Relations in degree d

(*The symbol c[k,j] stands for c_k(j)*)
(*Properties of generators*)
c[k_, j_]:=0/;k<0
c[0,1]=d;
c[0,0]=0;

(*Normalization*)
c[1,0]=0;
c[1,1]=0;

Truncated relations in C[T^+] with degree d - 1, d

(*A[s,n] and B[s,n] are A_s, B_s defined in Proposition 2.6*)
A[s_,n_]:=(-1)^(s+2)c[s+1,0]+(-1)^(s+1)(3-n-chi/d)c[s,1]+(-1)^s/(2d^2)((n-7/2)d+chi)((n-5/2)d+chi)c[s-1,2];
B[s_, n_]:=(-1)^(s+2)c[s+1,0]+(-1)^(s+1)(2-n-chi/d)c[s,1]+(-1)^s/(2d^2)((n-5/2)d+chi)((n-3/2)d+chi)c[s-1,2];

(*Given a partition m of l, term9[m,n] is the corresponding term in the sum (9)*)
term9[m_, n_]:=(Times@@Function[c, 1/c[[2]]!]/@Tally[m])*Times@@(Function[s, (s-1)!(A[s,n]-Sum[(-1)^i beta^i/i! B[s-i,n], {i,0,2}])]/@m)

(*rule for truncating equations*)
trunc=c[j_, k_]/;(j+k-d<-1)->0;

(*rule to simplify factorials, which mathematica does not do automatically*)
simpfactorials={(d+1)!->(d+1)d!, d!->d*(d-1)!, (d-1)!->(d-2)!(d-1),(d-3)!->(d-2)!/(d-2) };

(*relsproduct1 are the (truncated) relations in degree l=d+1 in the product M x P^2 produced by Proposition 2.6*)
relsproduct1[n_]=Simplify[1/(d-3)!Plus@@Function[m,term9[m,n]]/@{{d+1}, {d,1}, {d-1,1,1}, {d-1,2}, {d-2,2,1}, {d-2,3}, {d-2,1,1,1}}/.trunc//.simpfactorials];

(*relsproduct2 are the (truncated) relations in degree l=d+2 in the product M x P^2 produced by Proposition 2.6*)
relsproduct2[n_]=Simplify[1/(d-3)!Plus@@Function[m,term9[m,n]]/@{{d+2}, {d+1,1}, {d,1,1}, {d,2}, {d-1,2,1}, {d-1,3}, {d-1,1,1,1},{d-2,4}, {d-2,3,1}, {d-2,2,2}, {d-2,2,1,1}, {d-2,1,1,1,1}}/.trunc//.simpfactorials];

(*relations (a) are relations in degree d-1 obtained by pushing forward relations in M x P^2 in degree l=d+1*)
relationa[n_]:=Coefficient[relsproduct1[n], beta,2];

(*relations (b) are relations in degree d obtained from relations in M x P^2 in degree l=d+1 by multiplying by beta and pushing forward*)
relationb[n_]:=Coefficient[relsproduct1[n], beta,1];

(*relations (c) produces relations in degree d by pushing forward relations in M x P^2 in degree l=d+2*)
relationc[n_]:=Coefficient[relsproduct2[n], beta,2];

Verifying linear independence

(*Determinant appearing when expressing generators in degree d-1 in terms of lower generators*)
det1=Factor[Det[Table[Coefficient[relationa[n], c[d-j,j]], {n,1,3}, {j,0,2}]]]

(*Set of monomials Mon_2*)
monomials2=Join[{c[d+1,0], c[d,1], c[d-1,2]},Table[c[d-1,0]c[3-i, i], {i,0,2}]]

(*Determinant of 6x6 matrix of coefficients of monomials2 in relations R_(b), R_(c)*)
eqs=Join@@Table[{relationb[n], relationc[n]}, {n,1,3}];
det2=Factor[Det[Table[Coefficient[eqs[[i]], monomials2[[j]]], {i,1,6}, {j,1,6}]]//.{d!->d*(d-1)!, (d-1)!->(d-2)!(d-2),(d-3)!->(d-2)!/(d-2) }]

Truncated relations in C[T] with degree d - 1, d

(*Writes c[d,0],c[d-1,1], c[d-2,2] in terms of lower degree generators*)
sol1=Simplify[Solve[relationa[1]==0&&relationa[2]==0&&relationa[3]==0, {c[d,0],c[d-1,1], c[d-2,2]}]//.{d!->d*(d-1)!, (d-2)!->(d-1)!/(d-1)}][[1]];

(*Writes c[d+1,0],c[d,1], c[d-1,2] in terms of lower degree generators*)
sol2=Simplify[Solve[relationb[1]==0&&relationc[1]==0&&relationc[2]==0, {c[d+1,0],c[d,1], c[d-1,2]}]][[1]];

(*Basis of C[T]^2 x T^(d-2) ordered by ≺*)
basis=Join[Join@@Table[c[d-i,i-1]c[4-j, j-1], {i,1,3}, {j,1,3}], Join@@Table[{c[d-i,i-1]c[2,0]^2,c[d-i,i-1]c[2,0]c[0,2],c[d-i,i-1]c[0,2]^2}, {i,1,3}]]

(*Matrix in row-echelon form with the coefficients of the 3 relations R_1, R_2, R_3. Obtained by plugging sol1, sol2 into the remaining 3 equations*)
TruncRelations=Together[Simplify[Factor[RowReduce[Outer[Function[{rel, mon}, Coefficient[rel, mon]], {relationb[2], relationb[3], relationc[3]}/.sol2/.sol1, basis]]]]];

MatrixForm[TruncRelations]

(* Relations[d,chi] is same as the above TruncRelations but also keeps track of the dependence on d and chi. *)

