Schur Algebra S( 3 ,5) 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 66 .

The dimensions of the irreducible submodules modules are 5, 5, 3, 3, 1 .



The module M has radical filtration (Loewy series)
1, 2, 2, 2, 2, 4, 4, 5, 5, 5, 5, 5

3, 3, 4, 4, 4, 4, 5, 5

4, 4, 5, 5, 5, 5



The module M has socle filtration (socle series)
4, 4, 5, 5, 5, 5

3, 3, 4, 4, 4, 4, 5, 5

1, 2, 2, 2, 2, 4, 4, 5, 5, 5, 5, 5



The module M has simple direct summands:

1 copy of simple module number 1
4 copies of simple module number 2
1 copy of simple module number 5

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

1). 4 direct summands of the form:


radical layers
5
4
5



socle layers
5
4
5


2). 2 direct summands of the form:


radical layers
4
3, 5
4



socle layers
4
3, 5
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 5, 5, 6, 10, 5 .

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, 2 are simple.

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


Projective module number 3


radical layers
3
4



socle layers
3
4



Projective module number 4


radical layers
4
3, 5
4



socle layers
4
3, 5
4



Projective module number 5


radical layers
5
4
5



socle layers
5
4
5


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

The Basic Algebra H of the Schur Algebra



The dimension of H is 11 .

The dimensions of the irreducible H-modules are 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 .

The dimensions of the projective modules of H are 4, 1, 3, 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 2, 5 are simple.

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


Projective module number 1


radical layers
1
3, 4
1



socle layers
1
3, 4
1



Projective module number 3


radical layers
3
1
3



socle layers
3
1
3



Projective module number 4


radical layers
4
1



socle layers
4
1


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 is Projective.



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 is Projective.