PSL(2,23) Modulo 3

The Hecke algebra for the Projective Special Linear Group of dimension 2 over GF(23) 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 prime field of chacteristic 3, having point stablilizer of order 24 The dimension of M is 253 .

The dimensions of the irreducible submodules modules are 120, 22, 1 .

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

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

The module M has simple direct summands:

1 copy of simple module number 1 1 copy of simple module number 3

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

1).

radical layers
2
2
2

socle layers
2
2
2

2).

radical layers
2
2
2

socle layers
2
2
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 120, 66, 1 .

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 modules number 1, 3 are simple.

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

Projective module number 2

radical layers
2
2
2

socle layers
2
2
2

The degrees of the splitting fields are 5, 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 18 .

The dimensions of the irreducible H-modules are 5, 2, 1 .

The degrees of the splitting fields are 5, 1, 1 .

The dimensions of the projective modules of H are 5, 6, 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, 3 are simple.

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

Projective module number 2

radical layers
2
2
2

socle layers
2
2
2