A7 Modulo 2
The Hecke algebra for the Alternating Group on 7 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 8
The dimension of M is 315
.
The dimensions of the irreducible submodules modules are
20,
14,
6,
4,
4,
1
.
The module M has radical filtration (Loewy series)
1,
1,
1,
2,
2,
2,
3,
3,
4,
5,
6
2,
3,
3,
4,
5,
6,
6,
6,
6,
6
1,
1,
2,
2,
2,
3,
4,
5
6,
6,
6
1,
1,
2
The module M has socle filtration (socle series)
1,
1,
2
6,
6,
6
1,
1,
2,
2,
2,
3,
4,
5
2,
3,
3,
4,
5,
6,
6,
6,
6,
6
1,
1,
1,
2,
2,
2,
3,
3,
4,
5,
6
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
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
5
3
4
socle layers
5
3
4
2).
radical layers
4
3
5
socle layers
4
3
5
3).
radical layers
3
4,
5
3
socle layers
3
4,
5
3
4).
radical layers
1
6
2
6
1
socle layers
1
6
2
6
1
5).
radical layers
1
6
2
6
1
socle layers
1
6
2
6
1
6).
radical layers
2
2,
6
1
6
2
socle layers
2
6
1
2,
6
2
7).
radical layers
1,
2
6,
6
1,
2
socle layers
1,
2
6,
6
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
56,
64,
20,
14,
14,
71
.
The cartan matrix of A is
2,
1,
0,
0,
0,
2
1,
3,
0,
0,
0,
2
0,
0,
2,
1,
1,
0
0,
0,
1,
1,
1,
0
0,
0,
1,
1,
1,
0
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,
6
(2).
3,
4,
5
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
6
2
6
1
socle layers
1
6
2
6
1
Projective module number 2
radical layers
2
2,
6
1
6
2
socle layers
2
6
1
2,
6
2
Projective module number 3
radical layers
3
4,
5
3
socle layers
3
4,
5
3
Projective module number 4
radical layers
4
3
5
socle layers
4
3
5
Projective module number 5
radical layers
5
3
4
socle layers
5
3
4
Projective module number 6
radical layers
6
1,
2
6,
6
1,
2
socle layers
6
1,
2
6,
6
1,
2
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
49
.
The dimensions of the irreducible H-modules are
2,
1,
1,
1,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
7,
1,
2,
3,
8,
10,
3,
5,
3
.
The cartan matrix of H is
2,
0,
0,
0,
1,
2,
0,
0,
0
0,
1,
0,
0,
0,
0,
0,
0,
0
0,
0,
1,
0,
0,
0,
0,
1,
0
0,
0,
0,
1,
1,
1,
0,
0,
0
1,
0,
0,
1,
3,
2,
0,
0,
0
2,
0,
0,
1,
2,
3,
0,
0,
0
0,
0,
0,
0,
0,
0,
1,
1,
1
0,
0,
1,
0,
0,
0,
1,
2,
1
0,
0,
0,
0,
0,
0,
1,
1,
1
The determinant of the Cartan matrix is 0.
The blocks of H consist of the following irreducible
modules:
(1).
1,
4,
5,
6
(2).
2
(3).
3,
7,
8,
9
Projective module number 2 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 1
radical layers
1
6
5
6
1
socle layers
1
6
5
6
1
Projective module number 3
radical layers
3
8
socle layers
3
8
Projective module number 4
radical layers
4
6
5
socle layers
4
6
5
Projective module number 5
radical layers
5
6
1,
4,
5
6
5
socle layers
5
6
1,
4,
5
6
5
Projective module number 6
radical layers
6
1,
4,
5
6,
6
1,
5
socle layers
6
1,
4,
5
6,
6
1,
5
Projective module number 7
radical layers
7
8
9
socle layers
7
8
9
Projective module number 8
radical layers
8
3,
7,
9
8
socle layers
8
3,
7,
9
8
Projective module number 9
radical layers
9
8
7
socle layers
9
8
7