q-6j symbols for rank two quantum groups

E. Ardonne, J.K. Slingerland,
Clebsch-Gordan and 6j-coefficients for rank two quantum groups,
arXiv:1004.5456

Display of q-6j symbols

Manipulate[If[oldtype4≠type4,a4={0,0};b4={0,0};c4={0,0};d4={0,0};groupinit4[type4];calcfsym;,,groupinit4[type4];calcfsym;];If[Not[MemberQ[dvalues4[a4,b4,c4],d4]],d4=dvalues4[a4,b4,c4][[1]]];Column[DeleteCases[{Row[{Style[Subsuperscript[F,d4,{a4,b4,c4}]],Text[" = "],Flatten[Table[Simplify[Chop[fsym[a4,b4,c4,d4,e,f,{v1,v2,v3,v4},q]/.{q->If[kinf4,1,Exp[2. Pi I/(4(k4+dcox4))]]}]],{e,evalues4[a4,b4,c4,d4]},{f,fvalues4[a4,b4,c4,d4]},{v1,vvalues4[a4,b4,e]},{v2,vvalues4[e,c4,d4]},{v3,vvalues4[b4,c4,f]},{v4,vvalues4[a4,f,d4]}],{{1,3,4},{2,5,6}}]//MatrixForm}],Row[{Text["e takes the value(s): "],evalues4[a4,b4,c4,d4]}],Row[{Text["f takes the value(s): "],fvalues4[a4,b4,c4,d4]}],If[type4==1,If[MemberQ[Flatten[Table[{vvalues4[a4,b4,e],vvalues4[e,c4,d4],vvalues4[b4,c4,f],vvalues4[a4,f,d4]},{e,evalues4[a4,b4,c4,d4]},{f,fvalues4[a4,b4,c4,d4]}]],x_/;x>1],Sequence@@{Row[{Piecewise[{{Text["v1 and v2 have "],a4==b4==c4==d4=={1,1}},{Text["v1 has "],a4==b4=={1,1}},{Text["v2 has "],c4==d4=={1,1}}}],Text["vertex multiplicities when e={1,1}"]}],Row[{Piecewise[{{Text["v3 and v4 have "],a4==b4==c4==d4=={1,1}},{Text["v3 has "],b4==c4=={1,1}},{Text["v4 has "],a4==d4=={1,1}}}],Text["vertex multiplicities when f={1,1}"]}]},Row[{Text["There are no vertex multiplicities"]}]]],Row[If[checkpentagon,{Text["The pentagon equations are satisfied"]},{Text["The pentagon equations are not satisfied"]}]]},Null]],{{type4,1,""},{2->Style["su(3)_2"],1->Style["su(3)_3/Z_3"],3->Style["so(5)_1"],4->Style["G2_1"]}},{{a4,{1,1},"a"},irrepss[type4]},{{b4,{1,1},"b"},irrepss[type4]},{{c4,{1,1},"c"},irrepss[type4]},{{d4,{1,1},"d"},dvalues4[a4,b4,c4]},{{kinf4,False,"q"},{False->Exp[2 Pi I/kplusdcox4[type4]],True->1}},ControlType->SetterBar,ControlPlacement->Top,SaveDefinitions->True]

