Schur Algebra S(
4
,8) 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 4.
. The dimension of M is 8143
.
The dimensions of the irreducible submodules modules are
90,
35,
35,
28,
28,
21,
21,
13,
13,
7,
7,
1,
1
.
The simple module number 1 has dimension 90 and corresponds to the partition
[ 4, 2, 1, 1 ]
.
The simple module number 2 has dimension 35 and corresponds to the partition
[ 4, 2, 2 ]
.
The simple module number 3 has dimension 35 and corresponds to the partition
[ 5, 2, 1 ]
.
The simple module number 4 has dimension 28 and corresponds to the partition
[ 3, 2, 2, 1 ]
.
The simple module number 5 has dimension 28 and corresponds to the partition
[ 5, 3 ]
.
The simple module number 6 has dimension 21 and corresponds to the partition
[ 3, 3, 2 ]
.
The simple module number 7 has dimension 21 and corresponds to the partition
[ 6, 1, 1 ]
.
The simple module number 8 has dimension 13 and corresponds to the partition
[ 3, 3, 1, 1 ]
.
The simple module number 9 has dimension 13 and corresponds to the partition
[ 6, 2 ]
.
The simple module number 10 has dimension 7 and corresponds to the partition
[ 4, 3, 1 ]
.
The simple module number 11 has dimension 7 and corresponds to the partition
[ 7, 1 ]
.
The simple module number 12 has dimension 1 and corresponds to the partition
[ 4, 4 ]
.
The simple module number 13 has dimension 1 and corresponds to the partition
[ 8 ]
.
The module M has radical filtration (Loewy series)
1,
1,
1,
1,
1,
1,
1,
3,
3,
3,
3,
3,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13
2,
2,
2,
2,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
5,
5,
5,
5,
5,
6,
6,
6,
6,
6,
6,
6,
8,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
10,
10,
10,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
12,
12,
12,
12,
12,
12,
12,
12,
12,
12,
12,
12,
13,
13,
13,
13,
13
2,
2,
2,
2,
2,
2,
2,
2,
2,
3,
3,
3,
3,
3,
3,
3,
3,
4,
4,
4,
4,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
7,
7,
7,
7,
7,
7,
7,
8,
8,
8,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
12,
12,
12,
12,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13
2,
2,
2,
2,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
5,
5,
5,
8,
9,
9,
9,
9,
10,
10,
10,
11,
11,
11,
11,
11,
11,
11,
11,
11,
12,
12,
12,
12,
12,
12,
12,
12,
12,
13,
13,
13
3,
3,
3,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
9,
9,
9,
9,
9,
9,
9,
9,
9,
11,
11,
11,
11,
13
The module M has socle filtration (socle series)
3,
3,
3,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
9,
9,
9,
9,
9,
9,
9,
9,
9,
11,
11,
11,
11,
13
2,
2,
2,
2,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
5,
5,
5,
8,
9,
9,
9,
9,
10,
10,
10,
11,
11,
11,
11,
11,
11,
11,
11,
11,
12,
12,
12,
12,
12,
12,
12,
12,
12,
13,
13,
13
2,
2,
2,
2,
2,
2,
2,
2,
2,
3,
3,
3,
3,
3,
3,
3,
3,
4,
4,
4,
4,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
7,
7,
7,
7,
7,
7,
7,
8,
8,
8,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
10,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
12,
12,
12,
12,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13
2,
2,
2,
2,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
3,
5,
5,
5,
5,
5,
6,
6,
6,
6,
6,
6,
6,
8,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
10,
10,
10,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
12,
12,
12,
12,
12,
12,
12,
12,
12,
12,
12,
12,
13,
13,
13,
13,
13
1,
1,
1,
1,
1,
1,
1,
3,
3,
3,
3,
3,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
5,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
7,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
9,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
11,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13,
13
The module M has simple direct summands:
7 copies of simple module number 1
5 copies of simple module number 5
13 copies of simple module number 7
5 copies of simple module number 11
9 copies of simple module number 13
The remaining indecomposable components of M
have radical and socle filtrations as follows:
1). 7 direct summands of the form:
radical layers
7
6
7
socle layers
7
6
7
2). 20 direct summands of the form:
radical layers
11
9
11
socle layers
11
9
11
3). 2 direct summands of the form:
radical layers
3
5,
13
3
socle layers
3
5,
13
3
4). 3 direct summands of the form:
radical layers
9
11,
12
9
socle layers
9
11,
12
9
5). 5 direct summands of the form:
radical layers
5,
13
3
5,
13
socle layers
5,
13
3
5,
13
6). 19 direct summands of the form:
radical layers
5
3
5,
10,
13
3
5
socle layers
5
3
5,
10,
13
3
5
7). 9 direct summands of the form:
radical layers
9
11,
12
2,
9
11,
12
9
socle layers
9
11,
12
2,
9
11,
12
9
8). 4 direct summands of the form:
radical layers
11
2,
9
4,
11,
11,
12
2,
9
11
socle layers
11
2,
9
4,
11,
11,
12
2,
9
11
9). 1 direct summand of the form:
radical layers
13
3,
8
5,
10,
13,
13
3,
8
13
socle layers
13
3,
8
5,
10,
13,
13
3,
8
13
10). 3 direct summands of the form:
radical layers
3
5,
10,
13
3,
3,
8
5,
10,
13
3
socle layers
3
5,
10,
13
3,
3,
8
5,
10,
13
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
90,
126,
225,
70,
162,
42,
63,
70,
90,
126,
153,
70,
135
.
The cartan matrix of A is
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
0,
2,
0,
1,
0,
0,
0,
0,
1,
0,
2,
1,
0
0,
0,
4,
0,
2,
0,
0,
1,
0,
2,
0,
0,
2
0,
1,
0,
1,
0,
0,
0,
0,
0,
0,
1,
0,
0
0,
0,
2,
0,
3,
0,
0,
0,
0,
1,
0,
0,
1
0,
0,
0,
0,
0,
1,
1,
0,
0,
0,
0,
0,
0
0,
0,
0,
0,
0,
1,
2,
0,
0,
0,
0,
0,
0
0,
0,
1,
0,
0,
0,
0,
2,
0,
1,
0,
0,
2
0,
1,
0,
0,
0,
0,
0,
0,
3,
0,
2,
2,
0
0,
0,
2,
0,
1,
0,
0,
1,
0,
2,
0,
0,
1
0,
2,
0,
1,
0,
0,
0,
0,
2,
0,
4,
1,
0
0,
1,
0,
0,
0,
0,
0,
0,
2,
0,
1,
2,
0
0,
0,
2,
0,
1,
0,
0,
2,
0,
1,
0,
0,
4
The determinant of the Cartan matrix is 15.
The blocks of A consist of the following irreducible
modules:
(1).
1
(2).
2,
4,
9,
11,
12
(3).
3,
5,
8,
10,
13
(4).
