M12 Modulo 2
The Hecke algebra for the Mathieu Group M12 with point
stabilizer being the normalizer of a
Sylow 2-subgroup
in characteristic 2
.
The Module M
The module M is the permutation module over the
field of chacteristic 2, having point stablilizer
of order 64
. The dimension of M is 1485
.
The dimensions of the irreducible submodules modules are
144,
44,
32,
10,
1
.
The module M has radical filtration (Loewy series)
1,
1,
1,
2,
2,
2,
3,
4,
5
2,
3,
3,
4,
4,
4,
4,
5,
5,
5,
5,
5
1,
1,
2,
2,
2,
4,
4,
4,
5,
5
2,
2,
2,
4,
4,
4,
5,
5
2,
4,
4,
5
2
The module M has socle filtration (socle series)
2
2,
4,
4,
5
2,
2,
2,
4,
4,
4,
5,
5
1,
1,
2,
2,
2,
4,
4,
4,
5,
5
2,
3,
3,
4,
4,
4,
4,
5,
5,
5,
5,
5
1,
1,
1,
2,
2,
2,
3,
4,
5
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 3
1 copy of simple module number 5
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
1
3
1
socle layers
1
3
1
2).
radical layers
1
3
1
socle layers
1
3
1
3).
radical layers
2,
2
4,
4,
5,
5
2,
4,
5
2,
2,
4
4,
5
2
socle layers
2
4,
5
2,
2,
4
2,
4,
5
4,
4,
5,
5
2,
2
4).
radical layers
2,
4
2,
4,
4,
5,
5,
5
2,
2,
4,
4,
5
2,
4,
4,
5,
5
2,
4
socle layers
2,
4
2,
4,
4,
5,
5
2,
2,
4,
4,
5
2,
4,
4,
5,
5,
5
2,
4
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
320,
372,
320,
349,
228
.
The cartan matrix of A is
2,
0,
1,
0,
0
0,
7,
0,
6,
4
2,
0,
1,
0,
0
0,
6,
0,
8,
5
0,
4,
0,
5,
2
The determinant of the Cartan matrix is 0.
The blocks of A consist of the following irreducible
modules:
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
3
1
socle layers
1
3
1
Projective module number 2
radical layers
2
2,
4,
5
2,
4,
4,
5,
5
2,
2,
2,
4,
4
4,
5
2
socle layers
2
4,
5
2,
4
2,
2,
4
4,
4,
5,
5,
5
2,
2,
2,
4
Projective module number 3
radical layers
3
1,
1
socle layers
3
1,
1
Projective module number 4
radical layers
4
2,
4,
5,
5
2,
2,
4,
4,
5
2,
2,
4,
4,
4,
5,
5
2,
4
socle layers
4
2,
4,
5,
5
2,
2,
4,
4
2,
4,
4,
5,
5,
5
2,
2,
4,
4
Projective module number 5
radical layers
5
2,
4,
4
2,
2,
4,
5
4,
4
2
socle layers
5
4
2,
2,
4
2,
4,
4,
5
2,
4
The degrees of the splitting fields are
1,
1,
2,
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
44
.
The dimensions of the irreducible H-modules are
2,
2,
1,
1,
1,
1
.
The degrees of the splitting fields are
2,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
2,
5,
1,
13,
15,
3
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0
0,
2,
0,
0,
0,
1
0,
0,
1,
0,
0,
0
0,
0,
0,
7,
6,
0
0,
0,
0,
6,
9,
0
0,
1,
0,
0,
0,
1
The determinant of the Cartan matrix is 27.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2,
6
(3).
3
(4).
4,
5
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
6
2
socle layers
2
6
2
Projective module number 4
radical layers
4
4,
5
4,
5,
5
4,
4,
5
4,
5
4,
5
socle layers
4
5
4
4,
4,
5
4,
5,
5
4,
4,
5,
5
Projective module number 5
radical layers
5
4,
5,
5
4,
4,
5
4,
5,
5,
5
4,
5
4
5
socle layers
5
4
5
4
4,
5,
5
4,
4,
5,
5
4,
5,
5,
5
Projective module number 6
radical layers
6
2
socle layers
6
2