6 ,
3 ,
1
The Hecke algebra for the Symmetric Group
on 10 Letters, with the Partition
[ 6, 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 840
.
The dimensions of the irreducible submodules modules are
198,
160,
48,
26,
16,
8,
1
.
The module M has radical filtration (Loewy series)
2,
3,
6,
7
4,
4,
6,
7
3,
4,
5,
7,
7,
7
4,
4,
6,
7
1,
3,
6,
7
4,
7,
7
4,
5
4,
7
3,
6
The module M has socle filtration (socle series)
3,
6
4,
7
4,
5
4,
7,
7
1,
3,
6,
7
4,
4,
6,
7
3,
4,
5,
7,
7,
7
4,
4,
6,
7
2,
3,
6,
7
The module M has simple direct summands:
1 copy of simple module number 2
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
7
4,
6
3,
7,
7
4,
6
7
socle layers
7
4,
6
3,
7,
7
4,
6
7
2).
radical layers
6
7
4
7
6
7
4
7
6
socle layers
6
7
4
7
6
7
4
7
6
3).
radical layers
3
4
5,
7
4
1,
3
4,
7
5
4
3
socle layers
3
4
5
4,
7
1,
3
4
5,
7
4
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
315,
160,
480,
866,
405,
80,
507
.
The cartan matrix of A is
1,
0,
1,
2,
1,
0,
1
0,
1,
0,
0,
0,
0,
0
1,
0,
3,
4,
2,
0,
2
2,
0,
4,
8,
3,
2,
6
1,
0,
2,
3,
2,
0,
1
0,
0,
0,
2,
0,
3,
4
1,
0,
2,
6,
1,
4,
9
The determinant of the Cartan matrix is 3.
The blocks of A consist of the following irreducible
modules:
(1).
1,
3,
4,
5,
6,
7
(2).
2
Projective module number 2 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 1
radical layers
1
4,
7
5
4
3
socle layers
1
4
5,
7
4
3
Projective module number 3
radical layers
3
4
5,
7
4
1,
3
4,
7
5
4
3
socle layers
3
4
5
4,
7
1,
3
4
5,
7
4
3
Projective module number 4
radical layers
4
1,
3,
5,
7
4,
4,
4,
6,
7
1,
3,
5,
7,
7
4,
4,
4,
7
3,
5,
7
4,
6
3
socle layers
4
5
4,
7
1,
1,
3,
3,
6
4,
4,
7
4,
5,
5,
7,
7
4,
4,
4,
7
3,
3,
6,
7
Projective module number 5
radical layers
5
4
1,
3
4,
7
5
4
3
socle layers
5
4
1,
3
4
5,
7
4
3
Projective module number 6
radical layers
6
7
4
7
6
7
4
7
6
socle layers
6
7
4
7
6
7
4
7
6
Projective module number 7
radical layers
7
1,
4,
6,
7
3,
4,
7,
7,
7
4,
4,
5,
6
4,
7,
7,
7
3,
4,
6
7
6
socle layers
7
4
7
1,
6,
6
4,
7,
7
3,
4,
4,
5,
7,
7
4,
4,
7,
7
3,
6,
6,
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
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1
.
The dimensions of the projective modules of H are
1,
5,
7,
4
.
The cartan matrix of H is
1,
0,
0,
0
0,
3,
2,
0
0,
2,
4,
1
0,
0,
1,
3
The determinant of the Cartan matrix is 21.
The blocks of H consist of the following irreducible
modules:
Projective module number 1 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 2
radical layers
2
2,
3
3
2
socle layers
2
3
2,
3
2
Projective module number 3
radical layers
3
2,
3,
3,
4
2,
3
socle layers
3
3,
3
2,
2,
3,
4
Projective module number 4
radical layers
4
3,
4
4
socle layers
4
3,
4
4