Initialization

Initialization for su(3)_2

su32init4:=With[{},
k4=2;
rank4=2;
dcox4=3;
Clear[n4,nv4,fus4,weights4,irreptonum4];
irreps4={{0,0},{1,0},{0,1},{2,0},{1,1},{0,2}};
numirreps4=Length[irreps4];
weights4[{0,0}]={{0,0,1}};
weights4[{1,0}]={{1,0,1},{-1,1,1},{0,-1,1}};
weights4[{0,1}]={{0,1,1},{1,-1,1},{-1,0,1}};
weights4[{1,1}]={{1,1,1},{-1,2,1},{2,-1,1},{0,0,1},{0,0,2},{-2,1,1},{1,-2,1},{-1,-1,1}};
weights4[{2,0}]={{2,0,1},{0,1,1},{-2,2,1},{1,-1,1},{-1,0,1},{0,-2,1}};
weights4[{0,2}]={{0,2,1},{1,0,1},{2,-2,1},{-1,1,1},{0,-1,1},{-2,0,1}};
n4[1]=IdentityMatrix[6];
n4[2]={{0,1,0,0,0,0},{0,0,1,1,0,0},{1,0,0,0,1,0},{0,0,0,0,1,0},{0,1,0,0,0,1},{0,0,1,0,0,0}};
n4[3]={{0,0,1,0,0,0},{1,0,0,0,1,0},{0,1,0,0,0,1},{0,1,0,0,0,0},{0,0,1,1,0,0},{0,0,0,0,1,0}};
n4[4]={{0,0,0,1,0,0},{0,0,0,0,1,0},{0,1,0,0,0,0},{0,0,0,0,0,1},{0,0,1,0,0,0},{1,0,0,0,0,0}};
n4[5]={{0,0,0,0,1,0},{0,1,0,0,0,1},{0,0,1,1,0,0},{0,0,1,0,0,0},{1,0,0,0,1,0},{0,1,0,0,0,0}};
n4[6]={{0,0,0,0,0,1},{0,0,1,0,0,0},{0,0,0,0,1,0},{1,0,0,0,0,0},{0,1,0,0,0,0},{0,0,0,1,0,0}};
Do[irreptonum4[irreps4[[i]]]=i;,{i,1,numirreps4}];
Do[fus4[irreps4[[i]],irreps4[[j]]]=DeleteCases[Table[If[n4[i][[j,k]]>0,irreps4[[k]]],{k,1,numirreps4}],Null],{i,1,numirreps4},{j,1,numirreps4}];
Do[nv4[i,j,k]=Range[n4[irreptonum4[i]][[irreptonum4[j],irreptonum4[k]]]],{i,irreps4},{j,irreps4},{k,fus4[i,j]}];
pentlist=Flatten[Table[{a,b,c,d,e,f,g,fp,gp,{v1,v2,v3,v4,v5,v6}},{a,irreps4},{b,irreps4},{c,irreps4},{d,irreps4},{f,fus4[a,b]},{g,fus4[f,c]},{e,fus4[g,d]},{gp,Cases[fus4[c,d],x_/;MemberQ[fus4[f,x],e]]},{fp,Cases[fus4[b,gp],x_/;MemberQ[fus4[a,x],e]]},{v1,nv4[a,b,f]},{v2,nv4[f,c,g]},{v3,nv4[g,d,e]},{v4,nv4[c,d,gp]},{v5,nv4[b,gp,fp]},{v6,nv4[a,fp,e]}],14];
qCGlist4=Flatten[Table[{j1,m1,j2,m2,j3,m3,{v}},{j1,irreps4},{j2,irreps4},{j3,fus4[j1,j2]},{v,nv4[j1,j2,j3]},{m3,weights4[j3]},{m1,weights4[j1]},{m2,Cases[weights4[j2],x_/;m1[[1;;2]]+x[[1;;2]]==m3[[1;;2]]]}],6];
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(Sqrt[qn4[4,q]]))),((Sqrt[1+qn4[2,q]])/((Sqrt[2]) q (Sqrt[qn4[2,q]]) (Sqrt[qn4[4,q]]))),((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) q (Sqrt[qn4[2,q]]) (Sqrt[qn4[4,q]]))),-(1/((q^3) (Sqrt[qn4[4,q]]))),((q^2)/(Sqrt[qn4[3,q]])),-(1/(Sqrt[qn4[3,q]])),(1/((q^2) (Sqrt[qn4[3,q]]))),((q^2)/(Sqrt[qn4[3,q]])),-(1/(q (Sqrt[qn4[2,q]]) (Sqrt[qn4[3,q]]))),-(q/((Sqrt[qn4[2,q]]) (Sqrt[qn4[3,q]]))),(1/((q^2) (Sqrt[qn4[3,q]]))),((q^2)/(Sqrt[qn4[3,q]])),-(1/(Sqrt[qn4[3,q]])),(1/((q^2) (Sqrt[qn4[3,q]]))),((q^3)/((Sqrt[qn4[2,q]]) (Sqrt[qn4[3,q]]))),-(1/(Sqrt[qn4[3,q]])),-(1/(Sqrt[qn4[3,q]])),(1/((q^3) (Sqrt[qn4[2,q]]) (Sqrt[qn4[3,q]]))),((q^2)/(Sqrt[qn4[3,q]])),-(1/(q (Sqrt[qn4[2,q]]) (Sqrt[qn4[3,q]]))),-(q/((Sqrt[qn4[2,q]]) (Sqrt[qn4[3,q]]))),(1/((q^2) (Sqrt[qn4[3,q]]))),((q^2)/(Sqrt[qn4[3,q]])),-(1/(Sqrt[qn4[3,q]])),(1/((q^2) (Sqrt[qn4[3,q]])))};
flist=Sort[Flatten[Table[{a,b,c,d,e,f,{v1,v2,v3,v4}},{a,irreps4},{b,irreps4},{c,irreps4},{e,fus4[a,b]},{d,fus4[e,c]},{f,Cases[fus4[b,c],x_/;MemberQ[fus4[a,x],d]]},{v1,nv4[a,b,e]},{v2,nv4[e,c,d]},{v3,nv4[b,c,f]},{v4,nv4[a,f,d]}],9]];
Clear[qCG4];
Do[qCG4[Sequence@@qCGlist4[[i]],q]=qCGvalues4[[i]],{i,1,Length[qCGlist4]}];
]

