M11 Modulo 5

The Hecke algebra for the Mathieu Group M11 with point stabilizer being the normalizer of a Sylow 5-subgroup in characteristic 5 .

The Module M

The module M is the permutation module over the prime field of chacteristic 5, having point stablilizer of order 20 The dimension of M is 396 .

The dimensions of the irreducible submodules modules are 55, 45, 16, 16, 11, 10, 1 .

The module M has radical filtration (Loewy series)
1, 1, 2, 2, 3, 4, 5, 6, 6, 7
3, 5, 7
3, 4, 7
4, 5, 7
3, 4, 5


The module M has socle filtration (socle series)
3, 4, 5
3, 5, 7
3, 4, 7
4, 5, 7
1, 1, 2, 2, 3, 4, 5, 6, 6, 7


The module M has simple direct summands:

2 copies of simple module number 1
2 copies of simple module number 2
2 copies of simple module number 6
1 copy of simple module number 7

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

1).


radical layers
5
3
7
4
5



socle layers
5
3
7
4
5


2).


radical layers
3
7
4
5
3



socle layers
3
7
4
5
3


3).


radical layers
4
5
3
7
4



socle layers
4
5
3
7
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 55, 45, 60, 60, 55, 10, 44 .

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, 2, 6 are simple.

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


Projective module number 3


radical layers
3
7
4
5
3



socle layers
3
7
4
5
3



Projective module number 4


radical layers
4
5
3
7
4



socle layers
4
5
3
7
4



Projective module number 5


radical layers
5
3
7
4
5



socle layers
5
3
7
4
5



Projective module number 7


radical layers
7
4
5
3



socle layers
7
4
5
3


The degrees of the splitting fields are 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 25 .

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

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

The dimensions of the projective modules of H are 2, 2, 2, 1, 4, 4, 4 .

The cartan matrix of H is



The determinant of the Cartan matrix is 4.

The blocks of H consist of the following irreducible modules:

Projective modules number 1, 2, 3, 4 are simple.

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


Projective module number 5


radical layers
5
6
7
5



socle layers
5
6
7
5



Projective module number 6


radical layers
6
7
5
6



socle layers
6
7
5
6



Projective module number 7


radical layers
7
5
6
7



socle layers
7
5
6
7