# 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 222 .

The dimensions of the irreducible submodules modules are 9, 6, 4, 4, 1, 1 .

The simple module number 1 has dimension 9 and corresponds to the partition [ 4, 2 ] .
The simple module number 2 has dimension 6 and corresponds to the partition [ 4, 1, 1 ] .
The simple module number 3 has dimension 4 and corresponds to the partition [ 3, 2, 1 ] .
The simple module number 4 has dimension 4 and corresponds to the partition [ 5, 1 ] .
The simple module number 5 has dimension 1 and corresponds to the partition [ 6 ] .
The simple module number 6 has dimension 1 and corresponds to the partition [ 3, 3 ] .

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

2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6

2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6

2, 2, 3, 4, 5, 5, 6, 6

4, 4, 5

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

2, 2, 3, 4, 5, 5, 6, 6

2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6

2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6

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

#### The module M has simple direct summands:

8 copies of simple module number 1
2 copies of simple module number 5

5
4
5

socle layers
5
4
5

4
2, 5
4

socle layers
4
2, 5
4

4
5, 6
4

socle layers
4
5, 6
4

5
3, 4
2, 5, 5, 6
3, 4
5

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

4
2, 5, 6
3, 4, 4
2, 5, 6
4

socle layers
4
2, 5, 6
3, 4, 4
2, 5, 6
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 9, 26, 21, 36, 27, 21 .

#### The cartan matrix of A is

1, 0, 0, 0, 0, 0
0, 2, 1, 2, 1, 1
0, 1, 2, 1, 2, 1
0, 2, 1, 4, 2, 2
0, 1, 2, 2, 4, 1
0, 1, 1, 2, 1, 2

The determinant of the Cartan matrix is 6.

#### The blocks of A consist of the following irreducible modules:

(1). 1
(2). 2, 3, 4, 5, 6

2
3, 4
2, 5, 6
4

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

3
2, 5, 6
3, 4
5

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

4
2, 5, 6
3, 4, 4
2, 5, 6
4

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

5
3, 4
2, 5, 5, 6
3, 4
5

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

#### Projective module number 6

6
3, 4
2, 5, 6
4

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

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

## The Basic Algebra H of the Schur Algebra

The dimension of H is 48 .

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.

Simple H-module 1 corresponds to the direct summand of M isomorphic to simple A-module 1.
Simple H-module 2 corresponds to the direct summand of M isomorphic to simple A-module 5.
Simple H-module 3 corresponds to the direct summand of M isomorphic to the nonsimple A-module 1.
Simple H-module 4 corresponds to the direct summand of M isomorphic to the nonsimple A-module 2.
Simple H-module 5 corresponds to the direct summand of M isomorphic to the nonsimple A-module 3.
Simple H-module 6 corresponds to the direct summand of M isomorphic to the nonsimple A-module 4.
Simple H-module 7 corresponds to the direct summand of M isomorphic to the nonsimple A-module 5.

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, 3, 11, 11, 7, 7 .

#### The cartan matrix of H is

1, 0, 0, 0, 0, 0, 0
0, 2, 1, 1, 2, 1, 1
0, 1, 1, 0, 1, 0, 0
0, 1, 0, 4, 2, 2, 2
0, 2, 1, 2, 4, 1, 1
0, 1, 0, 2, 1, 2, 1
0, 1, 0, 2, 1, 1, 2

The determinant of the Cartan matrix is 1.