Initialization for su(3)_3 / Z_3

su33init4:=With[{},
k4=3;
rank4=2;
dcox4=3;
Clear[n4,nv4,fus4,weights4,irreptonum4];
irreps4={{0,0},{1,1},{3,0},{0,3}};
numirreps4=Length[irreps4];
weights4[{0,0}]={{0,0,1}};
weights4[{1,1}]={{1,1,1},{-1,2,1},{2,-1,1},{0,0,1},{0,0,2},{-2,1,1},{1,-2,1},{-1,-1,1}};
weights4[{3,0}]={{3,0,1},{1,1,1},{-1,2,1},{2,-1,1},{-3,3,1},{0,0,1},{-2,1,1},{1,-2,1},{-1,-1,1},{0,-3,1}};
weights4[{0,3}]={{0,3,1},{1,1,1},{-1,2,1},{2,-1,1},{0,0,1},{3,-3,1},{-2,1,1},{1,-2,1},{-1,-1,1},{-3,0,1}};
n4[1]=IdentityMatrix[4];
n4[2]={{0,1,0,0},{1,2,1,1},{0,1,0,0},{0,1,0,0}};
n4[3]={{0,0,1,0},{0,1,0,0},{0,0,0,1},{1,0,0,0}};
n4[4]={{0,0,0,1},{0,1,0,0},{1,0,0,0},{0,0,1,0}};
Do[
irreptonum4[irreps4[[i]]]=i;
,{i,1,numirreps4}];
Do[fus4[irreps4[[i]],irreps4[[j]]]=DeleteCases[Table[If[n4[i][[j,k]]>0,irreps4[[k]]],{k,1,numirreps4}],Null],{i,1,numirreps4},{j,1,numirreps4}];
Do[nv4[i,j,k]=Range[n4[irreptonum4[i]][[irreptonum4[j],irreptonum4[k]]]],{i,irreps4},{j,irreps4},{k,fus4[i,j]}];
pentlist=Flatten[Table[{a,b,c,d,e,f,g,fp,gp,{v1,v2,v3,v4,v5,v6}},{a,irreps4},{b,irreps4},{c,irreps4},{d,irreps4},{f,fus4[a,b]},{g,fus4[f,c]},{e,fus4[g,d]},{gp,Cases[fus4[c,d],x_/;MemberQ[fus4[f,x],e]]},{fp,Cases[fus4[b,gp],x_/;MemberQ[fus4[a,x],e]]},{v1,nv4[a,b,f]},{v2,nv4[f,c,g]},{v3,nv4[g,d,e]},{v4,nv4[c,d,gp]},{v5,nv4[b,gp,fp]},{v6,nv4[a,fp,e]}],14];
qCGlist4=Flatten[Table[{j1,m1,j2,m2,j3,m3,{v}},{j1,irreps4},{j2,irreps4},{j3,fus4[j1,j2]},{v,nv4[j1,j2,j3]},{m3,weights4[j3]},{m1,weights4[j1]},{m2,Cases[weights4[j2],x_/;m1[[1;;2]]+x[[1;;2]]==m3[[1;;2]]]}],6];
flist=Sort[Flatten[Table[{a,b,c,d,e,f,{v1,v2,v3,v4}},{a,irreps4},{b,irreps4},{c,irreps4},{e,fus4[a,b]},{d,fus4[e,c]},{f,Cases[fus4[b,c],x_/;MemberQ[fus4[a,x],d]]},{v1,nv4[a,b,e]},{v2,nv4[e,c,d]},{v3,nv4[b,c,f]},{v4,nv4[a,f,d]}],9]];
qCGvalues4={1,1,1,1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,1,((q^4)/(Sqrt[qn4[2,q] qn4[4,q]])),-((q^2)/(Sqrt[qn4[2,q] qn4[4,q]])),-((q^2)/(Sqrt[qn4[2,q] qn4[4,q]])),(1/(Sqrt[qn4[2,q] qn4[4,q]])),0,0,(1/(Sqrt[qn4[2,q] qn4[4,q]])),-(1/((q^2) (Sqrt[qn4[2,q] qn4[4,q]]))),-(1/((q^2) (Sqrt[qn4[2,q] qn4[4,q]]))),(1/((q^4) (Sqrt[qn4[2,q] qn4[4,q]]))),((q^3)/(Sqrt[1+qn4[4,q]])),0,-((Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))])/(Sqrt[2])),-((Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))])/(Sqrt[2])),(1/((q^3) (Sqrt[1+qn4[4,q]]))),0,(((q^2) (Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))]))/(Sqrt[2])),((-(1/q)-q+(q^3))/(2 (Sqrt[1+qn4[4,q]]))),-(1/2) q (Sqrt[((qn4[3,q])/(1+qn4[4,q]))]),(((1/(q^3))-(1/q)-q)/(2 (Sqrt[1+qn4[4,q]]))),-((Sqrt[((qn4[3,q])/(1+qn4[4,q]))])/(2 q)),((Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))])/((Sqrt[2]) (q^2))),(((q^2) (Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))]))/(Sqrt[2])),((-(1/q)-q+(q^3))/(2 (Sqrt[1+qn4[4,q]]))),(1/2) q (Sqrt[((qn4[3,q])/(1+qn4[4,q]))]),(((1/(q^3))-(1/q)-q)/(2 (Sqrt[1+qn4[4,q]]))),((Sqrt[((qn4[3,q])/(1+qn4[4,q]))])/(2 q)),((Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))])/((Sqrt[2]) (q^2))),(q/(Sqrt[1+qn4[4,q]])),((-(1/q)+q+(q^3))/(2 (Sqrt[1+qn4[4,q]]))),((-(1/q)+q+(q^3))/(2 (Sqrt[1+qn4[4,q]]))),(((1/(q^3))+(q^3)-2 ((1/q)+q))/(2 (Sqrt[1+qn4[4,q]]))),0,0,(((1/(q^3))+(q^3))/(2 (Sqrt[1+qn4[4,q]]))),(((1/(q^3))+(1/q)-q)/(2 (Sqrt[1+qn4[4,q]]))),(((1/(q^3))+(1/q)-q)/(2 (Sqrt[1+qn4[4,q]]))),(1/(q (Sqrt[1+qn4[4,q]]))),0,(1/2) q (Sqrt[((qn4[3,q])/(1+qn4[4,q]))]),-(1/2) q (Sqrt[((qn4[3,q])/(1+qn4[4,q]))]),0,(((1/(q^3))+(q^3))/(2 (Sqrt[1+qn4[4,q]]))),(((1/(q^3))+(q^3))/(2 (Sqrt[1+qn4[4,q]]))),0,-((Sqrt[((qn4[3,q])/(1+qn4[4,q]))])/(2 q)),((Sqrt[((qn4[3,q])/(1+qn4[4,q]))])/(2 