PSL(3,2) Modulo 3
The Hecke algebra for the Projective Special Linear Group of dimension 3
over GF(2) 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 6
The dimension of M is 28
.
The dimensions of the irreducible submodules modules are
7,
6,
1
.
The module M has radical filtration (Loewy series)
1,
2,
2,
3
3
1
The module M has socle filtration (socle series)
1
3
1,
2,
2,
3
The module M has simple direct summands:
2 copies of simple module number 2
1 copy of simple module number 3
The remaining indecomposable component of M has
radical and socle filtrations as follows:
1).
radical layers
1
3
1
socle layers
1
3
1
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
15,
6,
8
.
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 module number 2 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 1
radical layers
1
3
1
socle layers
1
3
1
Projective module number 3
radical layers
3
1
socle layers
3
1
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
7
.
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,
2
.
The cartan matrix of H is
The determinant of the Cartan matrix is 2.
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
socle layers
3
3