#This file contains all the necessary commands to compute the left coaction on the image via the section \gamma of each power of a root vector.

LoadPackage("GBNP");

GBNP.ConfigPrint("a_{2}","a_{12}","a_{beta}","a_{112}","a_{1112}","a_{1}","g_{1}","g_{2}","y_{2}","y_{12}","y_{beta}","y_{112}","y_{1112}","y_{1}");

s:=Indeterminate(Rationals,"s");;
t:=Indeterminate(Rationals,"t");;
q:=E(7);

q11:=q;;
q22:=q^3;;
q21:=q;;
q12:=q^3;;
F:=Field(s,t);
F1:= One(F);

#mute variables:
#[31]=[7,7,7,7,8]
#[32]=[6,6,6,7,8,8]
#[33]=[4,4,4]
#[34]=[6,7,8,8]
#[35]=[5,6,6,7,8,8]
#[36]=[6,7,7,7,7,7,8,8,8]
#[37]=[6,6,6,6,7,8,8]
#[38]=[6,6,7,8,8]
#[39]=[7,8,8]
#[40]=[5,6,7,8,8]
#[41]=[7,7,7,7,7,8,8,8]
#[42]=[2,2,2]
#[43]=[5,7,8,8]
#[44]=[4,7,8,8]
#[45]=[4,6,7,8,8]
#[46]=[2,6,7,8,8]
#[47]=[12,12,12]
#[48]=[10,10,10]


