PSL(2,31) Modulo 2
The Hecke algebra for the Projective Special Linear Group of dimension 2
over GF(31) with point stabilizer being the normalizer of a
Sylow 2-subgroup
in characteristic 2
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 2, having point stablilizer
of order 32
The dimension of M is 465
.
The dimensions of the irreducible submodules modules are
128,
64,
32,
15,
15,
1
.
The module M has radical filtration (Loewy series)
1,
2,
3,
4,
5,
6
4,
5
4,
5
4,
5
4,
5
4,
5
4,
5
4,
5
The module M has socle filtration (socle series)
4,
5
4,
5
4,
5
4,
5
4,
5
4,
5
4,
5
1,
2,
3,
4,
5,
6
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 2
1 copy of simple module number 3
1 copy of simple module number 6
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
4
5
4
5
4
5
4
5
socle layers
4
5
4
5
4
5
4
5
2).
radical layers
5
4
5
4
5
4
5
4
socle layers
5
4
5
4
5
4
5
4
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
128,
64,
32,
120,
120,
1
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0
0,
0,
1,
0,
0,
0
0,
0,
0,
4,
4,
0
0,
0,
0,
4,
4,
0
0,
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 0.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4,
5
(5).
6
Projective modules number
1,
2,
3,
6
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 4
radical layers
4
5
4
5
4
5
4
5
socle layers
4
5
4
5
4
5
4
5
Projective module number 5
radical layers
5
4
5
4
5
4
5
4
socle layers
5
4
5
4
5
4
5
4
The degrees of the splitting fields are
4,
2,
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
24
.
The dimensions of the irreducible H-modules are
4,
2,
1,
1,
1,
1
.
The degrees of the splitting fields are
4,
2,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
4,
2,
1,
1,
8,
8
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0
0,
0,
1,
0,
0,
0
0,
0,
0,
1,
0,
0
0,
0,
0,
0,
4,
4
0,
0,
0,
0,
4,
4
The determinant of the Cartan matrix is 0.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4
(5).
5,
6
Projective modules number
1,
2,
3,
4
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 5
radical layers
5
6
5
6
5
6
5
6
socle layers
5
6
5
6
5
6
5
6
Projective module number 6
radical layers
6
5
6
5
6
5
6
5
socle layers
6
5
6
5
6
5
6
5