A8 Modulo 3
The Hecke algebra for the Alternating Group on 8 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 72
The dimension of M is 280
.
The dimensions of the irreducible submodules modules are
35,
28,
13,
7,
1
.
The module M has radical filtration (Loewy series)
2,
3,
4,
5
1,
3,
4,
5
1,
2,
3,
4,
4,
5
1,
4,
5
2,
3
The module M has socle filtration (socle series)
2,
3
1,
4,
5
1,
2,
3,
4,
4,
5
1,
3,
4,
5
2,
3,
4,
5
The module M has simple direct summands:
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
1
2,
4,
5
1
2
socle layers
2
1
2,
4,
5
1
2
3).
radical layers
3
4,
5
1,
3
4,
5
3
socle layers
3
4,
5
1,
3
4,
5
3
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
147,
162,
90,
111,
98
.
The cartan matrix of A is
2,
2,
1,
1,
1
2,
3,
0,
1,
1
1,
0,
3,
2,
2
1,
1,
2,
3,
1
1,
1,
2,
1,
2
The determinant of the Cartan matrix is -3.
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
2,
4,
5
1,
3
2
socle layers
1
2,
4,
5
1
2,
3
Projective module number 2
radical layers
2
1
2,
4,
5
1
2
socle layers
2
1
2,
4,
5
1
2
Projective module number 3
radical layers
3
4,
5
1,
3
4,
5
3
socle layers
3
4,
5
1,
3
4,
5
3
Projective module number 4
radical layers
4
1,
3
2,
4,
4,
5
3
socle layers
4
1,
3
4,
5
2,
3,
4
Projective module number 5
radical layers
5
1,
3
2,
4,
5
3
socle layers
5
1,
3
4,
5
2,
3
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
11
.
The dimensions of the irreducible H-modules are
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1
.
The dimensions of the projective modules of H are
3,
1,
3,
4
.
The cartan matrix of H is
3,
0,
0,
0
0,
1,
0,
0
0,
0,
2,
1
0,
0,
1,
3
The determinant of the Cartan matrix is 15.
The blocks of H consist of the following irreducible
modules:
Projective module number 2 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 1
radical layers
1
1
1
socle layers
1
1
1
Projective module number 3
radical layers
3
3,
4
socle layers
3
3,
4
Projective module number 4
radical layers
4
3,
4
4
socle layers
4
3,
4
4