q)),0,(((q^2) (Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))]))/(Sqrt[2])),((-(1/q)-q+(q^3))/(2 (Sqrt[1+qn4[4,q]]))),(1/2) q (Sqrt[((qn4[3,q])/(1+qn4[4,q]))]),(((1/(q^3))-(1/q)-q)/(2 (Sqrt[1+qn4[4,q]]))),((Sqrt[((qn4[3,q])/(1+qn4[4,q]))])/(2 q)),((Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))])/((Sqrt[2]) (q^2))),(((q^2) (Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))]))/(Sqrt[2])),((-(1/q)-q+(q^3))/(2 (Sqrt[1+qn4[4,q]]))),-(1/2) q (Sqrt[((qn4[3,q])/(1+qn4[4,q]))]),(((1/(q^3))-(1/q)-q)/(2 (Sqrt[1+qn4[4,q]]))),-((Sqrt[((qn4[3,q])/(1+qn4[4,q]))])/(2 q)),((Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))])/((Sqrt[2]) (q^2))),((q^3)/(Sqrt[1+qn4[4,q]])),0,-((Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))])/(Sqrt[2])),-((Sqrt[((1+qn4[2,q])/(1+qn4[4,q]))])/(Sqrt[2])),(1/((q^3) (Sqrt[1+qn4[4,q]]))),0,0,((q^3)/(Sqrt[-1+qn4[4,q]])),-((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) (Sqrt[-1+qn4[4,q]]))),((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) (Sqrt[-1+qn4[4,q]]))),0,-(1/((q^3) (Sqrt[-1+qn4[4,q]]))),(((q^2) (Sqrt[-1+qn4[2,q]]))/((Sqrt[2]) (Sqrt[-1+qn4[4,q]]))),-((q (Sqrt[qn4[3,q]]))/(2 (Sqrt[-1+qn4[4,q]]))),((-(1/q)+q+(q^3))/(2 (Sqrt[-1+qn4[4,q]]))),((Sqrt[qn4[3,q]])/(2 q (Sqrt[-1+qn4[4,q]]))),-(((1/(q^3))+(1/q)-q)/(2 (Sqrt[-1+qn4[4,q]]))),-((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) (q^2) (Sqrt[-1+qn4[4,q]]))),-(((q^2) (Sqrt[-1+qn4[2,q]]))/((Sqrt[2]) (Sqrt[-1+qn4[4,q]]))),((q (Sqrt[qn4[3,q]]))/(2 (Sqrt[-1+qn4[4,q]]))),((-(1/q)+q+(q^3))/(2 (Sqrt[-1+qn4[4,q]]))),-((Sqrt[qn4[3,q]])/(2 q (Sqrt[-1+qn4[4,q]]))),-(((1/(q^3))+(1/q)-q)/(2 (Sqrt[-1+qn4[4,q]]))),((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) (q^2) (Sqrt[-1+qn4[4,q]]))),0,-((q (Sqrt[qn4[3,q]]))/(2 (Sqrt[-1+qn4[4,q]]))),((q (Sqrt[qn4[3,q]]))/(2 (Sqrt[-1+qn4[4,q]]))),0,-(((1/(q^3))-(q^3))/(2 (Sqrt[-1+qn4[4,q]]))),((-(1/(q^3))+(q^3))/(2 (Sqrt[-1+qn4[4,q]]))),0,-((Sqrt[qn4[3,q]])/(2 q (Sqrt[-1+qn4[4,q]]))),((Sqrt[qn4[3,q]])/(2 q (Sqrt[-1+qn4[4,q]]))),0,-(q/(Sqrt[-1+qn4[4,q]])),-((-(1/q)-q+(q^3))/(2 (Sqrt[-1+qn4[4,q]]))),-((-(1/q)-q+(q^3))/(2 (Sqrt[-1+qn4[4,q]]))),((-(1/(q^3))+(q^3))/(2 (Sqrt[-1+qn4[4,q]]))),0,0,((-(1/q)+q+(1/2) (-(1/(q^3))+(q^3)))/(Sqrt[-1+qn4[4,q]])),(((1/(q^3))-(1/q)-q)/(2 (Sqrt[-1+qn4[4,q]]))),(((1/(q^3))-(1/q)-q)/(2 (Sqrt[-1+qn4[4,q]]))),(1/(q (Sqrt[-1+qn4[4,q]]))),-(((q^2) (Sqrt[-1+qn4[2,q]]))/((Sqrt[2]) (Sqrt[-1+qn4[4,q]]))),((q (Sqrt[qn4[3,q]]))/(2 (Sqrt[-1+qn4[4,q]]))),((-(1/q)+q+(q^3))/(2 (Sqrt[-1+qn4[4,q]]))),-((Sqrt[qn4[3,q]])/(2 q (Sqrt[-1+qn4[4,q]]))),-(((1/(q^3))+(1/q)-q)/(2 (Sqrt[-1+qn4[4,q]]))),((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) (q^2) (Sqrt[-1+qn4[4,q]]))),(((q^2) (Sqrt[-1+qn4[2,q]]))/((Sqrt[2]) (Sqrt[-1+qn4[4,q]]))),-((q (Sqrt[qn4[3,q]]))/(2 (Sqrt[-1+qn4[4,q]]))),((-(1/q)+q+(q^3))/(2 (Sqrt[-1+qn4[4,q]]))),((Sqrt[qn4[3,q]])/(2 q (Sqrt[-1+qn4[4,q]]))),-(((1/(q^3))+(1/q)-q)/(2 (Sqrt[-1+qn4[4,q]]))),-((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) (q^2) (Sqrt[-1+qn4[4,q]]))),0,((q^3)/(Sqrt[-1+qn4[4,q]])),-((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) (Sqrt[-1+qn4[4,q]]))),((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) (Sqrt[-1+qn4[4,q]]))),0,-(1/((q^3) (Sqrt[-1+qn4[4,q]]))),(q/(Sqrt[qn4[2,q]])),-(1/(q (Sqrt[qn4[2,q]]))),((Sqrt[1+qn4[2,q]])/((Sqrt[2]) (Sqrt[qn4[2,q] qn4[3,q]]))),((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) (Sqrt[qn4[2,q] qn4[3,q]]))),((q^3)/(Sqrt[qn4[2,q] qn4[3,q]])),-(1/((q^3) (Sqrt[qn4[2,q] qn4[3,q]]))),-((Sqrt[1+qn4[2,q]])/((Sqrt[2]) (Sqrt[qn4[2,q] qn4[3,q]]))),-((Sqrt[-1+qn4[2,q]])/((Sqrt[2]) 