A8 Modulo 2
The Hecke algebra for the Alternating Group on 8 letters with point
stabilizer being the normalizer of a
Sylow 2-subgroup
in characteristic 2
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 2, having point stablilizer
of order 64
The dimension of M is 315
.
The dimensions of the irreducible submodules modules are
64,
20,
20,
14,
6,
4,
4,
1
.
The module M has radical filtration (Loewy series)
1,
2,
3,
4,
5,
6,
7,
8
4,
5,
5,
5,
6,
7,
8,
8,
8
2,
3,
4,
4,
5,
6,
7
8,
8,
8
2,
3,
4
The module M has socle filtration (socle series)
2,
3,
4
8,
8,
8
2,
3,
4,
4,
5,
6,
7
4,
5,
5,
5,
6,
7,
8,
8,
8
1,
2,
3,
4,
5,
6,
7,
8
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 8
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
6
5
7
socle layers
6
5
7
2).
radical layers
7
5
6
socle layers
7
5
6
3).
radical layers
3
8
4
8
2
socle layers
3
8
4
8
2
4).
radical layers
2
8
4
8
3
socle layers
2
8
4
8
3
5).
radical layers
5
4,
6,
7
5
socle layers
5
4,
6,
7
5
6).
radical layers
4
5,
8
2,
3
8
4
socle layers
4
8
2,
3
5,
8
4
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,
56,
56,
76,
34,
14,
14,
111
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0
0,
1,
1,
1,
0,
0,
0,
2
0,
1,
1,
1,
0,
0,
0,
2
0,
1,
1,
2,
1,
0,
0,
2
0,
0,
0,
1,
2,
1,
1,
0
0,
0,
0,
0,
1,
1,
1,
0
0,
0,
0,
0,
1,
1,
1,
0
0,
2,
2,
2,
0,
0,
0,
3
The determinant of the Cartan matrix is 0.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
3,
4,
5,
6,
7,
8
(3).
5,
6,
7
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
8
4
8
3
socle layers
2
8
4
8
3
Projective module number 3
radical layers
3
8
4
8
2
socle layers
3
8
4
8
2
Projective module number 4
radical layers
4
5,
8
2,
3
8
4
socle layers
4
8
2,
3
5,
8
4
Projective module number 5
radical layers
5
4,
6,
7
5
socle layers
5
4,
6,
7
5
Projective module number 6
radical layers
6
5
7
socle layers
6
5
7
Projective module number 7
radical layers
7
5
6
socle layers
7
5
6
Projective module number 8
radical layers
8
2,
3,
4
8,
8
2,
3,
4
socle layers
8
2,
3,
4
8,
8
2,
3,
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
24
.
The dimensions of the irreducible H-modules are
1,
1,
1,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
1,
1,
3,
3,
5,
3,
3,
5
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0,
0
0,
0,
1,
1,
1,
0,
0,
0
0,
0,
1,
1,
1,
0,
0,
0
0,
0,
1,
1,
2,
0,
0,
1
0,
0,
0,
0,
0,
1,
1,
1
0,
0,
0,
0,
0,
1,
1,
1
0,
0,
0,
0,
1,
1,
1,
2
The determinant of the Cartan matrix is 0.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3,
4,
5,
6,
7,
8
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
5
4
socle layers
3
5
4
Projective module number 4
radical layers
4
5
3
socle layers
4
5
3
Projective module number 5
radical layers
5
3,
4,
8
5
socle layers
5
3,
4,
8
5
Projective module number 6
radical layers
6
8
7
socle layers
6
8
7
Projective module number 7
radical layers
7
8
6
socle layers
7
8
6
Projective module number 8
radical layers
8
5,
6,
7
8
socle layers
8
5,
6,
7
8