needsPackage("LieAlgebraRepresentations");
debug LieAlgebraRepresentations;


PPstar = (GTP) -> (
    n:=#(GTP#"weight")+1;
    p:=1;
    N2:=1;
    D1:=1;
    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 = 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 = 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));
	    )
	);
    );
    N2/D1
)



end


load "Proposition_A.8_statement.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 -> sqrt(molevCoefficient(gtpA(Blambda_i))*molevCoefficient(gtpA(Blambdastar_i)))==PPstar(gtpA(Blambda_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 -> sqrt(molevCoefficient(gtpA(Blambda_i))*molevCoefficient(gtpA(Blambdastar_i)))==PPstar(gtpA(Blambda_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 -> sqrt(molevCoefficient(gtpA(Blambda_i))*molevCoefficient(gtpA(Blambdastar_i)))==PPstar(gtpA(Blambda_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 -> sqrt(molevCoefficient(gtpA(Blambda_i))*molevCoefficient(gtpA(Blambdastar_i)))==PPstar(gtpA(Blambda_i)))