Schur Algebra S(
4
,5) 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 126
.
The dimensions of the irreducible submodules modules are
5,
5,
3,
3,
1,
1
.
The simple module number 1 has dimension 5 and corresponds to the partition
[ 3, 2 ]
.
The simple module number 2 has dimension 5 and corresponds to the partition
[ 2, 2, 1 ]
.
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
[ 2, 1, 1, 1 ]
.
The simple module number 6 has dimension 1 and corresponds to the partition
[ 5 ]
.
The module M has radical filtration (Loewy series)
1,
1,
1,
1,
1,
1,
1,
2,
2,
2,
3,
4,
4,
4,
4,
6,
6,
6,
6,
6,
6
3,
3,
3,
3,
4,
4,
4,
4,
4,
4,
5,
6,
6,
6,
6
3,
4,
4,
4,
4,
6,
6,
6,
6,
6
The module M has socle filtration (socle series)
3,
4,
4,
4,
4,
6,
6,
6,
6,
6
3,
3,
3,
3,
4,
4,
4,
4,
4,
4,
5,
6,
6,
6,
6
1,
1,
1,
1,
1,
1,
1,
2,
2,
2,
3,
4,
4,
4,
4,
6,
6,
6,
6,
6,
6
The module M has simple direct summands:
7 copies of simple module number 1
3 copies of simple module number 2
1 copy of simple module number 6
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1). 5 direct summands of the form:
radical layers
6
4
6
socle layers
6
4
6
2). 4 direct summands of the form:
radical layers
4
3,
6
4
socle layers
4
3,
6
4
3). 1 direct summand of the form:
radical layers
3
4,
5
3
socle layers
3
4,
5
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
5,
5,
10,
10,
4,
5
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0
0,
1,
0,
0,
0,
0
0,
0,
2,
1,
1,
0
0,
0,
1,
2,
0,
1
0,
0,
1,
0,
1,
0
0,
0,
0,
1,
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).
3,
4,
5,
6
Projective modules number
1,
2
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,
5
3
socle layers
3
4,
5
3
Projective module number 4
radical layers
4
3,
6
4
socle layers
4
3,
6
4
Projective module number 5
radical layers
5
3
socle layers
5
3
Projective module number 6
radical layers
6
4
6
socle layers
6
4
6
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1
.
The Basic Algebra H of the Schur Algebra
The dimension of H is
15
.
The dimensions of the irreducible H-modules are
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 2.
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 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.
Simple H-module 6 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
.
The dimensions of the projective modules of H are
1,
4,
4,
1,
2,
3
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0
0,
2,
1,
0,
1,
0
0,
1,
2,
0,
0,
1
0,
0,
0,
1,
0,
0
0,
1,
0,
0,
1,
0
0,
0,
1,
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,
5,
6
(3).
4
Projective modules number
1,
4
are simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 2
radical layers
2
3,
5
2
socle layers
2
3,
5
2
Projective module number 3
radical layers
3
2,
6
3
socle layers
3
2,
6
3
Projective module number 5
radical layers
5
2
socle layers
5
2
Projective module number 6
radical layers
6
3
6
socle layers
6
3
6
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
,
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_4*z_6*z_4
,
z_6*z_4*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*z_1 + 2*z_1
,
b_2*z_2 + 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*z_1
,
b_3*z_2
,
b_3*z_3 + 3*z_3
,
b_3*z_4 + 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*z_1
,
b_4*z_2
,
b_4*z_3
,
b_4*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*z_1
,
b_5*z_2
,
b_5*z_3
,
b_5*z_4
,
b_5*z_5 + 2*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*z_1
,
b_6*z_2
,
b_6*z_3
,
b_6*z_4
,
b_6*z_5
,
b_6*z_6 + 3*z_6
,
z_1*b_2
,
z_1*b_3 + 3*z_1
,
z_1*b_4
,
z_1*b_5
,
z_1*b_6
,
z_1^2
,
z_1*z_2
,
z_1*z_3 + 2*z_2*z_5
,
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 + 2*z_2
,
z_2*b_6
,
z_2*z_1
,
z_2^2
,
z_2*z_3
,
z_2*z_4
,
z_2*z_6
,
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*z_1 + 4*z_4*z_6
,
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 + 3*z_4
,
z_4*z_1
,
z_4*z_2
,
z_4*z_3
,
z_4^2
,
z_4*z_5
,
z_5*b_2 + 2*z_5
,
z_5*b_3
,
z_5*b_4
,
z_5*b_5
,
z_5*b_6
,
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*z_1
,
z_6*z_2
,
z_6*z_3
,
z_6*z_5
,
z_6^2
,
b_1 + 4*b_2 + b_3 + b_4 + 4*b_5 + b_6 + 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:
3
5
Degree 2:
2
6
Degree 3:
3
5
Degree 4:
2
Degree 5:
5
The projective resolution
of simple module no. 2 is graded.
Simple Module Number 3
Degree 0:
3
Degree 1:
2
6
Degree 2:
3
5
Degree 3:
2
Degree 4:
5
The projective resolution
of simple module no. 3 is graded.
Simple Module Number 4 is Projective.
Simple Module Number 5
Degree 0:
5
Degree 1:
2
Degree 2:
3
5
Degree 3:
2
6
Degree 4:
3
5
Degree 5:
2
Degree 6:
5
The projective resolution
of simple module no. 5 is graded.
Simple Module Number 6
Degree 0:
6
Degree 1:
3
Degree 2:
2
Degree 3:
5
The projective resolution
of simple module no. 6 is graded.