PSL(2,13) Modulo 7

The Hecke algebra for the Projective Special Linear Group of dimension 2 over GF(13) 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 14 The dimension of M is 78 .

The dimensions of the irreducible submodules modules are 14, 12, 1 .

The module M has radical filtration (Loewy series)
1, 1, 2, 3
2, 3
2
2

The module M has socle filtration (socle series)
2
2
2, 3
1, 1, 2, 3

The module M has simple direct summands:

2 copies of simple module number 1 1 copy of simple module number 3

The remaining indecomposable component of M has radical and socle filtrations as follows:

1).

radical layers
2
2, 3
2
2

socle layers
2
2
2, 3
2

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 14, 49, 13 .

The cartan matrix of A is

The determinant of the Cartan matrix is 3.

The blocks of A consist of the following irreducible modules:

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
2, 3
2
2

socle layers
2
2
2, 3
2

Projective module number 3

radical layers
3
2

socle layers
3
2

The degrees of the splitting fields are 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 9 .

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

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

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

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

Projective module number 3

radical layers
3
3
3
3

socle layers
3
3
3
3