6,
7
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,
11,
12
2,
9
11
socle layers
2
4,
11,
12
2,
9
11
Projective module number 3
radical layers
3
5,
10,
13
3,
3,
8
5,
10,
13
3
socle layers
3
5,
10,
13
3,
3,
8
5,
10,
13
3
Projective module number 4
radical layers
4
2
11
socle layers
4
2
11
Projective module number 5
radical layers
5
3
5,
10,
13
3
5
socle layers
5
3
5,
10,
13
3
5
Projective module number 6
radical layers
6
7
socle layers
6
7
Projective module number 7
radical layers
7
6
7
socle layers
7
6
7
Projective module number 8
radical layers
8
10,
13
3,
8
13
socle layers
8
10,
13
3,
8
13
Projective module number 9
radical layers
9
11,
12
2,
9
11,
12
9
socle layers
9
11,
12
2,
9
11,
12
9
Projective module number 10
radical layers
10
3,
8
5,
10,
13
3
socle layers
10
3,
8
5,
10,
13
3
Projective module number 11
radical layers
11
2,
9
4,
11,
11,
12
2,
9
11
socle layers
11
2,
9
4,
11,
11,
12
2,
9
11
Projective module number 12
radical layers
12
2,
9
11,
12
9
socle layers
12
2,
9
11,
12
9
Projective module number 13
radical layers
13
3,
8
5,
10,
13,
13
3,
8
13
socle layers
13
3,
8
5,
10,
13,
13
3,
8
13
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
.
The Basic Algebra H of the Schur Algebra
The dimension of H is
96
.
The dimensions of the irreducible H-modules are
1,
1,
1,
1,
1,
1,
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 5.
Simple H-module 2 corresponds
to the direct summand of M isomorphic to simple A-module 7.
Simple H-module 3 corresponds
to the direct summand of M isomorphic to simple A-module 13.
Simple H-module 4 corresponds
to the direct summand of M isomorphic to simple A-module 1.
Simple H-module 5 corresponds
to the direct summand of M isomorphic to simple A-module 11.
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.
Simple H-module 8 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 3.
Simple H-module 9 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 4.
Simple H-module 10 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 5.
Simple H-module 11 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 6.
Simple H-module 12 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 7.
Simple H-module 13 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 8.
Simple H-module 14 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 9.
Simple H-module 15 corresponds
to the direct summand of M isomorphic to the
nonsimple A-module 10.
The degrees of the splitting fields are
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1
.
The dimensions of the projective modules of H are
7,
10,
2,
3,
8,
3,
1,
3,
11,
3,
10,
6,
11,
7,
11
.
The cartan matrix of H is
2,
0,
0,
0,
1,
0,
0,
0,
0,
1,
2,
1,
0,
0,
0
0,
3,
0,
0,
0,
0,
0,
1,
2,
0,
0,
0,
2,
1,
1
0,
0,
1,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0
0,
0,
0,
1,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
1
1,
0,
0,
0,
3,
0,
0,
0,
0,
0,
2,
2,
0,
0,
0
0,
0,
1,
0,
0,
2,
0,
0,
0,
0,
0,
0,
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,
1,
0,
0,
0,
0,
0,
0
0,
2,
0,
1,
0,
0,
0,
1,
3,
0,
0,
0,
1,
1,
2
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
1,
0,
0,
0,
0
2,
0,
0,
0,
2,
0,
0,
0,
0,
1,
4,
1,
0,
0,
0
1,
0,
0,
0,
2,
0,
0,
0,
0,
0,
1,
2,
0,
0,
0
0,
2,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
4,
2,
2
0,
1,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
2,
2,
1
0,
1,
0,
1,
0,
0,
0,
0,
2,
0,
0,
0,
2,
1,
4
The determinant of the Cartan matrix is 1.
The blocks of H consist of the following irreducible
modules:
(1).
1,
5,
10,
11,
12
(2).
2,
4,
8,
9,
13,
14,
15
(3).
3,
6
(4).
7
Projective module number 7 is simple.
The radical and socle filtrations of the remaining
projective modules for H are the following:
Projective module number 1
radical layers
1
10,
11,
12
1,
5
11
socle layers
1
10,
11,
12
1,
5
11
Projective module number 2
radical layers
2
9,
13
2,
8,
14,
15
9,
13
2
socle layers
2
9,
13
2,
8,
14,
15
9,
13
2
