PSU(3,3) Modulo 2
The Hecke algebra for the Projective Special Unitary Group of dimension 3
over GF(3) 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 32
The dimension of M is 189
.
The dimensions of the irreducible submodules modules are
32,
32,
14,
6,
1
.
The module M has radical filtration (Loewy series)
1,
2,
3,
4,
5
3,
4,
5
3,
4,
5
4,
5
4,
5
3,
4,
5
3,
4
5
4
The module M has socle filtration (socle series)
4
5
3,
4
3,
4,
5
4,
5
4,
5
3,
4,
5
3,
4,
5
1,
2,
3,
4,
5
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 2
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
3,
5
4
5
4
3
socle layers
3
4
5
4
3,
5
4
3
2).
radical layers
4
3,
5
4
5
4
3,
5
4
5
4
socle layers
4
5
4
3,
5
4
5
4
3,
5
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
32,
32,
62,
89,
62
.
The cartan matrix of A is
1,
0,
0,
0,
0
0,
1,
0,
0,
0
0,
0,
3,
3,
2
0,
0,
3,
7,
5
0,
0,
2,
5,
4
The determinant of the Cartan matrix is 5.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3,
4,
5
Projective modules number
1,
2
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 3
radical layers
3
4
3,
5
4
5
4
3
socle layers
3
4
5
4
3,
5
4
3
Projective module number 4
radical layers
4
3,
5
4,
4
3,
5,
5
4,
4
3,
5
4
5
4
socle layers
4
5
4
3,
5
4
5
4,
4
3,
3,
5,
5
4,
4
Projective module number 5
radical layers
5
4
3,
5
4,
4
3,
5
4
5
4
socle layers
5
4
5
4
5
4
3,
3,
5
4,
4
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
14
.
The dimensions of the irreducible H-modules are
1,
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
1,
1,
1,
5,
6
.
The cartan matrix of H is
1,
0,
0,
0,
0
0,
1,
0,
0,
0
0,
0,
1,
0,
0
0,
0,
0,
3,
2
0,
0,
0,
2,
4
The determinant of the Cartan matrix is 8.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4,
5
Projective modules number
1,
2,
3
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 4
radical layers
4
5
4
5
4
socle layers
4
5
4
5
4
Projective module number 5
radical layers
5
4,
5
5
4
5
socle layers
5
4
5
4
5,
5