J1 Modulo 3
The Hecke algebra for the Janko Group J1 with point stabilizer
being the normalizer of a
Sylow 3-subgroup
in characteristic 3
.
The Module M
The module M is the permutation module over the
field of chacteristic 3, having point stablilizer
of order 60
. The dimension of M is 2926
.
The dimensions of the irreducible submodules modules are
360,
154,
133,
112,
76,
76,
1
.
The module M has radical filtration (Loewy series)
1,
1,
3,
3,
3,
3,
4,
5,
5,
5,
7
2,
6,
6,
6,
6,
7,
7,
7
3,
3,
3,
3,
4,
5,
5,
5
The module M has socle filtration (socle series)
3,
3,
3,
3,
4,
5,
5,
5
2,
6,
6,
6,
6,
7,
7,
7
1,
1,
3,
3,
3,
3,
4,
5,
5,
5,
7
The module M has simple direct summands:
2 copies of simple module number 1
1 copy of simple module number 7
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
3
6
3
socle layers
3
6
3
2).
radical layers
3
6
3
socle layers
3
6
3
3).
radical layers
5
7
5
socle layers
5
7
5
4).
radical layers
3
6
3
socle layers
3
6
3
5).
radical layers
3
6
3
socle layers
3
6
3
6).
radical layers
5
7
5
socle layers
5
7
5
7).
radical layers
5
7
5
socle layers
5
7
5
8).
radical layers
4
2
4
socle layers
4
2
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
360,
266,
342,
378,
153,
209,
77
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0
0,
1,
0,
1,
0,
0,
0
0,
0,
2,
0,
0,
1,
0
0,
1,
0,
2,
0,
0,
0
0,
0,
0,
0,
2,
0,
1
0,
0,
1,
0,
0,
1,
0
0,
0,
0,
0,
1,
0,
1
The determinant of the Cartan matrix is 1.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
4
(3).
3,
6
(4).
5,
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
4
socle layers
2
4
Projective module number 3
radical layers
3
6
3
socle layers
3
6
3
Projective module number 4
radical layers
4
2
4
socle layers
4
2
4
Projective module number 5
radical layers
5
7
5
socle layers
5
7
5
Projective module number 6
radical layers
6
3
socle layers
6
3
Projective module number 7
radical layers
7
5
socle layers
7
5
The degrees of the splitting fields are
3,
2,
1,
2,
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
67
.
The dimensions of the irreducible H-modules are
6,
4,
3,
2,
1
.
The degrees of the splitting fields are
3,
1,
1,
2,
1
.
The dimensions of the projective modules of H are
6,
8,
6,
6,
1
.
The cartan matrix of H is
1,
0,
0,
0,
0
0,
2,
0,
0,
0
0,
0,
2,
0,
0
0,
0,
0,
3,
0
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 12.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4
(5).
5
Projective modules number
1,
5
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
Projective module number 4
radical layers
4
4,
4
socle layers
4
4,
4