J1 Modulo 11

The Hecke algebra for the Janko Group J1 with point stabilizer being the normalizer of a Sylow 11-subgroup in characteristic 11. .

The Module M

The module M is the permutation module over the field of chacteristic 11, having point stablilizer of order 110 . The dimension of M is 1596 .

The dimensions of the irreducible submodules modules are 209, 119, 106, 77, 69, 64, 56, 49, 27, 14, 7, 1 .

The module M has radical filtration (Loewy series)
1, 1, 2, 3, 4, 4, 5, 7, 8, 12
6, 6, 9, 9, 10, 10, 11, 11, 12
2, 3, 5, 7, 8


The module M has socle filtration (socle series)
2, 3, 5, 7, 8
6, 6, 9, 9, 10, 10, 11, 11, 12
1, 1, 2, 3, 4, 4, 5, 7, 8, 12


The module M has simple direct summands:

2 copies of simple module number 1
2 copies of simple module number 4
1 copy of simple module number 12

The remaining indecomposable components of M have radical and socle filtrations as follows:

1).


radical layers
7
6
7



socle layers
7
6
7


2).


radical layers
2
10, 12
2



socle layers
2
10, 12
2


3).


radical layers
3
9, 10
3



socle layers
3
9, 10
3


4).


radical layers
5
6, 11
5



socle layers
5
6, 11
5


5).


radical layers
8
9, 11
8



socle layers
8
9, 11
8


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 209, 253, 253, 77, 209, 189, 176, 132, 182, 239, 125, 120 .

The cartan matrix of A is



The determinant of the Cartan matrix is 1.

The blocks of A consist of the following irreducible modules:

Projective modules number 1, 4 are simple.

The radical and socle filtrations of the remaining projective modules for A are the following:


Projective module number 2


radical layers
2
10, 12
2



socle layers
2
10, 12
2



Projective module number 3


radical layers
3
9, 10
3



socle layers
3
9, 10
3



Projective module number 5


radical layers
5
6, 11
5



socle layers
5
6, 11
5



Projective module number 6


radical layers
6
5, 7



socle layers
6
5, 7



Projective module number 7


radical layers
7
6
7



socle layers
7
6
7



Projective module number 8


radical layers
8
9, 11
8



socle layers
8
9, 11
8



Projective module number 9


radical layers
9
3, 8



socle layers
9
3, 8



Projective module number 10


radical layers
10
2, 3



socle layers
10
2, 3



Projective module number 11


radical layers
11
5, 8



socle layers
11
5, 8



Projective module number 12


radical layers
12
2



socle layers
12
2


The degrees of the splitting fields are 1, 1, 1, 1, 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 19 .

The dimensions of the irreducible H-modules are 2, 2, 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 2, 2, 2, 1, 2, 2, 2, 2 .

The cartan matrix of H is



The determinant of the Cartan matrix is 32.

The blocks of H consist of the following irreducible modules:

Projective modules number 1, 2, 4 are simple.

The radical and socle filtrations of the remaining projective modules for H are the following:


Projective module number 3


radical layers
3
3



socle layers
3
3



Projective module number 5


radical layers
5
5



socle layers
5
5



Projective module number 6


radical layers
6
6



socle layers
6
6



Projective module number 7


radical layers
7
7



socle layers
7
7



Projective module number 8


radical layers
8
8



socle layers
8
8