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(Sqrt[qn4[3,q] qn4[5,q]]))),(((1+(q^2)) (Sqrt[-1+qn4[2,q]]) (Sqrt[qn4[2,q]]))/((Sqrt[2]) (Sqrt[qn4[3,q] qn4[5,q]]))),((q (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[5,q]])),-((Sqrt[qn4[2,q]])/((q^3) (Sqrt[qn4[5,q]]))),(((q^5) (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[5,q]])),-((q^2)/(Sqrt[qn4[3,q] qn4[5,q]])),-((q (Sqrt[qn4[2,q]]))/(Sqrt[qn4[5,q]])),(((Sqrt[qn4[2,q]]) (Sqrt[1+qn4[2,q]]))/((Sqrt[2]) (q^2) (Sqrt[qn4[3,q] qn4[5,q]]))),(((Sqrt[-1+qn4[2,q]]) (Sqrt[qn4[2,q]]))/((Sqrt[2]) (q^2) (Sqrt[qn4[3,q] qn4[5,q]]))),-((Sqrt[qn4[2,q]])/((q^5) (Sqrt[qn4[3,q] qn4[5,q]]))),((q^6)/(Sqrt[qn4[3,q] qn4[5,q]])),-(((q^3) (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[5,q]])),-(((q^3) (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[5,q]])),(((Sqrt[qn4[2,q]]) (Sqrt[1+qn4[2,q]]))/((Sqrt[2]) (Sqrt[qn4[3,q] qn4[5,q]]))),(((Sqrt[-1+qn4[2,q]]) (Sqrt[qn4[2,q]]))/((Sqrt[2]) (Sqrt[qn4[3,q] qn4[5,q]]))),-((Sqrt[qn4[2,q]])/((q^3) (Sqrt[qn4[5,q]]))),((q^6)/(Sqrt[(-1+qn4[3,q]) qn4[5,q]])),-((q^4)/(Sqrt[(-1+qn4[3,q]) qn4[5,q]])),((q^2)/(Sqrt[(-1+qn4[3,q]) qn4[5,q]])),((q^2)/(Sqrt[(-1+qn4[3,q]) qn4[5,q]])),-(1/(Sqrt[(-1+qn4[3,q]) qn4[5,q]])),-(1/(Sqrt[(-1+qn4[3,q]) qn4[5,q]])),(1/((q^2) (Sqrt[(-1+qn4[3,q]) qn4[5,q]]))),(1/((q^2) (Sqrt[(-1+qn4[3,q]) qn4[5,q]]))),-(1/((q^4) (Sqrt[(-1+qn4[3,q]) qn4[5,q]]))),(1/((q^6) (Sqrt[(-1+qn4[3,q]) qn4[5,q]]))),((q^3)/(Sqrt[qn4[4,q]])),-(q/(Sqrt[qn4[4,q]])),(1/(q (Sqrt[qn4[4,q]]))),-(1/((q^3) (Sqrt[qn4[4,q]]))),((q^3)/(Sqrt[qn4[4,q]])),-((Sqrt[qn4[2,q]])/(Sqrt[qn4[3,q] qn4[4,q]])),-((q^3)/(Sqrt[qn4[3,q] qn4[4,q]])),(1/((q^3) (Sqrt[qn4[3,q] qn4[4,q]]))),((Sqrt[qn4[2,q]])/(Sqrt[qn4[3,q] qn4[4,q]])),-(1/((q^3) (Sqrt[qn4[4,q]]))),((q^3)/(Sqrt[qn4[4,q]])),-(1/(q (Sqrt[qn4[3,q] qn4[4,q]]))),-(((q^2) (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[4,q]])),((Sqrt[qn4[2,q]])/((q^2) (Sqrt[qn4[3,q] qn4[4,q]]))),(q/(Sqrt[qn4[3,q] qn4[4,q]])),-(1/((q^3) (Sqrt[qn4[4,q]]))),((q^5)/(Sqrt[qn4[3,q] qn4[4,q]])),-(q/(Sqrt[qn4[4,q]])),-(((q^2) (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[4,q]])),((Sqrt[qn4[2,q]])/((q^2) (Sqrt[qn4[3,q] qn4[4,q]]))),(1/(q (Sqrt[qn4[4,q]]))),-(1/((q^5) (Sqrt[qn4[3,q] qn4[4,q]]))),((q^3)/(Sqrt[qn4[4,q]])),-(q/(Sqrt[qn4[4,q]])),(1/(q (Sqrt[qn4[4,q]]))),-(1/((q^3) (Sqrt[qn4[4,q]]))),(((q^4) (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[4,q]])),-((Sqrt[qn4[2,q]])/(Sqrt[qn4[3,q] qn4[4,q]])),-((Sqrt[qn4[2,q]])/(Sqrt[qn4[3,q] qn4[4,q]])),((((1/(q^2))-(q^2)) (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[4,q]])),((Sqrt[qn4[2,q]])/(Sqrt[qn4[3,q] qn4[4,q]])),((Sqrt[qn4[2,q]])/(Sqrt[qn4[3,q] qn4[4,q]])),-((Sqrt[qn4[2,q]])/((q^4) (Sqrt[qn4[3,q] qn4[4,q]]))),((q^3)/(Sqrt[qn4[4,q]])),-(1/(q (Sqrt[qn4[3,q] qn4[4,q]]))),-(((q^2) (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[4,q]])),((Sqrt[qn4[2,q]])/((q^2) (Sqrt[qn4[3,q] qn4[4,q]]))),(q/(Sqrt[qn4[3,q] qn4[4,q]])),-(1/((q^3) (Sqrt[qn4[4,q]]))),((q^5)/(Sqrt[qn4[3,q] qn4[4,q]])),-(((q^2) (Sqrt[qn4[2,q]]))/(Sqrt[qn4[3,q] qn4[4,q]])),-(q/(Sqrt[qn4[4,q]])),(1/(q (Sqrt[qn4[4,q]]))),((Sqrt[qn4[2,q]])/((q^2) (Sqrt[qn4[3,q] qn4[4,q]]))),-(1/((q^5) (Sqrt[qn4[3,q] qn4[4,q]]))),((q^3)/(Sqrt[qn4[4,q]])),-((Sqrt[qn4[2,q]])/(Sqrt[qn4[3,q] qn4[4,q]])),(1/((q^3) (Sqrt[qn4[3,q] qn4[4,q]]))),-((q^3)/(Sqrt[qn4[3,q] qn4[4,q]])),((Sqrt[qn4[2,q]])/(Sqrt[qn4[3,q] qn4[4,q]])),-(1/((q^3) (Sqrt[qn4[4,q]]))),((q^3)/(Sqrt[qn4[4,q]])),-(q/(Sqrt[qn4[4,q]])),(1/(q (Sqrt[qn4[4,q]]))),-(1/((q^3) (Sqrt[qn4[4,q]])))};
Clear[qCG4];
Do[qCG4[Sequence@@qCGlist4[[i]],q]=qCGvalues4[[i]],{i,1,Length[qCGlist4]}];
]

