4 ,
3 ,
2
The Hecke algebra for the Symmetric Group
on 9 Letters, with the Partition
[ 4, 3, 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 1260
.
The dimensions of the irreducible submodules modules are
160,
78,
48,
40,
26,
16,
8,
1
.
The module M has radical filtration (Loewy series)
2,
3,
3,
3,
5,
5,
7,
7,
8
1,
5,
6,
6,
8,
8,
8
2,
4,
5,
5,
5,
7,
8,
8
4,
5,
6,
8
2,
6,
8,
8,
8
2,
5,
8,
8,
8
4,
4,
5,
5
6,
8,
8
2,
8
5
The module M has socle filtration (socle series)
5
2,
8
6,
8,
8
4,
4,
5,
5
2,
5,
8,
8,
8
2,
6,
8,
8,
8
4,
5,
6,
8
2,
4,
5,
5,
5,
7,
8,
8
1,
5,
6,
6,
8,
8,
8
2,
3,
3,
3,
5,
5,
7,
7,
8
The module M has simple direct summands:
3 copies of simple module number 3
1 copy of simple module number 7
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
7
1
7
socle layers
7
1
7
2).
radical layers
8
5
8
socle layers
8
5
8
3).
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
4).
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
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
168,
387,
48,
366,
448,
288,
176,
920
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
1,
0
0,
3,
0,
2,
2,
1,
0,
5
0,
0,
1,
0,
0,
0,
0,
0
0,
2,
0,
3,
2,
2,
0,
6
0,
2,
0,
2,
6,
3,
0,
8
0,
1,
0,
2,
3,
3,
0,
4
1,
0,
0,
0,
0,
0,
2,
0
0,
5,
0,
6,
8,
4,
0,
18
The determinant of the Cartan matrix is 11.
The blocks of A consist of the following irreducible
modules:
(1).
1,
7
(2).
2,
4,
5,
6,
8
(3).
3
Projective module number 3 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 1
radical layers
1
7
socle layers
1
7
Projective module number 2
radical layers
2
8
4,
5
6,
8
2,
8
5,
8
4
8
2
socle layers
2
8
4
8
2
8
4,
5
6,
8,
8
2,
5
Projective module number 4
radical layers
4
6,
8
2,
4,
8
5,
6,
8
4,
8
8,
8
2,
5
socle layers
4
8
2,
6,
8
4,
5,
8
4,
8
6,
8,
8
2,
5
Projective module number 5
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
Projective module number 6
radical layers
6
4,
5
6,
8
2,
4,
8
5,
6
8
8
5
socle layers
6
4,
5
8
6,
8
4,
5
2,
8
6,
8
5
Projective module number 7
radical layers
7
1
7
socle layers
7
1
7
Projective module number 8
radical layers
8
2,
4,
5,
8
4,
5,
6,
8,
8,
8
2,
4,
6,
8,
8,
8,
8
2,
4,
5,
5,
8,
8,
8,
8
2,
4,
4,
5,
5,
6,
8
6,
8,
8,
8
2,
5,
8
5
socle layers
8
8
4,
4,
5,
5
2,
8,
8,
8,
8
2,
6,
6,
8,
8,
8,
8
4,
4,
5,
5,
8,
8
2,
4,
4,
5,
5,
8,
8
6,
6,
8,
8,
8,
8
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
39
.
The dimensions of the irreducible H-modules are
3,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
3,
2,
3,
4,
10,
11
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0
0,
1,
1,
0,
0,
0
0,
1,
2,
0,
0,
0
0,
0,
0,
2,
1,
1
0,
0,
0,
1,
5,
4
0,
0,
0,
1,
4,
6
The determinant of the Cartan matrix is 25.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2,
3
(3).
4,
5,
6
Projective module number 1 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 2
radical layers
2
3
socle layers
2
3
Projective module number 3
radical layers
3
2
3
socle layers
3
2
3
Projective module number 4
radical layers
4
4,
5
6
socle layers
4
5
4,
6
Projective module number 5
radical layers
5
4,
5,
5,
6
5,
6,
6
5,
6
socle layers
5
4,
5,
6
5,
5,
6,
6
5,
6
Projective module number 6
radical layers
6
5,
6,
6
4,
5,
5,
6
5,
6
6
socle layers
6
5,
6
4,
5,
5,
6
5,
6,
6
6
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