Relations[d_,chi_]:={{1,0,0,0,0,-(1/(8 (d^3)))chi (chi-d) (2 chi-d) (-2+d),((-4+d)/(8 (-2+d))),-((chi (chi-d) (2 chi-d))/(2 (-2+d) (d^3))),(1/(32 (-2+d) (d^4)))(-48 (chi^4)+96 (chi^3) d+24 (chi^4) d-48 (chi^2) (d^2)-48 (chi^3) (d^2)+18 (chi^2) (d^3)+3 (d^4)+6 chi (d^4)+6 (chi^2) (d^4)-(d^5)-6 chi (d^5)),(-6 (chi^2)+6 chi d-2 (d^2)+(d^3))/(2 chi (2 (chi^2)-3 chi d+(d^2))),(12 (chi^4)-6 (chi^2) d-24 (chi^3) d-12 (chi^4) d+6 chi (d^2)+24 (chi^2) (d^2)+24 (chi^3) (d^2)-12 chi (d^3)-30 (chi^2) (d^3)-4 (d^4)+18 chi (d^4)+6 (chi^2) (d^4)+4 (d^5)-6 chi (d^5)-(d^6))/(8 chi (chi-d) (2 chi-d) (-2+d) d),(64 (chi^6)-192 (chi^5) d-128 (chi^6) d+208 (chi^4) (d^2)+384 (chi^5) (d^2)+128 (chi^6) (d^2)-96 (chi^3) (d^3)-392 (chi^4) (d^3)-384 (chi^5) (d^3)-64 (chi^6) (d^3)-8 (chi^2) (d^4)+144 (chi^3) (d^4)+344 (chi^4) (d^4)+192 (chi^5) (d^4)+24 chi (d^5)+94 (chi^2) (d^5)-48 (chi^3) (d^5)-160 (chi^4) (d^5)-4 (d^6)-102 chi (d^6)-196 (chi^2) (d^6)+12 (d^7)+156 chi (d^7)+140 (chi^2) (d^7)-9 (d^8)-108 chi (d^8)-24 (chi^2) (d^8)+2 (d^9)+24 chi (d^9))/(128 chi (chi-d) (2 chi-d) (-2+d) (d^4)),((4-d)/(4 (-2+d) d)),((4-15 d+10 (d^2)-2 (d^3))/(16 (-2+d) d)),(1/(256 (-2+d) (d^5)))(192 (chi^4)-384 (chi^3) d-240 (chi^4) d+192 (chi^2) (d^2)+480 (chi^3) (d^2)+96 (chi^4) (d^2)-96 (chi^2) (d^3)-192 (chi^3) (d^3)-12 (d^4)-144 chi (d^4)-240 (chi^2) (d^4)+31 (d^5)+336 chi (d^5)+240 (chi^2) (d^5)-20 (d^6)-240 chi (d^6)-48 (chi^2) (d^6)+4 (d^7)+48 chi (d^7)),((2+d) (6 (chi^2)-6 chi d+2 (d^2)-(d^3)))/(16 chi (chi-d) (2 chi-d) (-2+d)),(-128 (chi^6)+384 (chi^5) d+160 (chi^6) d-464 (chi^4) (d^2)-480 (chi^5) (d^2)-80 (chi^6) (d^2)+288 (chi^3) (d^3)+688 (chi^4) (d^3)+240 (chi^5) (d^3)-20 (chi^2) (d^4)-576 (chi^3) (d^4)-476 (chi^4) (d^4)-60 chi (d^5)+118 (chi^2) (d^5)+552 (chi^3) (d^5)+84 (chi^4) (d^5)+16 (d^6)+90 chi (d^6)-188 (chi^2) (d^6)-168 (chi^3) (d^6)-24 (d^7)-48 chi (d^7)+90 (chi^2) (d^7)+16 (d^8)-6 chi (d^8)-6 (chi^2) (d^8)-6 (d^9)+6 chi (d^9)+(d^10))/(64 chi (chi-d) (2 chi-d) (((-2+d))^2) (d^4)),(-3072 (chi^8)+12288 (chi^7) d+3840 (chi^8) d-18688 (chi^6) (d^2)-15360 (chi^7) (d^2)-1152 (chi^8) (d^2)+13056 (chi^5) (d^3)+21312 (chi^6) (d^3)+4608 (chi^7) (d^3)-384 (chi^8) (d^3)-3904 (chi^4) (d^4)-10176 (chi^5) (d^4)-1952 (chi^6) (d^4)+1536 (chi^7) (d^4)+384 (chi^3) (d^5)-816 (chi^4) (d^5)-10272 (chi^5) (d^5)-6656 (chi^6) (d^5)-208 (chi^2) (d^6)+672 (chi^3) (d^6)+12712 (chi^4) (d^6)+14592 (chi^5) (d^6)+1280 (chi^6) (d^6)+144 chi (d^7)+972 (chi^2) (d^7)-2928 (chi^3) (d^7)-12680 (chi^4) (d^7)-3840 (chi^5) (d^7)-24 (d^8)-444 chi (d^8)-1622 (chi^2) (d^8)+2832 (chi^3) (d^8)+3680 (chi^4) (d^8)+44 (d^9)+606 chi (d^9)+1180 (chi^2) (d^9)-960 (chi^3) (d^9)-34 (d^10)-420 chi (d^10)-316 (chi^2) (d^10)+13 (d^11)+156 chi (d^11)+24 (chi^2) (d^11)-2 (d^12)-24 chi (d^12))/(1024 chi (chi-d) (2 chi-d) (((-2+d))^2) (d^6))},{0,1,0,1,0,(1/(8 (d^2)))(-12 (chi^2)+12 chi d+6 (chi^2) d-(d^2)-6 chi (d^2)),0,(24 (chi^2)-24 chi d+2 (d^2)-(d^3))/(8 (-2+d) (d^2)),-((chi (chi-d) (2 chi-d) (-16+8 d+(d^2)))/(8 (-2+d) (d^3))),((3 (-1+d) d)/(chi (chi-d) (-2+d))),(4 (chi^2)-4 chi d-14 (chi^2) d+14 chi (d^2)+16 (chi^2) (d^2)+3 (d^3)-16 chi (d^3)-2 (chi^2) (d^3)-3 (d^4)+2 chi (d^4))/(4 chi (chi-d) (-2+d) d),((-1+d) (48 (chi^4)-96 (chi^3) d-48 (chi^4) d+48 (chi^2) (d^2)+96 (chi^3) (d^2)+48 (chi^4) (d^2)-36 (chi^2) (d^3)-96 (chi^3) (d^3)-3 (d^4)-12 chi (d^4)-4 (chi^2) (d^4)+6 (d^5)+52 chi (d^5)+8 (chi^2) (d^5)-8 chi (d^6)))/(64 chi (chi-d) (-2+d) (d^3)),0,0,(chi (chi-d) (2 chi-d) (-8+10 d-(d^2)-4 (d^3)+(d^4)))/(8 (-2+d) (d^4)),(-8 (chi^2)+8 chi d+8 (chi^2) d+6 (d^2)-8 chi (d^2)+2 (chi^2) (d^2)-3 (d^3)-2 chi (d^3)-3 (d^4))/(8 chi (chi-d) (((-2+d))^2) d),(96 (chi^4)-192 (chi^3) d-144 (chi^4) d+72 (chi^2) (d^2)+288 (chi^3) (d^2)+96 (chi^4) (d^2)+24 chi (d^3)-60 (chi^2) (d^3)-192 (chi^3) (d^3)-12 (d^4)-84 chi (d^4)+44 (chi^2) (d^4)+18 (d^5)+52 chi (d^5)-16 (chi^2) (d^5)-9 (d^6)+16 chi (d^6)+2 (chi^2) (d^6)+3 (d^7)-2 chi (d^7))/(32 chi (chi-d) (((-2+d))^2) (d^3)),(1920 (chi^6)-5760 (chi^5) d-2304 (chi^6) d+6240 (chi^4) (d^2)+6912 (chi^5) (d^2)-288 (chi^6) (d^2)-2880 (chi^3) (d^3)-8016 (chi^4) (d^3)+864 (chi^5) (d^3)+1536 (chi^6) (d^3)+552 (chi^2) (d^4)+4512 (chi^3) (d^4)+576 (chi^4) (d^4)-4608 (chi^5) (d^4)-384 (chi^6) (d^4)-72 chi (d^5)-1368 (chi^2) (d^5)-2592 (chi^3) (d^5)+3936 (chi^4) (d^5)+1152 (chi^5) (d^5)+18 (d^6)+264 chi (d^6)+1774 (chi^2) (d^6)-192 (chi^3) (d^6)-1200 (chi^4) (d^6)-33 (d^7)-334 chi (d^7)-864 (chi^2) (d^7)+480 (chi^3) (d^7)+21 (d^8)+192 chi (d^8)+28 (chi^2) (d^8)-6 (d^9)-76 chi (d^9)-8 (chi^2) (d^9)+8 chi (d^10))/(512 chi (chi-d) (((-2+d))^2) (d^5))},{0,0,1,0,0,0,(2/(-2+d)),0,((12 (chi^2)-12 chi d-(d^2))/(8 (-2+d) d)),(4 (d^2))/(chi (2 (chi^2)-3 chi d+(d^2))),(-6 (chi^2)+6 chi d+6 (chi^2) d-6 chi (d^2)-(d^3))/(chi (chi-d) (2 chi-d)),-(((-1+2 d) (-12 (chi^2)+12 chi d+12 (chi^2) d-(d^2)-12 chi (d^2)))/(16 chi (chi-d) (2 chi-d))),0,((3-d)/(-2+d)),((-3+d) (12 (chi^2)-12 chi d-12 (chi^2) d+(d^2)+12 chi (d^2)))/(8 (-2+d) (d^2)),-(((d^2) (2+d))/(2 chi (chi-d) (2 chi-d) (-2+d))),(-48 (chi^4)+96 (chi^3) d+72 (chi^4) d-36 (chi^2) (d^2)-144 (chi^3) (d^2)-12 chi (d^3)+66 (chi^2) (d^3)+4 (d^4)+6 chi (d^4)-6 (chi^2) (d^4)-2 (d^5)+6 chi (d^5)+(d^6))/(8 chi (chi-d) (2 chi-d) (-2+d) (d^2)),(128 (chi^6)-384 (chi^5) d-1344 (chi^6) d+224 (chi^4) (d^2)+4032 (chi^5) (d^2)+1216 (chi^6) (d^2)+192 (chi^3) (d^3)-3792 (chi^4) (d^3)-3648 (chi^5) (d^3)-232 (chi^2) (d^4)+864 (chi^3) (d^4)+3520 (chi^4) (d^4)+72 chi (d^5)+372 (chi^2) (d^5)-960 (chi^3) (d^5)-6 (d^6)-132 chi (d^6)-212 (chi^2) (d^6)+5 (d^7)+84 chi (d^7)+24 (chi^2) (d^7)-2 (d^8)-24 chi (d^8))/(128 chi (chi-d) (2 chi-d) (-2+d) (d^4))}}

(* Put the Relations[d,chi] into the matrix form. TopRelations for T^(d-2)xT^(2) and ExtRelations for T^(d-2)xSym^2(T^1). *)