K:=[
[[[6,5],[5,6],[],[31]],[F1,-q11^3*q12,-t,t]],
[[[6,4],[4,6],[5]],[F1,-q12*q11^2,-F1]],
[[[6,3],[3,6],[4,4],[1],[32]],[F1,-q11^3*q12^2,-q12*q*(q^3-1)/(q+1),-q^4*(1-q)*(1+q^2+q^4)*t,-(2*q-2*q^2-q^4+q^6)*s]],
[[[6,2],[2,6],[4]],[F1,-q12*q11,-F1]],
[[[6,1],[1,6],[2]],[F1,-q12,-F1]],
[[[5,4],[4,5],[2],[37]],[F1,-q11^3*q12,-(1-q^4)*t,-(-1+3*q^2+q^3+q^4+3*q^5)*s]],
[[[5,3],[3,5],[33],[1,4],[2,2],[35],[4,32],[36],[34]],[F1,-q11^6*q12^3,-q12^2*q^3*(q^3-1)*(q-1)/(q+1),q^4*(q-1)^2*(q^2+1)*t,-(q^4-q^3)^2*t,-(-2*q^2+5*q^4+5*q^5+6*q^6)*s,-(-3*q-5*q^3-4*q^4-4*q^5-5*q^6)*s,-(3*q+2*q^2-2*q^3-q^4-q^5-q^6)*s*t,-(4*q-3*q^2-4*q^3-5*q^4-7*q^5+q^6)*s*t]],
[[[5,2],[2,5],[4,4],[1],[32]],[F1,-q11^3*q12^2,-q12*q*(q^3-1)/(q+1),-q^4*(1-q)*(1+q^2+q^4)*t,-(2*q-2*q^2-q^4+q^6)*s]],
[[[5,1],[1,5],[2,4],[3],[38]],[F1,-q12^3*q22,-q12^2*(q^5-q^2),-q12*(q^3-q^2-q),-q^4*(q^3-1)^2*s]],
[[[4,3],[3,4],[],[1,2],[40],[41],[39]],[F1,-q11^3*q12,-q^2*(q+1)*(3*q^4+q^3+2*q^2+1)*s*t,-q^5*(q+1)*(q-1)^2*t,-(-2*q-3*q^2-4*q^3-3*q^4-2*q^5)*s,-(2*q+2*q^2+q^3+3*q^4+4*q^5+2*q^6)*s*t,-(q+q^3-2*q^4-4*q^5-3*q^6)*s*t]],
[[[4,2],[2,4],[3]],[F1,-q11^2*q12,-F1]],
[[[4,1],[1,4],[2,2],[34]],[F1,-q12^2*q22,-q12*(q^3-q),-(q^3-1)*s]],
[[[3,2],[2,3],[1,1],[43],[45]],[F1,-q11^3*q12,-(q^3-1)^2*t,-(q^4-1)*s,-(q^4-q)^2*s]],
[[[3,1],[1,3],[42],[44],[46]],[F1,-q12^3*q22^2,-q12^2*q^3*(q^2-1)*(q-1),-q*(1-q^3)*s,-q^4*(q^4-1)^2*s]],
[[[2,1],[1,2],[],[39]],[F1,-q12*q22,-s,s]],
#####
[[[7,8],[8,7]],[F1,-F1]],[[[7,1],[1,7]],[F1,-q12]],[[[7,2],[2,7]],[F1,-q11*q12]],[[[7,3],[3,7]],[F1,-q11^3*q12^2]],[[[7,4],[4,7]],[F1,-q11^2*q12]],[[[7,5],[5,7]],[F1,-q11^3*q12]],[[[7,6],[6,7]],[F1,-q11]],[[[8,1],[1,8]],[F1,-q22]],[[[8,2],[2,8]],[F1,-q21*q22]],[[[8,3],[3,8]],[F1,-q21^3*q22^2]],[[[8,4],[4,8]],[F1,-q21^2*q22]],[[[8,5],[5,8]],[F1,-q21^3*q22]],[[[8,6],[6,8]],[F1,-q21]],
#####
[[[14,13],[13,14],[]],[F1,-q11^3*q12,-t]],
[[[14,12],[12,14],[13]],[F1,-q12*q11^2,-F1]],
[[[14,11],[11,14],[12,12],[9]],[F1,-q11^3*q12^2,-q12*q*(q^3-1)/(q+1),-q^4*(1-q)*(1+q^2+q^4)*t]],
[[[14,10],[10,14],[12]],[F1,-q12*q11,-F1]],
[[[14,9],[9,14],[10]],[F1,-q12,-F1]],
[[[13,12],[12,13],[10]],[F1,-q11^3*q12,-(1-q^4)*t]],
[[[13,11],[11,13],[47],[9,12],[10,10]],[F1,-q11^6*q12^3,-q12^2*q^3*(q^3-1)*(q-1)/(q+1),q^4*(q-1)^2*(q^2+1)*t,-(q^4-q^3)^2*t]],
[[[13,10],[10,13],[12,12],[9]],[F1,-q11^3*q12^2,-q12*q*(q^3-1)/(q+1),-q^4*(1-q)*(1+q^2+q^4)*t]],
[[[13,9],[9,13],[10,12],[11]],[F1,-q12^3*q22,-q12^2*(q^5-q^2),-q12*(q^3-q^2-q)]],
[[[12,11],[11,12],[],[9,10]],[F1,-q11^3*q12,-q^2*(q+1)*(3*q^4+q^3+2*q^2+1)*s*t,-q^5*(q+1)*(q-1)^2*t]],
[[[12,10],[10,12],[11]],[F1,-q11^2*q12,-F1]],
[[[12,9],[9,12],[10,10]],[F1,-q12^2*q22,-q12*(q^3-q)]],
[[[11,10],[10,11],[9,9]],[F1,-q11^3*q12,-(q^3-1)^2*t]],
[[[11,9],[9,11],[48]],[F1,-q12^3*q22^2,-q12^2*q^3*(q^2-1)*(q-1)]],
[[[10,9],[9,10],[]],[F1,-q12*q22,-s]],
####
[[[1,9],[9,1]],[F1,-F1]],
[[[1,10],[10,1]],[F1,-F1]],
[[[1,11],[11,1]],[F1,-F1]],
[[[1,12],[12,1]],[F1,-F1]],
[[[1,13],[13,1]],[F1,-F1]],
[[[1,14],[14,1]],[F1,-F1]],
[[[2,9],[9,2]],[F1,-F1]],
[[[2,10],[10,2]],[F1,-F1]],
[[[2,11],[11,2]],[F1,-F1]],
[[[2,12],[12,2]],[F1,-F1]],
[[[2,13],[13,2]],[F1,-F1]],
[[[2,14],[14,2]],[F1,-F1]],
[[[3,9],[9,3]],[F1,-F1]],
[[[3,10],[10,3]],[F1,-F1]],
[[[3,11],[11,3]],[F1,-F1]],
[[[3,12],[12,3]],[F1,-F1]],
[[[3,13],[13,3]],[F1,-F1]],
[[[3,14],[14,3]],[F1,-F1]],
[[[4,9],[9,4]],[F1,-F1]],
[[[4,10],[10,4]],[F1,-F1]],
[[[4,11],[11,4]],[F1,-F1]],
[[[4,12],[12,4]],[F1,-F1]],
[[[4,13],[13,4]],[F1,-F1]],
[[[4,14],[14,4]],[F1,-F1]],
[[[5,9],[9,5]],[F1,-F1]],
[[[5,10],[10,5]],[F1,-F1]],
[[[5,11],[11,5]],[F1,-F1]],
[[[5,12],[12,5]],[F1,-F1]],
[[[5,13],[13,5]],[F1,-F1]],
[[[5,14],[14,5]],[F1,-F1]],
[[[6,9],[9,6]],[F1,-F1]],
[[[6,10],[10,6]],[F1,-F1]],
[[[6,11],[11,6]],[F1,-F1]],
[[[6,12],[12,6]],[F1,-F1]],
[[[6,13],[13,6]],[F1,-F1]],
[[[6,14],[14,6]],[F1,-F1]],
[[[7,9],[9,7]],[F1,-F1]],
[[[7,10],[10,7]],[F1,-F1]],
[[[7,11],[11,7]],[F1,-F1]],
[[[7,12],[12,7]],[F1,-F1]],
[[[7,13],[13,7]],[F1,-F1]],
[[[7,14],[14,7]],[F1,-F1]],
[[[8,9],[9,8]],[F1,-F1]],
[[[8,10],[10,8]],[F1,-F1]],
[[[8,11],[11,8]],[F1,-F1]],
[[[8,12],[12,8]],[F1,-F1]],
[[[8,13],[13,8]],[F1,-F1]],
[[[8,14],[14,8]],[F1,-F1]],
];

