5 ,
4 ,
1
The Hecke algebra for the Symmetric Group
on 10 Letters, with the Partition
[ 5, 4, 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 1260
.
The dimensions of the irreducible submodules modules are
198,
160,
128,
48,
26,
16,
8,
1
.
The module M has radical filtration (Loewy series)
2,
4,
5,
7,
8
3,
4,
5,
6,
7,
8,
8
2,
5,
5,
5,
6,
7,
8,
8
4,
5,
6,
8,
8
1,
4,
5,
7
5,
8,
8
5,
6
5,
8
4,
7
The module M has socle filtration (socle series)
4,
7
5,
8
5,
6
5,
8,
8
1,
4,
5,
7
4,
5,
6,
8,
8
2,
5,
5,
5,
6,
7,
8,
8
3,
4,
5,
6,
7,
8,
8
2,
4,
5,
7,
8
The indecomposable components of M have radical and
socle filtrations as follows:
1).
radical layers
2
3
2
socle layers
2
3
2
2).
radical layers
7
8
5
8
7
8
5
8
7
socle layers
7
8
5
8
7
8
5
8
7
3).
radical layers
4
5
6,
8
5
1,
4
5,
8
6
5
4
socle layers
4
5
6
5,
8
1,
4
5
6,
8
5
4
4).
radical layers
5,
8
4,
6,
7,
8
5,
5,
7,
8
4,
6,
8
5
socle layers
5
4,
6,
8
5,
5,
7,
8
4,
6,
7,
8
5,
8
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,
448,
288,
480,
908,
447,
80,
506
.
The cartan matrix of A is
1,
0,
0,
1,
2,
1,
0,
1
0,
2,
1,
0,
0,
0,
0,
0
0,
1,
1,
0,
0,
0,
0,
0
1,
0,
0,
3,
4,
2,
0,
2
2,
0,
0,
4,
9,
4,
2,
6
1,
0,
0,
2,
4,
3,
0,
1
0,
0,
0,
0,
2,
0,
3,
4
1,
0,
0,
2,
6,
1,
4,
8
The determinant of the Cartan matrix is 5.
The blocks of A consist of the following irreducible
modules:
(1).
1,
4,
5,
6,
7,
8
(2).
2,
3
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
5,
8
6
5
4
socle layers
1
5
6,
8
5
4
Projective module number 2
radical layers
2
3
2
socle layers
2
3
2
Projective module number 3
radical layers
3
2
socle layers
3
2
Projective module number 4
radical layers
4
5
6,
8
5
1,
4
5,
8
6
5
4
socle layers
4
5
6
5,
8
1,
4
5
6,
8
5
4
Projective module number 5
radical layers
5
1,
4,
6,
8
5,
5,
5,
7,
8
1,
4,
6,
6,
8,
8
5,
5,
5,
5,
8
4,
6,
8
5,
7
4
socle layers
5
6
5,
8
1,
1,
4,
4,
7
5,
5,
8
5,
5,
6,
6,
8,
8
5,
5,
6,
8
4,
4,
5,
7,
8
Projective module number 6
radical layers
6
5,
6
1,
4
5,
5,
8
6
5
4
socle layers
6
5
1,
4
5
6,
8
5,
6
4,
5
Projective module number 7
radical layers
7
8
5
8
7
8
5
8
7
socle layers
7
8
5
8
7
8
5
8
7
Projective module number 8
radical layers
8
1,
5,
7,
8
4,
5,
8,
8,
8
5,
5,
6,
7
5,
8,
8
4,
5,
7
8
7
socle layers
8
5
8
1,
7,
7
5,
8,
8
5,
5,
6,
8
4,
5,
8,
8,
8
4,
5,
7,
7
The degrees of the splitting fields are
1,
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
22
.
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
10,
5,
5,
2
.
The cartan matrix of H is
6,
2,
2,
0
2,
3,
0,
0
2,
0,
3,
0
0,
0,
0,
2
The determinant of the Cartan matrix is 60.
The blocks of H consist of the following irreducible
modules:
The radical and socle filtrations of the projective
modules for H are the following:
Projective module number 1
radical layers
1
1,
1,
1,
2,
3
1,
1,
2,
3
socle layers
1
1,
1,
1
1,
1,
2,
2,
3,
3
Projective module number 2
radical layers
2
1,
2
1
2
socle layers
2
1
1,
2
2
Projective module number 3
radical layers
3
1,
3
1
3
socle layers
3
1
1,
3
3
Projective module number 4
radical layers
4
4
socle layers
4
4
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