#### The blocks of H consist of the following irreducible modules:

(1). 1
(2). 2, 3, 4, 5, 6, 7

2
3, 5, 6, 7
2, 4
5

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

3
2
5

socle layers
3
2
5

4
5, 6, 7
2, 4, 4
5, 6, 7
4

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

5
2, 4
3, 5, 5, 6, 7
2, 4
5

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

6
2, 4
5, 6, 7
4

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

7
2, 4
5, 6, 7
4

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

### 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:
z_10*z_8*z_14*z_7 ,
z_10*z_8*z_14*z_8 ,
z_12*z_8*z_14*z_6 ,
z_12*z_8*z_14*z_7 ,
z_12*z_8*z_14*z_8 ,
z_14*z_8*z_14*z_6 ,
z_14*z_8*z_14*z_7 ,
z_14*z_8*z_14*z_8 ,
z_4*z_14*z_7 ,
z_4*z_14*z_8 ,
z_7*z_12*z_6 + 2*z_8*z_14*z_6 ,
z_7*z_12*z_8 + z_8*z_14*z_8 ,
z_10*z_7*z_12 + 2*z_10*z_8*z_14 ,
z_12*z_8*z_13 ,
z_14*z_7*z_12 + z_14*z_8*z_14 ,
z_14*z_8*z_13 ,
b_2^2 + b_2 ,
b_2*b_3 ,
b_2*b_4 ,
b_2*b_5 ,
b_2*b_6 ,
b_2*b_7 ,
b_2*z_1 + z_1 ,
b_2*z_2 + z_2 ,
b_2*z_3 + z_3 ,
b_2*z_4 + z_4 ,
b_2*z_5 ,
b_2*z_6 ,
b_2*z_7 ,
b_2*z_8 ,
b_2*z_9 ,
b_2*z_10 ,
b_2*z_11 ,
b_2*z_12 ,
b_2*z_13 ,
b_2*z_14 ,
b_3*b_2 ,
b_3^2 + b_3 ,
b_3*b_4 ,
b_3*b_5 ,
b_3*b_6 ,
b_3*b_7 ,
b_3*z_1 ,
b_3*z_2 ,
b_3*z_3 ,
b_3*z_4 ,
b_3*z_5 + z_5 ,
b_3*z_6 ,
b_3*z_7 ,
b_3*z_8 ,
b_3*z_9 ,
b_3*z_10 ,
b_3*z_11 ,
b_3*z_12 ,
b_3*z_13 ,
b_3*z_14 ,
b_4*b_2 ,
b_4*b_3 ,
b_4^2 + b_4 ,
b_4*b_5 ,
b_4*b_6 ,
b_4*b_7 ,
b_4*z_1 ,
b_4*z_2 ,
b_4*z_3 ,
b_4*z_4 ,
b_4*z_5 ,
b_4*z_6 + z_6 ,
b_4*z_7 + z_7 ,
b_4*z_8 + z_8 ,
b_4*z_9 ,
b_4*z_10 ,
b_4*z_11 ,
b_4*z_12 ,
b_4*z_13 ,
b_4*z_14 ,
b_5*b_2 ,
b_5*b_3 ,
b_5*b_4 ,
b_5^2 + b_5 ,
b_5*b_6 ,
b_5*b_7 ,
b_5*z_1 ,
b_5*z_2 ,
b_5*z_3 ,
b_5*z_4 ,
b_5*z_5 ,
b_5*z_6 ,
b_5*z_7 ,
b_5*z_8 ,
b_5*z_9 + z_9 ,
b_5*z_10 + z_10 ,
b_5*z_11 ,
b_5*z_12 ,
b_5*z_13 ,
b_5*z_14 ,
b_6*b_2 ,
b_6*b_3 ,
b_6*b_4 ,
b_6*b_5 ,
b_6^2 + b_6 ,
b_6*b_7 ,
b_6*z_1 ,
b_6*z_2 ,
b_6*z_3 ,
b_6*z_4 ,
b_6*z_5 ,
b_6*z_6 ,
b_6*z_7 ,
b_6*z_8 ,
b_6*z_9 ,
b_6*z_10 ,
b_6*z_11 + z_11 ,
b_6*z_12 + z_12 ,
b_6*z_13 ,
b_6*z_14 ,
b_7*b_2 ,
b_7*b_3 ,
b_7*b_4 ,
b_7*b_5 ,
b_7*b_6 ,
b_7^2 + b_7 ,
b_7*z_1 ,
b_7*z_2 ,
b_7*z_3 ,
b_7*z_4 ,
b_7*z_5 ,
b_7*z_6 ,
b_7*z_7 ,
b_7*z_8 ,
b_7*z_9 ,
b_7*z_10 ,
b_7*z_11 ,
b_7*z_12 ,
b_7*z_13 + z_13 ,
b_7*z_14 + z_14 ,
z_1*b_2 ,
z_1*b_3 + z_1 ,
z_1*b_4 ,
z_1*b_5 ,
z_1*b_6 ,
z_1*b_7 ,
z_1^2 ,
z_1*z_2 ,
z_1*z_3 ,
z_1*z_4 ,
z_1*z_5 + z_4*z_13 ,
z_1*z_6 ,
z_1*z_7 ,
z_1*z_8 ,
z_1*z_9 ,
z_1*z_10 ,
z_1*z_11 ,
z_1*z_12 ,
z_1*z_13 ,
z_1*z_14 ,
z_2*b_2 ,
z_2*b_3 ,
z_2*b_4 ,
z_2*b_5 + z_2 ,
z_2*b_6 ,
z_2*b_7 ,
z_2*z_1 ,
z_2^2 ,
z_2*z_3 ,
z_2*z_4 ,
z_2*z_5 ,
z_2*z_6 ,
z_2*z_7 ,
z_2*z_8 ,
z_2*z_9 ,
z_2*z_10 + 2*z_4*z_14 ,