TopRelations[d_,chi_]:={{Relations[d,chi][[1,1;;3]], Relations[d,chi][[1,4;;6]], Relations[d,chi][[1,7;;9]]},
{Relations[d,chi][[2,1;;3]], Relations[d,chi][[2,4;;6]], Relations[d,chi][[2,7;;9]]},
{Relations[d,chi][[3,1;;3]], Relations[d,chi][[3,4;;6]], Relations[d,chi][[3,7;;9]]}}

ExtRelations[d_,chi_]:={{Relations[d,chi][[1,10;;12]], Relations[d,chi][[1,13;;15]], Relations[d,chi][[1,16;;18]]},
{Relations[d,chi][[2,10;;12]], Relations[d,chi][[2,13;;15]], Relations[d,chi][[2,16;;18]]},
{Relations[d,chi][[3,10;;12]], Relations[d,chi][[3,13;;15]], Relations[d,chi][[3,16;;18]]}}

Proof of main theorem

(* Automorphism matrices; AutA for SL(T^{d-2}), AutB for SL(T^2), AutU for Hom(T^2, Sym^2(T^1)) and AutV for GL(Sym^2(T^1)). *)
AutA:=Table[a[i,j],{i,1,3},{j,1,3}]
AutB:=Table[b[i,j],{i,1,3},{j,1,3}]
AutU:=Table[u[i,j],{i,1,3},{j,1,3}]
AutV:=Table[v[i,j],{i,1,3},{j,1,3}]

(* Shuffle matrix S which is obtained by AutA^tr(-)AutB. *)
S:=Table[s[i,j],{i,1,3},{j,1,3}]

Find the shuffle matrix

(* Determinant map defined by three dimensional span(TopRelations[d,chi]). *)
DET[d_,chi_]:=Simplify[Det[Sum[TopRelations[d,chi][[i]]*x[i],{i,1,3}]]]

DET[d,chi]

(* Cubic curve defined by DET[d,chi] has a unique nodal singularity at [0:0:1]. *)
Solve[DET[d,chi]==0&&D[DET[d,chi],x[1]]==0&&D[DET[d,chi],x[2]]==0&&D[DET[d,chi],x[3]]==0&&(x[1]==1 ∨ x[2]==1 ∨ x[3]==1),{x[1],x[2],x[3]}, Complexes]

(* Solve the shuffle matrix S using the fact that it sends the determinant curve of (d,chi1) to that of (d,chi2). *)
f:=CoefficientList[DET[d,chi1],{x[1],x[2],x[3]},{4,4,4}]
g:=CoefficientList[(DET[d,chi2]/.{x[1]->Sum[s[j,1]*x[j],{j,1,3}],x[2]->Sum[s[j,2]*x[j],{j,1,3}],x[3]->Sum[s[j,3]*x[j],{j,1,3}]}),{x[1],x[2],x[3]},{4,4,4}]
coeff[u_,v_,w_]:=Simplify[(f-g)[[u+1,v+1,w+1]]]

(* By the singularity consideration, we have the following. *)
s[3,1]:=0
s[3,2]:=0

(* The following equations allow us to divide the solutions into two cases. *)
coeff[0,2,1]
coeff[2,0,1]
coeff[1,1,1]

Solve[s[2,1] s[2,2] s[3,3]==0&&s[1,1] s[1,2] s[3,3]==0&&(s[1,2] s[2,1]+s[1,1] s[2,2]) s[3,3]!=0,{s[1,1],s[2,2],s[3,3],s[1,2],s[2,1]}]

Type I solutions

(* We start with the first case, namely s[1,1]=s[2,2]=0, which we call Type I. *)
s[1,1]:=0
s[2,2]:=0

coeff[0,3,0]
coeff[3,0,0]
coeff[1,1,1]

(* The above three equations imply that s[3,3]^3=1. By scaling the matrices using a cubic root of unity, if necessary, we may assume that s[3,3]=1. *)
s[3,3]:=1

(* From coeff[0,3,0]=0, s[2,1] is given by a cubic root. From coeff[1,1,1]=0 and s[3,3]=1, we also solve s[1,2]. *)
s[2,1]:=CubeRoot[-((2 chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))]*d

Solve[coeff[1,1,1]==0,s[1,2]]

s[1,2]:=-((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/((2^(1/3)) chi2 d (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))))

(* Solve the rest of S. We carefully check that the coefficients are nonzero before solving it. *)

Coefficient[coeff[1,2,0],s[2,3],1]
Coefficient[coeff[2,1,0],s[1,3],1]

Simplify[Solve[coeff[1,2,0]==0&&coeff[2,1,0]==0,{s[2,3],s[1,3]}]]

s[2,3]:=(1/(8 (2-d) (d^2)))(-2+d) (-48 chi1 (-2+d)-48 d+24 (d^2)+(2^(1/3)) (24 (chi2^2) (-2+d)-24 chi2 (-2+d) d-(d^2) (8-5 d+(d^2))) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3)))
s[1,3]:=(1/(8 (d^3)))(-2+d) (-24 (chi1^2)+24 chi1 d+(((d^2) (8-5 d+(d^2)))/(-2+d))+(24 (2^(2/3)) chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/((2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3)))-(12 (2^(2/3)) chi1 d (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))))

MatrixForm[S]

Type I - Solve for A, B

(* We have solved the shuffle matrix S uniquely up to a choice of a cubic root in the case of s[1,1]=s[2,2]=0. We claim that there is no compatible pair (AutA,AutB) such that AutA^t(-)AutB corresponds to this shuffle matrix. To linearize the problem, we use the inverse matrix AutB^(-1). *)

At:=Transpose[AutA]
Binv:=Table[bb[i,j],{i,1,3},{j,1,3}]
Diff[d_,chi1_,chi2_]:=Table[At.TopRelations[d,chi1][[i]]-Sum[S[[i,j]]*TopRelations[d,chi2][[j]].Binv,{j,1,3}],{i,1,3}]

(* Use Diff[[3]] first. *)
(* From now on, we skip checking that we are solving linear equations with nonzero leading coefficients, unless it is nontrivial, for simplicity of the code. The solutions have denominators which are evidently nonzero. *)

Solve[Diff[d,chi1,chi2][[3]]==0,{bb[3,1],bb[3,2],bb[3,3],bb[1,1],bb[1,2],bb[1,3],a[3,2],a[1,2]}]

bb[3,1]:=((2 a[3,1])/(-2+d))
bb[3,2]:=0
bb[3,3]:=-(1/(8 (-2+d) d))(16 d a[1,1]-8 (d^2) a[1,1]-12 (chi1^2) a[3,1]+12 chi1 d a[3,1]+(d^2) a[3,1])
bb[1,1]:=-(((12 (chi2^2)-12 chi2 d-(d^2)) a[3,1])/(8 (-2+d) d))+a[3,3]
bb[1,2]:=0
bb[1,3]:=(1/(128 (-2+d) (d^2)))(12 (chi2^2)-12 chi2 d-(d^2)) (16 d a[1,1]-8 (d^2) a[1,1]-12 (chi1^2) a[3,1]+12 chi1 d a[3,1]+(d^2) a[3,1])-(1/(16 d))(16 d a[1,3]-8 (d^2) a[1,3]-12 (chi1^2) a[3,3]+12 chi1 d a[3,3]+(d^2) a[3,3])
a[3,2]:=0
a[1,2]:=0

Simplify[Diff[d,chi1,chi2][[3]]]

(* Solve b[2,j] using Diff[[2]][3,j]. *)

