3 ,
3 ,
1 ,
1
The Hecke algebra for the Symmetric Group
on 8 Letters, with the Partition
[ 3, 3, 1, 1 ]
in characteristic 3
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 3, having point stablilizer
equal to the Young subgroup of the partition.
The dimension of M is 1120
.
The dimensions of the irreducible submodules modules are
90,
35,
35,
28,
21,
21,
13,
13,
7,
7,
1,
1
.
The module M has radical filtration (Loewy series)
1,
3,
4,
4,
4,
4,
6,
6,
6,
7,
7,
10,
10,
10,
12
3,
3,
3,
4,
5,
7,
7,
9,
10,
10,
11,
11,
12
2,
3,
3,
4,
4,
4,
6,
7,
7,
8,
9,
9,
9,
10,
10,
12,
12,
12
3,
3,
3,
4,
9,
10,
11,
12
3,
4,
4,
4,
7
The module M has socle filtration (socle series)
3,
4,
4,
4,
7
3,
3,
3,
4,
9,
10,
11,
12
2,
3,
3,
4,
4,
4,
6,
7,
7,
8,
9,
9,
9,
10,
10,
12,
12,
12
3,
3,
3,
4,
5,
7,
7,
9,
10,
10,
11,
11,
12
1,
3,
4,
4,
4,
4,
6,
6,
6,
7,
7,
10,
10,
10,
12
The module M has simple direct summands:
1 copy of simple module number 1
1 copy of simple module number 4
2 copies of simple module number 6
1 copy of simple module number 10
1 copy of simple module number 12
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
10
7
10
socle layers
10
7
10
2).
radical layers
6
5
6
socle layers
6
5
6
3).
radical layers
10
7
10
socle layers
10
7
10
4).
radical layers
7
10,
11
7
socle layers
7
10,
11
7
5).
radical layers
4
3
4,
9,
12
3
4
socle layers
4
3
4,
9,
12
3
4
6).
radical layers
4
3
4,
9,
12
3
4
socle layers
4
3
4,
9,
12
3
4
7).
radical layers
4
3
4,
9,
12
3
4
socle layers
4
3
4,
9,
12
3
4
8).
radical layers
7
10,
11
2,
7
10,
11
7
socle layers
7
10,
11
2,
7
10,
11
7
9).
radical layers
3
4,
9,
12
3,
3,
8
4,
9,
12
3
socle layers
3
4,
9,
12
3,
3,
8
4,
9,
12
3
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
90,
56,
225,
162,
42,
63,
90,
56,
126,
83,
70,
120
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
1,
0,
0,
1,
1,
0
0,
0,
4,
2,
0,
0,
0,
1,
2,
0,
0,
2
0,
0,
2,
3,
0,
0,
0,
0,
1,
0,
0,
1
0,
0,
0,
0,
1,
1,
0,
0,
0,
0,
0,
0
0,
0,
0,
0,
1,
2,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
3,
0,
0,
2,
2,
0
0,
0,
1,
0,
0,
0,
0,
1,
1,
0,
0,
1
0,
0,
2,
1,
0,
0,
0,
1,
2,
0,
0,
1
0,
1,
0,
0,
0,
0,
2,
0,
0,
3,
1,
0
0,
1,
0,
0,
0,
0,
2,
0,
0,
1,
2,
0
0,
0,
2,
1,
0,
0,
0,
1,
1,
0,
0,
2
The determinant of the Cartan matrix is 1.
Projective module number 1 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
10,
11
7
socle layers
2
10,
11
7
Projective module number 3
radical layers
3
4,
9,
12
3,
3,
8
4,
9,
12
3
socle layers
3
4,
9,
12
3,
3,
8
4,
9,
12
3
Projective module number 4
radical layers
4
3
4,
9,
12
3
4
socle layers
4
3
4,
9,
12
3
4
Projective module number 5
radical layers
5
6
socle layers
5
6
Projective module number 6
radical layers
6
5
6
socle layers
6
5
6
Projective module number 7
radical layers
7
10,
11
2,
7
10,
11
7
socle layers
7
10,
11
2,
7
10,
11
7
Projective module number 8
radical layers
8
9,
12
3
socle layers
8
9,
12
3
Projective module number 9
radical layers
9
3,
8
4,
9,
12
3
socle layers
9
3,
8
4,
9,
12
3
Projective module number 10
radical layers
10
2,
7
10,
10,
11
7
socle layers
10
2,
7
10,
11
7,
10
Projective module number 11
radical layers
11
2,
7
10,
11
7
socle layers
11
2,
7
10,
11
7
Projective module number 12
radical layers
12
3,
8
4,
9,
12
3
socle layers
12
3,
8
4,
9,
12
3
The degrees of the splitting fields are
1,
1,
1,
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
92
.
The dimensions of the irreducible H-modules are
3,
2,
2,
1,
1,
1,
1,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
12,
3,
7,
10,
1,
4,
4,
3,
6,
7,
1
.
The cartan matrix of H is
3,
0,
0,
2,
0,
1,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
1,
0,
0,
0,
0
0,
0,
2,
0,
0,
0,
0,
1,
1,
1,
0
2,
0,
0,
4,
0,
0,
0,
0,
0,
0,
0
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0
1,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
2,
0,
0,
0,
0
0,
0,
1,
0,
0,
0,
0,
1,
0,
0,
0
0,
0,
1,
0,
0,
0,
0,
0,
2,
2,
0
0,
0,
1,
0,
0,
0,
0,
0,
2,
3,
0
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 4.
Projective modules number
5,
11
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 1
radical layers
1
4,
6
1
4
1
socle layers
1
4
1
4,
6
1
Projective module number 2
radical layers
2
7
socle layers
2
7
Projective module number 3
radical layers
3
8,
9
3,
10
socle layers
3
8,
9
3,
10
Projective module number 4
radical layers
4
1,
4
4
1
4
socle layers
4
1
4
1,
4
4
Projective module number 6
radical layers
6
1
socle layers
6
1
Projective module number 7
radical layers
7
2
7
socle layers
7
2
7
Projective module number 8
radical layers
8
3
socle layers
8
3
Projective module number 9
radical layers
9
3,
10
9
10
socle layers
9
3,
10
9
10
Projective module number 10
radical layers
10
9
3,
10
9
10
socle layers
10
9
3,
10
9
10
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