Projective module number 3
radical layers
3
6
socle layers
3
6
Projective module number 4
radical layers
4
9
15
socle layers
4
9
15
Projective module number 5
radical layers
5
11,
12
1,
5
11,
12
5
socle layers
5
11,
12
1,
5
11,
12
5
Projective module number 6
radical layers
6
3
6
socle layers
6
3
6
Projective module number 8
radical layers
8
9
2
socle layers
8
9
2
Projective module number 9
radical layers
9
2,
4,
8,
14,
15
9,
9,
13
2,
15
socle layers
9
2,
4,
8,
14,
15
9,
9,
13
2,
15
Projective module number 10
radical layers
10
1
11
socle layers
10
1
11
Projective module number 11
radical layers
11
1,
5
10,
11,
11,
12
1,
5
11
socle layers
11
1,
5
10,
11,
11,
12
1,
5
11
Projective module number 12
radical layers
12
1,
5
11,
12
5
socle layers
12
1,
5
11,
12
5
Projective module number 13
radical layers
13
2,
14,
15
9,
13,
13
2,
14,
15
13
socle layers
13
2,
14,
15
9,
13,
13
2,
14,
15
13
Projective module number 14
radical layers
14
9,
13
2,
14,
15
13
socle layers
14
9,
13
2,
14,
15
13
Projective module number 15
radical layers
15
9,
13
2,
4,
14,
15,
15
9,
13
15
socle layers
15
9,
13
2,
4,
14,
15,
15
9,
13
15
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
,
b_10
,
b_11
,
b_12
,
b_13
,
b_14
,
b_15
,
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
,
z_15
,
z_16
,
z_17
,
z_18
,
z_19
,
z_20
,
z_21
,
z_22
,
z_23
,
z_24
,
z_25
,
z_26
,
z_27
,
z_28
,
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_5*z_24*z_27*z_16
,
z_5*z_24*z_28*z_22 + z_4*z_12 + 2*z_5*z_22
,
z_5*z_24*z_28*z_23
,
z_16*z_28*z_24*z_27
,
z_19*z_9*z_21*z_9
,
z_21*z_9*z_21*z_8
,
z_21*z_9*z_21*z_9
,
z_26*z_24*z_28*z_22
,
z_26*z_24*z_28*z_23
,
z_3*z_21*z_9
,
z_5*z_23*z_26 + z_5*z_24*z_28
,
z_7*z_16*z_27
,
z_7*z_16*z_28
,
z_15*z_25*z_15
,
z_16*z_27*z_13
,
z_16*z_27*z_16 + z_16*z_28*z_24
,
z_16*z_28*z_23
,
z_21*z_9*z_20
,
z_23*z_26*z_22 + z_24*z_28*z_22
,
z_23*z_26*z_24 + z_24*z_27*z_16
,
z_24*z_27*z_13
,
z_24*z_28*z_24
,
z_25*z_15*z_25
,
z_26*z_24*z_27
,
z_27*z_16*z_27 + z_28*z_24*z_27
,
z_27*z_16*z_28 + z_28*z_23*z_26
,
z_28*z_24*z_28
,
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*b_8
,
b_2*b_9
,
b_2*b_10
,
b_2*b_11
,
b_2*b_12
,
b_2*b_13
,
b_2*b_14
,
b_2*b_15
,
b_2*z_1
,
b_2*z_2
,
b_2*z_3
,
b_2*z_4 + z_4
,
b_2*z_5 + 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_2*z_15
,
b_2*z_16
,
b_2*z_17
,
b_2*z_18
,
b_2*z_19
,
b_2*z_20
,
b_2*z_21
,
b_2*z_22
,
b_2*z_23
,
b_2*z_24
,
b_2*z_25
,
b_2*z_26
,
b_2*z_27
,
b_2*z_28
,
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*b_8
,
b_3*b_9
,
b_3*b_10
,
b_3*b_11
,
b_3*b_12
,
b_3*b_13
,
b_3*b_14
,
b_3*b_15
,
b_3*z_1
,
b_3*z_2
,
b_3*z_3
,
b_3*z_4
,
b_3*z_5
,
b_3*z_6 + 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_3*z_15
,
b_3*z_16
,
b_3*z_17
,
b_3*z_18
,
b_3*z_19
,
b_3*z_20
,
b_3*z_21
,
b_3*z_22
,
b_3*z_23
,
b_3*z_24
,
b_3*z_25
,
b_3*z_26
,
b_3*z_27
,
b_3*z_28
,
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*b_8
,
b_4*b_9
,
b_4*b_10
,
b_4*b_11
,
b_4*b_12
,
b_4*b_13
,
b_4*b_14
,
b_4*b_15
,
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_4*z_7 + 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_4*z_15
,
b_4*z_16
,
b_4*z_17
,
b_4*z_18
,
b_4*z_19
,
b_4*z_20
,
b_4*z_21
,
b_4*z_22
,
b_4*z_23
,
b_4*z_24
,
b_4*z_25
,
b_4*z_26
,
b_4*z_27
,
b_4*z_28
,
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*b_8
,
b_5*b_9
,
b_5*b_10
,
b_5*b_11
,
b_5*b_12
,
b_5*b_13
,
b_5*b_14
,
b_5*b_15
,
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
,
b_5*z_8 + z_8
,
b_5*z_9 + z_9
,
b_5*z_10
,
b_5*z_11
,
b_5*z_12
,
b_5*z_13
,
b_5*z_14
,
b_5*z_15
,
b_5*z_16
,
b_5*z_17
,
b_5*z_18
,
b_5*z_19
,
b_5*z_20
,
b_5*z_21
,
b_5*z_22
,
b_5*z_23
,
b_5*z_24
,
b_5*z_25
,
b_5*z_26
,
b_5*z_27
,
b_5*z_28
,
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*b_8
,
b_6*b_9
,
b_6*b_10
,
b_6*b_11
,
b_6*b_12
,
b_6*b_13
,
b_6*b_14
,
b_6*b_15
,
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
,
b_6*z_10 + z_10
,
b_6*z_11
,
b_6*z_12
,
b_6*z_13
