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

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

5
4
5

socle layers
5
4
5

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

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

The determinant of the Cartan matrix is 1.

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

3
4

socle layers
3
4

4
3, 5
4

socle layers
4
3, 5
4

#### Projective module number 5

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.

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

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

#### The cartan matrix of H is

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

The determinant of the Cartan matrix is 1.

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

1
3, 4
1

socle layers
1
3, 4
1

3
1
3

socle layers
3
1
3

4
1

socle layers
4
1

### 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 , 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 + 3*b_2 ,
b_2*b_3 ,
b_2*b_4 ,
b_2*b_5 ,
b_2*z_1 ,
b_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*z_1 ,
b_3*z_2 ,
b_3*z_3 + 3*z_3 ,
b_3*z_4 ,
b_4*b_2 ,
b_4*b_3 ,
b_4^2 + 3*b_4 ,
b_4*b_5 ,
b_4*z_1 ,
b_4*z_2 ,
b_4*z_3 ,
b_4*z_4 + 3*z_4 ,
b_5*b_2 ,
b_5*b_3 ,
b_5*b_4 ,
b_5^2 + 3*b_5 ,
b_5*z_1 ,
b_5*z_2 ,
b_5*z_3 ,
b_5*z_4 ,
z_1*b_2 ,
z_1*b_3 + 3*z_1 ,
z_1*b_4 ,
z_1*b_5 ,
z_1^2 ,
z_1*z_2 ,
z_1*z_3 + 3*z_2*z_4 ,
z_1*z_4 ,
z_2*b_2 ,
z_2*b_3 ,
z_2*b_4 + 3*z_2 ,
z_2*b_5 ,
z_2*z_1 ,
z_2^2 ,
z_2*z_3 ,
z_3*b_2 ,
z_3*b_3 ,
z_3*b_4 ,
z_3*b_5 ,
z_3*z_2 ,
z_3^2 ,
z_3*z_4 ,
z_4*b_2 ,
z_4*b_3 ,
z_4*b_4 ,
z_4*b_5 ,
z_4*z_1 ,
z_4*z_2 ,
z_4*z_3 ,
z_4^2 ,
b_1 + b_2 + b_3 + b_4 + b_5 + 3 .

Degree 0:
1

Degree 1:
3 4

Degree 2:
1

Degree 3:
4

Degree 0:
3

Degree 1:
1

Degree 2:
4

Degree 0:
4

Degree 1:
1

Degree 2:
3 4

Degree 3:
1

Degree 4:
4