Initialization for so(5)_1

so51init4:=With[{},
k4=1;
rank4=2;
dcox4=3;
Clear[n4,nv4,fus4,weights4,irreptonum4];
irreps4={{0,0},{0,1},{1,0}};
numirreps4=Length[irreps4];
weights4[{0,0}]={{0,0,1}};
weights4[{0,1}]={{0,1,1},{1,-1,1},{-1,1,1},{0,-1,1}};
weights4[{1,0}]={{1,0,1},{-1,2,1},{0,0,1},{1,-2,1},{-1,0,1}};
n4[1]=IdentityMatrix[3];
n4[2]={{0,1,0},{1,0,1},{0,1,0}};
n4[3]={{0,0,1},{0,1,0},{1,0,0}};
Do[
irreptonum4[irreps4[[i]]]=i;
,{i,1,numirreps4}];
Do[fus4[irreps4[[i]],irreps4[[j]]]=DeleteCases[Table[If[n4[i][[j,k]]>0,irreps4[[k]]],{k,1,numirreps4}],Null],{i,1,numirreps4},{j,1,numirreps4}];
Do[nv4[i,j,k]=Range[n4[irreptonum4[i]][[irreptonum4[j],irreptonum4[k]]]],{i,irreps4},{j,irreps4},{k,fus4[i,j]}];
pentlist=Flatten[Table[{a,b,c,d,e,f,g,fp,gp,{v1,v2,v3,v4,v5,v6}},{a,irreps4},{b,irreps4},{c,irreps4},{d,irreps4},{f,fus4[a,b]},{g,fus4[f,c]},{e,fus4[g,d]},{gp,Cases[fus4[c,d],x_/;MemberQ[fus4[f,x],e]]},{fp,Cases[fus4[b,gp],x_/;MemberQ[fus4[a,x],e]]},{v1,nv4[a,b,f]},{v2,nv4[f,c,g]},{v3,nv4[g,d,e]},{v4,nv4[c,d,gp]},{v5,nv4[b,gp,fp]},{v6,nv4[a,fp,e]}],14];
qCGlist4=Flatten[Table[{j1,m1,j2,m2,j3,m3,{v}},{j1,irreps4},{j2,irreps4},{j3,fus4[j1,j2]},{v,nv4[j1,j2,j3]},{m3,weights4[j3]},{m1,weights4[j1]},{m2,Cases[weights4[j2],x_/;m1[[1;;2]]+x[[1;;2]]==m3[[1;;2]]]}],6];
flist=Sort[Flatten[Table[{a,b,c,d,e,f,{v1,v2,v3,v4}},{a,irreps4},{b,irreps4},{c,irreps4},{e,fus4[a,b]},{d,fus4[e,c]},{f,Cases[fus4[b,c],x_/;MemberQ[fus4[a,x],d]]},{v1,nv4[a,b,e]},{v2,nv4[e,c,d]},{v3,nv4[b,c,f]},{v4,nv4[a,f,d]}],9]];
qCGvalues4={1,1,1,1,1,1,1,1,1,1,1,1,1,1,((q^2)/(Sqrt[-1+qn4[5,2,q]])),-(q/(Sqrt[-1+qn4[5,2,q]])),(1/(q (Sqrt[-1+qn4[5,2,q]]))),-(1/((q^2) (Sqrt[-1+qn4[5,2,q]]))),((Sqrt[q])/(Sqrt[qn4[2,2,q]])),-(1/((Sqrt[q]) (Sqrt[qn4[2,2,q]]))),((Sqrt[q])/(Sqrt[qn4[2,2,q]])),-(1/((Sqrt[q]) (Sqrt[qn4[2,2,q]]))),(1/(qn4[2,2,q])),(q/(qn4[2,2,q])),-(1/(q qn4[2,2,q])),-(1/(qn4[2,2,q])),((Sqrt[q])/(Sqrt[qn4[2,2,q]])),-(1/((Sqrt[q]) (Sqrt[qn4[2,2,q]]))),((Sqrt[q])/(Sqrt[qn4[2,2,q]])),-(1/((Sqrt[q]) (Sqrt[qn4[2,2,q]]))),((q^2)/(Sqrt[qn4[5,2,q]])),-(((Sqrt[q]) (Sqrt[qn4[2,2,q]]))/(Sqrt[qn4[5,2,q]])),((Sqrt[qn4[2,2,q]])/((q^(3/2)) (Sqrt[qn4[5,2,q]]))),(((q^(3/2)) (Sqrt[qn4[2,2,q]]))/(Sqrt[qn4[5,2,q]])),-(1/(Sqrt[qn4[5,2,q]])),((Sqrt[qn4[2,2,q]])/((q^(3/2)) (Sqrt[qn4[5,2,q]]))),(((q^(3/2)) (Sqrt[qn4[2,2,q]]))/(Sqrt[qn4[5,2,q]])),-(1/(Sqrt[qn4[5,2,q]])),((Sqrt[qn4[2,2,q]])/((q^(3/2)) (Sqrt[qn4[5,2,q]]))),(((q^(3/2)) (Sqrt[qn4[2,2,q]]))/(Sqrt[qn4[5,2,q]])),-((Sqrt[qn4[2,2,q]])/((Sqrt[q]) (Sqrt[qn4[5,2,q]]))),(1/((q^2) (Sqrt[qn4[5,2,q]]))),1,1,1,1,1,(((q^(3/2)) (Sqrt[qn4[2,2,q]]))/(Sqrt[qn4[5,2,q]])),-((Sqrt[qn4[2,2,q]])/((Sqrt[q]) (Sqrt[qn4[5,2,q]]))),(1/((q^2) (Sqrt[qn4[5,2,q]]))),(((q^(3/2)) (Sqrt[qn4[2,2,q]]))/(Sqrt[qn4[5,2,q]])),-(1/(Sqrt[qn4[5,2,q]])),((Sqrt[qn4[2,2,q]])/((q^(3/2)) (Sqrt[qn4[5,2,q]]))),(((q^(3/2)) (Sqrt[qn4[2,2,q]]))/(Sqrt[qn4[5,2,q]])),-(1/(Sqrt[qn4[5,2,q]])),((Sqrt[qn4[2,2,q]])/((q^(3/2)) (Sqrt[qn4[5,2,q]]))),((q^2)/(Sqrt[qn4[5,2,q]])),-(((Sqrt[q]) (Sqrt[qn4[2,2,q]]))/(Sqrt[qn4[5,2,q]])),((Sqrt[qn4[2,2,q]])/((q^(3/2)) (Sqrt[qn4[5,2,q]]))),((q^3)/(Sqrt[1+qn4[4,q]])),-(q/(Sqrt[1+qn4[4,q]])),(1/(Sqrt[1+qn4[4,q]])),-(1/(q (Sqrt[1+qn4[4,q]]))),(1/((q^3) (Sqrt[1+qn4[4,q]])))};
Clear[qCG4];
Do[qCG4[Sequence@@qCGlist4[[i]],q]=qCGvalues4[[i]],{i,1,Length[qCGlist4]}];
]