z_2*z_11 ,
z_2*z_12 ,
z_2*z_13 ,
z_2*z_14 ,
z_3*b_2 ,
z_3*b_3 ,
z_3*b_4 ,
z_3*b_5 ,
z_3*b_6 + z_3 ,
z_3*b_7 ,
z_3*z_1 ,
z_3*z_2 ,
z_3^2 ,
z_3*z_4 ,
z_3*z_5 ,
z_3*z_6 ,
z_3*z_7 ,
z_3*z_8 ,
z_3*z_9 ,
z_3*z_10 ,
z_3*z_11 + z_4*z_13 ,
z_3*z_12 + 2*z_4*z_14 ,
z_3*z_13 ,
z_3*z_14 ,
z_4*b_2 ,
z_4*b_3 ,
z_4*b_4 ,
z_4*b_5 ,
z_4*b_6 ,
z_4*b_7 + z_4 ,
z_4*z_1 ,
z_4*z_2 ,
z_4*z_3 ,
z_4^2 ,
z_4*z_5 ,
z_4*z_6 ,
z_4*z_7 ,
z_4*z_8 ,
z_4*z_9 ,
z_4*z_10 ,
z_4*z_11 ,
z_4*z_12 ,
z_5*b_2 + z_5 ,
z_5*b_3 ,
z_5*b_4 ,
z_5*b_5 ,
z_5*b_6 ,
z_5*b_7 ,
z_5*z_1 ,
z_5*z_3 ,
z_5*z_4 ,
z_5^2 ,
z_5*z_6 ,
z_5*z_7 ,
z_5*z_8 ,
z_5*z_9 ,
z_5*z_10 ,
z_5*z_11 ,
z_5*z_12 ,
z_5*z_13 ,
z_5*z_14 ,
z_6*b_2 ,
z_6*b_3 ,
z_6*b_4 ,
z_6*b_5 + z_6 ,
z_6*b_6 ,
z_6*b_7 ,
z_6*z_1 ,
z_6*z_2 ,
z_6*z_3 ,
z_6*z_4 ,
z_6*z_5 ,
z_6^2 ,
z_6*z_7 ,
z_6*z_8 ,
z_6*z_9 + z_8*z_13 ,
z_6*z_10 + 2*z_7*z_12 + z_8*z_14 ,
z_6*z_11 ,
z_6*z_12 ,
z_6*z_13 ,
z_6*z_14 ,
z_7*b_2 ,
z_7*b_3 ,
z_7*b_4 ,
z_7*b_5 ,
z_7*b_6 + z_7 ,
z_7*b_7 ,
z_7*z_1 ,
z_7*z_2 ,
z_7*z_3 ,
z_7*z_4 ,
z_7*z_5 ,
z_7*z_6 ,
z_7^2 ,
z_7*z_8 ,
z_7*z_9 ,
z_7*z_10 ,
z_7*z_11 + z_8*z_13 ,
z_7*z_13 ,
z_7*z_14 ,
z_8*b_2 ,
z_8*b_3 ,
z_8*b_4 ,
z_8*b_5 ,
z_8*b_6 ,
z_8*b_7 + z_8 ,
z_8*z_1 ,
z_8*z_2 ,
z_8*z_3 ,
z_8*z_4 ,
z_8*z_5 ,
z_8*z_6 ,
z_8*z_7 ,
z_8^2 ,
z_8*z_9 ,
z_8*z_10 ,
z_8*z_11 ,
z_8*z_12 ,
z_9*b_2 + z_9 ,
z_9*b_3 ,
z_9*b_4 ,
z_9*b_5 ,
z_9*b_6 ,
z_9*b_7 ,
z_9*z_3 + 2*z_10*z_7 ,
z_9*z_4 + 2*z_10*z_8 ,
z_9*z_5 ,
z_9*z_6 ,
z_9*z_7 ,
z_9*z_8 ,
z_9^2 ,
z_9*z_10 ,
z_9*z_11 ,
z_9*z_12 ,
z_9*z_13 ,
z_9*z_14 ,
z_10*b_2 ,
z_10*b_3 ,
z_10*b_4 + z_10 ,
z_10*b_5 ,
z_10*b_6 ,
z_10*b_7 ,
z_10*z_1 ,
z_10*z_2 ,
z_10*z_3 ,
z_10*z_4 ,
z_10*z_5 ,
z_10*z_9 ,
z_10^2 ,
z_10*z_11 ,
z_10*z_12 ,
z_10*z_13 ,
z_10*z_14 ,
z_11*b_2 + z_11 ,
z_11*b_3 ,
z_11*b_4 ,
z_11*b_5 ,
z_11*b_6 ,
z_11*b_7 ,
z_11*z_1 ,
z_11*z_2 + 2*z_12*z_6 ,
z_11*z_4 + z_12*z_8 ,
z_11*z_5 ,
z_11*z_6 ,
z_11*z_7 ,
z_11*z_8 ,
z_11*z_9 ,
z_11*z_10 ,
z_11^2 ,
z_11*z_12 ,
z_11*z_13 ,
z_11*z_14 ,
z_12*b_2 ,
z_12*b_3 ,
z_12*b_4 + z_12 ,
z_12*b_5 ,
z_12*b_6 ,
z_12*b_7 ,
z_12*z_1 ,
z_12*z_2 ,
z_12*z_3 ,
z_12*z_4 ,
z_12*z_5 ,
z_12*z_7 ,
z_12*z_9 ,
z_12*z_10 ,
z_12*z_11 ,
z_12^2 ,
z_12*z_13 ,
z_12*z_14 ,
z_13*b_2 + z_13 ,
z_13*b_3 ,
z_13*b_4 ,
z_13*b_5 ,
z_13*b_6 ,
z_13*b_7 ,
z_13*z_1 ,
z_13*z_2 + z_14*z_6 ,
z_13*z_3 + 2*z_14*z_7 ,
z_13*z_4 + z_14*z_8 ,
z_13*z_5 ,
z_13*z_6 ,
z_13*z_7 ,
z_13*z_8 ,
z_13*z_9 ,
z_13*z_10 ,
z_13*z_11 ,
z_13*z_12 ,
z_13^2 ,
z_13*z_14 ,
z_14*b_2 ,
z_14*b_3 ,
z_14*b_4 + z_14 ,
z_14*b_5 ,
z_14*b_6 ,
z_14*b_7 ,
z_14*z_1 ,
z_14*z_2 ,
z_14*z_3 ,
z_14*z_4 ,
z_14*z_5 ,
z_14*z_9 ,
z_14*z_10 ,
z_14*z_11 ,
z_14*z_12 ,
z_14*z_13 ,
z_14^2 ,
b_1 + b_2 + b_3 + b_4 + b_5 + b_6 + b_7 + 1 .

