5 ,
4
The Hecke algebra for the Symmetric Group
on 9 Letters, with the Partition
[ 5, 4 ]
in characteristic 2
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 2, having point stablilizer
equal to the Young subgroup of the partition.
The dimension of M is 126
.
The dimensions of the irreducible submodules modules are
48,
26,
16,
8,
1
.
The module M has radical filtration (Loewy series)
1,
2,
4,
5
3,
5
2
The module M has socle filtration (socle series)
2
3,
5
1,
2,
4,
5
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 4
The remaining indecomposable component of M has
radical and socle filtrations as follows:
1).
radical layers
2,
5
3,
5
2
socle layers
2
3,
5
2,
5
The Action Algebra
The action algebra A is the image of kG in the
k-endomorphism ring of M. It's simple modules are the irreducible
submodules of M.
The dimensions of the projective modules are
48,
69,
42,
8,
27
.
The cartan matrix of A is
1,
0,
0,
0,
0
0,
2,
1,
0,
1
0,
1,
1,
0,
0
0,
0,
0,
1,
0
0,
1,
0,
0,
1
The determinant of the Cartan matrix is 0.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
3,
5
(3).
4
Projective modules number
1,
4
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
3,
5
2
socle layers
2
3
2,
5
Projective module number 3
radical layers
3
2
socle layers
3
2
Projective module number 5
radical layers
5
2
socle layers
5
2
The degrees of the splitting fields are
1,
1,
1,
1,
1
.
The Hecke Algebra
The Hecke algebra H of the module M is the A-endomorphism
ring of M.
The dimension of H is
5
.
The dimensions of the irreducible H-modules are
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1
.
The dimensions of the projective modules of H are
1,
1,
3
.
The cartan matrix of H is
The determinant of the Cartan matrix is 3.
The blocks of H consist of the following irreducible
modules:
Projective modules number
1,
2
are simple.
The radical and socle filtrations of the remaining
projective module for H are the following:
Projective module number 3
radical layers
3
3,
3
socle layers
3
3,
3
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