4 ,
2 ,
2 ,
1
The Hecke algebra for the Symmetric Group
on 9 Letters, with the Partition
[ 4, 2, 2, 1 ]
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 3780
.
The dimensions of the irreducible submodules modules are
160,
78,
48,
40,
26,
16,
8,
1
.
The module M has radical filtration (Loewy series)
1,
2,
2,
2,
3,
3,
3,
3,
3,
5,
5,
5,
5,
7,
7,
7,
8
1,
3,
5,
6,
6,
6,
7,
7,
8,
8,
8,
8,
8,
8,
8
1,
2,
2,
2,
3,
4,
4,
4,
5,
5,
5,
5,
5,
5,
7,
8,
8,
8,
8
4,
4,
5,
5,
5,
6,
6,
6,
7,
7,
8,
8,
8
1,
2,
2,
2,
3,
6,
6,
8,
8,
8,
8,
8,
8,
8,
8
2,
2,
5,
5,
5,
8,
8,
8,
8,
8,
8,
8,
8
4,
4,
4,
4,
4,
5,
5,
5,
5,
5,
5,
5
6,
6,
6,
6,
8,
8,
8,
8,
8,
8,
8
2,
2,
2,
5,
5,
8,
8
5,
5,
8,
8
4,
4
8,
8
2,
2
The module M has socle filtration (socle series)
2,
2
8,
8
4,
4
5,
5,
8,
8
2,
2,
2,
5,
5,
8,
8
6,
6,
6,
6,
8,
8,
8,
8,
8,
8,
8
4,
4,
4,
4,
4,
5,
5,
5,
5,
5,
5,
5
2,
2,
5,
5,
5,
8,
8,
8,
8,
8,
8,
8,
8
1,
2,
2,
2,
3,
6,
6,
8,
8,
8,
8,
8,
8,
8,
8
4,
4,
5,
5,
5,
6,
6,
6,
7,
7,
8,
8,
8
1,
2,
2,
2,
3,
4,
4,
4,
5,
5,
5,
5,
5,
5,
7,
8,
8,
8,
8
1,
3,
5,
6,
6,
6,
7,
7,
8,
8,
8,
8,
8,
8,
8
1,
2,
2,
2,
3,
3,
3,
3,
3,
5,
5,
5,
5,
7,
7,
7,
8
The module M has simple direct summands:
4 copies of simple module number 3
2 copies of simple module number 7
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
8
5
8
socle layers
8
5
8
2).
radical layers
7
1
7
socle layers
7
1
7
3).
radical layers
5
8
2,
8
5
8
8
5
socle layers
5
8
8
5
2,
8
8
5
4).
radical layers
1,
3
3,
7,
7
1,
3
7,
7
1,
3
socle layers
1,
3
7,
7
1,
3
3,
7,
7
1,
3
5).
radical layers
2,
5
6,
8,
8
4,
5,
5
6,
8
2,
8
5,
8
4
8
2
socle layers
2
8
4
5,
8
2,
8
6,
8
4,
5,
5
6,
8,
8
2,
5
6).
radical layers
5
6,
8
2,
5,
8
4,
5
6,
8,
8
2,
8,
8
4,
5,
5
6,
8
8
5
socle layers
5
8
6,
8
4,
5,
5
2,
8,
8
6,
8,
8
4,
5
2,
5,
8
6,
8
5
7).
radical layers
5
6,
8
2,
5,
8
4,
5
6,
8,
8
2,
8,
8
4,
5,
5
6,
8
8
5
socle layers
5
8
6,
8
4,
5,
5
2,
8,
8
6,
8,
8
4,
5
2,
5,
8
6,
8
5
8).
radical layers
2
8
4,
5
6,
8
2,
8
5,
8
4,
5
6,
8,
8
2,
5
8
4
8
2
socle layers
2
8
4
8
2,
5
6,
8,
8
4,
5
5,
8
2,
8
6,
8
4,
5
8
2
9).
radical layers
2
8
4,
5
6,
8
2,
8
5,
8
4,
5
6,
8,
8
2,
5
8
4
8
2
socle layers
2
8
4
8
2,
5
6,
8,
8
4,
5
5,
8
2,
8
6,
8
4,
5
8
2
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
600,
576,
536,
555,
826,
477,
656,
1487
.
The cartan matrix of A is
3,
0,
2,
0,
0,
0,
3,
0
0,
4,
0,
3,
4,
2,
0,
8
2,
0,
4,
0,
0,
0,
3,
0
0,
3,
0,
4,
4,
3,
0,
9
0,
4,
0,
4,
10,
5,
0,
14
0,
2,
0,
3,
5,
4,
0,
7
3,
0,
3,
0,
0,
0,
4,
0
0,
8,
0,
9,
14,
7,
0,
27
The determinant of the Cartan matrix is 105.
The blocks of A consist of the following irreducible
modules:
(1).
1,
3,
7
(2).
