Schur Algebra S( 4 ,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 4. . The dimension of M is 8143 .

The dimensions of the irreducible submodules modules are 90, 35, 35, 28, 28, 21, 21, 13, 13, 7, 7, 1, 1 .



The module M has radical filtration (Loewy series)
1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13

2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13

3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 13



The module M has socle filtration (socle series)
3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 13

2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13

2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13

1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13



The module M has simple direct summands:

7 copies of simple module number 1
5 copies of simple module number 5
13 copies of simple module number 7
5 copies of simple module number 11
9 copies of simple module number 13

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

1). 7 direct summands of the form:


radical layers
7
6
7



socle layers
7
6
7


2). 20 direct summands of the form:


radical layers
11
9
11



socle layers
11
9
11


3). 2 direct summands of the form:


radical layers
3
5, 13
3



socle layers
3
5, 13
3


4). 3 direct summands of the form:


radical layers
9
11, 12
9



socle layers
9
11, 12
9


5). 5 direct summands of the form:


radical layers
5, 13
3
5, 13



socle layers
5, 13
3
5, 13


6). 19 direct summands of the form:


radical layers
5
3
5, 10, 13
3
5



socle layers
5
3
5, 10, 13
3
5


7). 9 direct summands of the form:


radical layers
9
11, 12
2, 9
11, 12
9



socle layers
9
11, 12
2, 9
11, 12
9


8). 4 direct summands of the form:


radical layers
11
2, 9
4, 11, 11, 12
2, 9
11



socle layers
11
2, 9
4, 11, 11, 12
2, 9
11


9). 1 direct summand of the form:


radical layers
13
3, 8
5, 10, 13, 13
3, 8
13



socle layers
13
3, 8
5, 10, 13, 13
3, 8
13


10). 3 direct summands of the form:


radical layers
3
5, 10, 13
3, 3, 8
5, 10, 13
3



socle layers
3
5, 10, 13
3, 3, 8
5, 10, 13
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, 126, 225, 70, 162, 42, 63, 70, 90, 126, 153, 70, 135 .

The cartan matrix of A is



The determinant of the Cartan matrix is 15.

The blocks of A consist of the following irreducible modules:

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
4, 11, 12
2, 9
11



socle layers
2
4, 11, 12
2, 9
11



Projective module number 3


radical layers
3
5, 10, 13
3, 3, 8
5, 10, 13
3



socle layers
3
5, 10, 13
3, 3, 8
5, 10, 13
3



Projective module number 4


radical layers
4
2
11



socle layers
4
2
11



Projective module number 5


radical layers
5
3
5, 10, 13
3
5



socle layers
5
3
5, 10, 13
3
5



Projective module number 6


radical layers
6
7



socle layers
6
7



Projective module number 7


radical layers
7
6
7



socle layers
7
6
7



Projective module number 8


radical layers
8
10, 13
3, 8
13



socle layers
8
10, 13
3, 8
13



Projective module number 9


radical layers
9
11, 12
2, 9
11, 12
9



socle layers
9
11, 12
2, 9
11, 12
9



Projective module number 10


radical layers
10
3, 8
5, 10, 13
3



socle layers
10
3, 8
5, 10, 13
3



Projective module number 11


radical layers
11
2, 9
4, 11, 11, 12
2, 9
11



socle layers
11
2, 9
4, 11, 11, 12
2, 9
11



Projective module number 12


radical layers
12
2, 9
11, 12
9



socle layers
12
2, 9
11, 12
9



Projective module number 13


radical layers
13
3, 8
5, 10, 13, 13
3, 8
13



socle layers
13
3, 8
5, 10, 13, 13
3, 8
13


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

The Basic Algebra H of the Schur Algebra



The dimension of H is 96 .

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

The dimensions of the projective modules of H are 7, 10, 2, 3, 8, 3, 1, 3, 11, 3, 10, 6, 11, 7, 11 .

The cartan matrix of H is



The determinant of the Cartan matrix is 1.

The blocks of H consist of the following irreducible modules:

Projective module number 7 is simple.

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


Projective module number 1


radical layers
1
10, 11, 12
1, 5
11



socle layers
1
10, 11, 12
1, 5
11



Projective module number 2


radical layers
2
9, 13
2, 8, 14, 15
9, 13
2



socle layers
2
9, 13
2, 8, 14, 15
9, 13
2



Projective module number 3


radical layers
3
6



socle layers
3
6



Projective module number 4


radical layers
4
9
15



socle layers
4
9
15



Projective module number 5


radical layers
5
11, 12
1, 5
11, 12
5



socle layers
5
11, 12
1, 5
11, 12
5



Projective module number 6


radical layers
6
3
6



socle layers
6
3
6



Projective module number 8


radical layers
8
9
2



socle layers
8
9
2



Projective module number 9


radical layers
9
2, 4, 8, 14, 15
9, 9, 13
2, 15



socle layers
9
2, 4, 8, 14, 15
9, 9, 13
2, 15



Projective module number 10


radical layers
10
1
11



socle layers
10
1
11



Projective module number 11


radical layers
11
1, 5
10, 11, 11, 12
1, 5
11



socle layers
11
1, 5
10, 11, 11, 12
1, 5
11



Projective module number 12


radical layers
12
1, 5
11, 12
5



socle layers
12
1, 5
11, 12
5



Projective module number 13


radical layers
13
2, 14, 15
9, 13, 13
2, 14, 15
13



socle layers
13
2, 14, 15
9, 13, 13
2, 14, 15
13



Projective module number 14


radical layers
14
9, 13
2, 14, 15
13



socle layers
14
9, 13
2, 14, 15
13



Projective module number 15


radical layers
15
9, 13
2, 4, 14, 15, 15
9, 13
15



socle layers
15
9, 13
2, 4, 14, 15, 15
9, 13
15


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 not generated by the elements of degree at most 2. The following relation were not contained in the ideal generated by the relations of degree 2:

z_5*z_24*z_28*z_22 + z_4*z_12 + 2*z_5*z_22 ,

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 is Projective.



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.



Simple Module Number 11



The projective resolution of simple module no. 11 is graded.



Simple Module Number 12



The projective resolution of simple module no. 12 is graded.



Simple Module Number 13



The projective resolution of simple module no. 13 is graded.



Simple Module Number 14



The projective resolution of simple module no. 14 is graded.



Simple Module Number 15



The projective resolution of simple module no. 15 is graded.