PSL(3,4) Modulo 3
The Hecke algebra for the Projective Special Linear 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 prime
field of chacteristic 3, having point stablilizer
of order 72
The dimension of M is 280
.
The dimensions of the irreducible submodules modules are
90,
19,
15,
15,
15,
1
.
The module M has radical filtration (Loewy series)
1,
2,
6
3,
4,
5,
6,
6
2,
2,
2
3,
4,
5,
6,
6
2
The module M has socle filtration (socle series)
2
3,
4,
5,
6,
6
2,
2,
2
3,
4,
5,
6,
6
1,
2,
6
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 6
The remaining indecomposable component of M has
radical and socle filtrations as follows:
1).
radical layers
2
3,
4,
5,
6,
6
2,
2,
2
3,
4,
5,
6,
6
2
socle layers
2
3,
4,
5,
6,
6
2,
2,
2
3,
4,
5,
6,
6
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
90,
189,
84,
84,
84,
125
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0
0,
5,
2,
2,
2,
4
0,
2,
1,
1,
1,
1
0,
2,
1,
1,
1,
1
0,
2,
1,
1,
1,
1
0,
4,
1,
1,
1,
4
The determinant of the Cartan matrix is 0.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
3,
4,
5,
6
Projective module number 1 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
3,
4,
5,
6,
6
2,
2,
2
3,
4,
5,
6,
6
2
socle layers
2
3,
4,
5,
6,
6
2,
2,
2
3,
4,
5,
6,
6
2
Projective module number 3
radical layers
3
2
4,
5,
6
2
socle layers
3
2
4,
5,
6
2
Projective module number 4
radical layers
4
2
3,
5,
6
2
socle layers
4
2
3,
5,
6
2
Projective module number 5
radical layers
5
2
3,
4,
6
2
socle layers
5
2
3,
4,
6
2
Projective module number 6
radical layers
6
2,
2
3,
4,
5,
6,
6,
6
2,
2
socle layers
6
2,
2
3,
4,
5,
6,
6,
6
2,
2
The degrees of the splitting fields are
2,
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
8
.
The dimensions of the irreducible H-modules are
2,
1,
1
.
The degrees of the splitting fields are
2,
1,
1
.
The dimensions of the projective modules of H are
2,
5,
1
.
The cartan matrix of H is
The determinant of the Cartan matrix is 5.
The blocks of H consist of the following irreducible
modules:
Projective modules number
1,
3
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,
2,
2
2
socle layers
2
2,
2,
2
2