# Field k

Finite field of size 7

## 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 561 .

The dimensions of the irreducible submodules modules are 35, 21, 21, 14, 14, 10, 5, 1 .

The simple module number 1 has dimension 35 and corresponds to the partition [ 4, 2, 1 ] .
The simple module number 2 has dimension 21 and corresponds to the partition [ 3, 3, 1 ] .
The simple module number 3 has dimension 21 and corresponds to the partition [ 3, 2, 2 ] .
The simple module number 4 has dimension 14 and corresponds to the partition [ 4, 3 ] .
The simple module number 5 has dimension 14 and corresponds to the partition [ 5, 2 ] .
The simple module number 6 has dimension 10 and corresponds to the partition [ 5, 1, 1 ] .
The simple module number 7 has dimension 5 and corresponds to the partition [ 6, 1 ] .
The simple module number 8 has dimension 1 and corresponds to the partition [ 7 ] .

The module M has radical filtration (Loewy series)
1, 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8

6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8

7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8

The module M has socle filtration (socle series)
7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8

6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8

1, 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8

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

4 copies of simple module number 1
2 copies of simple module number 2
1 copy of simple module number 3
6 copies of simple module number 4
10 copies of simple module number 5
1 copy of simple module number 8

8
7
8

socle layers
8
7
8

7
6, 8
7

socle layers
7
6, 8
7

## 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 35, 21, 21, 14, 14, 15, 21, 7 .

#### The cartan matrix of A is

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

The determinant of the Cartan matrix is 1.

(1). 1
(2). 2
(3). 3
(4). 4
(5). 5
(6). 6, 7, 8

6
7

socle layers
6
7

7
6, 8
7

socle layers
7
6, 8
7

#### Projective module number 8

8
7
8

socle layers
8
7
8

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

## The Basic Algebra H of the Schur Algebra

The dimension of H is 14 .

The dimensions of the irreducible H-modules are 1, 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 5.
Simple H-module 2 corresponds to the direct summand of M isomorphic to simple A-module 3.
Simple H-module 3 corresponds to the direct summand of M isomorphic to simple A-module 4.
Simple H-module 4 corresponds to the direct summand of M isomorphic to simple A-module 1.
Simple H-module 5 corresponds to the direct summand of M isomorphic to simple A-module 8.
Simple H-module 6 corresponds to the direct summand of M isomorphic to simple A-module 2.
Simple H-module 7 corresponds to the direct summand of M isomorphic to the nonsimple A-module 1.
Simple H-module 8 corresponds to the direct summand of M isomorphic to the nonsimple A-module 2.

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

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

#### The cartan matrix of H is

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

The determinant of the Cartan matrix is 1.

(1). 1
(2). 2, 5, 7
(3). 3
(4). 4
(5). 6
(6). 8

2
5, 7
2

socle layers
2
5, 7
2

5
2
5

socle layers
5
2
5

7
2

socle layers
7
2

### 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 , b_8 , z_1 , z_2 , z_3 , z_4 , 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:
b_2^2 + 4*b_2 ,
b_2*b_3 ,
b_2*b_4 ,
b_2*b_5 ,
b_2*b_6 ,
b_2*b_7 ,
b_2*b_8 ,
b_2*z_1 + 4*z_1 ,
b_2*z_2 + 4*z_2 ,
b_2*z_3 ,
b_2*z_4 ,
b_3*b_2 ,
b_3^2 + 4*b_3 ,
b_3*b_4 ,
b_3*b_5 ,
b_3*b_6 ,
b_3*b_7 ,
b_3*b_8 ,
b_3*z_1 ,
b_3*z_2 ,
b_3*z_3 ,
b_3*z_4 ,
b_4*b_2 ,
b_4*b_3 ,
b_4^2 + 2*b_4 ,
b_4*b_5 ,
b_4*b_6 ,
b_4*b_7 ,
b_4*b_8 ,
b_4*z_1 ,
b_4*z_2 ,
b_4*z_3 ,
b_4*z_4 ,
b_5*b_2 ,
b_5*b_3 ,
b_5*b_4 ,
b_5^2 + 4*b_5 ,
b_5*b_6 ,
b_5*b_7 ,
b_5*b_8 ,
b_5*z_1 ,
b_5*z_2 ,
b_5*z_3 + 4*z_3 ,
b_5*z_4 ,
b_6*b_2 ,
b_6*b_3 ,
b_6*b_4 ,
b_6*b_5 ,
b_6^2 + 2*b_6 ,
b_6*b_7 ,
b_6*b_8 ,
b_6*z_1 ,
b_6*z_2 ,
b_6*z_3 ,
b_6*z_4 ,
b_7*b_2 ,
b_7*b_3 ,
b_7*b_4 ,
b_7*b_5 ,
b_7*b_6 ,
b_7^2 + 4*b_7 ,
b_7*b_8 ,
b_7*z_1 ,
b_7*z_2 ,
b_7*z_3 ,
b_7*z_4 + 4*z_4 ,
b_8*b_2 ,
b_8*b_3 ,
b_8*b_4 ,
b_8*b_5 ,
b_8*b_6 ,
b_8*b_7 ,
b_8^2 + 2*b_8 ,
b_8*z_1 ,
b_8*z_2 ,
b_8*z_3 ,
b_8*z_4 ,
z_1*b_2 ,
z_1*b_3 ,
z_1*b_4 ,
z_1*b_5 + 4*z_1 ,
z_1*b_6 ,
z_1*b_7 ,
z_1*b_8 ,
z_1^2 ,
z_1*z_2 ,
z_1*z_3 + z_2*z_4 ,
z_1*z_4 ,
z_2*b_2 ,
z_2*b_3 ,
z_2*b_4 ,
z_2*b_5 ,
z_2*b_6 ,
z_2*b_7 + 4*z_2 ,
z_2*b_8 ,
z_2*z_1 ,
z_2^2 ,
z_2*z_3 ,
z_3*b_2 + 4*z_3 ,
z_3*b_3 ,
z_3*b_4 ,
z_3*b_5 ,
z_3*b_6 ,
z_3*b_7 ,
z_3*b_8 ,
z_3*z_2 ,
z_3^2 ,
z_3*z_4 ,
z_4*b_2 + 4*z_4 ,
z_4*b_3 ,
z_4*b_4 ,
z_4*b_5 ,
z_4*b_6 ,
z_4*b_7 ,
z_4*b_8 ,
z_4*z_1 ,
z_4*z_2 ,
z_4*z_3 ,
z_4^2 ,
b_1 + b_2 + b_3 + 2*b_4 + b_5 + 2*b_6 + b_7 + 2*b_8 + 4 .

Degree 0:
2

Degree 1:
5 7

Degree 2:
2

Degree 3:
7

Degree 0:
5

Degree 1:
2

Degree 2:
7

Degree 0:
7

Degree 1:
2

Degree 2:
5 7

Degree 3:
2

Degree 4:
7