PSU(3,3) Modulo 7
The Hecke algebra for the Projective Special Unitary Group of dimension 3
over GF(3) with point stabilizer being the normalizer of a
Sylow 7-subgroup
in characteristic 7
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 7, having point stablilizer
of order 21
The dimension of M is 288
.
The dimensions of the irreducible submodules modules are
56,
42,
26,
21,
14,
7,
6,
1
.
The module M has radical filtration (Loewy series)
1,
1,
2,
3,
4,
5,
6,
8
7,
8
3
7
3
The module M has socle filtration (socle series)
3
7
3
7,
8
1,
1,
2,
3,
4,
5,
6,
8
The module M has simple direct summands:
2 copies of simple module number 1
1 copy of simple module number 2
1 copy of simple module number 4
1 copy of simple module number 5
1 copy of simple module number 6
1 copy of simple module number 8
The remaining indecomposable component of M has
radical and socle filtrations as follows:
1).
radical layers
3
7,
8
3
7
3
socle layers
3
7
3
7,
8
3
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
56,
42,
91,
21,
14,
7,
64,
27
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0,
0
0,
0,
3,
0,
0,
0,
2,
1
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,
2,
0,
0,
0,
2,
0
0,
0,
1,
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
(3).
3,
7,
8
(4).
4
(5).
5
(6).
6
Projective modules number
1,
2,
4,
5,
6
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 3
radical layers
3
7,
8
3
7
3
socle layers
3
7
3
7,
8
3
Projective module number 7
radical layers
7
3
7
3
socle layers
7
3
7
3
Projective module number 8
radical layers
8
3
socle layers
8
3
The degrees of the splitting fields are
2,
2,
1,
1,
2,
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
18
.
The dimensions of the irreducible H-modules are
4,
2,
2,
1,
1,
1,
1
.
The degrees of the splitting fields are
2,
2,
2,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
4,
2,
2,
3,
1,
1,
1
.
The cartan matrix of H 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,
3,
0,
0,
0
0,
0,
0,
0,
1,
0,
0
0,
0,
0,
0,
0,
1,
0
0,
0,
0,
0,
0,
0,
1
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
(6).
6
(7).
7
Projective modules number
1,
2,
3,
5,
6,
7
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
socle layers
4
4
4