Schur Algebra S(
4
,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 4.
. The dimension of M is 522
.
The dimensions of the irreducible submodules modules are
10,
10,
8,
8,
5,
5,
5,
1,
1
.
The simple module number 1 has dimension 10 and corresponds to the partition
[ 3, 1, 1, 1 ]
.
The simple module number 2 has dimension 10 and corresponds to the partition
[ 4, 1, 1 ]
.
The simple module number 3 has dimension 8 and corresponds to the partition
[ 3, 2, 1 ]
.
The simple module number 4 has dimension 8 and corresponds to the partition
[ 4, 2 ]
.
The simple module number 5 has dimension 5 and corresponds to the partition
[ 2, 2, 2 ]
.
The simple module number 6 has dimension 5 and corresponds to the partition
[ 5, 1 ]
.
The simple module number 7 has dimension 5 and corresponds to the partition
[ 3, 3 ]
.
The simple module number 8 has dimension 1 and corresponds to the partition
[ 6 ]
.
The simple module number 9 has dimension 1 and corresponds to the partition
[ 2, 2, 1, 1 ]
.
The module M has radical filtration (Loewy series)
1,
1,
2,
2,
2,
2,
2,
2,
2,
2,
2,
3,
4,
4,
4,
4,
4,
4,
4,
4,
5,
5,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
7,
7,
7,
7,
7,
7,
8,
8,
8,
8,
8,
8,
8,
8,
8
3,
3,
3,
3,
3,
3,
3,
3,
4,
4,
4,
4,
4,
4,
4,
4,
8,
8,
8,
8,
8,
8,
8,
8,
9
3,
4,
4,
4,
4,
4,
4,
4,
4,
8,
8,
8,
8,
8,
8,
8
The module M has socle filtration (socle series)
3,
4,
4,
4,
4,
4,
4,
4,
4,
8,
8,
8,
8,
8,
8,
8
3,
3,
3,
3,
3,
3,
3,
3,
4,
4,
4,
4,
4,
4,
4,
4,
8,
8,
8,
8,
8,
8,
8,
8,
9
1,
1,
2,
2,
2,
2,
2,
2,
2,
2,
2,
3,
4,
4,
4,
4,
4,
4,
4,
4,
5,
5,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
6,
7,
7,
7,
7,
7,
7,
8,
8,
8,
8,
8,
8,
8,
8,
8
The module M has simple direct summands:
2 copies of simple module number 1
9 copies of simple module number 2
2 copies of simple module number 5
15 copies of simple module number 6
6 copies of simple module number 7
2 copies of simple module number 8
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1). 7 direct summands of the form:
radical layers
8
4
8
socle layers
8
4
8
2). 8 direct summands of the form:
radical layers
4
3,
8
4
socle layers
4
3,
8
4
3). 1 direct summand of the form:
radical layers
3
4,
9
3
socle layers
3
4,
9
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,
10,
25,
25,
5,
5,
5,
10,
9
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0,
0,
0
0,
0,
2,
1,
0,
0,
0,
0,
1
0,
0,
1,
2,
0,
0,
0,
1,
0
0,
0,
0,
0,
1,
0,
0,
0,
0
0,
0,
0,
0,
0,
1,
0,
0,
0
0,
0,
0,
0,
0,
0,
1,
0,
0
0,
0,
0,
1,
0,
0,
0,
2,
0
0,
0,
1,
0,
0,
0,
0,
0,
1
The determinant of the Cartan matrix is 1.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3,
4,
8,
9
(4).
5
(5).
6
(6).
7
Projective modules number
1,
2,
5,
6,
7
are simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 3
radical layers
3
4,
9
3
socle layers
3
4,
9
3
Projective module number 4
radical layers
4
3,
8
4
socle layers
4
3,
8
4
Projective module number 8
radical layers
8
4
8
socle layers
8
4
8
Projective module number 9
radical layers
9
3
socle layers
9
3
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1,
1,
1
.
The Basic Algebra H of the Schur Algebra
The dimension of H is
18
.
The dimensions of the irreducible H-modules are
1,
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 7.
Simple H-module 2 corresponds
to the direct summand of M isomorphic to simple A-module 1.
Simple H-module 3 corresponds
to the direct summand of M isomorphic to simple A-module 6.
Simple H-module 4 corresponds
to the direct summand of M isomorphic to simple A-module 2.
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 5.
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.
Simple H-module 9 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 3.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
1,
1,
4,
4,
1,
1,
1,
2,
3
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0,
0,
0,
0
0,
0,
2,
1,
0,
0,
0,
0,
1
0,
0,
1,
2,
0,
0,
0,
1,
0
0,
0,
0,
0,
1,
0,
0,
0,
0
0,
0,
0,
0,
0,
1,
0,
0,
0
0,
0,
0,
0,
0,
0,
1,
0,
0
0,
0,
0,
1,
0,
0,
0,
1,
0
0,
0,
1,
0,
0,
0,
0,
0,
2
The determinant of the Cartan matrix is 1.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2
(3).
3,
4,
8,
9
(4).
5
(5).
6
(6).
