6 ,
1 ,
1
The Hecke algebra for the Symmetric Group
on 8 Letters, with the Partition
[ 6, 1, 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 56
.
The dimensions of the irreducible submodules modules are
14,
6,
1
.
The module M has radical filtration (Loewy series)
2,
3
1,
2,
3
2,
3,
3
1
2
The module M has socle filtration (socle series)
2
1
2,
3,
3
1,
2,
3
2,
3
The indecomposable components of M have radical and
socle filtrations as follows:
1).
radical layers
3
2
3
socle layers
3
2
3
2).
radical layers
2
1,
3
2,
3
1
2
socle layers
2
1
2,
3
1,
3
2
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
41,
48,
29
.
The cartan matrix of A is
The determinant of the Cartan matrix is 3.
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
2,
3
1
2
socle layers
1
2
1,
3
2
Projective module number 2
radical layers
2
1,
3
2,
3
1
2
socle layers
2
1
2,
3
1,
3
2
Projective module number 3
radical layers
3
1,
2,
3
2,
3
socle layers
3
1,
2,
3
2,
3
The degrees of the splitting fields are
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
.
The degrees of the splitting fields are
1,
1
.
The dimensions of the projective modules of H are
3,
4
.
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,
2
socle layers
1
1,
2
Projective module number 2
radical layers
2
1,
2
2
socle layers
2
1,
2
2
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