,
b_6*z_14
,
b_6*z_15
,
b_6*z_16
,
b_6*z_17
,
b_6*z_18
,
b_6*z_19
,
b_6*z_20
,
b_6*z_21
,
b_6*z_22
,
b_6*z_23
,
b_6*z_24
,
b_6*z_25
,
b_6*z_26
,
b_6*z_27
,
b_6*z_28
,
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*b_8
,
b_7*b_9
,
b_7*b_10
,
b_7*b_11
,
b_7*b_12
,
b_7*b_13
,
b_7*b_14
,
b_7*b_15
,
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
,
b_7*z_13
,
b_7*z_14
,
b_7*z_15
,
b_7*z_16
,
b_7*z_17
,
b_7*z_18
,
b_7*z_19
,
b_7*z_20
,
b_7*z_21
,
b_7*z_22
,
b_7*z_23
,
b_7*z_24
,
b_7*z_25
,
b_7*z_26
,
b_7*z_27
,
b_7*z_28
,
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 + b_8
,
b_8*b_9
,
b_8*b_10
,
b_8*b_11
,
b_8*b_12
,
b_8*b_13
,
b_8*b_14
,
b_8*b_15
,
b_8*z_1
,
b_8*z_2
,
b_8*z_3
,
b_8*z_4
,
b_8*z_5
,
b_8*z_6
,
b_8*z_7
,
b_8*z_8
,
b_8*z_9
,
b_8*z_10
,
b_8*z_11 + z_11
,
b_8*z_12
,
b_8*z_13
,
b_8*z_14
,
b_8*z_15
,
b_8*z_16
,
b_8*z_17
,
b_8*z_18
,
b_8*z_19
,
b_8*z_20
,
b_8*z_21
,
b_8*z_22
,
b_8*z_23
,
b_8*z_24
,
b_8*z_25
,
b_8*z_26
,
b_8*z_27
,
b_8*z_28
,
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 + b_9
,
b_9*b_10
,
b_9*b_11
,
b_9*b_12
,
b_9*b_13
,
b_9*b_14
,
b_9*b_15
,
b_9*z_1
,
b_9*z_2
,
b_9*z_3
,
b_9*z_4
,
b_9*z_5
,
b_9*z_6
,
b_9*z_7
,
b_9*z_8
,
b_9*z_9
,
b_9*z_10
,
b_9*z_11
,
b_9*z_12 + z_12
,
b_9*z_13 + z_13
,
b_9*z_14 + z_14
,
b_9*z_15 + z_15
,
b_9*z_16 + z_16
,
b_9*z_17
,
b_9*z_18
,
b_9*z_19
,
b_9*z_20
,
b_9*z_21
,
b_9*z_22
,
b_9*z_23
,
b_9*z_24
,
b_9*z_25
,
b_9*z_26
,
b_9*z_27
,
b_9*z_28
,
b_10*b_2
,
b_10*b_3
,
b_10*b_4
,
b_10*b_5
,
b_10*b_6
,
b_10*b_7
,
b_10*b_8
,
b_10*b_9
,
b_10^2 + b_10
,
b_10*b_11
,
b_10*b_12
,
b_10*b_13
,
b_10*b_14
,
b_10*b_15
,
b_10*z_1
,
b_10*z_2
,
b_10*z_3
,
b_10*z_4
,
b_10*z_5
,
b_10*z_6
,
b_10*z_7
,
b_10*z_8
,
b_10*z_9
,
b_10*z_10
,
b_10*z_11
,
b_10*z_12
,
b_10*z_13
,
b_10*z_14
,
b_10*z_15
,
b_10*z_16
,
b_10*z_17 + z_17
,
b_10*z_18
,
b_10*z_19
,
b_10*z_20
,
b_10*z_21
,
b_10*z_22
,
b_10*z_23
,
b_10*z_24
,
b_10*z_25
,
b_10*z_26
,
b_10*z_27
,
b_10*z_28
,
b_11*b_2
,
b_11*b_3
,
b_11*b_4
,
b_11*b_5
,
b_11*b_6
,
b_11*b_7
,
b_11*b_8
,
b_11*b_9
,
b_11*b_10
,
b_11^2 + b_11
,
b_11*b_12
,
b_11*b_13
,
b_11*b_14
,
b_11*b_15
,
b_11*z_1
,
b_11*z_2
,
b_11*z_3
,
b_11*z_4
,
b_11*z_5
,
b_11*z_6
,
b_11*z_7
,
b_11*z_8
,
b_11*z_9
,
b_11*z_10
,
b_11*z_11
,
b_11*z_12
,
b_11*z_13
,
b_11*z_14
,
b_11*z_15
,
b_11*z_16
,
b_11*z_17
,
b_11*z_18 + z_18
,
b_11*z_19 + z_19
,
b_11*z_20
,
b_11*z_21
,
b_11*z_22
,
b_11*z_23
,
b_11*z_24
,
b_11*z_25
,
b_11*z_26
,
b_11*z_27
,
b_11*z_28
,
b_12*b_2
,
b_12*b_3
,
b_12*b_4
,
b_12*b_5
,
b_12*b_6
,
b_12*b_7
,
b_12*b_8
,
b_12*b_9
,
b_12*b_10
,
b_12*b_11
,
b_12^2 + b_12
,
b_12*b_13
,
b_12*b_14
,
b_12*b_15
,
b_12*z_1
,
b_12*z_2
,
b_12*z_3
,
b_12*z_4
,
b_12*z_5
,
b_12*z_6
,
b_12*z_7
,
b_12*z_8
,
b_12*z_9
,
b_12*z_10
,
b_12*z_11
,
b_12*z_12
,
b_12*z_13
,
b_12*z_14
,
b_12*z_15
,
b_12*z_16
,
b_12*z_17
,
b_12*z_18
,
b_12*z_19
,
b_12*z_20 + z_20
,
b_12*z_21 + z_21
,
b_12*z_22
,
b_12*z_23
,
b_12*z_24
,
b_12*z_25
,
b_12*z_26
,
b_12*z_27
,
b_12*z_28
,
b_13*b_2
,
b_13*b_3
,
b_13*b_4
,
b_13*b_5
,
b_13*b_6
,
b_13*b_7
,
b_13*b_8
,
b_13*b_9
,
b_13*b_10
,
b_13*b_11
,
b_13*b_12
,
b_13^2 + b_13
,
b_13*b_14
,
b_13*b_15
,
b_13*z_1
,
b_13*z_2
,
b_13*z_3
,
b_13*z_4
,
b_13*z_5
,
b_13*z_6
,
b_13*z_7
,
b_13*z_8
,
b_13*z_9
,
b_13*z_10
,
b_13*z_11
,
b_13*z_12
,
b_13*z_13
,
b_13*z_14
,
b_13*z_15
,
b_13*z_16
,
b_13*z_17
,
b_13*z_18
,
b_13*z_19
,
b_13*z_20
,
b_13*z_21
,
b_13*z_22 + z_22
,
b_13*z_23 + z_23
,
b_13*z_24 + z_24
,
b_13*z_25
,
b_13*z_26
,
b_13*z_27
,
b_13*z_28
,
b_14*b_2
,
b_14*b_3
,
b_14*b_4
,
b_14*b_5
,
b_14*b_6
,
b_14*b_7
,
b_14*b_8
,
b_14*b_9
,
b_14*b_10
,
b_14*b_11
,
b_14*b_12
,
b_14*b_13
,
b_14^2 + b_14
,
b_14*b_15
,
b_14*z_1
,
b_14*z_2
,
b_14*z_3
,
b_14*z_4
,
b_14*z_5
,
b_14*z_6
,
b_14*z_7
,
b_14*z_8
,
b_14*z_9
,
b_14*z_10
,
b_14*z_11
,
b_14*z_12
,
b_14*z_13
,
b_14*z_14
,
b_14*z_15
,
b_14*z_16
,
b_14*z_17
,
b_14*z_18
,
b_14*z_19
,
b_14*z_20
,
b_14*z_21
,
b_14*z_22
,
b_14*z_23
,
b_14*z_24
,
