5 ,
2 ,
1
The Hecke algebra for the Symmetric Group
on 8 Letters, with the Partition
[ 5, 2, 1 ]
in characteristic 5
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 5, having point stablilizer
equal to the Young subgroup of the partition.
The dimension of M is 168
.
The dimensions of the irreducible submodules modules are
43,
21,
21,
20,
7,
1
.
The module M has radical filtration (Loewy series)
3,
4,
4,
5,
5,
6
1,
2
3,
5
The module M has socle filtration (socle series)
3,
5
1,
2
3,
4,
4,
5,
5,
6
The module M has simple direct summands:
2 copies of simple module number 4
1 copy of simple module number 5
1 copy of simple module number 6
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
3
1
3
socle layers
3
1
3
2).
radical layers
5
2
5
socle layers
5
2
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
64,
28,
85,
20,
35,
1
.
The cartan matrix of A is
1,
0,
1,
0,
0,
0
0,
1,
0,
0,
1,
0
1,
0,
2,
0,
0,
0
0,
0,
0,
1,
0,
0
0,
1,
0,
0,
2,
0
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,
3
(2).
2,
5
(3).
4
(4).
6
Projective modules number
4,
6
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 1
radical layers
1
3
socle layers
1
3
Projective module number 2
radical layers
2
5
socle layers
2
5
Projective module number 3
radical layers
3
1
3
socle layers
3
1
3
Projective module number 5
radical layers
5
2
5
socle layers
5
2
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
12
.
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,
1,
2,
2,
3
.
The cartan matrix of H is
1,
0,
0,
0,
0
0,
1,
0,
0,
0
0,
0,
2,
0,
0
0,
0,
0,
1,
1
0,
0,
0,
1,
2
The determinant of the Cartan matrix is 2.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4,
5
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
3
socle layers
3
3
Projective module number 4
radical layers
4
5
socle layers
4
5
Projective module number 5
radical layers
5
4
5
socle layers
5
4
5