Sz(8) Modulo 13
The Hecke algebra for the Suzuki Group over GF(8) 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
field of chacteristic 13, having point stablilizer
of order 52
. The dimension of M is 560
.
The dimensions of the irreducible submodules modules are
65,
65,
65,
35,
14,
14,
1
.
The module M has radical filtration (Loewy series)
1,
1,
2,
2,
3,
3,
4,
7
4,
5
4,
7
6
4
The module M has socle filtration (socle series)
4
5
4,
7
4,
6
1,
1,
2,
2,
3,
3,
4,
7
The module M has simple direct summands:
2 copies of simple module number 1
2 copies of simple module number 2
2 copies of simple module number 3
1 copy of simple module number 7
The remaining indecomposable component of M
has radical and socle filtrations as follows:
1).
radical layers
4
4,
5
4,
7
6
4
socle layers
4
5
4,
7
4,
6
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
65,
65,
65,
169,
64,
49,
50
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0
0,
0,
1,
0,
0,
0,
0
0,
0,
0,
4,
1,
1,
1
0,
0,
0,
1,
1,
1,
1
0,
0,
0,
1,
0,
1,
0
0,
0,
0,
1,
0,
1,
1
The determinant of the Cartan matrix is 3.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4,
5,
6,
7
Projective modules number
1,
2,
3
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 4
radical layers
4
4,
5
4,
7
6
4
socle layers
4
5
4,
7
4,
6
4
Projective module number 5
radical layers
5
7
6
4
socle layers
5
7
6
4
Projective module number 6
radical layers
6
4
socle layers
6
4
Projective module number 7
radical layers
7
6
4
socle layers
7
6
4
The degrees of the splitting fields are
1,
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
17
.
The dimensions of the irreducible H-modules are
2,
2,
2,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
2,
2,
2,
4,
1
.
The cartan matrix of H is
1,
0,
0,
0,
0
0,
1,
0,
0,
0
0,
0,
1,
0,
0
0,
0,
0,
4,
0
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 4.
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,
5
are simple.
The radical and socle filtrations of the remaining
projective module for H are the following:
Projective module number 4
radical layers
4
4
4
4
socle layers
4
4
4
4