M11 Modulo 11
The Hecke algebra for the Mathieu Group M11 with point
stabilizer being the normalizer of a
Sylow 11-subgroup
in characteristic 11
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 11, having point stablilizer
of order 55
The dimension of M is 144
.
The dimensions of the irreducible submodules modules are
55,
16,
11,
10,
10,
9,
1
.
The module M has radical filtration (Loewy series)
1,
2,
3,
7
2,
4
6
5
2
The module M has socle filtration (socle series)
2
4
6
2,
5
1,
2,
3,
7
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 7
The remaining indecomposable component of M has
radical and socle filtrations as follows:
1).
radical layers
2
2,
4
6
5
2
socle layers
2
4
6
2,
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
55,
77,
11,
45,
26,
35,
1
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0
0,
3,
0,
1,
1,
1,
0
0,
0,
1,
0,
0,
0,
0
0,
1,
0,
1,
1,
1,
0
0,
1,
0,
0,
1,
0,
0
0,
1,
0,
0,
1,
1,
0
0,
0,
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 2.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
4,
5,
6
(3).
3
(4).
7
Projective modules number
1,
3,
7
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
2,
4
6
5
2
socle layers
2
4
6
2,
5
2
Projective module number 4
radical layers
4
6
5
2
socle layers
4
6
5
2
Projective module number 5
radical layers
5
2
socle layers
5
2
Projective module number 6
radical layers
6
5
2
socle layers
6
5
2
The degrees of the splitting fields are
1,
1,
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
6
.
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,
1,
1
.
The cartan matrix of H is
3,
0,
0,
0
0,
1,
0,
0
0,
0,
1,
0
0,
0,
0,
1
The determinant of the Cartan matrix is 3.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4
Projective modules number
2,
3,
4
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