Initialization for G2_1

g21init4:=With[{},
k4=1;
rank4=2;
dcox4=4;
Clear[n4,nv4,fus4,weights4,irreptonum4];
irreps4={{0,0},{0,1}};
numirreps4=Length[irreps4];
weights4[{0,0}]={{0,0,1}};
weights4[{0,1}]={{0,1,1},{1,-1,1},{-1,2,1},{0,0,1},{1,-2,1},{-1,1,1},{0,-1,1}};
n4[1]=IdentityMatrix[2];
n4[2]={{0,1},{1,1}};
Do[
irreptonum4[irreps4[[i]]]=i;
,{i,1,numirreps4}];
Do[fus4[irreps4[[i]],irreps4[[j]]]=DeleteCases[Table[If[n4[i][[j,k]]>0,irreps4[[k]]],{k,1,numirreps4}],Null],{i,1,numirreps4},{j,1,numirreps4}];
Do[nv4[i,j,k]=Range[n4[irreptonum4[i]][[irreptonum4[j],irreptonum4[k]]]],{i,irreps4},{j,irreps4},{k,fus4[i,j]}];
pentlist=Flatten[Table[{a,b,c,d,e,f,g,fp,gp,{v1,v2,v3,v4,v5,v6}},{a,irreps4},{b,irreps4},{c,irreps4},{d,irreps4},{f,fus4[a,b]},{g,fus4[f,c]},{e,fus4[g,d]},{gp,Cases[fus4[c,d],x_/;MemberQ[fus4[f,x],e]]},{fp,Cases[fus4[b,gp],x_/;MemberQ[fus4[a,x],e]]},{v1,nv4[a,b,f]},{v2,nv4[f,c,g]},{v3,nv4[g,d,e]},{v4,nv4[c,d,gp]},{v5,nv4[b,gp,fp]},{v6,nv4[a,fp,e]}],14];
qCGlist4=Flatten[Table[{j1,m1,j2,m2,j3,m3,{v}},{j1,irreps4},{j2,irreps4},{j3,fus4[j1,j2]},{v,nv4[j1,j2,j3]},{m3,weights4[j3]},{m1,weights4[j1]},{m2,Cases[weights4[j2],x_/;m1[[1;;2]]+x[[1;;2]]==m3[[1;;2]]]}],6];
flist=Sort[Flatten[Table[{a,b,c,d,e,f,{v1,v2,v3,v4}},{a,irreps4},{b,irreps4},{c,irreps4},{e,fus4[a,b]},{d,fus4[e,c]},{f,Cases[fus4[b,c],x_/;MemberQ[fus4[a,x],d]]},{v1,nv4[a,b,e]},{v2,nv4[e,c,d]},{v3,nv4[b,c,f]},{v4,nv4[a,f,d]}],9]];
qCGvalues4={1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,((q^(10/3))/(Sqrt[qn4[3,3,q]-qn4[7,3,q]+qn4[11,3,q]])),-((q^(8/3))/(Sqrt[qn4[3,3,q]-qn4[7,3,q]+qn4[11,3,q]])),((q^(2/3))/(Sqrt[qn4[3,3,q]-qn4[7,3,q]+qn4[11,3,q]])),-(1/(Sqrt[qn4[3,3,q]-qn4[7,3,q]+qn4[11,3,q]])),(1/((q^(2/3)) (Sqrt[qn4[3,3,q]-qn4[7,3,q]+qn4[11,3,q]]))),-(1/((q^(8/3)) (Sqrt[qn4[3,3,q]-qn4[7,3,q]+qn4[11,3,q]]))),(1/((q^(10/3)) (Sqrt[qn4[3,3,q]-qn4[7,3,q]+qn4[11,3,q]]))),((q^2)/(Sqrt[-1+qn4[7,3,q]])),-((q (Sqrt[qn4[2,3,q]]))/(Sqrt[-1+qn4[7,3,q]])),((Sqrt[qn4[2,3,q]])/(q (Sqrt[-1+qn4[7,3,q]]))),-(1/((q^2) (Sqrt[-1+qn4[7,3,q]]))),(((q^(5/3)) (Sqrt[qn4[2,3,q]]))/(Sqrt[-1+qn4[7,3,q]])),-((q^(2/3))/(Sqrt[-1+qn4[7,3,q]])),(1/((q^(2/3)) (Sqrt[-1+qn4[7,3,q]]))),-((Sqrt[qn4[2,3,q]])/((q^(5/3)) (Sqrt[-1+qn4[7,3,q]]))),(((q^(5/3)) (Sqrt[qn4[2,3,q]]))/(Sqrt[-1+qn4[7,3,q]])),-((q^(2/3))/(Sqrt[-1+qn4[7,3,q]])),(1/((q^(2/3)) (Sqrt[-1+qn4[7,3,q]]))),-((Sqrt[qn4[2,3,q]])/((q^(5/3)) (Sqrt[-1+qn4[7,3,q]]))),((q^(4/3))/(Sqrt[-1+qn4[7,3,q]])),((q^2)/(Sqrt[-1+qn4[7,3,q]])),-(1/(Sqrt[-1+qn4[7,3,q]])),(((1/(q^(2/3)))-(q^(2/3)))/(Sqrt[-1+qn4[7,3,q]])),(1/(Sqrt[-1+qn4[7,3,q]])),-(1/((q^2) (Sqrt[-1+qn4[7,3,q]]))),-(1/((q^(4/3)) (Sqrt[-1+qn4[7,3,q]]))),(((q^(5/3)) (Sqrt[qn4[2,3,q]]))/(Sqrt[-1+qn4[7,3,q]])),-((q^(2/3))/(Sqrt[-1+qn4[7,3,q]])),(1/((q^(2/3)) (Sqrt[-1+qn4[7,3,q]]))),-((Sqrt[qn4[2,3,q]])/((q^(5/3)) (Sqrt[-1+qn4[7,3,q]]))),(((q^(5/3)) (Sqrt[qn4[2,3,q]]))/(Sqrt[-1+qn4[7,3,q]])),-((q^(2/3))/(Sqrt[-1+qn4[7,3,q]])),(1/((q^(2/3)) (Sqrt[-1+qn4[7,3,q]]))),-((Sqrt[qn4[2,3,q]])/((q^(5/3)) (Sqrt[-1+qn4[7,3,q]]))),((q^2)/(Sqrt[-1+qn4[7,3,q]])),-((q (Sqrt[qn4[2,3,q]]))/(Sqrt[-1+qn4[7,3,q]])),((Sqrt[qn4[2,3,q]])/(q (Sqrt[-1+qn4[7,3,q]]))),-(1/((q^2) (Sqrt[-1+qn4[7,3,q]])))};
Clear[qCG4];
Do[qCG4[Sequence@@qCGlist4[[i]],q]=qCGvalues4[[i]],{i,1,Length[qCGlist4]}];
]

