A7 Modulo 3
The Hecke algebra for the Alternating Group on 7 letters 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 36
The dimension of M is 70
.
The dimensions of the irreducible submodules modules are
20,
13,
6,
1
.
The module M has radical filtration (Loewy series)
2,
3,
4
4,
4
1,
2
4,
4
2
The module M has socle filtration (socle series)
2
4,
4
1,
2
4,
4
2,
3,
4
The module M has simple direct summands:
1 copy of simple module number 3
1 copy of simple module number 4
The remaining indecomposable component of M has
radical and socle filtrations as follows:
1).
radical layers
2
4,
4
1,
2
4,
4
2
socle layers
2
4,
4
1,
2
4,
4
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
48,
63,
6,
77
.
The cartan matrix of A is
1,
2,
0,
2
1,
3,
0,
4
0,
0,
1,
0
1,
4,
0,
5
The determinant of the Cartan matrix is -1.
The blocks of A consist of the following irreducible
modules:
Projective module number 3 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 1
radical layers
1
4,
4
2,
2
socle layers
1
4,
4
2,
2
Projective module number 2
radical layers
2
4,
4
1,
2
4,
4
2
socle layers
2
4,
4
1,
2
4,
4
2
Projective module number 4
radical layers
4
1,
2,
2
4,
4,
4,
4
2,
2
socle layers
4
1,
2,
2
4,
4,
4,
4
2,
2
The degrees of the splitting fields are
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
5
.
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
3,
1,
1
.
The cartan matrix of H is
The determinant of the Cartan matrix is 3.
The blocks of H consist of the following irreducible
modules:
Projective modules number
2,
3
are simple.
The radical and socle filtrations of the remaining
projective module for H are the following:
Projective module number 1
radical layers
1
1
1
socle layers
1
1
1