Sz(8) Modulo 5
The Hecke algebra for the Suzuki Group over GF(8) 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
field of chacteristic 5, having point stablilizer
of order 20
. The dimension of M is 1456
.
The dimensions of the irreducible submodules modules are
195,
63,
35,
35,
35,
14,
14,
1
.
The module M has radical filtration (Loewy series)
1,
1,
1,
1,
2,
2,
2,
3,
3,
4,
4,
5,
5,
8
6,
6,
6,
8,
8,
8
7,
7,
7
2,
2,
2
The module M has socle filtration (socle series)
2,
2,
2
6,
6,
6
7,
7,
7,
8,
8,
8
1,
1,
1,
1,
2,
2,
2,
3,
3,
4,
4,
5,
5,
8
The module M has simple direct summands:
4 copies of simple module number 1
2 copies of simple module number 3
2 copies of simple module number 4
2 copies of simple module number 5
1 copy of simple module number 8
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
2
6,
8
7
2
socle layers
2
6
7,
8
2
2).
radical layers
2
6,
8
7
2
socle layers
2
6
7,
8
2
3).
radical layers
2
6,
8
7
2
socle layers
2
6
7,
8
2
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
195,
155,
35,
35,
35,
91,
77,
64
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0
0,
2,
0,
0,
0,
1,
1,
1
0,
0,
1,
0,
0,
0,
0,
0
0,
0,
0,
1,
0,
0,
0,
0
0,
0,
0,
0,
1,
0,
0,
0
0,
1,
0,
0,
0,
1,
1,
0
0,
1,
0,
0,
0,
0,
1,
0
0,
1,
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,
6,
7,
8
(3).
3
(4).
4
(5).
5
Projective modules number
1,
3,
4,
5
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
6,
8
7
2
socle layers
2
6
7,
8
2
Projective module number 6
radical layers
6
7
2
socle layers
6
7
2
Projective module number 7
radical layers
7
2
socle layers
7
2
Projective module number 8
radical layers
8
2
socle layers
8
2
The degrees of the splitting fields are
3,
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
79
.
The dimensions of the irreducible H-modules are
12,
3,
2,
2,
2,
1
.
The degrees of the splitting fields are
3,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
12,
6,
2,
2,
2,
1
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0
0,
2,
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,
1
The determinant of the Cartan matrix is 2.
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,
3,
4,
5,
6
are simple.
The radical and socle filtrations of the remaining
projective module for H are the following:
Projective module number 2
radical layers
2
2
socle layers
2
2