(*relations from Proposition 4*) NN[D4]=1 GG3=(NN[A1,A1,A1,A1]==2*3^4) (*relations of the form (4.3)*) (*NN[?,A1,A1]=2NN[?,A1^2]+3NN[?,A2]*) GG1=(NN[A2,A1,A1]==2NN[A2,A1^2]+3NN[A2,A2]) GG2=(NN[A1^2,A1,A1]==2NN[A1^2,A1^2]+3NN[A2,A1^2]) (*relations of the form (4.2)*) (*rank 3*) NN[A3]=NN[A3,A1] NN[A2*A1]=NN[A2*A1,A1] NN[A1^3]=NN[A1^3,A1] NN[A2,A1]=NN[A2,A1,A1] NN[A1^2,A1]=NN[A1^2,A1,A1] NN[A1,A1,A1]=NN[A1,A1,A1,A1] (*rank 2*) NN[A2]=NN[A2,A2]+NN[A2,A1^2] NN[A1^2]=NN[A2,A1^2]+NN[A1^2,A1^2] NN[A1,A1]=NN[A2,A1,A1]+NN[A1^2,A1,A1] (*rank 1*) NN[A1]=12 (*rank 0*) NN[]=1 (*The zeta polynomials, Armstrong convention*) Z[A1,z_]:=z+1 Z[A2,z_]:=(3z+2)(3z+3)/6 Z[A3,z_]:=(4z+2)(4z+3)(4z+4)/24 Z[A4,z_]:=(5z+2)(5z+3)(5z+4)(5z+5)/120 Z[A5,z_]:=(6z+2)(6z+3)(6z+4)(6z+5)(6z+6)/720 Z[A6,z_]:=(7z+2)(7z+3)(7z+4)(7z+5)(7z+6)(7z+7)/5040 Z[A7,z_]:=(8z+2)(8z+3)(8z+4)(8z+5)(8z+6)(8z+7)(8z+8)/40320 Z[D4,z_]:=(6z+2)(6z+4)(6z+6)(6z+4)/(2 4 6 4) Z[D5,z_]:=(8z+2)(8z+4)(8z+6)(8z+8)(8z+5)/(2 4 6 8 5) Z[D6,z_]:=(10z+2)(10z+4)(10z+6)(10z+8)(10z+10)(10z+6)/(2 4 6 8 10 6) Z[E6,z_]:=(12z+2)(12z+5)(12z+6)(12z+8)(12z+9)(12z+12)/(2 5 6 8 9 12) Z[A4*A1,z_]:=Z[A4,z]Z[A1,z] Z[D4*A1,z_]:=Z[D4,z]Z[A1,z] Z[A3*A2,z_]:=Z[A3,z]Z[A2,z] Z[A3*A1^2,z_]:=Z[A3,z]Z[A1,z]^2 Z[A2^2*A1,z_]:=Z[A2,z]^2Z[A1,z] Z[A2*A1^3,z_]:=Z[A2,z]Z[A1,z]^3 Z[A1^5,z_]:=Z[A1,z]^5 Z[A3*A1,z_]:=Z[A3,z]Z[A1,z] Z[A2^2,z_]:=Z[A2,z]^2 Z[A2*A1^2,z_]:=Z[A2,z]Z[A1,z]^2 Z[A1^4,z_]:=Z[A1,z]^4 Z[A2*A1,z_]:=Z[A2,z]Z[A1,z] Z[A1^3,z_]:=Z[A1,z]^3 Z[A1^2,z_]:=Z[A1,z]^2 NNNN[x___]:=NN[x]Product[Z[{x}[[ii]],z-1],{ii,1,Length[{x}]}]* Binomial[m,Length[{x}]] (*right-hand side of (4.7)*) Expr=Expand[ NNNN[D4]+ 2NNNN[A3,A1]+ 2NNNN[A2*A1,A1]+ 2NNNN[A1^3,A1]+ NNNN[A2,A2]+ 2NNNN[A2,A1^2]+ NNNN[A1^2,A1^2]+ 3NNNN[A2,A1,A1]+ 3NNNN[A1^2,A1,A1]+ NNNN[A1,A1,A1,A1]+ NNNN[A3]+ NNNN[A2*A1]+ NNNN[A1^3]+ 2NNNN[A2,A1]+ 2NNNN[A1^2,A1]+ NNNN[A1,A1,A1]+ NNNN[A2]+ NNNN[A1^2]+ NNNN[A1,A1]+ NNNN[A1]+ 1] (*left-hand side of (4.7)*) Expr2=Expand[Z[D4,z m]] (*the variables*) Var={ NN[A3,A1], NN[A2*A1,A1], NN[A1^3,A1], NN[A2,A2], NN[A2,A1^2], NN[A1^2,A1^2], NN[A2,A1,A1], NN[A1^2,A1,A1], NN[A1,A1,A1,A1] } (*Solve the system of equations*) Gl1=Flatten[Table[Table[Coefficient[Coefficient[Expr-Expr2,z,ii],m,jj]==0,{ii,0,4}],{jj,0,4}]] Sol:=Solve[Union[Gl1,{GG1,GG2,GG3}],Var] Sol