4 ,
2 ,
1 ,
1
The Hecke algebra for the Symmetric Group
on 8 Letters, with the Partition
[ 4, 2, 1, 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 840
.
The dimensions of the irreducible submodules modules are
64,
40,
14,
8,
6,
1
.
The module M has radical filtration (Loewy series)
1,
1,
1,
1,
3,
3,
5,
5,
6
3,
3,
4,
5,
5,
5,
6,
6,
6
2,
3,
4,
4,
5,
5,
5,
6,
6,
6
2,
3,
3,
3,
4,
5,
5,
5,
6
3,
3,
3,
4,
5,
5,
5,
5,
6,
6
3,
3,
4,
4,
5,
6,
6,
6
2,
5,
5,
5
3,
4,
6
3,
5
6
2
6
3
The module M has socle filtration (socle series)
3
6
2
6
3,
5
3,
4,
6
2,
5,
5,
5
3,
3,
4,
4,
5,
6,
6,
6
3,
3,
3,
4,
5,
5,
5,
5,
6,
6
2,
3,
3,
3,
4,
5,
5,
5,
6
2,
3,
4,
4,
5,
5,
5,
6,
6,
6
3,
3,
4,
5,
5,
5,
6,
6,
6
1,
1,
1,
1,
3,
3,
5,
5,
6
The module M has simple direct summands:
4 copies of simple module number 1
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
6
5
6
socle layers
6
5
6
2).
radical layers
3
5
4
5
3
socle layers
3
5
4
5
3
3).
radical layers
5
3,
6
5,
6
3
5
socle layers
5
3
5,
6
3,
6
5
4).
radical layers
5
3,
4,
6
5,
5,
6
2,
3,
3,
4
5,
5,
6
3,
4,
6
5
socle layers
5
3,
4,
6
5,
5,
6
2,
3,
3,
4
5,
5,
6
3,
4,
6
5
5).
radical layers
3
5,
6
2,
3,
4
5,
5,
6
3,
3,
4,
5,
6
3,
4,
5,
6,
6
2,
5,
5
3,
4,
6
3,
5
6
2
6
3
socle layers
3
6
2
6
3,
5
3,
4,
6
2,
5,
5
3,
4,
5,
6,
6
3,
3,
4,
5,
6
5,
5,
6
2,
3,
4
5,
6
3
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
64,
216,
410,
237,
410,
495
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0
0,
3,
4,
2,
3,
6
0,
4,
10,
5,
10,
10
0,
2,
5,
5,
7,
5
0,
3,
10,
7,
14,
10
0,
6,
10,
5,
10,
15
The determinant of the Cartan matrix is 55.
Projective module number 1 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
4,
6
3,
6
3,
5,
6
2,
5
4,
6
3,
5
6
2
6
3
socle layers
2
6
3
6
2
4,
6,
6
3,
3,
5
5,
6
2,
4
5,
6
3
Projective module number 3
radical layers
3
3,
5,
6
2,
3,
4,
5,
6
2,
4,
5,
5,
6
3,
3,
4,
5,
5,
6,
6
3,
3,
4,
5,
6,
6
2,
5,
5
3,
4,
6
3,
5
6
2
6
3
socle layers
3
6
2
6
3,
5
3,
4,
6
2,
5,
5
3,
4,
5,
6,
6
3,
3,
3,
4,
5,
6
5,
5,
5,
6,
6
2,
2,
3,
4,
4
5,
5,
6,
6
3,
3
Projective module number 4
radical layers
4
2,
4,
5
3,
4,
5,
6
3,
4,
5,
6,
6
3,
5,
5,
6
2,
3,
4,
5
5,
6
3
socle layers
4
5
3,
4,
5
3,
4,
5,
6
2,
5,
5,
6
2,
3,
4,
6
3,
4,
5,
6,
6
3,
5
Projective module number 5
radical layers
5
3,
4,
5,
6
3,
4,
5,
5,
6,
6
2,
3,
3,
4,
5,
5,
5,
6
2,
3,
3,
4,
4,
5,
5,
6,
6
3,
4,
5,
5,
5,
6,
6,
6
2,
3,
3,
4,
5
5,
6
3
socle layers
5
3,
4
5,
5
3,
4,
5,
5,
6
3,
3,
3,
4,
4,
5,
6,
6,
6
2,
5,
5,
5,
5,
6,
6
2,
2,
3,
3,
4,
4,
5,
6
3,
4,
5,
5,
6,
6,
6
3,
3,
5
Projective module number 6
radical layers
6
2,
3,
5,
6
2,
3,
4,
5,
5,
6,
6,
6
2,
3,
4,
4,
5,
6,
6
3,
3,
4,
5,
5,
6,
6,
6
2,
3,
3,
5,
5,
5
4,
6,
6
2,
3,
5
6,
6
2,
3
6
3
socle layers
6
2
6
3,
3
6,
6
2,
2,
5
4,
6,
6,
6,
6
3,
3,
3,
3,
5,
5,
6
2,
5,
5,
5,
6,
6
2,
2,
4,
4,
4,
6
3,
4,
5,
5,
5,
6,
6,
6
3,
3,
3,
5
The degrees of the splitting fields are
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
68
.
The dimensions of the irreducible H-modules are
4,
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
4,
4,
10,
7,
16,
15
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0
0,
2,
1,
0,
1,
0
0,
1,
3,
1,
3,
2
0,
0,
1,
2,
2,
2
0,
1,
3,
2,
6,
4
0,
0,
2,
2,
4,
7
The determinant of the Cartan matrix is 73.
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
2,
3
5
socle layers
2
3
2,
5
Projective module number 3
radical layers
3
2,
3,
5
3,
4,
5,
6
5,
6
socle layers
3
2,
3,
5
3,
4,
5,
6
5,
6
Projective module number 4
radical layers
4
4,
5,
6
3
5
6
socle layers
4
5
3
4,
5,
6
6
Projective module number 5
radical layers
5
3,
4,
5,
6
2,
3,
5,
5,
6
3,
4,
5,
6
5,
6
socle layers
5
3,
4,
5
2,
3,
5,
5
3,
4,
5,
6,
6
5,
6,
6
Projective module number 6
radical layers
6
4,
5,
6,
6
3,
5,
6,
6
3,
5
4,
5,
6
6
socle layers
6
4,
5,
6
3,
5
3,
5,
6
4,
5,
6,
6
6,
6
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