4 ,
4
The Hecke algebra for the Symmetric Group
on 8 Letters, with the Partition
[ 4, 4 ]
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 70
.
The dimensions of the irreducible submodules modules are
14,
8,
6,
1
.
The module M has radical filtration (Loewy series)
3,
4
1,
2,
4
3,
3
1,
2
3
The module M has socle filtration (socle series)
3
1,
2
3,
3
1,
2,
4
3,
4
The module M is indecomposable
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
34,
42,
69,
7
.
The cartan matrix of A is
1,
1,
2,
0
1,
2,
2,
0
2,
2,
4,
1
0,
0,
1,
1
The determinant of the Cartan matrix is -1.
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
3
2
3
socle layers
1
3
2
3
Projective module number 2
radical layers
2
2,
3
1
3
socle layers
2
3
1,
2
3
Projective module number 3
radical layers
3
1,
2,
4
3,
3
1,
2
3
socle layers
3
1,
2
3,
3
1,
2
3,
4
Projective module number 4
radical layers
4
3
socle layers
4
3
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
5
.
The dimensions of the irreducible H-modules are
1
.
The degrees of the splitting fields are
1
.
The dimensions of the projective modules of H are
5
.
The cartan matrix of H is
The determinant of the Cartan matrix is 5.
The radical and socle filtrations of the projective
modules for H are the following:
Projective module number 1
radical layers
1
1,
1,
1
1
socle layers
1
1,
1
1,
1
<\HTML><\BODY>