needsPackage("LieAlgebraRepresentations");
debug LieAlgebraRepresentations;

-- Here we list the quantities whose factorial will be taken
-- We don't actually compute the factorials
molevFactors = (GTP) -> (
    n:=#(GTP#"weight")+1;
    p:=1;
    N1:={};
    N2:={};
    D1:={};
    D2:={};
    for k from 2 to n do (
        for i from 1 to k-1 do (
	    for j from i to k-1 do (
		N1 = append(N1,((GTP#(k,i)-i+1)-(GTP#(k-1,j)-j+1)));
		D1 = append(D1,((GTP#(k-1,i)-i+1)-(GTP#(k-1,j)-j+1)));
	    )
	);
        for i from 1 to k-1 do (
	    for j from i+1 to k do (
		N2 = append(N2,((GTP#(k,i)-i+1)-(GTP#(k,j)-j+1)-1));
		D2 = append(D2,((GTP#(k-1,i)-i+1)-(GTP#(k,j)-j+1)-1));
	    )
	);
    );
    {N1,N2,D1,D2}
)


matchedFactors = (L0,L1) -> (
   sort(L1_0)==sort(L0_3) and sort(L1_2)==sort(L0_2) and sort(L1_1)==sort(L0_1) and sort(L1_3)==sort(L0_0)
);


end


load "Proposition_A.8_proof.m2";



-- Example 1
g = simpleLieAlgebra("A",3);
lambda = {2,0,0};
lambdastar = reverse lambda;
Blambda = gtPatterns("A",dynkinToPartition("A",lambda));
Blambdastar = gtPatterns("A",dynkinToPartition("A",lambdastar));
N = #Blambda;
sigma = apply(N, i -> first select(N, k -> Blambdastar_k==dualGTPattern(Blambda_i)));
Blambdastar = apply(N, i -> Blambdastar_(sigma_i));

all(N, i -> matchedFactors(molevFactors(gtpA(Blambda_i)),molevFactors(gtpA(Blambdastar_i))))




-- Example 2
g = simpleLieAlgebra("A",3);
lambda = {3,2,1};
lambdastar = reverse lambda;
Blambda = gtPatterns("A",dynkinToPartition("A",lambda));
Blambdastar = gtPatterns("A",dynkinToPartition("A",lambdastar));
N = #Blambda;
sigma = apply(N, i -> first select(N, k -> Blambdastar_k==dualGTPattern(Blambda_i)));
Blambdastar = apply(N, i -> Blambdastar_(sigma_i));

all(N, i -> matchedFactors(molevFactors(gtpA(Blambda_i)),molevFactors(gtpA(Blambdastar_i))))




-- Example 3
g = simpleLieAlgebra("A",5);
lambda = {1,0,0,0,1};
lambdastar = reverse lambda;
Blambda = gtPatterns("A",dynkinToPartition("A",lambda));
Blambdastar = gtPatterns("A",dynkinToPartition("A",lambdastar));
N = #Blambda;
sigma = apply(N, i -> first select(N, k -> Blambdastar_k==dualGTPattern(Blambda_i)));
Blambdastar = apply(N, i -> Blambdastar_(sigma_i));

all(N, i -> matchedFactors(molevFactors(gtpA(Blambda_i)),molevFactors(gtpA(Blambdastar_i))))



-- Example 4
g = simpleLieAlgebra("A",6);
lambda = {2,0,0,0,1,0};
lambdastar = reverse lambda;
Blambda = gtPatterns("A",dynkinToPartition("A",lambda));
Blambdastar = gtPatterns("A",dynkinToPartition("A",lambdastar));
N = #Blambda;
sigma = apply(N, i -> first select(N, k -> Blambdastar_k==dualGTPattern(Blambda_i)));
Blambdastar = apply(N, i -> Blambdastar_(sigma_i));

all(N, i -> matchedFactors(molevFactors(gtpA(Blambda_i)),molevFactors(gtpA(Blambdastar_i))))