6 ,
1 ,
1
The Hecke algebra for the Symmetric Group
on 8 Letters, with the Partition
[ 6, 1, 1 ]
in characteristic 5
.
The Module M
The module M is the permutation module over the prime
field of chacteristic 5, having point stablilizer
equal to the Young subgroup of the partition.
The dimension of M is 56
.
The dimensions of the irreducible submodules modules are
21,
20,
7,
1
.
The module M has radical filtration (Loewy series)
1,
2,
3,
3,
4
The module M has socle filtration (socle series)
1,
2,
3,
3,
4
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
21,
20,
7,
1
.
The cartan matrix of A is
1,
0,
0,
0
0,
1,
0,
0
0,
0,
1,
0
0,
0,
0,
1
The determinant of the Cartan matrix is 1.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3
(4).
4