2,
4,
5,
6,
8
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
1,
7
3,
7
3,
7
1
socle layers
1
7
1,
3
7,
7
1,
3
Projective module number 2
radical layers
2
8
4,
5
6,
8
2,
8
5,
8
4,
5
6,
8,
8
2,
5
8
4
8
2
socle layers
2
8
4
8
2,
5
6,
8,
8
4,
5
5,
8
2,
8
6,
8
4,
5
8
2
Projective module number 3
radical layers
3
3,
3,
7
1,
7
1,
7
3
socle layers
3
7
1,
3
3,
7,
7
1,
3
Projective module number 4
radical layers
4
6,
8
2,
4,
8
5,
6,
8
4,
5,
8
6,
8,
8,
8
2,
5,
5
8
4
8
2
socle layers
4
8
2
6,
8,
8
4,
5
5,
8
2,
8
4,
6,
8
4,
5,
8
6,
8,
8
2,
5
Projective module number 5
radical layers
5
5,
6,
8
2,
5,
6,
8,
8
4,
5,
5,
5,
8
4,
6,
6,
8,
8,
8
2,
5,
8,
8,
8,
8
2,
4,
4,
5,
5
6,
8,
8
2,
8
5
socle layers
5
8
6,
8
4,
5,
5
2,
5,
5,
8,
8
6,
8,
8,
8,
8
4,
5,
6,
6,
8,
8
2,
4,
4,
5,
5,
5,
8
6,
8,
8,
8
2,
2,
5
Projective module number 6
radical layers
6
4,
5
5,
6,
8
2,
4,
6,
8,
8
5,
5,
6,
8
4,
8
8,
8
2,
5
socle layers
6
4,
5
5,
8
6,
8,
8
4,
5,
6,
8
2,
4,
5,
8
6,
8,
8
2,
5
Projective module number 7
radical layers
7
1,
3,
7
1,
3,
7,
7
1,
3
socle layers
7
1,
3
7,
7,
7
1,
1,
3,
3
Projective module number 8
radical layers
8
2,
4,
5,
8
4,
5,
5,
6,
8,
8,
8
2,
4,
5,
6,
6,
8,
8,
8,
8,
8
2,
4,
5,
5,
5,
6,
8,
8,
8,
8,
8,
8
2,
4,
4,
4,
5,
5,
5,
5,
6,
8
6,
6,
8,
8,
8,
8,
8,
8
2,
2,
4,
5,
5,
8
5,
8,
8
2,
4
8
2
socle layers
8
4
8
2,
2,
5
6,
8,
8,
8,
8
4,
4,
4,
5,
5
5,
5,
5,
8,
8,
8,
8,
8
2,
2,
6,
8,
8,
8,
8,
8,
8
4,
4,
5,
6,
6,
6,
8,
8,
8
2,
4,
4,
4,
5,
5,
5,
5,
5,
8,
8
6,
6,
8,
8,
8,
8,
8
2,
2,
2,
5,
5
The degrees of the splitting fields are
1,
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
163
.
The dimensions of the irreducible H-modules are
4,
2,
2,
2,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
5,
3,
16,
24,
6,
5,
14,
22,
10
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0,
0,
0,
1
0,
1,
0,
0,
0,
1,
0,
0,
0
0,
0,
4,
2,
0,
0,
1,
3,
0
0,
0,
2,
6,
1,
0,
3,
4,
0
0,
0,
0,
1,
2,
0,
1,
1,
0
0,
1,
0,
0,
0,
2,
0,
0,
1
0,
0,
1,
3,
1,
0,
3,
2,
0
0,
0,
3,
4,
1,
0,
2,
5,
0
1,
0,
0,
0,
0,
1,
0,
0,
5
The determinant of the Cartan matrix is 198.
The blocks of H consist of the following irreducible
modules:
(1).
1,
2,
6,
9
(2).
3,
4,
5,
7,
8
The radical and socle filtrations of the projective
modules for H are the following:
Projective module number 1
radical layers
1
9
socle layers
1
9
Projective module number 2
radical layers
2
6
socle layers
2
6
Projective module number 3
radical layers
3
8
3,
4
7,
8
3,
4
8
3
socle layers
3
8
3,
4
7,
8
3,
4
8
3
Projective module number 4
radical layers
4
4,
7,
8
3,
4,
5,
7,
8
4,
4,
8
3,
7,
8
4
socle layers
4
7,
8
4,
4,
8
4,
5,
7,
8
4,
7,
8
3,
3,
4
Projective module number 5
radical layers
5
5,
7,
8
4
socle layers
5
7,
8
4,
5
Projective module number 6
radical layers
6
2,
9
6
socle layers
6
2
6,
9
Projective module number 7
radical layers
7
4,
5,
7
4,
7,
8
3,
8
4
socle layers
7
4
5,
7,
8
4,
7,
8
3,
4
Projective module number 8
radical layers
8
3,
4,
5,
8
4,
7,
8,
8
3,
4,
7
4,
8
3
socle layers
8
3,
4
7,
8,
8
3,
4,
4,
5
7,
8,
8
3,
4
Projective module number 9
radical layers
9
1,
6,
9,
9
9,
9
socle layers
9
1,
6,
9,
9
9,
9
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