PSL(2,25) Modulo 13
The Hecke algebra for the Projective Special Linear Group of dimension 2
over GF(25) with point stabilizer being the normalizer of a
Sylow 13-subgroup
in characteristic 13
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 13, having point stablilizer
of order 26
The dimension of M is 300
.
The dimensions of the irreducible submodules modules are
26,
26,
24,
13,
13,
1
.
The module M has radical filtration (Loewy series)
1,
1,
2,
2,
3,
4,
5,
6
3,
6
3
3
3
3
3
The module M has socle filtration (socle series)
3
3
3
3
3
3,
6
1,
1,
2,
2,
3,
4,
5,
6
The module M has simple direct summands:
2 copies of simple module number 1
2 copies of simple module number 2
1 copy of simple module number 4
1 copy of simple module number 5
1 copy of simple module number 6
The remaining indecomposable component of M has
radical and socle filtrations as follows:
1).
radical layers
3
3,
6
3
3
3
3
3
socle layers
3
3
3
3
3
3,
6
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
26,
26,
169,
13,
13,
25
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0
0,
0,
7,
0,
0,
1
0,
0,
0,
1,
0,
0
0,
0,
0,
0,
1,
0
0,
0,
1,
0,
0,
1
The determinant of the Cartan matrix is 6.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3,
6
(4).
4
(5).
5
Projective modules number
1,
2,
4,
5
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 3
radical layers
3
3,
6
3
3
3
3
3
socle layers
3
3
3
3
3
3,
6
3
Projective module number 6
radical layers
6
3
socle layers
6
3
The degrees of the splitting fields are
1,
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
18
.
The dimensions of the irreducible H-modules are
2,
2,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
2,
2,
7,
1,
1,
1
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0
0,
0,
7,
0,
0,
0
0,
0,
0,
1,
0,
0
0,
0,
0,
0,
1,
0
0,
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 7.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4
(5).
5
(6).
6
Projective modules number
1,
2,
4,
5,
6
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
3
3
3
3
3
socle layers
3
3
3
3
3
3
3