7
Projective modules number
1,
2,
5,
6,
7
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 3
radical layers
3
4,
9
3
socle layers
3
4,
9
3
Projective module number 4
radical layers
4
3,
8
4
socle layers
4
3,
8
4
Projective module number 8
radical layers
8
4
socle layers
8
4
Projective module number 9
radical layers
9
3
9
socle layers
9
3
9
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
,
b_9
,
z_1
,
z_2
,
z_3
,
z_4
,
z_5
,
z_6
,
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_2*z_6*z_2
,
z_6*z_2*z_6
,
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*b_8
,
b_2*b_9
,
b_2*z_1
,
b_2*z_2
,
b_2*z_3
,
b_2*z_4
,
b_2*z_5
,
b_2*z_6
,
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*b_8
,
b_3*b_9
,
b_3*z_1 + 3*z_1
,
b_3*z_2 + 3*z_2
,
b_3*z_3
,
b_3*z_4
,
b_3*z_5
,
b_3*z_6
,
b_4*b_2
,
b_4*b_3
,
b_4^2 + 3*b_4
,
b_4*b_5
,
b_4*b_6
,
b_4*b_7
,
b_4*b_8
,
b_4*b_9
,
b_4*z_1
,
b_4*z_2
,
b_4*z_3 + 3*z_3
,
b_4*z_4 + 3*z_4
,
b_4*z_5
,
b_4*z_6
,
b_5*b_2
,
b_5*b_3
,
b_5*b_4
,
b_5^2 + 2*b_5
,
b_5*b_6
,
b_5*b_7
,
b_5*b_8
,
b_5*b_9
,
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_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*b_8
,
b_6*b_9
,
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_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*b_8
,
b_7*b_9
,
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_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*b_9
,
b_8*z_1
,
b_8*z_2
,
b_8*z_3
,
b_8*z_4
,
b_8*z_5 + 2*z_5
,
b_8*z_6
,
b_9*b_2
,
b_9*b_3
,
b_9*b_4
,
b_9*b_5
,
b_9*b_6
,
b_9*b_7
,
b_9*b_8
,
b_9^2 + 2*b_9
,
b_9*z_1
,
b_9*z_2
,
b_9*z_3
,
b_9*z_4
,
b_9*z_5
,
b_9*z_6 + 2*z_6
,
z_1*b_2
,
z_1*b_3
,
z_1*b_4 + 3*z_1
,
z_1*b_5
,
z_1*b_6
,
z_1*b_7
,
z_1*b_8
,
z_1*b_9
,
z_1^2
,
z_1*z_2
,
z_1*z_3 + 3*z_2*z_6
,
z_1*z_4
,
z_1*z_5
,
z_1*z_6
,
z_2*b_2
,
z_2*b_3
,
z_2*b_4
,
z_2*b_5
,
z_2*b_6
,
z_2*b_7
,
z_2*b_8
,
z_2*b_9 + 2*z_2
,
z_2*z_1
,
z_2^2
,
z_2*z_3
,
z_2*z_4
,
z_2*z_5
,
z_3*b_2
,
z_3*b_3 + 3*z_3
,
z_3*b_4
,
z_3*b_5
,
z_3*b_6
,
z_3*b_7
,
z_3*b_8
,
z_3*b_9
,
z_3*z_1 + z_4*z_5
,
z_3*z_2
,
z_3^2
,
z_3*z_4
,
z_3*z_5
,
z_3*z_6
,
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*b_8 + 2*z_4
,
z_4*b_9
,
z_4*z_1
,
z_4*z_2
,
z_4*z_3
,
z_4^2
,
z_4*z_6
,
z_5*b_2
,
z_5*b_3
,
z_5*b_4 + 3*z_5
,
z_5*b_5
,
z_5*b_6
,
z_5*b_7
,
z_5*b_8
,
z_5*b_9
,
z_5*z_1
,
z_5*z_2
,
z_5*z_3
,
z_5*z_4
,
z_5^2
,
z_5*z_6
,
z_6*b_2
,
z_6*b_3 + 3*z_6
,
z_6*b_4
,
z_6*b_5
,
z_6*b_6
,
z_6*b_7
,
z_6*b_8
,
z_6*b_9
,
z_6*z_1
,
z_6*z_3
,
z_6*z_4
,
z_6*z_5
,
z_6^2
,
b_1 + 4*b_2 + b_3 + b_4 + 4*b_5 + b_6 + b_7 + 4*b_8 + 4*b_9 + 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 is Projective.
Simple Module Number 3
Degree 0:
3
Degree 1:
4
9
Degree 2:
3
8
Degree 3:
4
Degree 4:
8
The projective resolution
of simple module no. 3 is graded.
Simple Module Number 4
Degree 0:
4
Degree 1:
3
8
Degree 2:
4
9
Degree 3:
3
8
Degree 4:
4
Degree 5:
8
The projective resolution
of simple module no. 4 is graded.
Simple Module Number 5 is Projective.
Simple Module Number 6 is Projective.
Simple Module Number 7 is Projective.
Simple Module Number 8
Degree 0:
8
Degree 1:
4
Degree 2:
3
8
Degree 3:
4
9
Degree 4:
3
8
Degree 5:
4
Degree 6:
8
The projective resolution
of simple module no. 8 is graded.
Simple Module Number 9
Degree 0:
9
Degree 1:
3
Degree 2:
4
Degree 3:
8
The projective resolution
of simple module no. 9 is graded.