3 ,
3 ,
3
The Hecke algebra for the Symmetric Group
on 9 Letters, with the Partition
[ 3, 3, 3 ]
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 1680
.
The dimensions of the irreducible submodules modules are
162,
41,
35,
35,
27,
21,
21,
7,
7,
1,
1
.
The module M has radical filtration (Loewy series)
1,
1,
1,
2,
2,
5,
5,
5,
7,
9,
9,
10
2,
2,
4,
4,
6,
9,
9,
9,
10,
10,
11,
11
2,
2,
2,
2,
3,
3,
7,
7,
7,
8,
8,
9,
9,
10,
10
4,
4,
4,
6,
6,
9,
9,
9,
9,
11,
11,
11
2,
2,
2,
3,
7,
7,
8,
10
4,
9,
11
2
The module M has socle filtration (socle series)
2
4,
9,
11
2,
2,
2,
3,
7,
7,
8,
10
4,
4,
4,
6,
6,
9,
9,
9,
9,
11,
11,
11
2,
2,
2,
2,
3,
3,
7,
7,
7,
8,
8,
9,
9,
10,
10
2,
2,
4,
4,
6,
9,
9,
9,
10,
10,
11,
11
1,
1,
1,
2,
2,
5,
5,
5,
7,
9,
9,
10
The module M has simple direct summands:
3 copies of simple module number 1
3 copies of simple module number 5
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
9,
10
2,
10
9
socle layers
9
2,
10
9,
10
2).
radical layers
2,
7,
9
2,
4,
6,
9,
9,
10,
11
2,
2,
3,
7,
7,
8,
9,
10
4,
6,
9,
9,
11
2,
7
socle layers
2,
7
4,
6,
9,
9,
11
2,
2,
3,
7,
7,
8,
9,
10
2,
4,
6,
9,
9,
10,
11
2,
7,
9
3).
radical layers
2
4,
9,
11
2,
2,
3,
7,
8,
10
4,
4,
6,
9,
9,
11,
11
2,
2,
3,
7,
8,
10
4,
9,
11
2
socle layers
2
4,
9,
11
2,
2,
3,
7,
8,
10
4,
4,
6,
9,
9,
11,
11
2,
2,
3,
7,
8,
10
4,
9,
11
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
162,
567,
288,
477,
27,
210,
343,
252,
469,
225,
414
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
0,
6,
2,
4,
0,
1,
2,
2,
4,
2,
4
0,
2,
2,
2,
0,
1,
1,
1,
2,
1,
2
0,
4,
2,
4,
0,
1,
2,
2,
3,
2,
3
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0
0,
1,
1,
1,
0,
2,
2,
1,
1,
0,
1
0,
2,
1,
2,
0,
2,
4,
1,
3,
1,
1
0,
2,
1,
2,
0,
1,
1,
2,
1,
0,
2
0,
4,
2,
3,
0,
1,
3,
1,
5,
2,
2
0,
2,
1,
2,
0,
0,
1,
0,
2,
2,
1
0,
4,
2,
3,
0,
1,
1,
2,
2,
1,
4
The determinant of the Cartan matrix is 0.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
3,
4,
6,
7,
8,
9,
10,
11
(3).
5
Projective modules number
1,
5
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
4,
9,
11
2,
2,
3,
7,
8,
10
4,
4,
6,
9,
9,
11,
11
2,
2,
3,
7,
8,
10
4,
9,
11
2
socle layers
2
4,
9,
11
2,
2,
3,
7,
8,
10
4,
4,
6,
9,
9,
11,
11
2,
2,
3,
7,
8,
10
4,
9,
11
2
Projective module number 3
radical layers
3
4,
6,
9,
11
2,
3,
7,
8,
10
4,
9,
11
2
socle layers
3
4,
6,
9,
11
2,
3,
7,
8,
10
4,
9,
11
2
Projective module number 4
radical layers
4
2,
3,
7,
8,
10
4,
4,
6,
9,
9,
11,
11
2,
2,
3,
7,
8,
10
4,
9,
11
2
socle layers
4
2,
3,
7,
8,
10
4,
4,
6,
9,
9,
11,
11
2,
2,
3,
7,
8,
10
4,
9,
11
2
Projective module number 6
radical layers
6
3,
7,
8
4,
6,
9,
11
2,
7
socle layers
6
3,
7,
8
4,
6,
9,
11
2,
7
Projective module number 7
radical layers
7
4,
6,
9
2,
3,
7,
7,
8,
10
4,
6,
9,
9,
11
2,
7
socle layers
7
4,
6,
9
2,
3,
7,
7,
8,
10
4,
6,
9,
11
2,
7,
9
Projective module number 8
radical layers
8
4,
6,
11
2,
3,
7,
8
4,
9,
11
2
socle layers
8
4,
6,
11
2,
3,
7,
8
4,
9,
11
2
Projective module number 9
radical layers
9
2,
3,
7,
10
4,
4,
6,
9,
9,
9,
11
2,
2,
3,
7,
7,
8,
10
4,
9,
11
2
socle layers
9
2,
3,
7,
10
4,
4,
6,
9,
9,
11
2,
2,
3,
7,
8,
10
4,
9,
9,
11
2,
7
Projective module number 10
radical layers
10
4,
9
2,
3,
7,
10
4,
9,
11
2
socle layers
10
4,
9
2,
3,
7,
10
4,
9,
11
2
Projective module number 11
radical layers
11
2,
3,
8
4,
4,
6,
9,
11,
11
2,
2,
3,
7,
8,
10
4,
9,
11
2
socle layers
11
2,
3,
8
4,
4,
6,
9,
11,
11
2,
2,
3,
7,
8,
10
4,
9,
11
2
The degrees of the splitting fields are
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
55
.
The dimensions of the irreducible H-modules are
3,
3,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
3,
3,
7,
18,
12
.
The cartan matrix of H is
1,
0,
0,
0,
0
0,
1,
0,
0,
0
0,
0,
3,
3,
1
0,
0,
3,
10,
5
0,
0,
1,
5,
6
The determinant of the Cartan matrix is 71.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3,
4,
5
Projective modules number
1,
2
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 3
radical layers
3
4
3,
4
3,
5
4
socle layers
3
4
4
3
3,
4,
5
Projective module number 4
radical layers
4
3,
4,
4,
5
3,
4,
4,
5,
5
4,
4,
4
3,
4,
5
4,
5
socle layers
4
3,
4
3,
4,
5
4,
4,
4,
5
3,
4,
4,
5
4,
4,
5,
5
Projective module number 5
radical layers
5
4,
5
4,
4,
5
3,
5,
5
4
4
5
socle layers
5
4
4
3,
5,
5
4,
4,
5
4,
5
5