## The projective resolutions of the simple modules.

#### Simple Module Number 2

Degree 0:
2

Degree 1:
3 5 6 7

Degree 2:
2 2 2 4 4

Degree 3:
3 3 5 6 6 7 7

Degree 4:
2 2 2 4

Degree 5:
3 3 6 7

Degree 6:
2

Degree 7:
3

Degree 0:
3

Degree 1:
2

Degree 2:
3 6 7

Degree 3:
2 2 4

Degree 4:
3 3 5 6 7

Degree 5:
2 2 4

Degree 6:
3 6 7

Degree 7:
2

Degree 8:
3

Degree 0:
4

Degree 1:
5 6 7

Degree 2:
2 2 4

Degree 3:
3 6 7

Degree 4:
2

Degree 5:
3

Degree 0:
5

Degree 1:
2 4

Degree 2:
6 7

Degree 3:
2

Degree 4:
3

Degree 0:
6

Degree 1:
2 4

Degree 2:
3 5 6 7

Degree 3:
2 2 4

Degree 4:
3 6 7

Degree 5:
2

Degree 6:
3

Degree 0:
7

Degree 1:
2 4

Degree 2:
3 5 6 7

Degree 3:
2 2 4

Degree 4:
3 6 7

Degree 5:
2

Degree 6:
3