b_14*z_25 + z_25
,
b_14*z_26 + z_26
,
b_14*z_27
,
b_14*z_28
,
b_15*b_2
,
b_15*b_3
,
b_15*b_4
,
b_15*b_5
,
b_15*b_6
,
b_15*b_7
,
b_15*b_8
,
b_15*b_9
,
b_15*b_10
,
b_15*b_11
,
b_15*b_12
,
b_15*b_13
,
b_15*b_14
,
b_15^2 + b_15
,
b_15*z_1
,
b_15*z_2
,
b_15*z_3
,
b_15*z_4
,
b_15*z_5
,
b_15*z_6
,
b_15*z_7
,
b_15*z_8
,
b_15*z_9
,
b_15*z_10
,
b_15*z_11
,
b_15*z_12
,
b_15*z_13
,
b_15*z_14
,
b_15*z_15
,
b_15*z_16
,
b_15*z_17
,
b_15*z_18
,
b_15*z_19
,
b_15*z_20
,
b_15*z_21
,
b_15*z_22
,
b_15*z_23
,
b_15*z_24
,
b_15*z_25
,
b_15*z_26
,
b_15*z_27 + z_27
,
b_15*z_28 + z_28
,
z_1*b_2
,
z_1*b_3
,
z_1*b_4
,
z_1*b_5
,
z_1*b_6
,
z_1*b_7
,
z_1*b_8
,
z_1*b_9
,
z_1*b_10 + z_1
,
z_1*b_11
,
z_1*b_12
,
z_1*b_13
,
z_1*b_14
,
z_1*b_15
,
z_1^2
,
z_1*z_2
,
z_1*z_3
,
z_1*z_4
,
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_1*z_15
,
z_1*z_16
,
z_1*z_17 + z_3*z_20
,
z_1*z_18
,
z_1*z_19
,
z_1*z_20
,
z_1*z_21
,
z_1*z_22
,
z_1*z_23
,
z_1*z_24
,
z_1*z_25
,
z_1*z_26
,
z_1*z_27
,
z_1*z_28
,
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
,
z_2*b_10
,
z_2*b_11 + z_2
,
z_2*b_12
,
z_2*b_13
,
z_2*b_14
,
z_2*b_15
,
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_10
,
z_2*z_11
,
z_2*z_12
,
z_2*z_13
,
z_2*z_14
,
z_2*z_15
,
z_2*z_16
,
z_2*z_17
,
z_2*z_18
,
z_2*z_19 + z_3*z_21
,
z_2*z_20
,
z_2*z_21
,
z_2*z_22
,
z_2*z_23
,
z_2*z_24
,
z_2*z_25
,
z_2*z_26
,
z_2*z_27
,
z_2*z_28
,
z_3*b_2
,
z_3*b_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*b_10
,
z_3*b_11
,
z_3*b_12 + z_3
,
z_3*b_13
,
z_3*b_14
,
z_3*b_15
,
z_3*z_1
,
z_3*z_2
,
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_3*z_15
,
z_3*z_16
,
z_3*z_17
,
z_3*z_18
,
z_3*z_19
,
z_3*z_22
,
z_3*z_23
,
z_3*z_24
,
z_3*z_25
,
z_3*z_26
,
z_3*z_27
,
z_3*z_28
,
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
,
z_4*b_9 + z_4
,
z_4*b_10
,
z_4*b_11
,
z_4*b_12
,
z_4*b_13
,
z_4*b_14
,
z_4*b_15
,
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_4*z_15 + z_5*z_23
,
z_4*z_16 + z_5*z_24
,
z_4*z_17
,
z_4*z_18
,
z_4*z_19
,
z_4*z_20
,
z_4*z_21
,
z_4*z_22
,
z_4*z_23
,
z_4*z_24
,
z_4*z_25
,
z_4*z_26
,
z_4*z_27
,
z_4*z_28
,
z_5*b_2
,
z_5*b_3
,
z_5*b_4
,
z_5*b_5
,
z_5*b_6
,
z_5*b_7
,
z_5*b_8
,
z_5*b_9
,
z_5*b_10
,
z_5*b_11
,
z_5*b_12
,
z_5*b_13 + z_5
,
z_5*b_14
,
z_5*b_15
,
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_5*z_9
,
z_5*z_10
,
z_5*z_11
,
z_5*z_12
,
z_5*z_13
,
z_5*z_14
,
z_5*z_15
,
z_5*z_16
,
z_5*z_17
,
z_5*z_18
,
z_5*z_19
,
z_5*z_20
,
z_5*z_21
,
z_5*z_25
,
z_5*z_26
,
z_5*z_27
,
z_5*z_28
,
z_6*b_2
,
z_6*b_3
,
z_6*b_4
,
z_6*b_5
,
z_6*b_6 + z_6
,
z_6*b_7
,
z_6*b_8
,
z_6*b_9
,
z_6*b_10
,
z_6*b_11
,
z_6*b_12
,
z_6*b_13
,
z_6*b_14
,
z_6*b_15
,
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_6*z_14
,
z_6*z_15
,
z_6*z_16
,
z_6*z_17
,
z_6*z_18
,
z_6*z_19
,
z_6*z_20
,
z_6*z_21
,
z_6*z_22
,
z_6*z_23
,
z_6*z_24
,
z_6*z_25
,
z_6*z_26
,
z_6*z_27
,
z_6*z_28
,
z_7*b_2
,
z_7*b_3
,
z_7*b_4
,
z_7*b_5
,
z_7*b_6
,
z_7*b_7
,
z_7*b_8
,
z_7*b_9 + z_7
,
z_7*b_10
,
z_7*b_11
,
z_7*b_12
,
z_7*b_13
,
z_7*b_14
,
z_7*b_15
,
z_7*z_1
,
z_7*z_2
,
z_7*z_3
,
z_7*z_4
,
z_7*z_5
,
z_7*z_6
,
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_7*z_15
,
z_7*z_17
,
z_7*z_18
,
z_7*z_19
,
z_7*z_20
,
z_7*z_21
,
z_7*z_22
,
z_7*z_23
,
z_7*z_24
,
z_7*z_25
,
z_7*z_26
,
z_7*z_27
,
z_7*z_28
,
z_8*b_2
,
z_8*b_3
,
z_8*b_4
,
z_8*b_5
,
z_8*b_6
,
z_8*b_7
,
z_8*b_8
,
z_8*b_9
,
z_8*b_10
,
z_8*b_11 + z_8
,
z_8*b_12
,
z_8*b_13
,
z_8*b_14
,
z_8*b_15
,
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_9
,
z_8*z_10
,
z_8*z_11
,
z_8*z_12
,
z_8*z_13
,
z_8*z_14
,
z_8*z_15
,
z_8*z_16
,
z_8*z_17
,
z_8*z_18 + 2*z_9*z_20
,
z_8*z_19 + z_9*z_21
,
z_8*z_20
,
z_8*z_21
,
z_8*z_22
,
z_8*z_23
,
z_8*z_24
,
z_8*z_25
,
z_8*z_26
,
z_8*z_27
,
z_8*z_28
,
z_9*b_2
,
z_9*b_3
,
z_9*b_4
,
z_9*b_5
,
z_9*b_6
,
z_9*b_7
,
z_9*b_8
,
z_9*b_9
,
z_9*b_10
,
z_9*b_11
,
z_9*b_12 + z_9
,
z_9*b_13
,
z_9*b_14
,
z_9*b_15
,
z_9*z_1
,
z_9*z_2
,
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_9*z_15
