5 ,
2 ,
2
The Hecke algebra for the Symmetric Group
on 9 Letters, with the Partition
[ 5, 2, 2 ]
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 756
.
The dimensions of the irreducible submodules modules are
78,
48,
40,
26,
16,
8,
1
.
The module M has radical filtration (Loewy series)
1,
2,
2,
4,
4,
6,
6,
7
4,
5,
7,
7,
7
1,
3,
4,
4,
7,
7
4,
5,
7
1,
7,
7
4,
7,
7
3,
4
7
1
The module M has socle filtration (socle series)
1
7
3,
4
4,
7,
7
1,
7,
7
4,
5,
7
1,
3,
4,
4,
7,
7
4,
5,
7,
7,
7
1,
2,
2,
4,
4,
6,
6,
7
The module M has simple direct summands:
2 copies of simple module number 2
2 copies of simple module number 6
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
7
4
7
socle layers
7
4
7
2).
radical layers
4
7
1,
7
4
7
7
4
socle layers
4
7
7
4
1,
7
7
4
3).
radical layers
1,
4
5,
7,
7
3,
4,
4
5,
7
1,
7
4,
7
3
7
1
socle layers
1
7
3
4,
7
1,
7
5,
7
3,
4,
4
5,
7,
7
1,
4
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
387,
48,
282,
321,
161,
8,
709
.
The cartan matrix of A is
3,
0,
2,
2,
1,
0,
5
0,
1,
0,
0,
0,
0,
0
2,
0,
2,
1,
1,
0,
4
2,
0,
1,
4,
1,
0,
5
1,
0,
1,
1,
1,
0,
1
0,
0,
0,
0,
0,
1,
0
5,
0,
4,
5,
1,
0,
13
The determinant of the Cartan matrix is -5.
The blocks of A consist of the following irreducible
modules:
(1).
1,
3,
4,
5,
7
(2).
2
(3).
6
Projective modules number
2,
6
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 1
radical layers
1
7
3,
4
5,
7
1,
7
4,
7
3
7
1
socle layers
1
7
3
7
1
7
3,
4
5,
7,
7
1,
4
Projective module number 3
radical layers
3
5,
7
1,
7
4,
7
3
7
1
socle layers
3
7
1
7
3
5,
7,
7
1,
4
Projective module number 4
radical layers
4
5,
7
1,
4,
7
3,
4
7,
7
1,
7
4
socle layers
4
7
7
4,
5
1,
3,
4,
7
7,
7
1,
4
Projective module number 5
radical layers
5
3,
4
7
1
socle layers
5
3,
4
7
1
Projective module number 7
radical layers
7
1,
3,
4,
7
3,
4,
5,
7,
7,
7
1,
3,
7,
7,
7,
7
1,
4,
4,
7,
7,
7
1,
3,
4
7
1
socle layers
7
3
7
1,
1,
7
4,
4,
7,
7
3,
3,
3,
7,
7
5,
7,
7,
7,
7,
7,
7
1,
1,
1,
4,
4,
4
The degrees of the splitting fields are
1,
1,
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
26
.
The dimensions of the irreducible H-modules are
2,
2,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
2,
2,
4,
8,
6
.
The cartan matrix of H is
1,
0,
0,
0,
0
0,
1,
0,
0,
0
0,
0,
2,
1,
1
0,
0,
1,
5,
2
0,
0,
1,
2,
3
The determinant of the Cartan matrix is 18.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3,
4,
5
Projective modules number
1,
2
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 3
radical layers
3
3,
4,
5
socle layers
3
3,
4,
5
Projective module number 4
radical layers
4
3,
4,
4,
5
4,
4,
5
socle layers
4
3,
4,
4,
5
4,
4,
5
Projective module number 5
radical layers
5
3,
4,
5
4,
5
socle layers
5
3,
4,
5
4,
5
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