PSL(2,11) Modulo 5
The Hecke algebra for the Projective Special Linear Group of dimension 2
over GF(11) with point stabilizer being the normalizer of a
Sylow 5-subgroup
in characteristic 5
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 5, having point stablilizer
of order 10
The dimension of M is 66
.
The dimensions of the irreducible submodules modules are
11,
10,
5,
5,
1
.
The module M has radical filtration (Loewy series)
1,
2,
2,
3,
4,
5
5
1
5
1
The module M has socle filtration (socle series)
1
5
1
5
1,
2,
2,
3,
4,
5
The module M has simple direct summands:
2 copies of simple module number 2
1 copy of simple module number 3
1 copy of simple module number 4
1 copy of simple module number 5
The remaining indecomposable component of M has
radical and socle filtrations as follows:
1).
radical layers
1
5
1
5
1
socle layers
1
5
1
5
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
35,
10,
5,
5,
24
.
The cartan matrix of A is
3,
0,
0,
0,
2
0,
1,
0,
0,
0
0,
0,
1,
0,
0
0,
0,
0,
1,
0
2,
0,
0,
0,
2
The determinant of the Cartan matrix is 2.
The blocks of A consist of the following irreducible
modules:
(1).
1,
5
(2).
2
(3).
3
(4).
4
Projective modules number
2,
3,
4
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 1
radical layers
1
5
1
5
1
socle layers
1
5
1
5
1
Projective module number 5
radical layers
5
1
5
1
socle layers
5
1
5
1
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
10
.
The dimensions of the irreducible H-modules are
2,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
2,
1,
1,
1,
3
.
The cartan matrix of H is
1,
0,
0,
0,
0
0,
1,
0,
0,
0
0,
0,
1,
0,
0
0,
0,
0,
1,
0
0,
0,
0,
0,
3
The determinant of the Cartan matrix is 3.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4
(5).
5
Projective modules number
1,
2,
3,
4
are simple.
The radical and socle filtrations of the remaining
projective module for H are the following:
Projective module number 5
radical layers
5
5
5
socle layers
5
5
5