4 ,
2 ,
2
The Hecke algebra for the Symmetric Group
on 8 Letters, with the Partition
[ 4, 2, 2 ]
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 420
.
The dimensions of the irreducible submodules modules are
35,
35,
28,
21,
13,
7,
7,
1,
1
.
The module M has radical filtration (Loewy series)
3,
3,
4,
5,
7,
7,
9
2,
2,
5,
5,
7,
8
1,
3,
3,
5,
6,
7,
7,
9,
9
2,
7,
8
3,
5
The module M has socle filtration (socle series)
3,
5
2,
7,
8
1,
3,
3,
5,
6,
7,
7,
9,
9
2,
2,
5,
5,
7,
8
3,
3,
4,
5,
7,
7,
9
The module M has simple direct summands:
1 copy of simple module number 4
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1).
radical layers
7
5
7
socle layers
7
5
7
2).
radical layers
7
5
7
socle layers
7
5
7
3).
radical layers
3,
9
2
3,
9
socle layers
3,
9
2
3,
9
4).
radical layers
3
2
3,
6,
9
2
3
socle layers
3
2
3,
6,
9
2
3
5).
radical layers
5
7,
8
1,
5
7,
8
5
socle layers
5
7,
8
1,
5
7,
8
5
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
56,
134,
162,
21,
90,
70,
83,
70,
65
.
The cartan matrix of A is
1,
0,
0,
0,
1,
0,
1,
1,
0
0,
2,
2,
0,
0,
1,
0,
0,
1
0,
2,
3,
0,
0,
1,
0,
0,
1
0,
0,
0,
1,
0,
0,
0,
0,
0
1,
0,
0,
0,
3,
0,
2,
2,
0
0,
1,
1,
0,
0,
1,
0,
0,
0
1,
0,
0,
0,
2,
0,
3,
1,
0
1,
0,
0,
0,
2,
0,
1,
2,
0
0,
1,
1,
0,
0,
0,
0,
0,
2
The determinant of the Cartan matrix is 1.
Projective module number 4 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 1
radical layers
1
7,
8
5
socle layers
1
7,
8
5
Projective module number 2
radical layers
2
3,
6,
9
2
3
socle layers
2
3,
6,
9
2
3
Projective module number 3
radical layers
3
2
3,
6,
9
2
3
socle layers
3
2
3,
6,
9
2
3
Projective module number 5
radical layers
5
7,
8
1,
5
7,
8
5
socle layers
5
7,
8
1,
5
7,
8
5
Projective module number 6
radical layers
6
2
3
socle layers
6
2
3
Projective module number 7
radical layers
7
1,
5
7,
7,
8
5
socle layers
7
1,
5
7,
8
5,
7
Projective module number 8
radical layers
8
1,
5
7,
8
5
socle layers
8
1,
5
7,
8
5
Projective module number 9
radical layers
9
2
3,
9
socle layers
9
2
3,
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
26
.
The dimensions of the irreducible H-modules are
2,
1,
1,
1,
1
.
The degrees of the splitting fields are
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
5,
5,
5,
1,
5
.
The cartan matrix of H is
2,
0,
0,
0,
1
0,
3,
2,
0,
0
0,
2,
3,
0,
0
0,
0,
0,
1,
0
1,
0,
0,
0,
3
The determinant of the Cartan matrix is 25.
Projective module number 4 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 1
radical layers
1
1,
5
socle layers
1
1,
5
Projective module number 2
radical layers
2
2,
3
2
3
socle layers
2
3
2
2,
3
Projective module number 3
radical layers
3
2
3
2
3
socle layers
3
2
3
2
3
Projective module number 5
radical layers
5
1,
5
5
socle layers
5
1,
5
5
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