needsPackage("LieAlgebraRepresentations");
debug LieAlgebraRepresentations;

GTPplusdelta = (GTP,i,j) -> (
    gtpA(gtpPMDeltakiEntries(GTP,1,i,j))
);


aIJXkFactors = (GTP, i,k) -> (
    n:=#(GTP#"weight")+1;
    num1 := product apply(toList(1..k-1), q -> ((GTP#(k,i)-i+1)-(GTP#(k-1,q)-q+1)+1));
    num2 := product apply(toList(1..k+1), q -> ((GTP#(k,i)-i+1)-(GTP#(k+1,q)-q+1)));
    denom1 := product apply(toList(1..k), q -> ((GTP#(k,i)-i+1)-(GTP#(k,q)-q+1)+1));
    denom2 := product apply(toList(1..k), q -> if q==i then 1 else ((GTP#(k,i)-i+1)-(GTP#(k,q)-q+1)));
    {num1,num2,denom1,denom2}
);


aIJYkFactors = (GTP, i,k) -> (
    n:=#(GTP#"weight")+1;
    num1 := product apply(toList(1..k-1), q -> ((GTP#(k,i)-i+1)-(GTP#(k-1,q)-q+1)));
    num2 := product apply(toList(1..k+1), q -> ((GTP#(k,i)-i+1)-(GTP#(k+1,q)-q+1)-1));
    denom1 := product apply(toList(1..k), q -> ((GTP#(k,i)-i+1)-(GTP#(k,q)-q+1)-1));
    denom2 :=product apply(toList(1..k), q -> if q==i then 1 else ((GTP#(k,i)-i+1)-(GTP#(k,q)-q+1)));
    {num1,num2,denom1,denom2}
);



end


load "Proposition_A.4_proof.m2"

-- Example 1
g = simpleLieAlgebra("A",3);
lambda = {2,0,0};
Blambda = gtPatterns("A",dynkinToPartition("A",lambda));
N = #Blambda;

-- Test Xk
LX = {};
for z from 0 to N-1 do (
  P = gtpA(Blambda_z);   
  for k from 1 to #lambda do (
    for i from 1 to k do (
      GTPPlusDeltakiEntries:=gtpPMDeltakiEntries(P,1,k,i);
      if not isValidEntryList("A",GTPPlusDeltakiEntries) then continue;
      Q = dualGTPattern(GTPPlusDeltakiEntries);
      v1=aIJXkFactors(P,i,k);
      v2=aIJXkFactors(gtpA(Q),k+1-i,k);
      LX = append(LX,{z,k,i,v1,v2,v2==(-1)^(k-1)*{v1_0,v1_1,v1_3,v1_2}})
)));
all(LX, x -> last x)

-- Test Yk
LY = {};
for z from 0 to N-1 do (
  P = gtpA(Blambda_z);   
  for k from 1 to #lambda do (
    for i from 1 to k do (
      GTPMinusDeltakiEntries:=gtpPMDeltakiEntries(P,-1,k,i);
      if not isValidEntryList("A",GTPMinusDeltakiEntries) then continue;
      Q = dualGTPattern(GTPMinusDeltakiEntries);
      v1=aIJYkFactors(P,i,k);
      v2=aIJYkFactors(gtpA(Q),k+1-i,k);
      LY = append(LY,{z,k,i,v1,v2,v2==(-1)^(k-1)*{v1_0,v1_1,-v1_3,-v1_2}})
)));
all(LY, x -> last x)





-- Example 2
g = simpleLieAlgebra("A",3);
lambda = {3,2,1};
Blambda = gtPatterns("A",dynkinToPartition("A",lambda));
N = #Blambda;

-- Test Xk
LX = {};
for z from 0 to N-1 do (
  P = gtpA(Blambda_z);   
  for k from 1 to #lambda do (
    for i from 1 to k do (
      GTPPlusDeltakiEntries:=gtpPMDeltakiEntries(P,1,k,i);
      if not isValidEntryList("A",GTPPlusDeltakiEntries) then continue;
      Q = dualGTPattern(GTPPlusDeltakiEntries);
      v1=aIJXkFactors(P,i,k);
      v2=aIJXkFactors(gtpA(Q),k+1-i,k);
      LX = append(LX,{z,k,i,v1,v2,v2==(-1)^(k-1)*{v1_0,v1_1,v1_3,v1_2}})
)));
all(LX, x -> last x)

