Schur Algebra S( 3 ,6) in characteristic 5

Field k

Finite field of size 5

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 222 .

The dimensions of the irreducible submodules modules are 10, 8, 8, 5, 5, 5, 1 .



The module M has radical filtration (Loewy series)
1, 1, 1, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7

2, 2, 2, 3, 3, 3, 3, 3, 7, 7, 7

3, 3, 3, 7, 7, 7, 7, 7



The module M has socle filtration (socle series)
3, 3, 3, 7, 7, 7, 7, 7

2, 2, 2, 3, 3, 3, 3, 3, 7, 7, 7

1, 1, 1, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7



The module M has simple direct summands:

3 copies of simple module number 1
1 copy of simple module number 4
3 copies of simple module number 5
9 copies of simple module number 6
2 copies of simple module number 7

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

1). 5 direct summands of the form:


radical layers
7
3
7



socle layers
7
3
7


2). 3 direct summands of the form:


radical layers
3
2, 7
3



socle layers
3
2, 7
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 10, 16, 25, 5, 5, 5, 10 .

The cartan matrix of A is



The determinant of the Cartan matrix is 1.

The blocks of A consist of the following irreducible modules:

Projective modules number 1, 4, 5, 6 are simple.

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


Projective module number 2


radical layers
2
3



socle layers
2
3



Projective module number 3


radical layers
3
2, 7
3



socle layers
3
2, 7
3



Projective module number 7


radical layers
7
3
7



socle layers
7
3
7


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

The Basic Algebra H of the Schur Algebra



The dimension of H is 13 .

The dimensions of the irreducible H-modules are 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 .

The dimensions of the projective modules of H are 1, 4, 1, 3, 1, 2, 1 .

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 modules number 1, 3, 5, 7 are simple.

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


Projective module number 2


radical layers
2
4, 6
2



socle layers
2
4, 6
2



Projective module number 4


radical layers
4
2
4



socle layers
4
2
4



Projective module number 6


radical layers
6
2



socle layers
6
2


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 is Projective.



Simple Module Number 2



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



Simple Module Number 3 is Projective.



Simple Module Number 4



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



Simple Module Number 5 is Projective.



Simple Module Number 6



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



Simple Module Number 7 is Projective.