Simplify[Solve[Diff[d,chi1,chi2][[2]][[3,1]]==0&&Diff[d,chi1,chi2][[2]][[3,2]]==0&&Diff[d,chi1,chi2][[2]][[3,3]]==0,{bb[2,1],bb[2,2],bb[2,3]}]]

bb[2,1]:=((-2+d) (d^2) (-16 a[2,3]+(1/((((-2+d))^2) (d^3)))(2^(1/3)) (-24 (chi2^4) (-2+d)+48 (chi2^3) (-2+d) d+(-3+d) (d^4)+6 chi2 (-1+d) (d^4)-6 (chi2^2) (d^2) (-8+3 d+(d^2))) a[3,1] (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))-(1/(2 (2^(2/3)) (((-2+d))^2)))(-4+d) (-12 (chi2^2) a[3,1]+12 chi2 d a[3,1]+d (-16 a[3,3]+d (a[3,1]+8 a[3,3]))) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))-(1/((2-d) (d^2)))4 a[3,3] (48 chi1 (-2+d)+48 d-24 (d^2)-(2^(1/3)) (24 (chi2^2) (-2+d)-24 chi2 (-2+d) d-(d^2) (8-5 d+(d^2))) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3)))))/(8 (2^(1/3)) chi2 (chi2-d) (-2 chi2+d) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3)))
bb[2,2]:=((d^3) (8 a[1,3]-a[3,3])+24 (chi1^2) a[3,3]-24 chi1 d a[3,3]+2 (d^2) (-8 a[1,3]+a[3,3]))/(4 (2^(1/3)) chi2 (chi2-d) (2 chi2-d) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3)))
bb[2,3]:=((-2+d) (32 (d^2) (-6 (chi1^2) (-2+d)+6 chi1 (-2+d) d+(d^2)) a[2,3]+(1/(-2+d))32 chi1 (chi1-d) (2 chi1-d) d (-16+8 d+(d^2)) a[3,3]+(1/(((-2+d))^2))(2^(1/3)) (-24 (chi2^4) (-2+d)+48 (chi2^3) (-2+d) d+(-3+d) (d^4)+6 chi2 (-1+d) (d^4)-6 (chi2^2) (d^2) (-8+3 d+(d^2))) ((d^2) (8 a[1,1]-a[3,1])+12 (chi1^2) a[3,1]-4 d (4 a[1,1]+3 chi1 a[3,1])) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))+(1/(2 (2^(2/3)) (((-2+d))^2)))(4-d) (d^3) ((12 (chi2^2)-12 chi2 d-(d^2)) (-12 (chi1^2) a[3,1]+(d^2) (-8 a[1,1]+a[3,1])+4 d (4 a[1,1]+3 chi1 a[3,1]))+8 (-2+d) d ((d^2) (8 a[1,3]-a[3,3])+12 (chi1^2) a[3,3]-4 d (4 a[1,3]+3 chi1 a[3,3]))) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))+(1/(2-d))4 d (-12 (chi1^2) a[3,3]+(d^2) (-8 a[1,3]+a[3,3])+4 d (4 a[1,3]+3 chi1 a[3,3])) (48 chi1 (-2+d)+48 d-24 (d^2)-(2^(1/3)) (24 (chi2^2) (-2+d)-24 chi2 (-2+d) d-(d^2) (8-5 d+(d^2))) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3)))))/(128 (2^(1/3)) chi2 (chi2-d) (d^2) (-2 chi2+d) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3)))

(* Use the rest of Diff[[2]]. *)

Simplify[Solve[Diff[d,chi1,chi2][[2]][[1,1]]==0&&Diff[d,chi1,chi2][[2]][[2,1]]==0 &&Diff[d,chi1,chi2][[2]][[1,2]]==0,{a[2,1],a[2,2],a[1,1]}]]

a[2,1]:=-(2^(1/3)) d (((-12 (chi2^2)+12 chi2 d+(d^2)) a[3,1])/(8 (-2+d) d)+a[3,3]) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))+(1/(4 (2-d) (d^2)))a[3,1] (-48 chi1 (-2+d)-48 d+24 (d^2)+(2^(1/3)) (24 (chi2^2) (-2+d)-24 chi2 (-2+d) d-(d^2) (8-5 d+(d^2))) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3)))
a[2,2]:=(1/(2 (2^(2/3)) (d^2)))chi2 (chi2-d) (2 chi2-d) a[3,1] (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))
a[1,1]:=((-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) a[3,1])/(8 (-2+d) (d^2))

Simplify[Solve[Diff[d,chi1,chi2][[2]][[1,3]]==0,a[3,1]]]

a[3,1]:=((-2+d) (d^2) ((d^3) (8 a[1,3]-a[3,3])+24 (chi1^2) a[3,3]-24 chi1 d a[3,3]+2 (d^2) (-8 a[1,3]+a[3,3])) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3)))/((2^(2/3)) chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))

Simplify[Diff[d,chi1,chi2][[2]]]

(* We would like to use Diff[[1]] to solve a[1,3] and a[2,3]. We first check that the leading coefficients of the linear equations are nonzero. *)

Simplify[Coefficient[Diff[d,chi1,chi2][[1]][[2,1]],a[1,3],1]]
Simplify[Coefficient[Diff[d,chi1,chi2][[1]][[1,1]],a[2,3],1]]

(* Check that (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) is nonzero if d is coprime to chi1 and chi2. Note that coprimality forces d to divide 24 for such solutions to exist. *)
Divlist:=Divisors[24]

For[i=1, i<Length[Divlist]+1, i++, 
For[chi2=1,chi2<Divlist[[i]]/2 ,chi2++,
If[(
(-24 (chi2^2)+24 chi2 d+(-2+d) (d^2))/.d->Divlist[[i]])==0, Print[{Divlist[[i]],chi1,chi2}]]
]
]

Clear[d,chi1,chi2]

(* Therefore we solve a[1,3] and a[2,3] using Diff[[1]][[2,1]] and Diff[[1]][[1,1]]. *)

Simplify[Solve[Diff[d,chi1,chi2][[1]][[2,1]]==0&&Diff[d,chi1,chi2][[1]][[1,1]]==0,{a[1,3],a[2,3]}]]

a[1,3]:=(1/(8 (-2+d) (d^2)))a[3,3] (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)-(8 (2^(2/3)) chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))))
a[2,3]:=(a[3,3] (24 chi1 (chi1-d) (-2+d) (((-2 chi1+d))^2) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2))+(2^(1/3)) chi1 (-2+d) (2 (chi1^2)-3 chi1 d+(d^2)) (48 (chi2^4) (4+d)-96 (chi2^3) d (4+d)+(d^4) (8-6 d+(d^2))-6 chi2 (d^3) (8-2 d+(d^2))+6 (chi2^2) (d^2) (40+6 d+(d^2))) (((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))+8 (2^(2/3)) (chi2^2) (-4+d) (6 (chi1^2)-6 chi1 d+(d^2)) (((2 (chi2^2)-3 chi2 d+(d^2)))^2) ((((chi1 (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2)))/((chi2^2) (-2+d) (((2 (chi2^2)-3 chi2 d+(d^2)))^2)))^(1/3))^2)))/(4 chi1 (chi1-d) (2 chi1-d) (-2+d) (d^2) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)))

(* Conclude a[3,3]=0 from Diff[[1]][1,2]. *)

Simplify[Diff[d,chi1,chi2][[1]][[1,2]]]

a[3,3]:=0

(* This forces AutA and Binv to be zero, hence a contradiction with Det[AutA]=Det[AutB]=1. *)

AutA
Binv

Type II

Clear[s,a,bb]

(* Now we consider the type II case, namely s[1,2]=s[2,1]=0. Recall that we also had s[3,1]=s[3,2]=0 from the singulariy consideration. *)
s[3,1]:=0
s[3,2]:=0
s[1,2]:=0
s[2,1]:=0

(* We solve the matrix S similar to the previous case. Consider the following three equations involving s[1,1],s[2,2],s[3,3].*)

coeff[3,0,0]
coeff[0,3,0]
coeff[1,1,1]

(* The above three equations imply that s[3,3]^3=1. By scaling the matrices using a cubic root of unity, if necessary, we may assume that s[3,3]=1. *)

s[3,3]:=1

(* From coeff[0,3,0]=0, s[2,2] is given by a cubic root. From coeff[1,1,1]=0 and s[3,3]=1, we also solve s[1,1]=s[2,2]^2. *)

s[2,2]:=CubeRoot[(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2)))]
s[1,1]:=s[2,2]^2

(* Solve the rest of S. *)

Simplify[Solve[coeff[2,1,0]==0&&coeff[1,2,0]==0,{s[1,3],s[2,3]}]]

s[1,3]:=(1/(8 (d^3)))(-24 (chi1^2) (-2+d)+24 chi1 (-2+d) d+(d^2) (8-5 d+(d^2))+(24 (chi2^2) (-2+d)-24 chi2 (-2+d) d-(d^2) (8-5 d+(d^2))) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))
s[2,3]:=(1/(d^2))3 (-2+d) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))

MatrixForm[S]

Type II - Solve for A, B

(* We have solved the shuffle matrix S uniquely up to a choice of a cubic root for s[2,2]. We claim that for each S there is a unique pair (AutA,AutB) up to scaling such that AutA^t(-)AutB corresponds to this shuffle matrix. To linearize the problem, we again use the inverse matrix AutB^(-1). *)

At:=Transpose[AutA]
Binv:=Table[bb[i,j],{i,1,3},{j,1,3}]
Diff'[d_,chi1_,chi2_]:=Table[At.TopRelations[d,chi1][[i]]-Sum[S[[i,j]]*TopRelations[d,chi2][[j]].Binv,{j,1,3}],{i,1,3}]

