Schur Algebra S(
3
,6) in characteristic 5
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
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1). 5 direct summands of the form:
radical layers
7
3
7
socle layers
7
3
7
2). 3 direct summands of the form:
radical layers
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.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
3,
7
(3).
4
(4).
5
(5).
6
Projective modules number
1,
4,
5,
6
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
3
socle layers
2
3
Projective module number 3
radical layers
3
2,
7
3
socle layers
3
2,
7
3
Projective module number 7
radical layers
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.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2,
4,
6
(3).
3
(4).
5
(5).
7
Projective modules number
1,
3,
5,
7
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 2
radical layers
2
4,
6
2
socle layers
2
4,
6
2
Projective module number 4
radical layers
4
2
4
socle layers
4
2
4
Projective module number 6
radical layers
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
.
The ideal of relations is generated by the elements
of degree at most 2.
The projective resolutions of the simple modules.
Simple Module Number 1 is Projective.
Simple Module Number 2
Degree 0:
2
Degree 1:
4
6
Degree 2:
2
Degree 3:
6
The projective resolution
of simple module no. 2 is graded.
Simple Module Number 3 is Projective.
Simple Module Number 4
Degree 0:
4
Degree 1:
2
Degree 2:
6
The projective resolution
of simple module no. 4 is graded.
Simple Module Number 5 is Projective.
Simple Module Number 6
Degree 0:
6
Degree 1:
2
Degree 2:
4
6
Degree 3:
2
Degree 4:
6
The projective resolution
of simple module no. 6 is graded.
Simple Module Number 7 is Projective.