PSL(3,3) Modulo 13
The Hecke algebra for the Projective Special Linear Group of dimension 3
over GF(3) with point stabilizer being the normalizer of a
Sylow 3-subgroup
in characteristic 13
.
The Module M
The module M is the permutation module over the
field of chacteristic 13, having point stablilizer
of order 39
. The dimension of M is 144
.
The dimensions of the irreducible submodules modules are
39,
16,
13,
11,
1
.
The module M has radical filtration (Loewy series)
1,
2,
3,
5
2,
4
2
2
2
The module M has socle filtration (socle series)
2
2
2
2,
4
1,
2,
3,
5
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 3
1 copy of simple module number 5
The remaining indecomposable component of M
has radical and socle filtrations as follows:
1).
radical layers
2
2,
4
2
2
2
socle layers
2
2
2
2,
4
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
39,
91,
13,
27,
1
.
The cartan matrix of A is
1,
0,
0,
0,
0
0,
5,
0,
1,
0
0,
0,
1,
0,
0
0,
1,
0,
1,
0
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 4.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
4
(3).
3
(4).
5
Projective modules number
1,
3,
5
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
2,
4
2
2
2
socle layers
2
2
2
2,
4
2
Projective module number 4
radical layers
4
2
socle layers
4
2
The degrees of the splitting fields are
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
8
.
The dimensions of the irreducible H-modules are
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1
.
The dimensions of the projective modules of H are
5,
1,
1,
1
.
The cartan matrix of H is
5,
0,
0,
0
0,
1,
0,
0
0,
0,
1,
0
0,
0,
0,
1
The determinant of the Cartan matrix is 5.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4
Projective modules number
2,
3,
4
are simple.
The radical and socle filtrations of the remaining
projective module for H are the following:
Projective module number 1
radical layers
1
1
1
1
1
socle layers
1
1
1
1
1