(* Use Diff'[[3]] first. *)

Solve[Diff'[d,chi1,chi2][[3]]==0,{bb[1,1],bb[1,2],bb[1,3],bb[3,1],bb[3,2],bb[3,3],a[3,2],a[1,2]}]

bb[1,1]:=-(((12 (chi2^2)-12 chi2 d-(d^2)) a[3,1])/(8 (-2+d) d))+a[3,3]
bb[1,2]:=0
bb[1,3]:=(1/(128 (-2+d) (d^2)))(12 (chi2^2)-12 chi2 d-(d^2)) (16 d a[1,1]-8 (d^2) a[1,1]-12 (chi1^2) a[3,1]+12 chi1 d a[3,1]+(d^2) a[3,1])-(1/(16 d))(16 d a[1,3]-8 (d^2) a[1,3]-12 (chi1^2) a[3,3]+12 chi1 d a[3,3]+(d^2) a[3,3])
bb[3,1]:=((2 a[3,1])/(-2+d))
bb[3,2]:=0
bb[3,3]:=-(1/(8 (-2+d) d))(16 d a[1,1]-8 (d^2) a[1,1]-12 (chi1^2) a[3,1]+12 chi1 d a[3,1]+(d^2) a[3,1])
a[3,2]:=0
a[1,2]:=0

Simplify[Diff'[d,chi1,chi2][[3]]]

(* Solve b[2,j] using Diff'[[2]][1,j]. *)

Solve[Diff'[d,chi1,chi2][[2]][[1,1]]==0&&Diff'[d,chi1,chi2][[2]][[1,2]]==0&&Diff'[d,chi1,chi2][[2]][[1,3]]==0,{bb[2,1],bb[2,2],bb[2,3]}]

bb[2,1]:=((d^2) a[2,1]-12 chi1 a[3,1]+6 d a[3,1]+12 chi2 a[3,1] (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))-6 d a[3,1] (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/((d^2) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))
bb[2,2]:=(-16 (d^2) a[1,1]+8 (d^3) a[1,1]+24 (chi1^2) a[3,1]-24 chi1 d a[3,1]+2 (d^2) a[3,1]-(d^3) a[3,1])/(8 (-2+d) (d^2) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))
bb[2,3]:=-((-(1/(8 (d^2)))(-12 (chi1^2)+(12 chi1+6 (chi1^2)) d+(-1-6 chi1) (d^2)) a[2,1]+(chi1 (-2 chi1+d) (-chi1+d) (-16+8 d+(d^2)) a[3,1])/(8 (-2+d) (d^3))-(1/(8 (d^3)))3 (16 d a[1,1]-8 (d^2) a[1,1]-12 (chi1^2) a[3,1]+12 chi1 d a[3,1]+(d^2) a[3,1]) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))))/((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))))

(* Use the rest of Diff'[[2]]. *)

Simplify[Solve[Diff'[d,chi1,chi2][[2]][[2,1]]==0&&Diff'[d,chi1,chi2][[2]][[3,2]]==0&&Diff'[d,chi1,chi2][[2]][[3,1]]==0,{a[2,2],a[1,3],a[2,3]}]]

a[2,2]:=(1/(8 (-2+d) (d^2)))(-24 (chi2^2) a[3,1]+24 chi2 d a[3,1]+(-2+d) (d^2) (a[3,1]+8 a[3,3])) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))
a[1,3]:=(1/(64 (((-2+d))^2) (d^4)))((24 (chi2^2)-24 chi2 d-(-2+d) (d^2)) ((d^3) (8 a[1,1]-a[3,1])+24 (chi1^2) a[3,1]-24 chi1 d a[3,1]+2 (d^2) (-8 a[1,1]+a[3,1]))+8 (-2+d) (d^2) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) a[3,3])
a[2,3]:=(1/(d^2))6 a[3,3] (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+(1/(8 (((-2+d))^2) (d^4) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))))(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (-2 chi2 (chi2-d) (2 chi2-d) d (-16+8 d+(d^2)) a[3,1] (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))-(-2+d) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) ((d^2) a[2,1]-12 chi1 a[3,1]+6 d a[3,1]+6 (2 chi2-d) a[3,1] (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))))

Simplify[Solve[Diff'[d,chi1,chi2][[2]][[3,3]]==0,{a[3,3]}]]

a[3,3]:=-((6 (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) ((d^3) (8 a[1,1]-a[3,1])+24 (chi1^2) a[3,1]-24 chi1 d a[3,1]+2 (d^2) (-8 a[1,1]+a[3,1])) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))-3 d (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) ((d^3) (8 a[1,1]-a[3,1])+24 (chi1^2) a[3,1]-24 chi1 d a[3,1]+2 (d^2) (-8 a[1,1]+a[3,1])) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))-(6 (chi1^2) (-2+d)-6 chi1 (-2+d) d-(d^2)) (-2 chi2 (chi2-d) (2 chi2-d) d (-16+8 d+(d^2)) a[3,1] (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))-(-2+d) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) ((d^2) a[2,1]-12 chi1 a[3,1]+6 d a[3,1]+6 (2 chi2-d) a[3,1] (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))))+(d^2) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (chi2 (-2 chi2+d) (-chi2+d) (-16+8 d+(d^2)) (-12 (chi1^2) a[3,1]+(d^2) (-8 a[1,1]+a[3,1])+4 d (4 a[1,1]+3 chi1 a[3,1]))-((-2+d) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)) (d (-6 (chi1^2) (-2+d)+6 chi1 (-2+d) d+(d^2)) a[2,1]+(1/(-2+d))chi1 (-2 chi1+d) (-chi1+d) (-16+8 d+(d^2)) a[3,1]-3 (-12 (chi1^2) a[3,1]+(d^2) (-8 a[1,1]+a[3,1])+4 d (4 a[1,1]+3 chi1 a[3,1])) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))))/(d (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))))/(8 chi1 (-2+d) (d^3) (-16+8 d+(d^2)) (2 (chi1^2)-3 chi1 d+(d^2))))

Simplify[Diff'[d,chi1,chi2][[2]]]

(* Use Diff'[[1]] to solve the rest of AutA. *)

Solve[Diff'[d,chi1,chi2][[1]][[1,2]]==0,a[3,1]]

a[3,1]:=0

Simplify[Solve[Diff'[d,chi1,chi2][[1]][[3,1]]==0,a[2,1]]]

a[2,1]:=-(((-4+d) a[1,1] (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))))/(4 chi1 chi2 (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2)))

Simplify[Diff'[d,chi1,chi2][[1]]]

(* By rescaling AutA by a cubic root of unity and AutB by the inverse of it so that S stays unchanged, we may assume that a[1,1]=s[2,2]. *)

Simplify[Det[AutA]]

a[1,1]:=s[2,2]

(* We record the unique solutions AutA, AutB, S up to a choice of a cubic root. *)

Simplify[AutA]
Simplify[Adjugate[Binv]]
Simplify[S]

solA[d_,chi1_,chi2_]:={{(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)),0,(1/(8 (-2+d) (d^2)))(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2))))},{-(((-4+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(4 chi1 chi2 (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))),1,((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) ((1/(-16+8 d+(d^2)))192 (chi2^2) (2 (chi2^2)-3 chi2 d+(d^2)) (32-32 d+6 (d^2)+(d^3)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+((-4+d) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/((-2 chi2+d) (-chi2+d))))/(32 chi1 chi2 (-2+d) (d^2) (2 (chi1^2)-3 chi1 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))},{0,0,(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))}}

solB[d_,chi1_,chi2_]:={{((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2)/((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))),0,-((((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-6 (chi2^2) (-2+d)+6 chi2 (-2+d) d+(d^2))-chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-6 (chi1^2) (-2+d)+6 chi1 (-2+d) d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))))/(8 chi1 (chi1-d) (2 chi1-d) (d^2) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))))},{((-4+d) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(4 chi1 chi2 (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2)),1,(((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (-(((-4+d) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-6 (chi2^2) (-2+d)+6 chi2 (-2+d) d+(d^2))-chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-6 (chi1^2) (-2+d)+6 chi1 (-2+d) d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(chi1 (chi1-d) (2 chi1-d) (-2 chi2+d) (-chi2+d)))+((chi2^2) (2 (chi2^2)-3 chi2 d+(d^2)) (32-32 d+6 (d^2)+(d^3)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (96 (-2+d) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+((-4+d) (6 (chi1^2) (-2+d)-6 chi1 (-2+d) d-(d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(chi1 chi2 (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))))/((-2+d) (-16+8 d+(d^2)))))/(32 chi1 chi2 (d^2) (2 (chi1^2)-3 chi1 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))},{0,0,(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))}}

solS[d_,chi1_,chi2_]:={{((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2),0,(1/(8 (d^3)))(-24 (chi1^2) (-2+d)+24 chi1 (-2+d) d+(d^2) (8-5 d+(d^2))+(24 (chi2^2) (-2+d)-24 chi2 (-2+d) d-(d^2) (8-5 d+(d^2))) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))},{0,(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)),(1/(d^2))3 (-2+d) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))},{0,0,1}}

