A8 Modulo 5
The Hecke algebra for the Alternating Group on 8 letters with point
stabilizer being the normalizer of a
Sylow 5-subgroup
in characteristic 5
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 5, having point stablilizer
of order 60
The dimension of M is 336
.
The dimensions of the irreducible submodules modules are
90,
70,
43,
21,
21,
20,
13,
7,
1
.
The module M has radical filtration (Loewy series)
1,
2,
3,
6,
8,
9
4,
5,
7
3,
8
The module M has socle filtration (socle series)
3,
8
4,
5,
7
1,
2,
3,
6,
8,
9
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 2
1 copy of simple module number 6
1 copy of simple module number 9
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
8
5
8
socle layers
8
5
8
2).
radical layers
3
4,
7
3
socle layers
3
4,
7
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
90,
70,
120,
64,
28,
20,
56,
35,
1
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0,
0,
0
0,
0,
2,
1,
0,
0,
1,
0,
0
0,
0,
1,
1,
0,
0,
0,
0,
0
0,
0,
0,
0,
1,
0,
0,
1,
0
0,
0,
0,
0,
0,
1,
0,
0,
0
0,
0,
1,
0,
0,
0,
1,
0,
0
0,
0,
0,
0,
1,
0,
0,
2,
0
0,
0,
0,
0,
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 0.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3,
4,
7
(4).
5,
8
(5).
6
(6).
9
Projective modules number
1,
2,
6,
9
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 3
radical layers
3
4,
7
3
socle layers
3
4,
7
3
Projective module number 4
radical layers
4
3
socle layers
4
3
Projective module number 5
radical layers
5
8
socle layers
5
8
Projective module number 7
radical layers
7
3
socle layers
7
3
Projective module number 8
radical layers
8
5
8
socle layers
8
5
8
The degrees of the splitting fields are
2,
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
9
.
The dimensions of the irreducible H-modules are
2,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
2,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
2,
2,
2,
1,
1,
1
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0
0,
2,
0,
0,
0,
0
0,
0,
2,
0,
0,
0
0,
0,
0,
1,
0,
0
0,
0,
0,
0,
1,
0
0,
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 4.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4
(5).
5
(6).
6
Projective modules number
1,
4,
5,
6
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
socle layers
2
2
Projective module number 3
radical layers
3
3
socle layers
3
3