6 ,
4 ,
1
The Hecke algebra for the Symmetric Group
on 11 Letters, with the Partition
[ 6, 4, 1 ]
in characteristic 2
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 2, having point stablilizer
equal to the Young subgroup of the partition.
The dimension of M is 2310
.
The dimensions of the irreducible submodules modules are
198,
186,
164,
144,
100,
44,
32,
10,
1
.
The module M has radical filtration (Loewy series)
3,
5,
6,
6,
8,
8,
9
3,
5,
6,
7,
8,
9,
9
1,
2,
5,
8,
9
2,
9,
9
3,
4,
6
2,
9
1
9
3
The module M has socle filtration (socle series)
3
9
1
2,
9
3,
4,
6
2,
9,
9
1,
2,
5,
8,
9
3,
5,
6,
7,
8,
9,
9
3,
5,
6,
6,
8,
8,
9
The module M has simple direct summands:
1 copy of simple module number 6
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
9
3
9
socle layers
9
3
9
2).
radical layers
8
5
8
socle layers
8
5
8
3).
radical layers
5,
8
7,
8
5
socle layers
5
7,
8
5,
8
4).
radical layers
6
6,
9
2
9
6
socle layers
6
9
2
6,
9
6
5).
radical layers
3
9
1
2,
9
3,
4
2,
9
1
9
3
socle layers
3
9
1
2,
9
3,
4
2,
9
1
9
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
1243,
1287,
1408,
693,
242,
320,
132,
120,
2047
.
The cartan matrix of A is
2,
2,
2,
1,
0,
0,
0,
0,
3
2,
2,
2,
1,
0,
1,
0,
0,
3
2,
2,
3,
1,
0,
0,
0,
0,
4
1,
1,
1,
1,
0,
0,
0,
0,
1
0,
0,
0,
0,
2,
0,
1,
1,
0
0,
1,
0,
0,
0,
3,
0,
0,
2
0,
0,
0,
0,
1,
0,
1,
0,
0
0,
0,
0,
0,
1,
0,
0,
2,
0
3,
3,
4,
1,
0,
2,
0,
0,
7
The determinant of the Cartan matrix is -1.
The blocks of A consist of the following irreducible
modules:
(1).
1,
2,
3,
4,
6,
9
(2).
5,
7,
8
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
2,
9
3,
4
2,
9
1
9
3
socle layers
1
2,
9
3,
4
2,
9
1
9
3
Projective module number 2
radical layers
2
1,
4,
9
2,
6,
9
1,
3
9
3
socle layers
2
4
2,
9
1,
1
9,
9
3,
3,
6
Projective module number 3
radical layers
3
9
1
2,
9
3,
4
2,
9
1
9
3
socle layers
3
9
1
2,
9
3,
4
2,
9
1
9
3
Projective module number 4
radical layers
4
2
1
9
3
socle layers
4
2
1
9
3
Projective module number 5
radical layers
5
7,
8
5
socle layers
5
7
5,
8
Projective module number 6
radical layers
6
6,
9
2
9
6
socle layers
6
9
2
6,
9
6
Projective module number 7
radical layers
7
5
socle layers
7
5
Projective module number 8
radical layers
8
5
8
socle layers
8
5
8
Projective module number 9
radical layers
9
1,
2,
3,
6
2,
9,
9,
9
1,
3,
4,
6
2,
9,
9
1,
3
9
3
socle layers
9
1
2,
9
3,
3,
4
2,
2,
9,
9
1,
1
9,
9,
9
3,
3,
6,
6
The degrees of the splitting fields are
1,
1,
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
22
.
The dimensions of the irreducible H-modules are
1,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
3,
4,
5,
4,
2,
4
.
The cartan matrix of H is
2,
1,
0,
0,
0,
0
1,
3,
0,
0,
0,
0
0,
0,
3,
2,
0,
0
0,
0,
2,
2,
0,
0
0,
0,
0,
0,
1,
1
0,
0,
0,
0,
1,
3
The determinant of the Cartan matrix is 20.
The blocks of H consist of the following irreducible
modules:
(1).
1,
2
(2).
3,
4
(3).
5,
6
The radical and socle filtrations of the projective
modules for H are the following:
Projective module number 1
radical layers
1
1,
2
socle layers
1
1,
2
Projective module number 2
radical layers
2
1,
2
2
socle layers
2
1,
2
2
Projective module number 3
radical layers
3
3,
4
3,
4
socle layers
3
3,
4
3,
4
Projective module number 4
radical layers
4
3
3
4
socle layers
4
3
3
4
Projective module number 5
radical layers
5
6
socle layers
5
6
Projective module number 6
radical layers
6
5,
6
6
socle layers
6
5,
6
6