Schur Algebra S(
3
,6) in characteristic 2
Field k
Finite field of size 2
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
16,
4,
4,
1
.
The simple module number 1 has dimension 16 and corresponds to the partition
[ 3, 2, 1 ]
.
The simple module number 2 has dimension 4 and corresponds to the partition
[ 5, 1 ]
.
The simple module number 3 has dimension 4 and corresponds to the partition
[ 4, 2 ]
.
The simple module number 4 has dimension 1 and corresponds to the partition
[ 6 ]
.
The module M has radical filtration (Loewy series)
1,
1,
1,
2,
2,
2,
2,
2,
3,
4,
4,
4,
4,
4,
4,
4
2,
2,
2,
2,
3,
3,
3,
4,
4,
4,
4,
4,
4
2,
3,
3,
3,
3,
3,
4,
4,
4,
4,
4,
4,
4
2,
2,
3,
3,
4,
4,
4,
4,
4,
4
2,
2,
2,
2,
2,
3,
4,
4
4,
4,
4
2,
3,
3
4,
4,
4
2,
2,
3
The module M has socle filtration (socle series)
2,
2,
3
4,
4,
4
2,
3,
3
4,
4,
4
2,
2,
2,
2,
2,
3,
4,
4
2,
2,
3,
3,
4,
4,
4,
4,
4,
4
2,
3,
3,
3,
3,
3,
4,
4,
4,
4,
4,
4,
4
2,
2,
2,
2,
3,
3,
3,
4,
4,
4,
4,
4,
4
1,
1,
1,
2,
2,
2,
2,
2,
3,
4,
4,
4,
4,
4,
4,
4
The module M has simple direct summands:
3 copies of simple module number 1
2 copies of simple module number 4
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1). 2 direct summands of the form:
radical layers
4
2
4
socle layers
4
2
4
2). 3 direct summands of the form:
radical layers
2
4
3
4
2
socle layers
2
4
3
4
2
3). 2 direct summands of the form:
radical layers
4
2,
3
4,
4
2,
3
4
socle layers
4
2,
3
4,
4
2,
3
4
4). 2 direct summands of the form:
radical layers
2
4
3
4
2
4
3
4
2
socle layers
2
4
3
4
2
4
3
4
2
5). 1 direct summand of the form:
radical layers
3,
4
3,
4
2,
4
4
3
4
2
4
3
socle layers
3
4
2
4
3
4
2,
4
3,
4
3,
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
16,
24,
29,
45
.
The cartan matrix of A is
1,
0,
0,
0
0,
3,
2,
4
0,
2,
4,
5
0,
4,
5,
9
The determinant of the Cartan matrix is 13.
The blocks of A consist of the following irreducible
modules:
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
3
4
2
4
3
4
2
socle layers
2
4
3
4
2
4
3
4
2
Projective module number 3
radical layers
3
3,
4
2,
4
4
3
4
2
4
3
socle layers
3
4
2
4
3
4
2
3,
4
3,
4
Projective module number 4
radical layers
4
2,
3,
4
3,
4,
4
2,
3,
4
4,
4
2,
3
4,
4
2,
3
socle layers
4
2,
3
4,
4
2,
3
4,
4
2,
3,
4
3,
4,
4
2,
3,
4
The degrees of the splitting fields are
1,
1,
1,
1
.
The Basic Algebra H of the Schur Algebra
The dimension of H is
58
.
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 1.
Simple H-module 2 corresponds
to the direct summand of M isomorphic to simple A-module 4.
Simple H-module 3 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 1.
Simple H-module 4 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 2.
Simple H-module 5 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 3.
Simple H-module 6 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 4.
Simple H-module 7 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 5.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
1,
8,
10,
4,
8,
14,
13
.
The cartan matrix of H is
1,
0,
0,
0,
0,
0,
0
0,
2,
2,
0,
1,
2,
1
0,
2,
3,
0,
1,
2,
2
0,
0,
0,
1,
1,
1,
1
0,
1,
1,
1,
2,
2,
1
0,
2,
2,
1,
2,
4,
3
0,
1,
2,
1,
1,
3,
5
The determinant of the Cartan matrix is 1.
The blocks of H consist of the following irreducible
modules:
(1).
1
(2).
