A7 Modulo 5
The Hecke algebra for the Alternating Group on 7 letters 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 prime
field of chacteristic 5, having point stablilizer
of order 20
The dimension of M is 126
.
The dimensions of the irreducible submodules modules are
35,
13,
8,
6,
1
.
The module M has radical filtration (Loewy series)
1,
1,
2,
4,
5
3,
3,
5
2,
4
The module M has socle filtration (socle series)
2,
4
3,
3,
5
1,
1,
2,
4,
5
The module M has simple direct summands:
2 copies of simple module number 1
1 copy of simple module number 5
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
4
3
4
socle layers
4
3
4
2).
radical layers
2
3,
5
2
socle layers
2
3,
5
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
35,
35,
27,
20,
14
.
The cartan matrix of A is
1,
0,
0,
0,
0
0,
2,
1,
0,
1
0,
1,
1,
1,
0
0,
0,
1,
2,
0
0,
1,
0,
0,
1
The determinant of the Cartan matrix is -1.
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
3,
5
2
socle layers
2
3,
5
2
Projective module number 3
radical layers
3
2,
4
socle layers
3
2,
4
Projective module number 4
radical layers
4
3
4
socle layers
4
3
4
Projective module number 5
radical layers
5
2
socle layers
5
2
The degrees of the splitting fields are
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
9
.
The dimensions of the irreducible H-modules are
2,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1
.
The dimensions of the projective modules of H are
2,
2,
1,
2
.
The cartan matrix of H is
1,
0,
0,
0
0,
2,
0,
0
0,
0,
1,
0
0,
0,
0,
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
Projective modules number
1,
3
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 2
radical layers
2
2
socle layers
2
2
Projective module number 4
radical layers
4
4
socle layers
4
4