(* Solutions solA, solB, solS are just sign matrices when chi1=chi2 and chi1+chi2=d. *)

Simplify[MatrixForm[solA[d,chi,chi]]]
Simplify[MatrixForm[solB[d,chi,chi]]]
Simplify[MatrixForm[solS[d,chi,chi]]]

Simplify[MatrixForm[solA[d,chi,d-chi]]]
Simplify[MatrixForm[solB[d,chi,d-chi]]]
Simplify[MatrixForm[solS[d,chi,d-chi]]]

Type II - Solve for U, V

(* Using the solutions above, we find AutU and AutV. Existence of the solution, which we assumed, will lead to a contradiction later. *)

ExtDiff[d_,chi1_,chi2_]:=Table[Transpose[solA[d,chi1,chi2]].TopRelations[d,chi1][[i]].AutU+Transpose[solA[d,chi1,chi2]].ExtRelations[d,chi1][[i]].AutV-Sum[solS[d,chi1,chi2][[i,j]]*ExtRelations[d,chi2][[j]],{j,1,3}],{i,1,3}]

(* We solve the linear equation for variables u[i,1] and v[i,1] by looking at the first columns of ExtDiff[[1]],ExtDiff[[2]],ExtDiff[[3]]. *)

MatrixForm[Simplify[Coefficient[ExtDiff[d,chi1,chi2][[3]][[1;;3,1]],u[1,1],1]]]

Solve[ExtDiff[d,chi1,chi2][[3]][[3,1]]==0,{u[1,1]}]

u[1,1]:=(chi1 (-2+d) (2 (chi1^2)-3 chi1 d+(d^2)) (-(((d^2) (2+d))/(2 chi2 (chi2-d) (2 chi2-d) (-2+d)))-((chi2 (12 (chi1^2)-12 chi1 d-(d^2)) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(8 chi1 (-2+d) d (2 (chi1^2)-3 chi1 d+(d^2)))+(1/(8 (-2+d) (d^2)))(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2))))) u[3,1]-(-((chi2 (d^2) (2+d) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(2 (chi1^2) (chi1-d) (2 chi1-d) (-2+d) (2 (chi1^2)-3 chi1 d+(d^2))))+((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))))/(2 chi1 (chi1-d) (2 chi1-d) (-2+d))) v[1,1]-((chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-48 (chi1^4)+(96 (chi1^3)+72 (chi1^4)) d+(-36 (chi1^2)-144 (chi1^3)) (d^2)+(-12 chi1+66 (chi1^2)) (d^3)+(4+6 chi1-6 (chi1^2)) (d^4)+(-2+6 chi1) (d^5)+(d^6)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(8 (chi1^2) (-2+d) (d^2) (-2 chi1+d) (-chi1+d) (2 (chi1^2)-3 chi1 d+(d^2)))+((-(d^3)+(chi1^2) (-6+6 d)+chi1 (6 d-6 (d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))))/(8 chi1 (-2+d) (d^2) (-2 chi1+d) (-chi1+d))+((3-d) ((1/(-16+8 d+(d^2)))192 (chi2^2) (2 (chi2^2)-3 chi2 d+(d^2)) (32-32 d+6 (d^2)+(d^3)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+((-4+d) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/((-2 chi2+d) (-chi2+d))))/(32 chi1 chi2 (((-2+d))^2) (d^2) (2 (chi1^2)-3 chi1 d+(d^2)))) v[2,1]-((chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (128 (chi1^6)+(-384 (chi1^5)-1344 (chi1^6)) d+(224 (chi1^4)+4032 (chi1^5)+1216 (chi1^6)) (d^2)+(192 (chi1^3)-3792 (chi1^4)-3648 (chi1^5)) (d^3)+(-232 (chi1^2)+864 (chi1^3)+3520 (chi1^4)) (d^4)+(72 chi1+372 (chi1^2)-960 (chi1^3)) (d^5)+(-6-132 chi1-212 (chi1^2)) (d^6)+(5+84 chi1+24 (chi1^2)) (d^7)+(-2-24 chi1) (d^8)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(128 (chi1^2) (-2+d) (d^4) (-2 chi1+d) (-chi1+d) (2 (chi1^2)-3 chi1 d+(d^2)))+(3 (-(1/2)+d) ((chi1^2) (1-d)+((d^2)/12)+chi1 (-d+(d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))))/(16 chi1 (-2+d) (d^2) (-2 chi1+d) (-chi1+d))+(3 (-3+d) ((chi1^2)+(-chi1-(chi1^2)) d+((1/12)+chi1) (d^2)) ((1/(-16+8 d+(d^2)))192 (chi2^2) (2 (chi2^2)-3 chi2 d+(d^2)) (32-32 d+6 (d^2)+(d^3)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+((-4+d) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/((-2 chi2+d) (-chi2+d))))/(64 chi1 chi2 (((-2+d))^2) (d^4) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1]))/(2 chi2 (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))

MatrixForm[Simplify[Coefficient[ExtDiff[d,chi1,chi2][[3]][[1;;3,1]],u[3,1],1]]]

Solve[ExtDiff[d,chi1,chi2][[3]][[1,1]]==0,{u[3,1]}]

u[3,1]:=((4 (d^2))/(chi2 (chi2-d) (2 chi2-d))-(4 (d^2) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) v[1,1])/(chi1 (chi1-d) (2 chi1-d))-(((-(d^3)+(chi1^2) (-6+6 d)+chi1 (6 d-6 (d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (-2 chi1+d) (-chi1+d))-((3-d) (-4+d) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(4 chi1 chi2 (-2+d) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[2,1]-((3 (-(1/2)+d) ((chi1^2) (1-d)+((d^2)/12)+chi1 (-d+(d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(2 chi1 (-2 chi1+d) (-chi1+d))-(3 (-4+d) (-3+d) ((chi1^2)+(-chi1-(chi1^2)) d+((1/12)+chi1) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(8 chi1 chi2 (-2+d) (d^2) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1])/((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))

MatrixForm[Simplify[Coefficient[ExtDiff[d,chi1,chi2][[2]][[1;;3,1]],u[2,1],1]]]

Solve[ExtDiff[d,chi1,chi2][[2]][[1,1]]==0,{u[2,1]}]

