5 ,
3 ,
1
The Hecke algebra for the Symmetric Group
on 9 Letters, with the Partition
[ 5, 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 504
.
The dimensions of the irreducible submodules modules are
162,
41,
35,
27,
21,
7,
1,
1
.
The module M has radical filtration (Loewy series)
1,
2,
4,
4,
6,
7
2,
5,
6,
6,
7,
8
2,
3,
6,
6,
7
2,
5,
7
6
The module M has socle filtration (socle series)
6
2,
5,
7
2,
3,
6,
6,
7
2,
5,
6,
6,
7,
8
1,
2,
4,
4,
6,
7
The module M has simple direct summands:
1 copy of simple module number 1
2 copies of simple module number 4
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
7
6
7
socle layers
7
6
7
2).
radical layers
2
6,
8
2
socle layers
2
6,
8
2
3).
radical layers
6
2,
5,
7
3,
6,
6
2,
5,
7
6
socle layers
6
2,
5,
7
3,
6,
6
2,
5,
7
6
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
162,
195,
105,
27,
133,
189,
114,
42
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0
0,
3,
1,
0,
1,
2,
1,
1
0,
1,
1,
0,
1,
1,
1,
0
0,
0,
0,
1,
0,
0,
0,
0
0,
1,
1,
0,
2,
2,
1,
0
0,
2,
1,
0,
2,
4,
2,
0
0,
1,
1,
0,
1,
2,
3,
0
0,
1,
0,
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 1.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
3,
5,
6,
7,
8
(3).
4
Projective modules number
1,
4
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
3,
6,
8
2,
2,
5,
7
6
socle layers
2
3,
6
2,
5,
7,
8
2,
6
Projective module number 3
radical layers
3
2,
5,
7
6
socle layers
3
2,
5,
7
6
Projective module number 5
radical layers
5
3,
6
2,
5,
7
6
socle layers
5
3,
6
2,
5,
7
6
Projective module number 6
radical layers
6
2,
5,
7
3,
6,
6
2,
5,
7
6
socle layers
6
2,
5,
7
3,
6,
6
2,
5,
7
6
Projective module number 7
radical layers
7
3,
6
2,
5,
7,
7
6
socle layers
7
3,
6
2,
5,
7
6,
7
Projective module number 8
radical layers
8
2
socle layers
8
2
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
2,
1,
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,
3,
6,
1,
3
.
The cartan matrix of H is
1,
0,
0,
0,
0
0,
2,
1,
0,
0
0,
1,
4,
0,
1
0,
0,
0,
1,
0
0,
0,
1,
0,
2
The determinant of the Cartan matrix is 12.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2,
3,
5
(3).
4
Projective modules number
1,
4
are 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
socle layers
2
2,
3
Projective module number 3
radical layers
3
2,
3,
3,
5
3
socle layers
3
2,
3,
3,
5
3
Projective module number 5
radical layers
5
3,
5
socle layers
5
3,
5