7 ,
3 ,
1
The Hecke algebra for the Symmetric Group
on 11 Letters, with the Partition
[ 7, 3, 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 1320
.
The dimensions of the irreducible submodules modules are
198,
186,
164,
100,
44,
10,
1
.
The module M has radical filtration (Loewy series)
5,
5,
6,
6,
7
2,
3,
4,
4,
5,
7
6,
6,
7,
7
1,
5
2,
7
3
7
The module M has socle filtration (socle series)
7
3
2,
7
1,
5
6,
6,
7,
7
2,
3,
4,
4,
5,
7
5,
5,
6,
6,
7
The indecomposable components of M have radical and
socle filtrations as follows:
1).
radical layers
6
4
6
socle layers
6
4
6
2).
radical layers
6
4
6
socle layers
6
4
6
3).
radical layers
5,
5,
7
2,
3,
5,
7
7,
7
1,
5
2,
7
3
7
socle layers
7
3
2,
7
1,
5
7,
7
2,
3,
5,
7
5,
5,
7
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
550,
781,
715,
110,
276,
120,
1542
.
The cartan matrix of A is
1,
1,
1,
0,
0,
0,
2
1,
2,
1,
0,
1,
0,
3
1,
1,
2,
0,
0,
0,
3
0,
0,
0,
1,
0,
1,
0
0,
1,
0,
0,
2,
0,
2
0,
0,
0,
1,
0,
2,
0
2,
3,
3,
0,
2,
0,
8
The determinant of the Cartan matrix is 1.
The blocks of A consist of the following irreducible
modules:
(1).
1,
2,
3,
5,
7
(2).
4,
6
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
2,
7
3
7
socle layers
1
7
2,
3
7
Projective module number 2
radical layers
2
1,
7
2,
5,
7
3
7
socle layers
2
1
7
2,
3,
7
5,
7
Projective module number 3
radical layers
3
7
1
2,
7
3
7
socle layers
3
7
1
7
2,
3
7
Projective module number 4
radical layers
4
6
socle layers
4
6
Projective module number 5
radical layers
5
5,
7
2
7
socle layers
5
7
2
5,
7
Projective module number 6
radical layers
6
4
6
socle layers
6
4
6
Projective module number 7
radical layers
7
1,
2,
3,
5,
7
2,
7,
7,
7
1,
3,
5
2,
7,
7
3
7
socle layers
7
3
2,
7
1,
1
7,
7,
7
2,
2,
3,
3,
7
5,
5,
7,
7
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
17
.
The dimensions of the irreducible H-modules are
2,
1
.
The degrees of the splitting fields are
1,
1
.
The dimensions of the projective modules of H are
4,
9
.
The cartan matrix of H is
The determinant of the Cartan matrix is 18.
The blocks of H consist of the following irreducible
modules:
The radical and socle filtrations of the projective
modules for H are the following:
Projective module number 1
radical layers
1
1
socle layers
1
1
Projective module number 2
radical layers
2
2,
2,
2,
2
2,
2,
2
2
socle layers
2
2
2,
2,
2
2,
2,
2,
2