u[2,1]:=((3 (-1+d) d (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi2 (chi2-d) (-2+d))+(12 (-2+d) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))))/(chi2 (chi2-d) (2 chi2-d))-(3 (-1+d) d (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) v[1,1])/(chi1 (chi1-d) (-2+d))-((-4 (chi1^2)+(4 chi1+14 (chi1^2)) d+(-14 chi1-16 (chi1^2)) (d^2)+(-3+16 chi1+2 (chi1^2)) (d^3)+(3-2 chi1) (d^4)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) v[2,1])/(4 chi1 (-2+d) d (-chi1+d))-(((-1+d) (-6 (chi1^4)+(12 (chi1^3)+6 (chi1^4)) d-6 (chi1^2) (((1+chi1))^2) (d^2)+(((9 (chi1^2))/2)+12 (chi1^3)) (d^3)+((3/8)+((3 chi1)/2)+((chi1^2)/2)) (d^4)+(-(3/4)-((13 chi1)/2)-(chi1^2)) (d^5)+chi1 (d^6)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(8 chi1 (-2+d) (d^3) (-chi1+d))-((-4+d) (-2 chi1+d) (-chi1+d) (-8+10 d-(d^2)-4 (d^3)+(d^4)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(32 chi2 (-2+d) (d^4) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1]+((-4+d) (-12 (chi1^2)+(12 chi1+6 (chi1^2)) d+(-1-6 chi1) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))) ((4 (d^2))/(chi2 (chi2-d) (2 chi2-d))-(4 (d^2) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) v[1,1])/(chi1 (chi1-d) (2 chi1-d))-(((-(d^3)+(chi1^2) (-6+6 d)+chi1 (6 d-6 (d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (-2 chi1+d) (-chi1+d))-((3-d) (-4+d) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(4 chi1 chi2 (-2+d) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[2,1]-((3 (-(1/2)+d) ((chi1^2) (1-d)+((d^2)/12)+chi1 (-d+(d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(2 chi1 (-2 chi1+d) (-chi1+d))-(3 (-4+d) (-3+d) ((chi1^2)+(-chi1-(chi1^2)) d+((1/12)+chi1) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(8 chi1 chi2 (-2+d) (d^2) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1]))/(32 chi1 chi2 (d^2) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+((-4+d) (-2+d) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))) (-(((d^2) (2+d))/(2 chi2 (chi2-d) (2 chi2-d) (-2+d)))-(-((chi2 (d^2) (2+d) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(2 (chi1^2) (chi1-d) (2 chi1-d) (-2+d) (2 (chi1^2)-3 chi1 d+(d^2))))+((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))))/(2 chi1 (chi1-d) (2 chi1-d) (-2+d))) v[1,1]-((chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-48 (chi1^4)+(96 (chi1^3)+72 (chi1^4)) d+(-36 (chi1^2)-144 (chi1^3)) (d^2)+(-12 chi1+66 (chi1^2)) (d^3)+(4+6 chi1-6 (chi1^2)) (d^4)+(-2+6 chi1) (d^5)+(d^6)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(8 (chi1^2) (-2+d) (d^2) (-2 chi1+d) (-chi1+d) (2 (chi1^2)-3 chi1 d+(d^2)))+((-(d^3)+(chi1^2) (-6+6 d)+chi1 (6 d-6 (d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))))/(8 chi1 (-2+d) (d^2) (-2 chi1+d) (-chi1+d))+((3-d) ((1/(-16+8 d+(d^2)))192 (chi2^2) (2 (chi2^2)-3 chi2 d+(d^2)) (32-32 d+6 (d^2)+(d^3)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+((-4+d) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/((-2 chi2+d) (-chi2+d))))/(32 chi1 chi2 (((-2+d))^2) (d^2) (2 (chi1^2)-3 chi1 d+(d^2)))) v[2,1]-((chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (128 (chi1^6)+(-384 (chi1^5)-1344 (chi1^6)) d+(224 (chi1^4)+4032 (chi1^5)+1216 (chi1^6)) (d^2)+(192 (chi1^3)-3792 (chi1^4)-3648 (chi1^5)) (d^3)+(-232 (chi1^2)+864 (chi1^3)+3520 (chi1^4)) (d^4)+(72 chi1+372 (chi1^2)-960 (chi1^3)) (d^5)+(-6-132 chi1-212 (chi1^2)) (d^6)+(5+84 chi1+24 (chi1^2)) (d^7)+(-2-24 chi1) (d^8)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(128 (chi1^2) (-2+d) (d^4) (-2 chi1+d) (-chi1+d) (2 (chi1^2)-3 chi1 d+(d^2)))+(3 (-(1/2)+d) ((chi1^2) (1-d)+((d^2)/12)+chi1 (-d+(d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))))/(16 chi1 (-2+d) (d^2) (-2 chi1+d) (-chi1+d))+(3 (-3+d) ((chi1^2)+(-chi1-(chi1^2)) d+((1/12)+chi1) (d^2)) ((1/(-16+8 d+(d^2)))192 (chi2^2) (2 (chi2^2)-3 chi2 d+(d^2)) (32-32 d+6 (d^2)+(d^3)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+((-4+d) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/((-2 chi2+d) (-chi2+d))))/(64 chi1 chi2 (((-2+d))^2) (d^4) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1]-(((chi2 (12 (chi1^2)-12 chi1 d-(d^2)) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(8 chi1 (-2+d) d (2 (chi1^2)-3 chi1 d+(d^2)))+(1/(8 (-2+d) (d^2)))(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2))))) ((4 (d^2))/(chi2 (chi2-d) (2 chi2-d))-(4 (d^2) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) v[1,1])/(chi1 (chi1-d) (2 chi1-d))-(((-(d^3)+(chi1^2) (-6+6 d)+chi1 (6 d-6 (d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (-2 chi1+d) (-chi1+d))-((3-d) (-4+d) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(4 chi1 chi2 (-2+d) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[2,1]-((3 (-(1/2)+d) ((chi1^2) (1-d)+((d^2)/12)+chi1 (-d+(d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(2 chi1 (-2 chi1+d) (-chi1+d))-(3 (-4+d) (-3+d) ((chi1^2)+(-chi1-(chi1^2)) d+((1/12)+chi1) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(8 chi1 chi2 (-2+d) (d^2) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1]))/((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))))/(8 (chi2^2) (-2 chi2+d) (-chi2+d) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2)))/((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))

MatrixForm[Simplify[Coefficient[ExtDiff[d,chi1,chi2][[3]][[1;;3,1]],v[2,1],1]]]

Solve[ExtDiff[d,chi1,chi2][[3]][[2,1]]==0,v[2,1]]

v[2,1]:=-(1/(8 (d^2)))(-12 (chi1^2)+12 chi1 d+12 (chi1^2) d-(d^2)-12 chi1 (d^2)) v[3,1]

MatrixForm[Simplify[Coefficient[ExtDiff[d,chi1,chi2][[2]][[1;;3,1]],v[1,1],1]]]

Solve[ExtDiff[d,chi1,chi2][[2]][[2,1]]==0,v[1,1]]