G:=SGrobner(K);

#d12

d12:=[[[2],[7,8,10],[6,8,9]],[1,1,1-q^4]];

na2:=MulNP(d12,d12);

na3:=StrongNormalFormNP(MulNP(na2,d12),G);

na4:=StrongNormalFormNP(MulNP(na3,d12),G);

na7:=StrongNormalFormNP(MulNP(na4,na3),G);


#d112

d112:=[[[4],[7,7,8,12],[6,6,8,9],[6,7,8,10]],[F1,F1, (1-q^-3)*(1-q^-2), (1-q^-2)*(1+q)]];

na2:=MulNP(d112,d112);

na3:=StrongNormalFormNP(MulNP(na2,d112),G);

na4:=StrongNormalFormNP(MulNP(na3,d112),G);

na5:=StrongNormalFormNP(MulNP(na4,d112),G);

na6:=StrongNormalFormNP(MulNP(na5,d112),G);

na7:=StrongNormalFormNP(MulNP(na6,d112),G);


#remain of 112

d2:=[[[1],[8,9]],[1,1]];

d12:=[[[2],[7,8,10],[6,8,9]],[1,1,1-q^4]];

dbeta:=[[[3],[7,7,7,8,8,11],[4,6,8,9],[5,8,9],[6,6,6,8,8,9,9],[6,6,7,8,8,9,10],[4,7,8,10],[6,7,7,8,8,10,10]],[1,1,q^2*(1-q^4)^2,q*(1-q^4)*(q^3-q^2-q),q*(1-q^4)^2*(1-q^5)*(1-q^6),q^2*(1-q^4)^2*(1-q^5),q^2*(1-q^4),q^2*(1-q^4)*(1-q^5)]];

S1:=AddNP(MulNP(MulNP(MulNP(d12,d12),d12),dbeta),MulNP(MulNP(d2,dbeta),dbeta),(8*q+9*q^2+3*q^3+4*q^4+5*q^5-q^6)*t^2,(5*q+8*q^2+9*q^3+8*q^4+5*q^5)*t^2);

S2:=AddNP(MulNP(MulNP(MulNP(d2,d2),d12),d12),MulNP(d2,d12),-(7*q^3+14*q^4+7*q^5)*t^3,(2*q+4*q^2+13*q^3+22*q^4+17*q^5+5*q^6)*s*t^3);

S3:=AddNP(S1,S2,1,1);

S:=AddNP(S3,d,1,1);

Print(StrongNormalFormNP(S,G));


#d1112

d1112:=[[[5],[7,7,7,8,13],[6,6,6,8,9],[6,6,7,8,10],[6,7,7,8,12]],[F1,F1, (1-q^-3)*(1-q^-2)*(1-q^-1), q^2*(1-q^-3)*(1-q^-2), q^2*(1-q^-3)]]; 

na2:=MulNP(d1112,d1112);

na3:=StrongNormalFormNP(MulNP(na2,d1112),G);

na4:=StrongNormalFormNP(MulNP(na3,d1112),G);

na5:=StrongNormalFormNP(MulNP(na4,d1112),G);

na6:=StrongNormalFormNP(MulNP(na5,d1112),G);

na7:=StrongNormalFormNP(MulNP(na6,d1112),G);


#dbeta

dbeta:=[ [ [ 6, 7, 7, 8, 8, 10, 10 ], [ 6, 6, 7, 8, 8, 9, 10 ], [ 6, 6, 6, 8, 8, 9, 9 ], [ 7, 7, 7, 8, 8, 11 ], [ 6, 6, 7, 8, 8 ],
      [ 4, 7, 8, 10 ], [ 4, 6, 8, 9 ], [ 5, 8, 9 ], [ 3 ] ], [ (E(7)+2*E(7)^2+E(7)^3+2*E(7)^4+E(7)^5), (2*E(7)^2+2*E(7)^3+3*E(7)^4+E(7)^5-E(7)^6),
      (2*E(7)+E(7)^2+4*E(7)^3+4*E(7)^4+E(7)^5+2*E(7)^6), 1, (E(7)^2+2*E(7)^3+E(7)^4+2*E(7)^5+E(7)^6)*s, (E(7)^2-E(7)^6), (E(7)^2+E(7)^3-2*E(7)^6),
      (-2*E(7)-2*E(7)^2-2*E(7)^3-E(7)^5), 1 ] ];

na2:=MulNP(dbeta,dbeta);

na3:=StrongNormalFormNP(MulNP(na2,dbeta),G);

na4:=StrongNormalFormNP(MulNP(na3,dbeta),G);

na5:=StrongNormalFormNP(MulNP(na4,dbeta),G);

na6:=StrongNormalFormNP(MulNP(na5,dbeta),G);

na7:=StrongNormalFormNP(MulNP(na6,dbeta),G);