General

qn4[n_,q_]:=Sum[q^(4((n+1)/2-i)),{i,1,n}];
qn4[n_,d_,q_]:=Sum[q^(4/d((n+1)/2-i)),{i,1,n}];

dvalues4[a_,b_,c_]:=Sort[Union[Flatten[Table[fus4[i,c],{i,fus4[a,b]}],1]],Position[irreps4,#1][[1,1]]<Position[irreps4,#2][[1,1]]&]
evalues4[a_,b_,c_,d_]:=Cases[fus4[a,b],x_/;MemberQ[fus4[x,c],d]]
fvalues4[a_,b_,c_,d_]:=Cases[fus4[b,c],x_/;MemberQ[fus4[a,x],d]]
vvalues4[a_,b_,e_]:=nv4[a,b,e]

groupinit4[type_]:=With[{},
If[Not[oldtype4==type],
Piecewise[{{su33init4,type==1},{su32init4,type==2},{so51init4,type==3},{g21init4,type==4}}];oldtype4=type;]
];

irrepss[type_]:=Piecewise[{{{{0,0},{1,1},{3,0},{0,3}},type==1},{{{0,0},{1,0},{0,1},{2,0},{1,1},{0,2}},type==2},{{{0,0},{0,1},{1,0}},type==3},{{{0,0},{0,1}},type==4}}]
kplusdcox4[type_]:=Piecewise[{{6,type==1},{5,type==2},{4,type==3},{5,type==4}}];

F-symbol initialization

calcfsym:=With[{},
Clear[fsym,fsymn];
Do[fsym[Sequence@@flist[[i]],q]=Sum[qCG4[flist[[i,1]],m1,flist[[i,2]],m2,flist[[i,5]],m12,{flist[[i,7,1]]},q]qCG4[flist[[i,5]],m12,flist[[i,3]],m3,flist[[i,4]],weights4[flist[[i,4]]][[1]],{flist[[i,7,2]]},q]qCG4[flist[[i,2]],m2,flist[[i,3]],m3,flist[[i,6]],m23,{flist[[i,7,3]]},q]qCG4[flist[[i,1]],m1,flist[[i,6]],m23,flist[[i,4]],weights4[flist[[i,4]]][[1]],{flist[[i,7,4]]},q],{m1,weights4[flist[[i,1]]]},{m2,weights4[flist[[i,2]]]},{m12,Cases[weights4[flist[[i,5]]],x_/;(m1+m2)[[1;;rank4]]==x[[1;;rank4]]]},{m3,Cases[weights4[flist[[i,3]]],x_/;(m12+x)[[1;;rank4]]==flist[[i,4]]]},{m23,Cases[weights4[flist[[i,6]]],x_/;(m2+m3)[[1;;rank4]]==x[[1;;rank4]]]}],{i,1,Length[flist]}];Do[fsymn[Sequence@@flist[[i]]]=Chop[fsym[Sequence@@flist[[i]],Exp[2. Pi I/(4(k4+dcox4))]]],{i,1,Length[flist]}];
]

checkpentagon:=Catch[With[{},
pentholds=True;
Do[If[Chop[Sum[(fsymn[i[[6]],i[[3]],i[[4]],i[[5]],i[[7]],i[[9]],{i[[10,2]],i[[10,3]],i[[10,4]],v1}]fsymn[i[[1]],i[[2]],i[[9]],i[[5]],i[[6]],i[[8]],{i[[10,1]],v1,i[[10,5]],i[[10,6]]}]),{v1,nv4[i[[6]],i[[9]],i[[5]]]}]-Sum[(fsymn[i[[1]],i[[2]],i[[3]],i[[7]],i[[6]],h,{i[[10,1]],i[[10,2]],v2,v3}]fsymn[i[[1]],h,i[[4]],i[[5]],i[[7]],i[[8]],{v3,i[[10,3]],v4,i[[10,6]]}]fsymn[i[[2]],i[[3]],i[[4]],i[[8]],h,i[[9]],{v2,v4,i[[10,4]],i[[10,5]]}]),{h,Quiet[Cases[fus4[i[[2]],i[[3]]],x_/;MemberQ[fus4[i[[1]],x],i[[7]]]&&MemberQ[fus4[x,i[[4]]],i[[8]]]]]},{v2,nv4[i[[2]],i[[3]],h]},{v3,nv4[i[[1]],h,i[[7]]]},{v4,nv4[h,i[[4]],i[[8]]]}],10^-10]==0.,,pentholds=False;Hold[Print["False ",i]];Hold[Abort[]];,Hold[Print["Undecided ",i]];Hold[Abort[]]],{i,pentlist}];
];Throw[pentholds]]

Start

oldtype4=0;
groupinit4[1];
type4=1;
calcfsym;