,
z_9*z_16
,
z_9*z_17
,
z_9*z_18
,
z_9*z_19
,
z_9*z_22
,
z_9*z_23
,
z_9*z_24
,
z_9*z_25
,
z_9*z_26
,
z_9*z_27
,
z_9*z_28
,
z_10*b_2
,
z_10*b_3 + z_10
,
z_10*b_4
,
z_10*b_5
,
z_10*b_6
,
z_10*b_7
,
z_10*b_8
,
z_10*b_9
,
z_10*b_10
,
z_10*b_11
,
z_10*b_12
,
z_10*b_13
,
z_10*b_14
,
z_10*b_15
,
z_10*z_1
,
z_10*z_2
,
z_10*z_3
,
z_10*z_4
,
z_10*z_5
,
z_10*z_7
,
z_10*z_8
,
z_10*z_9
,
z_10^2
,
z_10*z_11
,
z_10*z_12
,
z_10*z_13
,
z_10*z_14
,
z_10*z_15
,
z_10*z_16
,
z_10*z_17
,
z_10*z_18
,
z_10*z_19
,
z_10*z_20
,
z_10*z_21
,
z_10*z_22
,
z_10*z_23
,
z_10*z_24
,
z_10*z_25
,
z_10*z_26
,
z_10*z_27
,
z_10*z_28
,
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*b_8
,
z_11*b_9 + z_11
,
z_11*b_10
,
z_11*b_11
,
z_11*b_12
,
z_11*b_13
,
z_11*b_14
,
z_11*b_15
,
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_11*z_13
,
z_11*z_14
,
z_11*z_15
,
z_11*z_16
,
z_11*z_17
,
z_11*z_18
,
z_11*z_19
,
z_11*z_20
,
z_11*z_21
,
z_11*z_22
,
z_11*z_23
,
z_11*z_24
,
z_11*z_25
,
z_11*z_26
,
z_11*z_27
,
z_11*z_28
,
z_12*b_2 + z_12
,
z_12*b_3
,
z_12*b_4
,
z_12*b_5
,
z_12*b_6
,
z_12*b_7
,
z_12*b_8
,
z_12*b_9
,
z_12*b_10
,
z_12*b_11
,
z_12*b_12
,
z_12*b_13
,
z_12*b_14
,
z_12*b_15
,
z_12*z_1
,
z_12*z_2
,
z_12*z_3
,
z_12*z_4 + z_16*z_27
,
z_12*z_5 + z_16*z_28
,
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_12*z_15
,
z_12*z_16
,
z_12*z_17
,
z_12*z_18
,
z_12*z_19
,
z_12*z_20
,
z_12*z_21
,
z_12*z_22
,
z_12*z_23
,
z_12*z_24
,
z_12*z_25
,
z_12*z_26
,
z_12*z_27
,
z_12*z_28
,
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*b_8
,
z_13*b_9
,
z_13*b_10
,
z_13*b_11
,
z_13*b_12
,
z_13*b_13
,
z_13*b_14
,
z_13*b_15
,
z_13*z_1
,
z_13*z_2
,
z_13*z_3
,
z_13*z_4
,
z_13*z_5
,
z_13*z_6
,
z_13*z_7 + z_15*z_25 + 2*z_16*z_27
,
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_13*z_15
,
z_13*z_16
,
z_13*z_17
,
z_13*z_18
,
z_13*z_19
,
z_13*z_20
,
z_13*z_21
,
z_13*z_22
,
z_13*z_23
,
z_13*z_24
,
z_13*z_25
,
z_13*z_26
,
z_13*z_27
,
z_13*z_28
,
z_14*b_2
,
z_14*b_3
,
z_14*b_4
,
z_14*b_5
,
z_14*b_6
,
z_14*b_7
,
z_14*b_8 + z_14
,
z_14*b_9
,
z_14*b_10
,
z_14*b_11
,
z_14*b_12
,
z_14*b_13
,
z_14*b_14
,
z_14*b_15
,
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_9
,
z_14*z_10
,
z_14*z_11 + 2*z_15*z_25 + 2*z_16*z_27
,
z_14*z_12
,
z_14*z_13
,
z_14^2
,
z_14*z_15
,
z_14*z_16
,
z_14*z_17
,
z_14*z_18
,
z_14*z_19
,
z_14*z_20
,
z_14*z_21
,
z_14*z_22
,
z_14*z_23
,
z_14*z_24
,
z_14*z_25
,
z_14*z_26
,
z_14*z_27
,
z_14*z_28
,
z_15*b_2
,
z_15*b_3
,
z_15*b_4
,
z_15*b_5
,
z_15*b_6
,
z_15*b_7
,
z_15*b_8
,
z_15*b_9
,
z_15*b_10
,
z_15*b_11
,
z_15*b_12
,
z_15*b_13
,
z_15*b_14 + z_15
,
z_15*b_15
,
z_15*z_1
,
z_15*z_2
,
z_15*z_3
,
z_15*z_4
,
z_15*z_5
,
z_15*z_6
,
z_15*z_7
,
z_15*z_8
,
z_15*z_9
,
z_15*z_10
,
z_15*z_11
,
z_15*z_12
,
z_15*z_13
,
z_15*z_14
,
z_15^2
,
z_15*z_16
,
z_15*z_17
,
z_15*z_18
,
z_15*z_19
,
z_15*z_20
,
z_15*z_21
,
z_15*z_22
,
z_15*z_23
,
z_15*z_24
,
z_15*z_26 + z_16*z_28
,
z_15*z_27
,
z_15*z_28
,
z_16*b_2
,
z_16*b_3
,
z_16*b_4
,
z_16*b_5
,
z_16*b_6
,
z_16*b_7
,
z_16*b_8
,
z_16*b_9
,
z_16*b_10
,
z_16*b_11
,
z_16*b_12
,
z_16*b_13
,
z_16*b_14
,
z_16*b_15 + z_16
,
z_16*z_1
,
z_16*z_2
,
z_16*z_3
,
z_16*z_4
,
z_16*z_5
,
z_16*z_6
,
z_16*z_7
,
z_16*z_8
,
z_16*z_9
,
z_16*z_10
,
z_16*z_11
,
z_16*z_12
,
z_16*z_13
,
z_16*z_14
,
z_16*z_15
,
z_16^2
,
z_16*z_17
,
z_16*z_18
,
z_16*z_19
,
z_16*z_20
,
z_16*z_21
,
z_16*z_22
,
z_16*z_23
,
z_16*z_24
,
z_16*z_25
,
z_16*z_26
,
z_17*b_2
,
z_17*b_3
,
z_17*b_4
,
z_17*b_5
,
z_17*b_6
,
z_17*b_7
,
z_17*b_8
,
z_17*b_9
,
z_17*b_10
,
z_17*b_11
,
z_17*b_12
,
z_17*b_13
,
z_17*b_14
,
z_17*b_15
,
z_17*z_1
,
z_17*z_3
,
z_17*z_4
,
z_17*z_5
,
z_17*z_6
,
z_17*z_7
,
z_17*z_8
,
z_17*z_9
,
z_17*z_10
,
z_17*z_11
,
z_17*z_12
,
z_17*z_13
,
z_17*z_14
,
z_17*z_15
,
z_17*z_16
,
z_17^2
,
z_17*z_18
,
z_17*z_19
,
z_17*z_20
,
z_17*z_21
,
z_17*z_22
,
z_17*z_23
,
z_17*z_24
,
z_17*z_25
,
z_17*z_26
,
z_17*z_27
,
z_17*z_28
,
z_18*b_2
,
z_18*b_3
,
z_18*b_4
,
z_18*b_5
,
z_18*b_6
,
z_18*b_7
,
z_18*b_8
