7 ,
1 ,
1
The Hecke algebra for the Symmetric Group
on 9 Letters, with the Partition
[ 7, 1, 1 ]
in characteristic 3
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 3, having point stablilizer
equal to the Young subgroup of the partition.
The dimension of M is 72
.
The dimensions of the irreducible submodules modules are
27,
21,
7,
1
.
The module M has radical filtration (Loewy series)
1,
3,
4
2,
3,
4
3,
4
The module M has socle filtration (socle series)
3,
4
2,
3,
4
1,
3,
4
The module M has simple direct summands:
1 copy of simple module number 1
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
4
3
4
socle layers
4
3
4
2).
radical layers
3
2,
4
3
socle layers
3
2,
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
27,
28,
36,
9
.
The cartan matrix of A is
1,
0,
0,
0
0,
1,
1,
0
0,
1,
2,
1
0,
0,
1,
2
The determinant of the Cartan matrix is 1.
The blocks of A consist of the following irreducible
modules:
Projective module number 1 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
3
socle layers
2
3
Projective module number 3
radical layers
3
2,
4
3
socle layers
3
2,
4
3
Projective module number 4
radical layers
4
3
4
socle layers
4
3
4
The degrees of the splitting fields are
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
7
.
The dimensions of the irreducible H-modules are
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1
.
The dimensions of the projective modules of H are
3,
3,
1
.
The cartan matrix of H is
The determinant of the Cartan matrix is 3.
The blocks of H consist of the following irreducible
modules:
Projective module number 3 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 1
radical layers
1
2
1
socle layers
1
2
1
Projective module number 2
radical layers
2
1
2
socle layers
2
1
2