2,
3,
4,
5,
6,
7
Projective module number 1 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 2
radical layers
2
3,
6
5,
7
6
2
3
socle layers
2
6
5
3,
6
2,
7
3
Projective module number 3
radical layers
3
2,
7
3,
6
5,
7
6
2
3
socle layers
3
2
6
5,
7
3,
6
2,
7
3
Projective module number 4
radical layers
4
5,
7
6
socle layers
4
5,
7
6
Projective module number 5
radical layers
5
4,
6
2,
5,
7
3,
6
socle layers
5
4,
6
2,
5,
7
3,
6
Projective module number 6
radical layers
6
2,
5,
7
3,
4,
6,
6
2,
5,
7,
7
3,
6
socle layers
6
2,
5,
7
4,
6,
6
2,
3,
5,
7
3,
6,
7
Projective module number 7
radical layers
7
3,
4,
6
2,
5,
7,
7,
7
3,
6,
6
7
socle layers
7
3,
6
2,
4,
7,
7
3,
5,
6,
7
6,
7
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
,
z_5
,
z_6
,
z_7
,
z_8
,
z_9
,
z_10
,
z_11
,
z_12
,
z_13
,
z_14
,
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_10*z_8*z_9 + z_1*z_3
,
z_3*z_2*z_10*z_8 + z_4*z_14
,
z_4*z_14*z_9*z_1 + z_3*z_1 + z_4*z_12
,
z_8*z_11*z_14*z_10
,
z_8*z_11*z_14*z_11
,
z_10*z_8*z_9*z_1 + z_11*z_12
,
z_11*z_14*z_11*z_14
,
z_14*z_10*z_8*z_9
,
z_14*z_11*z_14*z_10
,
z_1*z_3*z_1 + z_2*z_11*z_12
,
z_1*z_3*z_2
,
z_2*z_11*z_13
,
z_2*z_11*z_14
,
z_3*z_1*z_3 + z_4*z_14*z_9
,
z_3*z_2*z_11 + z_4*z_12*z_4
,
z_4*z_14*z_10
,
z_4*z_14*z_11
,
z_6*z_14*z_9
,
z_6*z_14*z_10
,
z_6*z_14*z_11
,
z_8*z_11*z_12
,
z_8*z_11*z_13
,
z_9*z_1*z_3
,
z_10*z_8*z_11 + z_11*z_13*z_6
,
z_11*z_12*z_4
,
z_11*z_14*z_9
,
z_12*z_4*z_12 + z_14*z_9*z_1
,
z_12*z_4*z_14
,
z_13*z_6*z_14 + z_14*z_10*z_8
,
z_14*z_11*z_12
,
z_14*z_11*z_13
,
b_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 + z_1
,
b_2*z_2 + z_2
,
b_2*z_3
,
b_2*z_4
,
b_2*z_5
,
b_2*z_6
,
b_2*z_7
,
b_2*z_8
,
b_2*z_9
,
b_2*z_10
,
b_2*z_11
,
b_2*z_12
,
b_2*z_13
,
b_2*z_14
,
b_3*b_2
,
b_3^2 + 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 + z_3
,
b_3*z_4 + z_4
,
b_3*z_5
,
b_3*z_6
,
b_3*z_7
,
b_3*z_8
,
b_3*z_9
,
b_3*z_10
,
b_3*z_11
,
b_3*z_12
,
b_3*z_13
,
b_3*z_14
,
b_4*b_2
,
b_4*b_3
,
b_4^2 + 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
,
b_4*z_4
,
b_4*z_5 + z_5
,
b_4*z_6 + z_6
,
b_4*z_7
,
b_4*z_8
,
b_4*z_9
,
b_4*z_10
,
b_4*z_11
,
b_4*z_12
,
b_4*z_13
,
b_4*z_14
,
b_5*b_2
,
b_5*b_3
,
b_5*b_4
,
b_5^2 + 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_5*z_5
,
b_5*z_6
,
b_5*z_7 + z_7
,
b_5*z_8 + z_8
,
b_5*z_9
,
b_5*z_10
,
b_5*z_11
,
b_5*z_12
,
b_5*z_13
,
b_5*z_14
,
b_6*b_2
,
b_6*b_3
,
b_6*b_4
,
b_6*b_5
,
b_6^2 + b_6
,
