PSU(3,4) Modulo 3
The Hecke algebra for the Projective Special Unitary Group of dimension 3
over GF(4) with point stabilizer being the normalizer of a
Sylow 3-subgroup
in characteristic 3
.
The Module M
The module M is the permutation module over the
field of chacteristic 3, having point stablilizer
of order 30
. The dimension of M is 2080
.
The dimensions of the irreducible submodules modules are
208,
78,
75,
75,
75,
75,
64,
52,
1
.
The module M has radical filtration (Loewy series)
1,
1,
2,
2,
3,
3,
4,
4,
5,
5,
6,
6,
7,
7,
7,
9
8,
8,
9,
9,
9
1,
1,
7,
7,
7
The module M has socle filtration (socle series)
1,
1,
7,
7,
7
8,
8,
9,
9,
9
1,
1,
2,
2,
3,
3,
4,
4,
5,
5,
6,
6,
7,
7,
7,
9
The module M has simple direct summands:
2 copies of simple module number 2
2 copies of simple module number 3
2 copies of simple module number 4
2 copies of simple module number 5
2 copies of simple module number 6
1 copy of simple module number 9
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
7
9
7
socle layers
7
9
7
2).
radical layers
7
9
7
socle layers
7
9
7
3).
radical layers
7
9
7
socle layers
7
9
7
4).
radical layers
1
8
1
socle layers
1
8
1
5).
radical layers
1
8
1
socle layers
1
8
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
468,
78,
75,
75,
75,
75,
129,
260,
65
.
The cartan matrix of A is
2,
0,
0,
0,
0,
0,
0,
1,
0
0,
1,
0,
0,
0,
0,
0,
0,
0
0,
0,
1,
0,
0,
0,
0,
0,
0
0,
0,
0,
1,
0,
0,
0,
0,
0
0,
0,
0,
0,
1,
0,
0,
0,
0
0,
0,
0,
0,
0,
1,
0,
0,
0
0,
0,
0,
0,
0,
0,
2,
0,
1
1,
0,
0,
0,
0,
0,
0,
1,
0
0,
0,
0,
0,
0,
0,
1,
0,
1
The determinant of the Cartan matrix is 1.
The blocks of A consist of the following irreducible
modules:
(1).
1,
8
(2).
2
(3).
3
(4).
4
(5).
5
(6).
6
(7).
7,
9
Projective modules number
2,
3,
4,
5,
6
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 1
radical layers
1
8
1
socle layers
1
8
1
Projective module number 7
radical layers
7
9
7
socle layers
7
9
7
Projective module number 8
radical layers
8
1
socle layers
8
1
Projective module number 9
radical layers
9
7
socle layers
9
7
The degrees of the splitting fields are
4,
2,
1,
1,
1,
1,
1,
4,
1
.
The Hecke Algebra
The Hecke algebra H of the module M is the A-endomorphism
ring of M.
The dimension of H is
75
.
The dimensions of the irreducible H-modules are
8,
4,
3,
2,
2,
2,
2,
1
.
The degrees of the splitting fields are
4,
2,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
40,
4,
6,
2,
2,
2,
2,
1
.
The cartan matrix of H is
5,
0,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0,
0
0,
0,
2,
0,
0,
0,
0,
0
0,
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,
0,
0,
0,
0,
1,
0
0,
0,
0,
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 10.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4
(5).
5
(6).
6
(7).
7
(8).
8
Projective modules number
2,
4,
5,
6,
7,
8
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 1
radical layers
1
1,
1,
1,
1
socle layers
1
1,
1,
1,
1
Projective module number 3
radical layers
3
3
socle layers
3
3