Schur Algebra S( 3 ,8) in characteristic 3

Field k

Finite field of size 3

The Module M

The module M is the direct sum of permutation module with point stabilizers being the Young subgroups corresponding to partitions of lenght at most 3. . The dimension of M is 1647 .

The dimensions of the irreducible submodules modules are 35, 35, 28, 21, 21, 13, 7, 7, 1, 1 .



The module M has radical filtration (Loewy series)
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 9, 9, 9, 9

1, 1, 3, 3, 3, 3, 3, 3, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10

2, 2, 2, 2, 8, 8, 9, 9

3, 3, 3, 3, 6, 6



The module M has socle filtration (socle series)
3, 3, 3, 3, 6, 6

2, 2, 2, 2, 8, 8, 9, 9

1, 1, 3, 3, 3, 3, 3, 3, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10

2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 9, 9, 9, 9

3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10



The module M has simple direct summands:

4 copies of simple module number 3
4 copies of simple module number 5
4 copies of simple module number 8
8 copies of simple module number 10

The remaining indecomposable components of M have radical and socle filtrations as follows:

1). 10 direct summands of the form:


radical layers
8
6
8



socle layers
8
6
8


2). 1 direct summand of the form:


radical layers
5
4
5



socle layers
5
4
5


3). 2 direct summands of the form:


radical layers
6
8, 9
6



socle layers
6
8, 9
6


4). 2 direct summands of the form:


radical layers
3, 10
2
3, 10



socle layers
3, 10
2
3, 10


5). 4 direct summands of the form:


radical layers
3
2
3, 7, 10
2
3



socle layers
3
2
3, 7, 10
2
3


6). 2 direct summands of the form:


radical layers
6
8, 9
1, 6
8, 9
6



socle layers
6
8, 9
1, 6
8, 9
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 56, 134, 162, 42, 63, 90, 70, 83, 70, 65 .

The cartan matrix of A is



The determinant of the Cartan matrix is 1.

The blocks of A consist of the following irreducible modules:

The radical and socle filtrations of the projective modules for A are the following:


Projective module number 1


radical layers
1
8, 9
6



socle layers
1
8, 9
6



Projective module number 2


radical layers
2
3, 7, 10
2
3



socle layers
2
3, 7, 10
2
3



Projective module number 3


radical layers
3
2
3, 7, 10
2
3



socle layers
3
2
3, 7, 10
2
3



Projective module number 4


radical layers
4
5



socle layers
4
5



Projective module number 5


radical layers
5
4
5



socle layers
5
4
5



Projective module number 6


radical layers
6
8, 9
1, 6
8, 9
6



socle layers
6
8, 9
1, 6
8, 9
6



Projective module number 7


radical layers
7
2
3



socle layers
7
2
3



Projective module number 8


radical layers
8
1, 6
8, 8, 9
6



socle layers
8
1, 6
8, 9
6, 8



Projective module number 9


radical layers
9
1, 6
8, 9
6



socle layers
9
1, 6
8, 9
6



Projective module number 10


radical layers
10
2
3, 10



socle layers
10
2
3, 10


The degrees of the splitting fields are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 .

The Basic Algebra H of the Schur Algebra



The dimension of H is 41 .

The dimensions of the irreducible H-modules are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 .

The Simple modules for H correspond to the following direct summands of the module M.


The degrees of the splitting fields are 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 .

The dimensions of the projective modules of H are 5, 2, 2, 3, 2, 6, 7, 5, 6, 3 .

The cartan matrix of H is



The determinant of the Cartan matrix is 1.

The blocks of H consist of the following irreducible modules:

The radical and socle filtrations of the projective modules for H are the following:


Projective module number 1


radical layers
1
5, 8
1, 9



socle layers
1
5, 8
1, 9



Projective module number 2


radical layers
2
7



socle layers
2
7



Projective module number 3


radical layers
3
10



socle layers
3
10



Projective module number 4


radical layers
4
7
6



socle layers
4
7
6



Projective module number 5


radical layers
5
1



socle layers
5
1



Projective module number 6


radical layers
6
7
4, 6
7
6



socle layers
6
7
4, 6
7
6



Projective module number 7


radical layers
7
2, 4, 6
7, 7
6



socle layers
7
4, 6
2, 7
6, 7



Projective module number 8


radical layers
8
1, 9
8
9



socle layers
8
1, 9
8
9



Projective module number 9


radical layers
9
8
1, 9
8
9



socle layers
9
8
1, 9
8
9



Projective module number 10


radical layers
10
3
10



socle layers
10
3
10


A presentation for H is the quotient of a polynomial ring P in noncommuting variables

The generators designated by a subscripted 'b' are generators for subspaces determined by primitive idempotents. The generators given by subscripted 'z' are generators for the radical.

A Groebner basis for the ideal of relation consists of the elements:


The ideal of relations is generated by the elements of degree at most 2.


The projective resolutions of the simple modules.


Simple Module Number 1



The projective resolution of simple module no. 1 is graded.



Simple Module Number 2



The projective resolution of simple module no. 2 is graded.



Simple Module Number 3



The projective resolution of simple module no. 3 is graded.



Simple Module Number 4



The projective resolution of simple module no. 4 is graded.



Simple Module Number 5



The projective resolution of simple module no. 5 is graded.



Simple Module Number 6



The projective resolution of simple module no. 6 is graded.



Simple Module Number 7



The projective resolution of simple module no. 7 is graded.



Simple Module Number 8



The projective resolution of simple module no. 8 is graded.



Simple Module Number 9



The projective resolution of simple module no. 9 is graded.



Simple Module Number 10



The projective resolution of simple module no. 10 is graded.