A8 Modulo 7

The Hecke algebra for the Alternating Group on 8 letters with point stabilizer being the normalizer of a Sylow 7-subgroup in characteristic 7 .

The Module M

The module M is the permutation module over the prime field of chacteristic 7, having point stablilizer of order 21 The dimension of M is 960 .

The dimensions of the irreducible submodules modules are 70, 56, 45, 42, 35, 28, 21, 19, 14, 7, 1 .

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


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


The module M has simple direct summands:

4 copies of simple module number 1
2 copies of simple module number 2
1 copy of simple module number 4
3 copies of simple module number 5
2 copies of simple module number 6
1 copy of simple module number 7
2 copies of simple module number 9
1 copy of simple module number 10
1 copy of simple module number 11

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

1).


radical layers
3
3, 8
3



socle layers
3
3, 8
3


2).


radical layers
3
3, 8
3



socle layers
3
3, 8
3


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 70, 56, 154, 42, 35, 28, 21, 64, 14, 7, 1 .

The cartan matrix of A is



The determinant of the Cartan matrix is 2.

The blocks of A consist of the following irreducible modules:

Projective modules number 1, 2, 4, 5, 6, 7, 9, 10, 11 are simple.

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


Projective module number 3


radical layers
3
3, 8
3



socle layers
3
3, 8
3



Projective module number 8


radical layers
8
3



socle layers
8
3


The degrees of the splitting fields are 1, 1, 1, 2, 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 54 .

The dimensions of the irreducible H-modules are 4, 3, 2, 2, 2, 2, 2, 1, 1, 1 .

The degrees of the splitting fields are 1, 1, 2, 1, 1, 1, 1, 1, 1, 1 .

The dimensions of the projective modules of H are 4, 3, 2, 6, 2, 2, 2, 1, 1, 1 .

The cartan matrix of H is



The determinant of the Cartan matrix is 3.

The blocks of H consist of the following irreducible modules:

Projective modules number 1, 2, 3, 5, 6, 7, 8, 9, 10 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



socle layers
4
4
4