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

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

7
3
7

socle layers
7
3
7

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

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

The determinant of the Cartan matrix is 1.

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

2
3

socle layers
2
3

3
2, 7
3

socle layers
3
2, 7
3

#### Projective module number 7

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.

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

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

#### The cartan matrix of H is

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

The determinant of the Cartan matrix is 1.

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

2
4, 6
2

socle layers
2
4, 6
2

4
2
4

socle layers
4
2
4

6
2

socle layers
6
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 , 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 + 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 + 2*z_1 ,
b_2*z_2 + 2*z_2 ,
b_2*z_3 ,
b_2*z_4 ,
b_3*b_2 ,
b_3^2 + 3*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_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*z_1 ,
b_4*z_2 ,
b_4*z_3 + 2*z_3 ,
b_4*z_4 ,
b_5*b_2 ,
b_5*b_3 ,
b_5*b_4 ,
b_5^2 + 3*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_6*b_2 ,
b_6*b_3 ,
b_6*b_4 ,
b_6*b_5 ,
b_6^2 + 3*b_6 ,
b_6*b_7 ,
b_6*z_1 ,
b_6*z_2 ,
b_6*z_3 ,
b_6*z_4 + 3*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 + 3*b_7 ,
b_7*z_1 ,
b_7*z_2 ,
b_7*z_3 ,
b_7*z_4 ,
z_1*b_2 ,
z_1*b_3 ,
z_1*b_4 + 2*z_1 ,
z_1*b_5 ,
z_1*b_6 ,
z_1*b_7 ,
z_1^2 ,
z_1*z_2 ,
z_1*z_3 + 2*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 + 3*z_2 ,
z_2*b_7 ,
z_2*z_1 ,
z_2^2 ,
z_2*z_3 ,
z_3*b_2 + 2*z_3 ,
z_3*b_3 ,
z_3*b_4 ,
z_3*b_5 ,
z_3*b_6 ,
z_3*b_7 ,
z_3*z_2 ,
z_3^2 ,
z_3*z_4 ,
z_4*b_2 + 2*z_4 ,
z_4*b_3 ,
z_4*b_4 ,
z_4*b_5 ,
z_4*b_6 ,
z_4*b_7 ,
z_4*z_1 ,
z_4*z_2 ,
z_4*z_3 ,
z_4^2 ,
b_1 + 4*b_2 + b_3 + 4*b_4 + b_5 + b_6 + b_7 + 3 .

Degree 0:
2

Degree 1:
4 6

Degree 2:
2

Degree 3:
6

Degree 0:
4

Degree 1:
2

Degree 2:
6

Degree 0:
6

Degree 1:
2

Degree 2:
4 6

Degree 3:
2

Degree 4:
6