J1 Modulo 2
The Hecke algebra for the Janko Group J1 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
field of chacteristic 2, having point stablilizer
of order 168
. The dimension of M is 1045
.
The dimensions of the irreducible submodules modules are
360,
112,
112,
76,
76,
20,
1
.
The module M has radical filtration (Loewy series)
1,
2,
5,
6,
7
3
6,
7,
7
3,
4
6,
7,
7
3
6
The module M has socle filtration (socle series)
6
3
6,
7,
7
3,
4
6,
7,
7
3
1,
2,
5,
6,
7
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 5
1 copy of simple module number 7
The remaining indecomposable component of M
has radical and socle filtrations as follows:
1).
radical layers
6
3
6,
7,
7
3,
4
6,
7,
7
3
6
socle layers
6
3
6,
7,
7
3,
4
6,
7,
7
3
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
360,
112,
838,
209,
76,
496,
496
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0
0,
0,
5,
2,
0,
6,
6
0,
0,
1,
1,
0,
1,
1
0,
0,
0,
0,
1,
0,
0
0,
0,
3,
1,
0,
4,
4
0,
0,
3,
1,
0,
4,
4
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,
6,
7
(4).
5
Projective modules number
1,
2,
5
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 3
radical layers
3
6,
6,
7,
7
3,
3,
4,
4
6,
6,
7,
7,
7,
7
3,
3
6,
6
socle layers
3
6,
6,
7,
7
3,
3,
4,
4
6,
6,
7,
7,
7,
7
3,
3
6,
6
Projective module number 4
radical layers
4
7
3
6
socle layers
4
7
3
6
Projective module number 6
radical layers
6
3
6,
7,
7
3,
4
6,
7,
7
3
6
socle layers
6
3
6,
7,
7
3,
4
6,
7,
7
3
6
Projective module number 7
radical layers
7
3,
4
6,
6,
7,
7,
7
3,
3
6,
6
socle layers
7
3,
4
6,
6,
7,
7,
7
3,
3
6,
6
The degrees of the splitting fields are
3,
2,
2,
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
11
.
The dimensions of the irreducible H-modules are
3,
2,
1,
1,
1
.
The degrees of the splitting fields are
3,
2,
1,
1,
1
.
The dimensions of the projective modules of H are
3,
2,
1,
4,
1
.
The cartan matrix of H is
1,
0,
0,
0,
0
0,
1,
0,
0,
0
0,
0,
1,
0,
0
0,
0,
0,
4,
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
Projective modules number
1,
2,
3,
5
are simple.
The radical and socle filtrations of the remaining
projective module for H are the following:
Projective module number 4
radical layers
4
4
4
4
socle layers
4
4
4
4