PSU(3,4) Modulo 5
The Hecke algebra for the Projective Special Unitary Group of dimension 3
over GF(4) with point stabilizer being the normalizer of a
Sylow 5-subgroup
in characteristic 5
.
The Module M
The module M is the permutation module over the
field of chacteristic 5, having point stablilizer
of order 150
. The dimension of M is 416
.
The dimensions of the irreducible submodules modules are
65,
39,
12,
1
.
The module M has radical filtration (Loewy series)
1,
2,
4
2,
3
2,
4
3
2,
4
3
2,
4
3
2,
4
3
4
3
2
The module M has socle filtration (socle series)
2
3
4
3
2,
4
3
2,
4
3
2,
4
3
2,
4
2,
3
1,
2,
4
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 4
The remaining indecomposable component of M
has radical and socle filtrations as follows:
1).
radical layers
2
2,
3
2,
4
3
2,
4
3
2,
4
3
2,
4
3
4
3
2
socle layers
2
3
4
3
2,
4
3
2,
4
3
2,
4
3
2,
4
2,
3
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
65,
350,
336,
272
.
The cartan matrix of A is
1,
0,
0,
0
0,
7,
6,
5
0,
6,
8,
6
0,
5,
6,
5
The determinant of the Cartan matrix is 8.
The blocks of A consist of the following irreducible
modules:
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
2,
3
2,
4
3
2,
4
3
2,
4
3
2,
4
3
4
3
2
socle layers
2
3
4
3
2,
4
3
2,
4
3
2,
4
3
2,
4
2,
3
2
Projective module number 3
radical layers
3
2,
4
3,
3
2,
4,
4
3,
3
2,
4
3
2,
4
3
2,
4
3
2
socle layers
3
4
3
4
3
2,
4
3
2,
4
3,
3
2,
4,
4
2,
3,
3
2,
2
Projective module number 4
radical layers
4
3
2,
4
3,
3
2,
4
3
2,
4
3
2,
4
3
2
socle layers
4
3
4
3
4
3
2,
4
3
2,
4
2,
3,
3
2,
2
The degrees of the splitting fields are
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
9
.
The dimensions of the irreducible H-modules are
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1
.
The dimensions of the projective modules of H are
1,
7,
1
.
The cartan matrix of H is
The determinant of the Cartan matrix is 7.
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
2
2
socle layers
2
2
2,
2
2,
2
2