v[1,1]:=((chi1 (d^2) (2+d) (2 (chi1^2)-3 chi1 d+(d^2)))/(4 (chi2^2) (chi2-d) (2 chi2-d) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))-(-12 (chi1^2)+(12 chi1+6 (chi1^2)) d+(-1-6 chi1) (d^2))/(2 chi2 (chi2-d) (2 chi2-d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+(2 (-2+d) (d^2) ((chi2 (12 (chi1^2)-12 chi1 d-(d^2)) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(8 chi1 (-2+d) d (2 (chi1^2)-3 chi1 d+(d^2)))+(1/(8 (-2+d) (d^2)))(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2))))))/(chi2 (chi2-d) (2 chi2-d))-(1/(8 (-2+d) (d^4)))chi1 (-2 chi1+d) (-chi1+d) (-8+10 d-(d^2)-4 (d^3)+(d^4)) v[3,1]-((-12 (chi1^2)+(12 chi1+6 (chi1^2)) d+(-1-6 chi1) (d^2)) (-12 (chi1^2)+12 chi1 d+12 (chi1^2) d-(d^2)-12 chi1 (d^2)) (((-(d^3)+(chi1^2) (-6+6 d)+chi1 (6 d-6 (d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (-2 chi1+d) (-chi1+d))-((3-d) (-4+d) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(4 chi1 chi2 (-2+d) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1])/(64 (d^4) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+(1/(16 (d^2)))(-2+d) (-12 (chi1^2)+12 chi1 d+12 (chi1^2) d-(d^2)-12 chi1 (d^2)) ((chi2 (12 (chi1^2)-12 chi1 d-(d^2)) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(8 chi1 (-2+d) d (2 (chi1^2)-3 chi1 d+(d^2)))+(1/(8 (-2+d) (d^2)))(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2))))) (((-(d^3)+(chi1^2) (-6+6 d)+chi1 (6 d-6 (d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (-2 chi1+d) (-chi1+d))-((3-d) (-4+d) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(4 chi1 chi2 (-2+d) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1]+((-12 (chi1^2)+(12 chi1+6 (chi1^2)) d+(-1-6 chi1) (d^2)) ((3 (-(1/2)+d) ((chi1^2) (1-d)+((d^2)/12)+chi1 (-d+(d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(2 chi1 (-2 chi1+d) (-chi1+d))-(3 (-4+d) (-3+d) ((chi1^2)+(-chi1-(chi1^2)) d+((1/12)+chi1) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(8 chi1 chi2 (-2+d) (d^2) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1])/(8 (d^2) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))-(1/2) (-2+d) ((chi2 (12 (chi1^2)-12 chi1 d-(d^2)) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(8 chi1 (-2+d) d (2 (chi1^2)-3 chi1 d+(d^2)))+(1/(8 (-2+d) (d^2)))(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2))))) ((3 (-(1/2)+d) ((chi1^2) (1-d)+((d^2)/12)+chi1 (-d+(d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(2 chi1 (-2 chi1+d) (-chi1+d))-(3 (-4+d) (-3+d) ((chi1^2)+(-chi1-(chi1^2)) d+((1/12)+chi1) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/(8 chi1 chi2 (-2+d) (d^2) (-2 chi2+d) (-chi2+d) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1]-(chi1 (-2+d) (2 (chi1^2)-3 chi1 d+(d^2)) (-12 (chi1^2)+12 chi1 d+12 (chi1^2) d-(d^2)-12 chi1 (d^2)) ((chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-48 (chi1^4)+(96 (chi1^3)+72 (chi1^4)) d+(-36 (chi1^2)-144 (chi1^3)) (d^2)+(-12 chi1+66 (chi1^2)) (d^3)+(4+6 chi1-6 (chi1^2)) (d^4)+(-2+6 chi1) (d^5)+(d^6)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(8 (chi1^2) (-2+d) (d^2) (-2 chi1+d) (-chi1+d) (2 (chi1^2)-3 chi1 d+(d^2)))+((-(d^3)+(chi1^2) (-6+6 d)+chi1 (6 d-6 (d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))))/(8 chi1 (-2+d) (d^2) (-2 chi1+d) (-chi1+d))+((3-d) ((1/(-16+8 d+(d^2)))192 (chi2^2) (2 (chi2^2)-3 chi2 d+(d^2)) (32-32 d+6 (d^2)+(d^3)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+((-4+d) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/((-2 chi2+d) (-chi2+d))))/(32 chi1 chi2 (((-2+d))^2) (d^2) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1])/(16 chi2 (d^2) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))+(chi1 (-2+d) (2 (chi1^2)-3 chi1 d+(d^2)) ((chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (128 (chi1^6)+(-384 (chi1^5)-1344 (chi1^6)) d+(224 (chi1^4)+4032 (chi1^5)+1216 (chi1^6)) (d^2)+(192 (chi1^3)-3792 (chi1^4)-3648 (chi1^5)) (d^3)+(-232 (chi1^2)+864 (chi1^3)+3520 (chi1^4)) (d^4)+(72 chi1+372 (chi1^2)-960 (chi1^3)) (d^5)+(-6-132 chi1-212 (chi1^2)) (d^6)+(5+84 chi1+24 (chi1^2)) (d^7)+(-2-24 chi1) (d^8)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(128 (chi1^2) (-2+d) (d^4) (-2 chi1+d) (-chi1+d) (2 (chi1^2)-3 chi1 d+(d^2)))+(3 (-(1/2)+d) ((chi1^2) (1-d)+((d^2)/12)+chi1 (-d+(d^2))) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))))/(16 chi1 (-2+d) (d^2) (-2 chi1+d) (-chi1+d))+(3 (-3+d) ((chi1^2)+(-chi1-(chi1^2)) d+((1/12)+chi1) (d^2)) ((1/(-16+8 d+(d^2)))192 (chi2^2) (2 (chi2^2)-3 chi2 d+(d^2)) (32-32 d+6 (d^2)+(d^3)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2) (2 chi1-d+(-2 chi2+d) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))+((-4+d) (-24 (chi2^2)+24 chi2 d+(-2+d) (d^2)) (chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (-24 (chi2^2)+24 chi2 d+(-4+d) (d^2))+chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (4 (6 (chi1^2)-6 chi1 d+(d^2))-(d^3) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))))/((-2 chi2+d) (-chi2+d))))/(64 chi1 chi2 (((-2+d))^2) (d^4) (2 (chi1^2)-3 chi1 d+(d^2)))) v[3,1])/(2 chi2 (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2)))/(-((-12 (chi1^2)+(12 chi1+6 (chi1^2)) d+(-1-6 chi1) (d^2))/(2 chi1 (chi1-d) (2 chi1-d)))-(chi1 (-2+d) (2 (chi1^2)-3 chi1 d+(d^2)) (-((chi2 (d^2) (2+d) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(2 (chi1^2) (chi1-d) (2 chi1-d) (-2+d) (2 (chi1^2)-3 chi1 d+(d^2))))+((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2)))))/(2 chi1 (chi1-d) (2 chi1-d) (-2+d))))/(2 chi2 (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))+(2 (-2+d) (d^2) (2 (chi1^2)-3 chi1 d+(d^2)) ((chi2 (12 (chi1^2)-12 chi1 d-(d^2)) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2))/(8 chi1 (-2+d) d (2 (chi1^2)-3 chi1 d+(d^2)))+(1/(8 (-2+d) (d^2)))(((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)) (24 (chi2^2)-24 chi2 d-(-2+d) (d^2)+(chi2 (2 (chi2^2)-3 chi2 d+(d^2)) (-24 (chi1^2)+24 chi1 d+(-2+d) (d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))/(chi1 (2 (chi1^2)-3 chi1 d+(d^2))))))/(chi2 (chi1-d) (2 chi1-d) (2 (chi2^2)-3 chi2 d+(d^2)) ((((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3))^2)))

(* We are left with only v[3,1] variable. We use collections of two linear equations of v[3,1] (from ExtDiff[d,chi1,chi2][[1]][[1,1]] and ExtDiff[d,chi1,chi2][[2]][[1,1]]) to obtain constraints for the existence of the solution. *)

AA[d_,chi1_,chi2_]:=Coefficient[ExtDiff[d,chi1,chi2][[1]][[1,1]],v[3,1],1]
BB[d_,chi1_,chi2_]:=Coefficient[ExtDiff[d,chi1,chi2][[1]][[1,1]],v[3,1],0]
CC[d_,chi1_,chi2_]:=Coefficient[ExtDiff[d,chi1,chi2][[1]][[2,1]],v[3,1],1]
DD[d_,chi1_,chi2_]:=Coefficient[ExtDiff[d,chi1,chi2][[1]][[2,1]],v[3,1],0]

Constraint[d_,chi1_,chi2_]:=Simplify[Together[AA[d,chi1,chi2]*DD[d,chi1,chi2]-BB[d,chi1,chi2]*CC[d,chi1,chi2]]]

(* Show that there are no (d,chi1,chi2) with coprime 0<chi1,chi2<d unless chi2=chi1 or chi2=d-chi1. *)

AA[d,chi1,chi2]

Constraint[d,chi1,chi2]

Type II - Analysing constraints to existence of solutions

(* We can ignore chi1 (chi1-d) (12-9d+2 (d^2)) and the denominator. The rest is a product of two parts P1 and P2. *)

P1[d_,chi1_]:=6 (((-1+d))^2) (d^2) (32-26 d+5 (d^2))+4 (chi1^4) (-120+150 d+3 (d^2)-47 (d^3)+11 (d^4))-8 (chi1^3) d (-120+150 d+3 (d^2)-47 (d^3)+11 (d^4))+chi1 d (-768+2256 d-2184 (d^2)+822 (d^3)-153 (d^4)+50 (d^5)-11 (d^6))+(chi1^2) (768-2256 d+1704 (d^2)-222 (d^3)+165 (d^4)-238 (d^5)+55 (d^6))
P2[d_,chi1_,chi2_]:=(chi1 (2 (chi1^2)-3 chi1 d+(d^2)) (6 (chi2^2)-6 chi2 d+(d^2))-chi2 (6 (chi1^2)-6 chi1 d+(d^2)) (2 (chi2^2)-3 chi2 d+(d^2)) (((chi1 (2 (chi1^2)-3 chi1 d+(d^2)))/(chi2 (2 (chi2^2)-3 chi2 d+(d^2))))^(1/3)))

(* We first show that P2 is nonzero unless chi1=chi2 or chi1+chi2=d. Note that P2 vanishing is equivalent to the vanishing of the following by taking cubes. By the argument in the paper involving coprimality, this is nonzero unless chi1=chi2 or chi1+chi2=d. *)

Factor[chi1 ^2(2 (chi1^2)-3 chi1 d+(d^2))^2 (6 (chi2^2)-6 chi2 d+(d^2))^3-chi2^2 (6 (chi1^2)-6 chi1 d+(d^2))^3 (2 (chi2^2)-3 chi2 d+(d^2))^2 ]

(* Therefore, the Constraint vanishing is equivalent to P1[d,chi1]=0. By the symmetric role of chi1 and chi2, we also have P1[d,chi2]=0. *)

P1[d,chi1]
P1[d,chi2]

(* P1[d,x] is a degree four polynomial in x with a symmetry. Therefore it has four distinct roots chi1,chi2,d-chi1,d-chi2. *)

Simplify[P1[d,x]-P1[d,d-x]]

(* We check that P1[d,0]>0 and P1[d,1]<0. This implies that P1[d,x] must have one more additional root between 0 and 1, contradicting that it is a degree four polynomial.  *)

Factor[P1[d,0]]
Factor[P1[d,1]]

Solve[P1[d,0]==0,d]
Solve[P1[d,1]==0,d]

(* This proves that there are no ring isomorphisms. *)