Schur Algebra S( 3 ,6) in characteristic 2

Field k

Finite field of size 2

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 16, 4, 4, 1 .

The module M has radical filtration (Loewy series)
1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4

2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4

2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4

2, 2, 3, 3, 4, 4, 4, 4, 4, 4

2, 2, 2, 2, 2, 3, 4, 4

4, 4, 4

2, 3, 3

4, 4, 4

2, 2, 3

The module M has socle filtration (socle series)
2, 2, 3

4, 4, 4

2, 3, 3

4, 4, 4

2, 2, 2, 2, 2, 3, 4, 4

2, 2, 3, 3, 4, 4, 4, 4, 4, 4

2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4

2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4

1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4

The module M has simple direct summands:

3 copies of simple module number 1 2 copies of simple module number 4

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

1). 2 direct summands of the form:

radical layers
4
2
4

socle layers
4
2
4

2). 3 direct summands of the form:

radical layers
2
4
3
4
2

socle layers
2
4
3
4
2

3). 2 direct summands of the form:

radical layers
4
2, 3
4, 4
2, 3
4

socle layers
4
2, 3
4, 4
2, 3
4

4). 2 direct summands of the form:

radical layers
2
4
3
4
2
4
3
4
2

socle layers
2
4
3
4
2
4
3
4
2

5). 1 direct summand of the form:

radical layers
3, 4
3, 4
2, 4
4
3
4
2
4
3

socle layers
3
4
2
4
3
4
2, 4
3, 4
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 16, 24, 29, 45 .

The cartan matrix of A is

The determinant of the Cartan matrix is 13.

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
3
4
2
4
3
4
2

socle layers
2
4
3
4
2
4
3
4
2

Projective module number 3

radical layers
3
3, 4
2, 4
4
3
4
2
4
3

socle layers
3
4
2
4
3
4
2
3, 4
3, 4

Projective module number 4

radical layers
4
2, 3, 4
3, 4, 4
2, 3, 4
4, 4
2, 3
4, 4
2, 3

socle layers
4
2, 3
4, 4
2, 3
4, 4
2, 3, 4
3, 4, 4
2, 3, 4

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

The Basic Algebra H of the Schur Algebra

The dimension of H is 58 .

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, 8, 10, 4, 8, 14, 13 .

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 1 is simple.

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

Projective module number 2

radical layers
2
3, 6
5, 7
6
2
3

socle layers
2
6
5
3, 6
2, 7
3

Projective module number 3

radical layers
3
2, 7
3, 6
5, 7
6
2
3

socle layers
3
2
6
5, 7
3, 6
2, 7
3

Projective module number 4

radical layers
4
5, 7
6

socle layers
4
5, 7
6

Projective module number 5

radical layers
5
4, 6
2, 5, 7
3, 6

socle layers
5
4, 6
2, 5, 7
3, 6

Projective module number 6

radical layers
6
2, 5, 7
3, 4, 6, 6
2, 5, 7, 7
3, 6

socle layers
6
2, 5, 7
4, 6, 6
2, 3, 5, 7
3, 6, 7

Projective module number 7

radical layers
7
3, 4, 6
2, 5, 7, 7, 7
3, 6, 6
7

socle layers
7
3, 6
2, 4, 7, 7
3, 5, 6, 7
6, 7

A presentation for H is the quotient of a polynomial ring P in noncommuting variables
b_1 , b_2 , b_3 , b_4 , b_5 , b_6 , b_7 ,
z_1 , z_2 , z_3 , z_4 , z_5 , z_6 , z_7 , z_8 , z_9 , z_10 , z_11 , z_12 , z_13 , z_14 ,
by an ideal of relations.

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_2*z_10*z_8*z_9 + z_1*z_3 ,
z_3*z_2*z_10*z_8 + z_4*z_14 ,
z_4*z_14*z_9*z_1 + z_3*z_1 + z_4*z_12 ,
z_10*z_8*z_9*z_1 + z_11*z_12 ,
z_1*z_3*z_1 + z_2*z_11*z_12 ,
z_1*z_3*z_2 ,
z_3*z_1*z_3 + z_4*z_14*z_9 ,
z_3*z_2*z_11 + z_4*z_12*z_4 ,
z_4*z_14*z_11 ,
z_9*z_1*z_3 ,
z_11*z_12*z_4 ,
z_12*z_4*z_12 + z_14*z_9*z_1 ,
z_12*z_4*z_14 ,
z_14*z_11*z_12 ,

The projective resolutions of the simple modules.

Simple Module Number 1 is Projective.

Simple Module Number 2

3 6

2 2 7

3 6

2 4

The projective resolution of simple module no. 2 is not graded.

Simple Module Number 3

2 7

3 4 6

2 5

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

Simple Module Number 4

5 7

3 4 4 6

5 6

2 4

The projective resolution of simple module no. 4 is not graded.

Simple Module Number 5

4 6

5 7

3 4

The projective resolution of simple module no. 5 is not graded.

Simple Module Number 6

2 5 7

3 4 6

2 4

The projective resolution of simple module no. 6 is not graded.

Simple Module Number 7

3 4 6

2 5

Schur Algebra S( 3 ,6) in characteristic 2

Field k

The Module M

The module M has simple direct summands:

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

1). 2 direct summands of the form:

2). 3 direct summands of the form:

3). 2 direct summands of the form:

4). 2 direct summands of the form:

5). 1 direct summand of the form:

The Action Algebra

The cartan matrix of A is

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

Projective module number 3

Projective module number 4

The Basic Algebra H of the Schur Algebra

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

The cartan matrix of H is

The blocks of H consist of the following irreducible modules:

Projective module number 1 is simple.

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

Projective module number 2

Projective module number 3

Projective module number 4

Projective module number 5

Projective module number 6

Projective module number 7

A presentation for H is the quotient of a polynomial ring P in noncommuting variables b_1 , b_2 , b_3 , b_4 , b_5 , b_6 , b_7 , z_1 , z_2 , z_3 , z_4 , z_5 , z_6 , z_7 , z_8 , z_9 , z_10 , z_11 , z_12 , z_13 , z_14 , by an ideal of relations.

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:

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 not 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 not graded.

Simple Module Number 5

The projective resolution of simple module no. 5 is not graded.

Simple Module Number 6

The projective resolution of simple module no. 6 is not graded.

Simple Module Number 7

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

A presentation for H is the quotient of a polynomial ring P in noncommuting variables
b_1 , b_2 , b_3 , b_4 , b_5 , b_6 , b_7 ,
z_1 , z_2 , z_3 , z_4 , z_5 , z_6 , z_7 , z_8 , z_9 , z_10 , z_11 , z_12 , z_13 , z_14 ,
by an ideal of relations.