,
z_18*b_9
,
z_18*b_10
,
z_18*b_11
,
z_18*b_12
,
z_18*b_13
,
z_18*b_14
,
z_18*b_15
,
z_18*z_3 + z_19*z_9
,
z_18*z_4
,
z_18*z_5
,
z_18*z_6
,
z_18*z_7
,
z_18*z_8
,
z_18*z_9
,
z_18*z_10
,
z_18*z_11
,
z_18*z_12
,
z_18*z_13
,
z_18*z_14
,
z_18*z_15
,
z_18*z_16
,
z_18*z_17
,
z_18^2
,
z_18*z_19
,
z_18*z_20
,
z_18*z_21
,
z_18*z_22
,
z_18*z_23
,
z_18*z_24
,
z_18*z_25
,
z_18*z_26
,
z_18*z_27
,
z_18*z_28
,
z_19*b_2
,
z_19*b_3
,
z_19*b_4
,
z_19*b_5 + z_19
,
z_19*b_6
,
z_19*b_7
,
z_19*b_8
,
z_19*b_9
,
z_19*b_10
,
z_19*b_11
,
z_19*b_12
,
z_19*b_13
,
z_19*b_14
,
z_19*b_15
,
z_19*z_1
,
z_19*z_2
,
z_19*z_3
,
z_19*z_4
,
z_19*z_5
,
z_19*z_6
,
z_19*z_7
,
z_19*z_10
,
z_19*z_11
,
z_19*z_12
,
z_19*z_13
,
z_19*z_14
,
z_19*z_15
,
z_19*z_16
,
z_19*z_17
,
z_19*z_18
,
z_19^2
,
z_19*z_20
,
z_19*z_21
,
z_19*z_22
,
z_19*z_23
,
z_19*z_24
,
z_19*z_25
,
z_19*z_26
,
z_19*z_27
,
z_19*z_28
,
z_20*b_2
,
z_20*b_3
,
z_20*b_4
,
z_20*b_5
,
z_20*b_6
,
z_20*b_7
,
z_20*b_8
,
z_20*b_9
,
z_20*b_10
,
z_20*b_11
,
z_20*b_12
,
z_20*b_13
,
z_20*b_14
,
z_20*b_15
,
z_20*z_1
,
z_20*z_2 + 2*z_21*z_8
,
z_20*z_3 + 2*z_21*z_9
,
z_20*z_4
,
z_20*z_5
,
z_20*z_6
,
z_20*z_7
,
z_20*z_8
,
z_20*z_9
,
z_20*z_10
,
z_20*z_11
,
z_20*z_12
,
z_20*z_13
,
z_20*z_14
,
z_20*z_15
,
z_20*z_16
,
z_20*z_17
,
z_20*z_18
,
z_20*z_19
,
z_20^2
,
z_20*z_21
,
z_20*z_22
,
z_20*z_23
,
z_20*z_24
,
z_20*z_25
,
z_20*z_26
,
z_20*z_27
,
z_20*z_28
,
z_21*b_2
,
z_21*b_3
,
z_21*b_4
,
z_21*b_5 + z_21
,
z_21*b_6
,
z_21*b_7
,
z_21*b_8
,
z_21*b_9
,
z_21*b_10
,
z_21*b_11
,
z_21*b_12
,
z_21*b_13
,
z_21*b_14
,
z_21*b_15
,
z_21*z_1
,
z_21*z_2
,
z_21*z_3
,
z_21*z_4
,
z_21*z_5
,
z_21*z_6
,
z_21*z_7
,
z_21*z_10
,
z_21*z_11
,
z_21*z_12
,
z_21*z_13
,
z_21*z_14
,
z_21*z_15
,
z_21*z_16
,
z_21*z_17
,
z_21*z_18
,
z_21*z_19
,
z_21*z_20
,
z_21^2
,
z_21*z_22
,
z_21*z_23
,
z_21*z_24
,
z_21*z_25
,
z_21*z_26
,
z_21*z_27
,
z_21*z_28
,
z_22*b_2 + z_22
,
z_22*b_3
,
z_22*b_4
,
z_22*b_5
,
z_22*b_6
,
z_22*b_7
,
z_22*b_8
,
z_22*b_9
,
z_22*b_10
,
z_22*b_11
,
z_22*b_12
,
z_22*b_13
,
z_22*b_14
,
z_22*b_15
,
z_22*z_1
,
z_22*z_2
,
z_22*z_3
,
z_22*z_4 + 2*z_24*z_27
,
z_22*z_5 + 2*z_23*z_26 + z_24*z_28
,
z_22*z_6
,
z_22*z_7
,
z_22*z_8
,
z_22*z_9
,
z_22*z_10
,
z_22*z_11
,
z_22*z_12
,
z_22*z_13
,
z_22*z_14
,
z_22*z_15
,
z_22*z_16
,
z_22*z_17
,
z_22*z_18
,
z_22*z_19
,
z_22*z_20
,
z_22*z_21
,
z_22^2
,
z_22*z_23
,
z_22*z_24
,
z_22*z_25
,
z_22*z_26
,
z_22*z_27
,
z_22*z_28
,
z_23*b_2
,
z_23*b_3
,
z_23*b_4
,
z_23*b_5
,
z_23*b_6
,
z_23*b_7
,
z_23*b_8
,
z_23*b_9
,
z_23*b_10
,
z_23*b_11
,
z_23*b_12
,
z_23*b_13
,
z_23*b_14 + z_23
,
z_23*b_15
,
z_23*z_1
,
z_23*z_2
,
z_23*z_3
,
z_23*z_4
,
z_23*z_5
,
z_23*z_6
,
z_23*z_7
,
z_23*z_8
,
z_23*z_9
,
z_23*z_10
,
z_23*z_11
,
z_23*z_12
,
z_23*z_13
,
z_23*z_14
,
z_23*z_15
,
z_23*z_16
,
z_23*z_17
,
z_23*z_18
,
z_23*z_19
,
z_23*z_20
,
z_23*z_21
,
z_23*z_22
,
z_23^2
,
z_23*z_24
,
z_23*z_25 + 2*z_24*z_27
,
z_23*z_27
,
z_23*z_28
,
z_24*b_2
,
z_24*b_3
,
z_24*b_4
,
z_24*b_5
,
z_24*b_6
,
z_24*b_7
,
z_24*b_8
,
z_24*b_9
,
z_24*b_10
,
z_24*b_11
,
z_24*b_12
,
z_24*b_13
,
z_24*b_14
,
z_24*b_15 + z_24
,
z_24*z_1
,
z_24*z_2
,
z_24*z_3
,
z_24*z_4
,
z_24*z_5
,
z_24*z_6
,
z_24*z_7
,
z_24*z_8
,
z_24*z_9
,
z_24*z_10
,
z_24*z_11
,
z_24*z_12
,
z_24*z_13
,
z_24*z_14
,
z_24*z_15
,
z_24*z_16
,
z_24*z_17
,
z_24*z_18
,
z_24*z_19
,
z_24*z_20
,
z_24*z_21
,
z_24*z_22
,
z_24*z_23
,
z_24^2
,
z_24*z_25
,
z_24*z_26
,
z_25*b_2
,
z_25*b_3
,
z_25*b_4
,
z_25*b_5
,
z_25*b_6
,
z_25*b_7
,
z_25*b_8
,
z_25*b_9 + z_25
,
z_25*b_10
,
z_25*b_11
,
z_25*b_12
,
z_25*b_13
,
z_25*b_14
,
z_25*b_15
,
z_25*z_1
,
z_25*z_2
,
z_25*z_3
,
z_25*z_4
,
z_25*z_5
,
z_25*z_6
,
z_25*z_7
,
z_25*z_8
,
z_25*z_9
,
z_25*z_10
,
z_25*z_11
,
z_25*z_12 + z_26*z_22
,
z_25*z_13
,
z_25*z_14
,
z_25*z_16 + z_26*z_24
,
z_25*z_17
,
z_25*z_18
,
z_25*z_19
,
z_25*z_20
,
z_25*z_21
,
z_25*z_22
,
z_25*z_23
,
z_25*z_24
,
z_25^2
,
z_25*z_26
,
z_25*z_27
,
z_25*z_28
,
z_26*b_2
,
z_26*b_3
,
z_26*b_4
,
z_26*b_5
,
z_26*b_6
,
z_26*b_7
,
