponding to specializations for A being semisimple.