-- Test Yk
LY = {};
for z from 0 to N-1 do (
  P = gtpA(Blambda_z);   
  for k from 1 to #lambda do (
    for i from 1 to k do (
      GTPMinusDeltakiEntries:=gtpPMDeltakiEntries(P,-1,k,i);
      if not isValidEntryList("A",GTPMinusDeltakiEntries) then continue;
      Q = dualGTPattern(GTPMinusDeltakiEntries);
      v1=aIJYkFactors(P,i,k);
      v2=aIJYkFactors(gtpA(Q),k+1-i,k);
      LY = append(LY,{z,k,i,v1,v2,v2==(-1)^(k-1)*{v1_0,v1_1,-v1_3,-v1_2}})
)));
all(LY, x -> last x)




-- Example 3
g = simpleLieAlgebra("A",5);
lambda = {1,0,0,0,1};
Blambda = gtPatterns("A",dynkinToPartition("A",lambda));
N = #Blambda;

-- Test Xk
LX = {};
for z from 0 to N-1 do (
  P = gtpA(Blambda_z);   
  for k from 1 to #lambda do (
    for i from 1 to k do (
      GTPPlusDeltakiEntries:=gtpPMDeltakiEntries(P,1,k,i);
      if not isValidEntryList("A",GTPPlusDeltakiEntries) then continue;
      Q = dualGTPattern(GTPPlusDeltakiEntries);
      v1=aIJXkFactors(P,i,k);
      v2=aIJXkFactors(gtpA(Q),k+1-i,k);
      LX = append(LX,{z,k,i,v1,v2,v2==(-1)^(k-1)*{v1_0,v1_1,v1_3,v1_2}})
)));
all(LX, x -> last x)

-- Test Yk
LY = {};
for z from 0 to N-1 do (
  P = gtpA(Blambda_z);   
  for k from 1 to #lambda do (
    for i from 1 to k do (
      GTPMinusDeltakiEntries:=gtpPMDeltakiEntries(P,-1,k,i);
      if not isValidEntryList("A",GTPMinusDeltakiEntries) then continue;
      Q = dualGTPattern(GTPMinusDeltakiEntries);
      v1=aIJYkFactors(P,i,k);
      v2=aIJYkFactors(gtpA(Q),k+1-i,k);
      LY = append(LY,{z,k,i,v1,v2,v2==(-1)^(k-1)*{v1_0,v1_1,-v1_3,-v1_2}})
)));
all(LY, x -> last x)



-- Example 4
g = simpleLieAlgebra("A",6);
lambda = {2,0,0,0,1,0};
Blambda = gtPatterns("A",dynkinToPartition("A",lambda));
N = #Blambda;

-- Test Xk
LX = {};
for z from 0 to N-1 do (
  P = gtpA(Blambda_z);   
  for k from 1 to #lambda do (
    for i from 1 to k do (
      GTPPlusDeltakiEntries:=gtpPMDeltakiEntries(P,1,k,i);
      if not isValidEntryList("A",GTPPlusDeltakiEntries) then continue;
      Q = dualGTPattern(GTPPlusDeltakiEntries);
      v1=aIJXkFactors(P,i,k);
      v2=aIJXkFactors(gtpA(Q),k+1-i,k);
      LX = append(LX,{z,k,i,v1,v2,v2==(-1)^(k-1)*{v1_0,v1_1,v1_3,v1_2}})
)));
all(LX, x -> last x)

-- Test Yk
LY = {};
for z from 0 to N-1 do (
  P = gtpA(Blambda_z);   
  for k from 1 to #lambda do (
    for i from 1 to k do (
      GTPMinusDeltakiEntries:=gtpPMDeltakiEntries(P,-1,k,i);
      if not isValidEntryList("A",GTPMinusDeltakiEntries) then continue;
      Q = dualGTPattern(GTPMinusDeltakiEntries);
      v1=aIJYkFactors(P,i,k);
      v2=aIJYkFactors(gtpA(Q),k+1-i,k);
      LY = append(LY,{z,k,i,v1,v2,v2==(-1)^(k-1)*{v1_0,v1_1,-v1_3,-v1_2}})
)));
all(LY, x -> last x)