z_26*b_8
,
z_26*b_9
,
z_26*b_10
,
z_26*b_11
,
z_26*b_12
,
z_26*b_13 + z_26
,
z_26*b_14
,
z_26*b_15
,
z_26*z_1
,
z_26*z_2
,
z_26*z_3
,
z_26*z_4
,
z_26*z_5
,
z_26*z_6
,
z_26*z_7
,
z_26*z_8
,
z_26*z_9
,
z_26*z_10
,
z_26*z_11
,
z_26*z_12
,
z_26*z_13
,
z_26*z_14
,
z_26*z_15
,
z_26*z_16
,
z_26*z_17
,
z_26*z_18
,
z_26*z_19
,
z_26*z_20
,
z_26*z_21
,
z_26*z_23
,
z_26*z_25
,
z_26^2
,
z_26*z_27
,
z_26*z_28
,
z_27*b_2
,
z_27*b_3
,
z_27*b_4
,
z_27*b_5
,
z_27*b_6
,
z_27*b_7
,
z_27*b_8
,
z_27*b_9 + z_27
,
z_27*b_10
,
z_27*b_11
,
z_27*b_12
,
z_27*b_13
,
z_27*b_14
,
z_27*b_15
,
z_27*z_1
,
z_27*z_2
,
z_27*z_3
,
z_27*z_4
,
z_27*z_5
,
z_27*z_6
,
z_27*z_7
,
z_27*z_8
,
z_27*z_9
,
z_27*z_10
,
z_27*z_11
,
z_27*z_12 + 2*z_28*z_22
,
z_27*z_14
,
z_27*z_15 + 2*z_28*z_23
,
z_27*z_17
,
z_27*z_18
,
z_27*z_19
,
z_27*z_20
,
z_27*z_21
,
z_27*z_22
,
z_27*z_23
,
z_27*z_24
,
z_27*z_25
,
z_27*z_26
,
z_27^2
,
z_27*z_28
,
z_28*b_2
,
z_28*b_3
,
z_28*b_4
,
z_28*b_5
,
z_28*b_6
,
z_28*b_7
,
z_28*b_8
,
z_28*b_9
,
z_28*b_10
,
z_28*b_11
,
z_28*b_12
,
z_28*b_13 + z_28
,
z_28*b_14
,
z_28*b_15
,
z_28*z_1
,
z_28*z_2
,
z_28*z_3
,
z_28*z_4
,
z_28*z_5
,
z_28*z_6
,
z_28*z_7
,
z_28*z_8
,
z_28*z_9
,
z_28*z_10
,
z_28*z_11
,
z_28*z_12
,
z_28*z_13
,
z_28*z_14
,
z_28*z_15
,
z_28*z_16
,
z_28*z_17
,
z_28*z_18
,
z_28*z_19
,
z_28*z_20
,
z_28*z_21
,
z_28*z_25
,
z_28*z_26
,
z_28*z_27
,
z_28^2
,
b_1 + b_2 + b_3 + b_4 + b_5 + b_6 + b_7 + b_8 + b_9 + b_10 + b_11 + b_12 + b_13
+ b_14 + b_15 + 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_5*z_24*z_28*z_22 + z_4*z_12 + 2*z_5*z_22
,
The projective resolutions of the simple modules.
Simple Module Number 1
Degree 0:
1
Degree 1:
10
11
12
Degree 2:
1
1
5
Degree 3:
10
12
The projective resolution
of simple module no. 1 is graded.
Simple Module Number 2
Degree 0:
2
Degree 1:
9
13
Degree 2:
2
4
14
15
Degree 3:
9
9
13
Degree 4:
2
4
8
14
Degree 5:
9
Degree 6:
4
The projective resolution
of simple module no. 2 is graded.
Simple Module Number 3
Degree 0:
3
Degree 1:
6
Degree 2:
3
The projective resolution
of simple module no. 3 is graded.
Simple Module Number 4
Degree 0:
4
Degree 1:
9
Degree 2:
2
4
8
14
Degree 3:
9
9
13
Degree 4:
2
4
4
8
14
15
Degree 5:
9
9
13
Degree 6:
2
4
8
14
Degree 7:
9
Degree 8:
4
The projective resolution
of simple module no. 4 is graded.
Simple Module Number 5
Degree 0:
5
Degree 1:
11
12
Degree 2:
1
5
Degree 3:
12
The projective resolution
of simple module no. 5 is graded.
Simple Module Number 6
The projective resolution
of simple module no. 6 is graded.
Simple Module Number 7 is Projective.
Simple Module Number 8
Degree 0:
8
Degree 1:
9
Degree 2:
4
8
14
15
Degree 3:
9
9
13
Degree 4:
2
4
8
8
14
Degree 5:
9
Degree 6:
4
The projective resolution
of simple module no. 8 is graded.
Simple Module Number 9
Degree 0:
9
Degree 1:
2
4
8
14
15
Degree 2:
9
9
9
13
13
Degree 3:
2
2
4
4
8
8
14
14
15
Degree 4:
9
9
9
13
Degree 5:
2
4
4
8
14
Degree 6:
9
Degree 7:
4
The projective resolution
of simple module no. 9 is graded.
Simple Module Number 10
Degree 0:
10
Degree 1:
1
Degree 2:
10
12
Degree 3:
1
Degree 4:
10
The projective resolution
of simple module no. 10 is graded.
Simple Module Number 11
Degree 0:
11
Degree 1:
1
5
Degree 2:
12
The projective resolution
of simple module no. 11 is graded.
Simple Module Number 12
Degree 0:
12
Degree 1:
1
5
Degree 2:
10
11
12
Degree 3:
1
5
Degree 4:
12
The projective resolution
of simple module no. 12 is graded.
Simple Module Number 13
Degree 0:
13
Degree 1:
2
14
15
Degree 2:
9
9
13
Degree 3:
2
4
8
14
Degree 4:
9
Degree 5:
4
The projective resolution
of simple module no. 13 is graded.
Simple Module Number 14
Degree 0:
14
Degree 1:
9
13
Degree 2:
2
4
8
14
15
Degree 3:
9
9
13
Degree 4:
2
4
8
14
Degree 5:
9
Degree 6:
4
The projective resolution
of simple module no. 14 is graded.
Simple Module Number 15
Degree 0:
15
Degree 1:
9
13
Degree 2:
2
8
14
Degree 3:
9
Degree 4:
4
The projective resolution
of simple module no. 15 is graded.