4 ,
3 ,
1
The Hecke algebra for the Symmetric Group
on 8 Letters, with the Partition
[ 4, 3, 1 ]
in characteristic 3
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 3, having point stablilizer
equal to the Young subgroup of the partition.
The dimension of M is 280
.
The dimensions of the irreducible submodules modules are
35,
28,
21,
13,
7,
7,
1,
1
.
The module M has radical filtration (Loewy series)
2,
2,
3,
4,
6,
6,
7
1,
4,
6,
8
2,
4,
5,
6,
7
1
2
The module M has socle filtration (socle series)
2
1
2,
4,
5,
6,
7
1,
4,
6,
8
2,
2,
3,
4,
6,
6,
7
The module M has simple direct summands:
1 copy of simple module number 2
1 copy of simple module number 3
1 copy of simple module number 6
1 copy of simple module number 7
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
6
4
6
socle layers
6
4
6
2).
radical layers
4
6,
8
4
socle layers
4
6,
8
4
3).
radical layers
2
1
2,
5,
7
1
2
socle layers
2
1
2,
5,
7
1
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
134,
162,
21,
34,
70,
27,
64,
14
.
The cartan matrix of A is
2,
2,
0,
0,
1,
0,
1,
0
2,
3,
0,
0,
1,
0,
1,
0
0,
0,
1,
0,
0,
0,
0,
0
0,
0,
0,
2,
0,
1,
0,
1
1,
1,
0,
0,
1,
0,
0,
0
0,
0,
0,
1,
0,
2,
0,
0
1,
1,
0,
0,
0,
0,
1,
0
0,
0,
0,
1,
0,
0,
0,
1
The determinant of the Cartan matrix is 0.
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
2,
5,
7
1
2
socle layers
1
2,
5,
7
1
2
Projective module number 2
radical layers
2
1
2,
5,
7
1
2
socle layers
2
1
2,
5,
7
1
2
Projective module number 4
radical layers
4
6,
8
4
socle layers
4
6,
8
4
Projective module number 5
radical layers
5
1
2
socle layers
5
1
2
Projective module number 6
radical layers
6
4
6
socle layers
6
4
6
Projective module number 7
radical layers
7
1
2
socle layers
7
1
2
Projective module number 8
radical layers
8
4
socle layers
8
4
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
17
.
The dimensions of the irreducible H-modules are
1,
1,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
1,
1,
2,
2,
4,
3,
4
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0
0,
0,
1,
0,
0,
0,
1
0,
0,
0,
1,
1,
0,
0
0,
0,
0,
1,
3,
0,
0
0,
0,
0,
0,
0,
2,
1
0,
0,
1,
0,
0,
1,
2
The determinant of the Cartan matrix is 2.
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
7
socle layers
3
7
Projective module number 4
radical layers
4
5
socle layers
4
5
Projective module number 5
radical layers
5
4,
5
5
socle layers
5
4,
5
5
Projective module number 6
radical layers
6
7
6
socle layers
6
7
6
Projective module number 7
radical layers
7
3,
6
7
socle layers
7
3,
6
7
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