b_6*b_7
,
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_6*z_7
,
b_6*z_8
,
b_6*z_9 + z_9
,
b_6*z_10 + z_10
,
b_6*z_11 + z_11
,
b_6*z_12
,
b_6*z_13
,
b_6*z_14
,
b_7*b_2
,
b_7*b_3
,
b_7*b_4
,
b_7*b_5
,
b_7*b_6
,
b_7^2 + b_7
,
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_7*z_7
,
b_7*z_8
,
b_7*z_9
,
b_7*z_10
,
b_7*z_11
,
b_7*z_12 + z_12
,
b_7*z_13 + z_13
,
b_7*z_14 + z_14
,
z_1*b_2
,
z_1*b_3 + z_1
,
z_1*b_4
,
z_1*b_5
,
z_1*b_6
,
z_1*b_7
,
z_1^2
,
z_1*z_2
,
z_1*z_4 + z_2*z_11
,
z_1*z_5
,
z_1*z_6
,
z_1*z_7
,
z_1*z_8
,
z_1*z_9
,
z_1*z_10
,
z_1*z_11
,
z_1*z_12
,
z_1*z_13
,
z_1*z_14
,
z_2*b_2
,
z_2*b_3
,
z_2*b_4
,
z_2*b_5
,
z_2*b_6 + z_2
,
z_2*b_7
,
z_2*z_1
,
z_2^2
,
z_2*z_3
,
z_2*z_4
,
z_2*z_5
,
z_2*z_6
,
z_2*z_7
,
z_2*z_8
,
z_2*z_9
,
z_2*z_12
,
z_2*z_13
,
z_2*z_14
,
z_3*b_2 + z_3
,
z_3*b_3
,
z_3*b_4
,
z_3*b_5
,
z_3*b_6
,
z_3*b_7
,
z_3^2
,
z_3*z_4
,
z_3*z_5
,
z_3*z_6
,
z_3*z_7
,
z_3*z_8
,
z_3*z_9
,
z_3*z_10
,
z_3*z_11
,
z_3*z_12
,
z_3*z_13
,
z_3*z_14
,
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
,
z_4*z_1
,
z_4*z_2
,
z_4*z_3
,
z_4^2
,
z_4*z_5
,
z_4*z_6
,
z_4*z_7
,
z_4*z_8
,
z_4*z_9
,
z_4*z_10
,
z_4*z_11
,
z_4*z_13
,
z_5*b_2
,
z_5*b_3
,
z_5*b_4
,
z_5*b_5 + z_5
,
z_5*b_6
,
z_5*b_7
,
z_5*z_1
,
z_5*z_2
,
z_5*z_3
,
z_5*z_4
,
z_5^2
,
z_5*z_6
,
z_5*z_7
,
z_5*z_8 + z_6*z_14
,
z_5*z_9
,
z_5*z_10
,
z_5*z_11
,
z_5*z_12
,
z_5*z_13
,
z_5*z_14
,
z_6*b_2
,
z_6*b_3
,
z_6*b_4
,
z_6*b_5
,
z_6*b_6
,
z_6*b_7 + z_6
,
z_6*z_1
,
z_6*z_2
,
z_6*z_3
,
z_6*z_4
,
z_6*z_5
,
z_6^2
,
z_6*z_7
,
z_6*z_8
,
z_6*z_9
,
z_6*z_10
,
z_6*z_11
,
z_6*z_12
,
z_6*z_13
,
z_7*b_2
,
z_7*b_3
,
z_7*b_4 + z_7
,
z_7*b_5
,
z_7*b_6
,
z_7*b_7
,
z_7*z_1
,
z_7*z_2
,
z_7*z_3
,
z_7*z_4
,
z_7*z_6 + z_8*z_11
,
z_7^2
,
z_7*z_8
,
z_7*z_9
,
z_7*z_10
,
z_7*z_11
,
z_7*z_12
,
z_7*z_13
,
z_7*z_14
,
z_8*b_2
,
z_8*b_3
,
z_8*b_4
,
z_8*b_5
,
z_8*b_6 + z_8
,
z_8*b_7
,
z_8*z_1
,
z_8*z_2
,
z_8*z_3
,
z_8*z_4
,
z_8*z_5
,
z_8*z_6
,
z_8*z_7
,
z_8^2
,
z_8*z_10
,
z_8*z_12
,
z_8*z_13
,
z_8*z_14
,
z_9*b_2 + z_9
,
z_9*b_3
,
z_9*b_4
,
z_9*b_5
,
z_9*b_6
,
z_9*b_7
,
z_9*z_2 + z_11*z_14
,
z_9*z_3
,
z_9*z_4
,
z_9*z_5
,
z_9*z_6
,
z_9*z_7
,
z_9*z_8
,
z_9^2
,
z_9*z_10
,
z_9*z_11
,
z_9*z_12
,
z_9*z_13
,
z_9*z_14
,
z_10*b_2
,
z_10*b_3
,
z_10*b_4
,
z_10*b_5 + z_10
,
z_10*b_6
,
z_10*b_7
,
z_10*z_1
,
z_10*z_2
,
z_10*z_3
,
z_10*z_4
,
z_10*z_5
,
z_10*z_6
,
z_10*z_7 + z_11*z_13
,
z_10*z_9
,
z_10^2
,
z_10*z_11
,
z_10*z_12
,
z_10*z_13
,
z_10*z_14
,
