PSL(2,23) Modulo 2
The Hecke algebra for the Projective Special Linear Group of dimension 2
over GF(23) 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 8
The dimension of M is 759
.
The dimensions of the irreducible submodules modules are
120,
22,
11,
11,
1
.
The module M has radical filtration (Loewy series)
1,
1,
1,
2,
2,
2,
2,
3,
3,
4,
4,
5
2,
2,
2,
2,
3,
3,
4,
4,
5,
5
2,
2,
3,
4
2,
3,
4
3,
4
The module M has socle filtration (socle series)
3,
4
2,
3,
4
2,
2,
3,
4
2,
2,
2,
2,
3,
3,
4,
4,
5,
5
1,
1,
1,
2,
2,
2,
2,
3,
3,
4,
4,
5
The module M has simple direct summands:
3 copies of simple module number 1
1 copy of simple module number 5
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
3
4
socle layers
3
4
2).
radical layers
4
3
socle layers
4
3
3).
radical layers
2
2
socle layers
2
2
4).
radical layers
2
2
socle layers
2
2
5).
radical layers
2
2
2
socle layers
2
2
2
6).
radical layers
2
2
2
2
socle layers
2
2
2
2
7).
radical layers
4
3,
5
4
3
4
socle layers
4
3
4
3,
5
4
8).
radical layers
3
4,
5
3
4
3
socle layers
3
4
3
4,
5
3
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,
88,
56,
56,
23
.
The cartan matrix of A is
1,
0,
0,
0,
0
0,
4,
0,
0,
0
0,
0,
3,
2,
1
0,
0,
2,
3,
1
0,
0,
1,
1,
1
The determinant of the Cartan matrix is 12.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3,
4,
5
Projective module number 1 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
2
2
2
socle layers
2
2
2
2
Projective module number 3
radical layers
3
4,
5
3
4
3
socle layers
3
4
3
4,
5
3
Projective module number 4
radical layers
4
3,
5
4
3
4
socle layers
4
3
4
3,
5
4
Projective module number 5
radical layers
5
3,
4
socle layers
5
3,
4
The degrees of the splitting fields are
5,
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
105
.
The dimensions of the irreducible H-modules are
15,
2,
1,
1,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
5,
1,
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
15,
8,
1,
10,
11,
4,
4,
7,
7
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0,
0,
0,
0
0,
2,
0,
2,
2,
0,
0,
0,
0
0,
0,
1,
0,
0,
0,
0,
0,
0
0,
2,
0,
3,
3,
0,
0,
0,
0
0,
2,
0,
3,
4,
0,
0,
0,
0
0,
0,
0,
0,
0,
1,
1,
1,
1
0,
0,
0,
0,
0,
1,
1,
1,
1
0,
0,
0,
0,
0,
1,
1,
3,
2
0,
0,
0,
0,
0,
1,
1,
2,
3
The determinant of the Cartan matrix is 0.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2,
4,
5
(3).
3
(4).
6,
7,
8,
9
Projective modules number
1,
3
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 2
radical layers
2
4
2,
5
4
5
socle layers
2
4
2,
5
4
5
Projective module number 4
radical layers
4
2,
5
4
2,
5
4
5
socle layers
4
2,
5
4
2,
5
4
5
Projective module number 5
radical layers
5
4
2,
5
4
2,
5
4
5
socle layers
5
4
2,
5
4
2,
5
4
5
Projective module number 6
radical layers
6
7,
9
8
socle layers
6
7,
9
8
Projective module number 7
radical layers
7
6,
8
9
socle layers
7
6,
8
9
Projective module number 8
radical layers
8
6,
9
7,
8
9
8
socle layers
8
9
6,
8
7,
9
8
Projective module number 9
radical layers
9
7,
8
6,
9
8
9
socle layers
9
8
7,
9
6,
8
9