5 ,
3 ,
2
The Hecke algebra for the Symmetric Group
on 10 Letters, with the Partition
[ 5, 3, 2 ]
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 2520
.
The dimensions of the irreducible submodules modules are
200,
198,
160,
128,
48,
26,
16,
8,
1
.
The module M has radical filtration (Loewy series)
2,
3,
3,
5,
6,
8,
9
4,
5,
6,
6,
6,
7,
8,
9,
9,
9
3,
5,
6,
6,
6,
7,
7,
8,
9,
9,
9
2,
5,
6,
6,
6,
7,
8,
9,
9,
9
1,
2,
5,
5,
6,
6,
8,
9
6,
7,
9,
9,
9
2,
6,
7,
8
6,
9,
9
5,
6,
8
The module M has socle filtration (socle series)
5,
6,
8
6,
9,
9
2,
6,
7,
8
6,
7,
9,
9,
9
1,
2,
5,
5,
6,
6,
8,
9
2,
5,
6,
6,
6,
7,
8,
9,
9,
9
3,
5,
6,
6,
6,
7,
7,
8,
9,
9,
9
4,
5,
6,
6,
6,
7,
8,
9,
9,
9
2,
3,
3,
5,
6,
8,
9
The module M has simple direct summands:
1 copy of simple module number 3
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
3
4
3
socle layers
3
4
3
2).
radical layers
9
6,
8
5,
9,
9
6,
8
9
socle layers
9
6,
8
5,
9,
9
6,
8
9
3).
radical layers
8
9
6
9
8
9
6
9
8
socle layers
8
9
6
9
8
9
6
9
8
4).
radical layers
5
6
7,
9
6
2,
5
6,
9
7
6
5
socle layers
5
6
7
6,
9
2,
5
6
7,
9
6
5
5).
radical layers
2,
6
5,
6,
7,
9,
9
6,
6,
7,
8
2,
5,
6,
7,
9,
9
1,
5,
6,
6
7,
9
2,
8
9
6
socle layers
6
9
2,
8
7,
9
1,
5,
6,
6
2,
5,
6,
7,
9,
9
6,
6,
7,
8
5,
6,
7,
9,
9
2,
6
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
450,
990,
448,
288,
480,
1556,
897,
530,
1605
.
The cartan matrix of A is
1,
1,
0,
0,
0,
1,
1,
1,
2
1,
3,
0,
0,
1,
4,
2,
1,
4
0,
0,
2,
1,
0,
0,
0,
0,
0
0,
0,
1,
1,
0,
0,
0,
0,
0
0,
1,
0,
0,
3,
4,
2,
0,
2
1,
4,
0,
0,
4,
10,
5,
3,
8
1,
2,
0,
0,
2,
5,
4,
1,
3
1,
1,
0,
0,
0,
3,
1,
4,
6
2,
4,
0,
0,
2,
8,
3,
6,
13
The determinant of the Cartan matrix is -1.
The blocks of A consist of the following irreducible
modules:
(1).
1,
2,
5,
6,
7,
8,
9
(2).
3,
4
The radical and socle filtrations of the projective
modules for A are the following:
Projective module number 1
radical layers
1
7,
9
2,
8
9
6
socle layers
1
9
8
7,
9
2,
6
Projective module number 2
radical layers
2
6,
9,
9
1,
2,
7
6,
6,
9
2,
5,
7,
8
9
6
socle layers
2
9
1,
6
2,
7,
9,
9
6,
6,
8
5,
7,
9
2,
6
Projective module number 3
radical layers
3
4
3
socle layers
3
4
3
Projective module number 4
radical layers
4
3
socle layers
4
3
Projective module number 5
radical layers
5
6
7,
9
6
2,
5
6,
9
7
6
5
socle layers
5
6
7
6,
9
2,
5
6
7,
9
6
5
Projective module number 6
radical layers
6
2,
5,
7,
9
6,
6,
6,
8,
9
2,
2,
5,
7,
7,
9,
9
1,
6,
6,
6,
6,
9
5,
7,
7,
9,
9
2,
6,
8,
8
5,
9
6
socle layers
6
9
7,
8
6,
9
1,
2,
2,
5,
5
6,
6,
6,
9
6,
7,
7,
7,
8,
9,
9
6,
6,
6,
7,
9,
9,
9
2,
2,
5,
5,
6,
8
Projective module number 7
radical layers
7
1,
6,
7
2,
5,
7,
9
6,
6,
8,
9
2,
7,
9
6,
6
5
socle layers
7
6
1,
2,
5
6,
9
7,
7,
8,
9
6,
6,
7,
9
2,
5,
6
Projective module number 8
radical layers
8
9
1,
6
7,
9,
9
2,
8,
8
9,
9
6,
6
9
8
socle layers
8
9
6
9
1,
8
9,
9
6,
8
7,
9,
9
2,
6,
8
Projective module number 9
radical layers
9
1,
2,
2,
6,
8,
9
5,
6,
7,
9,
9,
9,
9
1,
2,
6,
6,
7,
8,
8
6,
7,
9,
9,
9,
9,
9
2,
5,
6,
6,
8,
8
9,
9
6,
8
socle layers
9
6,
8
9,
9
1,
1,
2,
8
6,
9,
9,
9
5,
6,
6,
7,
8,
8,
9,
9
6,
6,
7,
7,
9,
9,
9,
9
2,
2,
2,
5,
6,
6,
8,
8,
9
The degrees of the splitting fields are
1,
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
40
.
The dimensions of the irreducible H-modules are
1,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
2,
6,
9,
3,
7,
13
.
The cartan matrix of H is
1,
0,
0,
1,
0,
0
0,
3,
2,
0,
0,
1
0,
2,
4,
0,
1,
2
1,
0,
0,
2,
0,
0
0,
0,
1,
0,
3,
3
0,
1,
2,
0,
3,
7
The determinant of the Cartan matrix is 76.
The blocks of H consist of the following irreducible
modules:
(1).
1,
4
(2).
2,
3,
5,
6
The radical and socle filtrations of the projective
modules for H are the following:
Projective module number 1
radical layers
1
4
socle layers
1
4
Projective module number 2
radical layers
2
2,
3
3,
6
2
socle layers
2
3
2,
3
2,
6
Projective module number 3
radical layers
3
2,
3,
3,
6
2,
3,
5,
6
socle layers
3
3,
3,
6
2,
2,
3,
5,
6
Projective module number 4
radical layers
4
1
4
socle layers
4
1
4
Projective module number 5
radical layers
5
5,
6
3,
6
6
5
socle layers
5
6
3,
6
5,
6
5
Projective module number 6
radical layers
6
3,
5,
6,
6
2,
3,
5,
6,
6,
6
5,
6
socle layers
6
3,
6,
6
3,
5,
6,
6
2,
5,
5,
6,
6
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