z_11*b_2
,
z_11*b_3
,
z_11*b_4
,
z_11*b_5
,
z_11*b_6
,
z_11*b_7 + z_11
,
z_11*z_1
,
z_11*z_2
,
z_11*z_3
,
z_11*z_4
,
z_11*z_5
,
z_11*z_6
,
z_11*z_7
,
z_11*z_8
,
z_11*z_9
,
z_11*z_10
,
z_11^2
,
z_12*b_2
,
z_12*b_3 + z_12
,
z_12*b_4
,
z_12*b_5
,
z_12*b_6
,
z_12*b_7
,
z_12*z_1
,
z_12*z_2
,
z_12*z_3 + z_14*z_9
,
z_12*z_5
,
z_12*z_6
,
z_12*z_7
,
z_12*z_8
,
z_12*z_9
,
z_12*z_10
,
z_12*z_11
,
z_12^2
,
z_12*z_13
,
z_12*z_14
,
z_13*b_2
,
z_13*b_3
,
z_13*b_4 + z_13
,
z_13*b_5
,
z_13*b_6
,
z_13*b_7
,
z_13*z_1
,
z_13*z_2
,
z_13*z_3
,
z_13*z_4
,
z_13*z_5 + z_14*z_10
,
z_13*z_7
,
z_13*z_8
,
z_13*z_9
,
z_13*z_10
,
z_13*z_11
,
z_13*z_12
,
z_13^2
,
z_13*z_14
,
z_14*b_2
,
z_14*b_3
,
z_14*b_4
,
z_14*b_5
,
z_14*b_6 + z_14
,
z_14*b_7
,
z_14*z_1
,
z_14*z_2
,
z_14*z_3
,
z_14*z_4
,
z_14*z_5
,
z_14*z_6
,
z_14*z_7
,
z_14*z_8
,
z_14*z_12
,
z_14*z_13
,
z_14^2
,
b_1 + b_2 + b_3 + b_4 + b_5 + b_6 + b_7 + 1
.
The ideal of relations is not generated by the elements
of degree at most 2. The following relation were not contained in the ideal
generated by the relations of degree 2:
z_2*z_10*z_8*z_9 + z_1*z_3
,
z_3*z_2*z_10*z_8 + z_4*z_14
,
z_4*z_14*z_9*z_1 + z_3*z_1 + z_4*z_12
,
z_10*z_8*z_9*z_1 + z_11*z_12
,
z_1*z_3*z_1 + z_2*z_11*z_12
,
z_1*z_3*z_2
,
z_3*z_1*z_3 + z_4*z_14*z_9
,
z_3*z_2*z_11 + z_4*z_12*z_4
,
z_4*z_14*z_11
,
z_9*z_1*z_3
,
z_11*z_12*z_4
,
z_12*z_4*z_12 + z_14*z_9*z_1
,
z_12*z_4*z_14
,
z_14*z_11*z_12
,
The projective resolutions of the simple modules.
Simple Module Number 1 is Projective.
Simple Module Number 2
Degree 0:
2
Degree 1:
3
6
Degree 2:
2
2
7
Degree 3:
3
6
Degree 4:
2
4
The projective resolution
of simple module no. 2 is not graded.
Simple Module Number 3
Degree 0:
3
Degree 1:
2
7
Degree 2:
3
4
6
Degree 3:
2
5
The projective resolution
of simple module no. 3 is graded.
Simple Module Number 4
Degree 0:
4
Degree 1:
5
7
Degree 2:
3
4
4
6
Degree 3:
5
6
Degree 4:
2
4
Degree 5:
5
Degree 6:
4
The projective resolution
of simple module no. 4 is not graded.
Simple Module Number 5
Degree 0:
5
Degree 1:
4
6
Degree 2:
5
7
Degree 3:
3
4
Degree 4:
5
Degree 5:
4
The projective resolution
of simple module no. 5 is not graded.
Simple Module Number 6
Degree 0:
6
Degree 1:
2
5
7
Degree 2:
3
4
6
Degree 3:
2
4
The projective resolution
of simple module no. 6 is not graded.
Simple Module Number 7
Degree 0:
7
Degree 1:
3
4
6
Degree 2:
2
5
The projective resolution
of simple module no. 7 is graded.