PSL(2,16) Modulo 3
The Hecke algebra for the Projective Special Linear Group of dimension 2
over GF(16) 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 30
The dimension of M is 136
.
The dimensions of the irreducible submodules modules are
34,
16,
1
.
The module M has radical filtration (Loewy series)
1,
2,
3
1,
3
1,
2
The module M has socle filtration (socle series)
1,
2
1,
3
1,
2,
3
The module M has simple direct summands:
1 copy of simple module number 3
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
2
3
2
socle layers
2
3
2
2).
radical layers
1
1
1
socle layers
1
1
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
102,
33,
17
.
The cartan matrix of A is
The determinant of the Cartan matrix is 3.
The blocks of A consist of the following irreducible
modules:
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
1
1
socle layers
1
1
1
Projective module number 2
radical layers
2
3
2
socle layers
2
3
2
Projective module number 3
radical layers
3
2
socle layers
3
2
The degrees of the splitting fields are
2,
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
2,
1,
1
.
The dimensions of the projective modules of H are
6,
1,
2
.
The cartan matrix of H is
The determinant of the Cartan matrix is 6.
The blocks of H consist of the following irreducible
modules:
Projective module number 2 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 1
radical layers
1
1
1
socle layers
1
1
1
Projective module number 3
radical layers
3
3
socle layers
3
3