Schur Algebra S(
3
,5) in characteristic 3
Field k
Finite field of size 3
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
6,
4,
4,
1,
1
.
The simple module number 1 has dimension 6 and corresponds to the partition
[ 3, 1, 1 ]
.
The simple module number 2 has dimension 4 and corresponds to the partition
[ 4, 1 ]
.
The simple module number 3 has dimension 4 and corresponds to the partition
[ 2, 2, 1 ]
.
The simple module number 4 has dimension 1 and corresponds to the partition
[ 3, 2 ]
.
The simple module number 5 has dimension 1 and corresponds to the partition
[ 5 ]
.
The module M has radical filtration (Loewy series)
1,
1,
2,
2,
2,
2,
2,
2,
5,
5,
5,
5,
5
3,
4,
4,
4,
4
2,
2,
2,
2,
5
The module M has socle filtration (socle series)
2,
2,
2,
2,
5
3,
4,
4,
4,
4
1,
1,
2,
2,
2,
2,
2,
2,
5,
5,
5,
5,
5
The module M has simple direct summands:
2 copies of simple module number 1
2 copies of simple module number 2
4 copies of simple module number 5
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1). 1 direct summand of the form:
radical layers
5
3
5
socle layers
5
3
5
2). 4 direct summands of the form:
radical layers
2
4
2
socle layers
2
4
2
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
6,
9,
5,
5,
6
.
The cartan matrix of A is
1,
0,
0,
0,
0
0,
2,
0,
1,
0
0,
0,
1,
0,
1
0,
1,
0,
1,
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,
4
(3).
3,
5
Projective module number 1 is simple.
The radical and socle filtrations of the remaining
projective modules for A are the following:
Projective module number 2
radical layers
2
4
2
socle layers
2
4
2
Projective module number 3
radical layers
3
5
socle layers
3
5
Projective module number 4
radical layers
4
2
socle layers
4
2
Projective module number 5
radical layers
5
3
5
socle layers
5
3
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
3,
2,
1,
2,
3
.
The cartan matrix of H is
2,
0,
0,
1,
0
0,
1,
0,
0,
1
0,
0,
1,
0,
0
1,
0,
0,
1,
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,
4
(2).
2,
5
(3).
3
Projective module number 3 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 1
radical layers
1
4
1
socle layers
1
4
1
Projective module number 2
radical layers
2
5
socle layers
2
5
Projective module number 4
radical layers
4
1
socle layers
4
1
Projective module number 5
radical layers
5
2
5
socle layers
5
2
5
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 + b_2
,
b_2*b_3
,
b_2*b_4
,
b_2*b_5
,
b_2*z_1
,
b_2*z_2 + z_2
,
b_2*z_3
,
b_2*z_4
,
b_3*b_2
,
b_3^2 + b_3
,
b_3*b_4
,
b_3*b_5
,
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 + b_4
,
b_4*b_5
,
b_4*z_1
,
b_4*z_2
,
b_4*z_3 + z_3
,
b_4*z_4
,
b_5*b_2
,
b_5*b_3
,
b_5*b_4
,
b_5^2 + b_5
,
b_5*z_1
,
b_5*z_2
,
b_5*z_3
,
b_5*z_4 + z_4
,
z_1*b_2
,
z_1*b_3
,
z_1*b_4 + z_1
,
z_1*b_5
,
z_1^2
,
z_1*z_2
,
z_1*z_4
,
z_2*b_2
,
z_2*b_3
,
z_2*b_4
,
z_2*b_5 + z_2
,
z_2*z_1
,
z_2^2
,
z_2*z_3
,
z_2*z_4
,
z_3*b_2
,
z_3*b_3
,
z_3*b_4
,
z_3*b_5
,
z_3*z_1
,
z_3*z_2
,
z_3^2
,
z_3*z_4
,
z_4*b_2 + z_4
,
z_4*b_3
,
z_4*b_4
,
z_4*b_5
,
z_4*z_1
,
z_4*z_3
,
z_4^2
,
b_1 + b_2 + b_3 + b_4 + b_5 + 1
.
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
The projective resolution
of simple module no. 1 is graded.
Simple Module Number 2
Degree 0:
2
Degree 1:
5
Degree 2:
2
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:
1
Degree 2:
4
The projective resolution
of simple module no. 4 is graded.
Simple Module Number 5
The projective resolution
of simple module no. 5 is graded.