PSU(3,5) Modulo 3
The Hecke algebra for the Projective Special Unitary Group of dimension 3
over GF(5) 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 72
. The dimension of M is 1750
.
The dimensions of the irreducible submodules modules are
288,
126,
126,
126,
84,
28,
28,
28,
21,
20,
1
.
The module M has radical filtration (Loewy series)
1,
1,
2,
3,
4,
5,
6,
7,
8,
9,
9,
11
5,
9,
10,
10,
10
5,
6,
6,
7,
7,
8,
8,
9,
11,
11,
11
10,
10,
10
6,
7,
8
The module M has socle filtration (socle series)
6,
7,
8
10,
10,
10
5,
6,
6,
7,
7,
8,
8,
9,
11,
11,
11
5,
9,
10,
10,
10
1,
1,
2,
3,
4,
5,
6,
7,
8,
9,
9,
11
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 3
1 copy of simple module number 4
1 copy of simple module number 9
1 copy of simple module number 11
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
9
5
9
socle layers
9
5
9
2).
radical layers
5
9
5
socle layers
5
9
5
3).
radical layers
6
10
7,
8,
11
10
6
socle layers
6
10
7,
8,
11
10
6
4).
radical layers
7
10
6,
8,
11
10
7
socle layers
7
10
6,
8,
11
10
7
5).
radical layers
8
10
6,
7,
11
10
8
socle layers
8
10
6,
7,
11
10
8
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
288,
126,
126,
126,
189,
153,
153,
153,
126,
249,
105
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0
0,
0,
0,
0,
2,
0,
0,
0,
1,
0,
0
0,
0,
0,
0,
0,
2,
1,
1,
0,
2,
1
0,
0,
0,
0,
0,
1,
2,
1,
0,
2,
1
0,
0,
0,
0,
0,
1,
1,
2,
0,
2,
1
0,
0,
0,
0,
1,
0,
0,
0,
2,
0,
0
0,
0,
0,
0,
0,
2,
2,
2,
0,
4,
1
0,
0,
0,
0,
0,
1,
1,
1,
0,
1,
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
(4).
4
(5).
5,
9
(6).
6,
7,
8,
10,
11
Projective modules number
1,
2,
3,
4
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 5
radical layers
5
9
5
socle layers
5
9
5
Projective module number 6
radical layers
6
10
7,
8,
11
10
6
socle layers
6
10
7,
8,
11
10
6
Projective module number 7
radical layers
7
10
6,
8,
11
10
7
socle layers
7
10
6,
8,
11
10
7
Projective module number 8
radical layers
8
10
6,
7,
11
10
8
socle layers
8
10
6,
7,
11
10
8
Projective module number 9
radical layers
9
5
9
socle layers
9
5
9
Projective module number 10
radical layers
10
6,
7,
8,
11
10,
10,
10
6,
7,
8
socle layers
10
6,
7,
8,
11
10,
10,
10
6,
7,
8
Projective module number 11
radical layers
11
10
6,
7,
8
socle layers
11
10
6,
7,
8
The degrees of the splitting fields are
2,
1,
1,
1,
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
33
.
The dimensions of the irreducible H-modules are
4,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
2,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
4,
1,
1,
1,
1,
3,
4,
4,
4,
2,
4
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0
0,
0,
0,
0,
0,
2,
0,
0,
0,
0,
1
0,
0,
0,
0,
0,
0,
2,
1,
1,
0,
0
0,
0,
0,
0,
0,
0,
1,
2,
1,
0,
0
0,
0,
0,
0,
0,
0,
1,
1,
2,
0,
0
0,
0,
0,
0,
0,
0,
0,
0,
0,
1,
1
0,
0,
0,
0,
0,
1,
0,
0,
0,
1,
2
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
(6).
6,
10,
11
(7).
7,
8,
9
Projective modules number
1,
2,
3,
4,
5
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 6
radical layers
6
11
6
socle layers
6
11
6
Projective module number 7
radical layers
7
8,
9
7
socle layers
7
8,
9
7
Projective module number 8
radical layers
8
7,
9
8
socle layers
8
7,
9
8
Projective module number 9
radical layers
9
7,
8
9
socle layers
9
7,
8
9
Projective module number 10
radical layers
10
11
socle layers
10
11
Projective module number 11
radical layers
11
6,
10
11
socle layers
11
6,
10
11