5 ,
3
The Hecke algebra for the Symmetric Group
on 8 Letters, with the Partition
[ 5, 3 ]
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 56
.
The dimensions of the irreducible submodules modules are
14,
8,
6,
1
.
The module M has radical filtration (Loewy series)
1,
4
3,
3
2,
4
3
1
The module M has socle filtration (socle series)
1
3
2,
4
3,
3
1,
4
The 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
1
3
2
3
1
socle layers
1
3
2
3
1
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
48,
28,
49,
8
.
The cartan matrix of A is
2,
1,
2,
0
1,
1,
1,
0
2,
1,
2,
1
0,
0,
1,
2
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
1
socle layers
1
3
2
3
1
Projective module number 2
radical layers
2
3
1
socle layers
2
3
1
Projective module number 3
radical layers
3
1,
2,
4
3
1
socle layers
3
2
3
1,
1,
4
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
4
.
The dimensions of the irreducible H-modules are
1,
1
.
The degrees of the splitting fields are
1,
1
.
The dimensions of the projective modules of H are
2,
2
.
The cartan matrix of H is
The determinant of the Cartan matrix is 4.
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
socle layers
2
2
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