// See BCN, pp.199-200
// The eigenmatrix of GH(s,t)
// u=(st)^{1/2}
eigenmatrixOfGeneralizedHexagon:=function(s,t,u)
F:=Parent(s);
k:=s*(t+1);
b0:=k;
lambda:=s-1;
a1:=lambda;
c1:=1;
c2:=1;
c3:=t+1;
b1:=k-(lambda+1)*c1;
b2:=k-(lambda+1)*c2;
a2:=k-b2-c2;
a3:=k-c3;
B:=Matrix(F,4,4,[
 0,b0,0,0,
c1,a1,b1,0,
 0,c2,a2,b2,
 0, 0,c3,a3]);
// Factorization(CharacteristicPolynomial(B));
theta:=[s*(t+1),s-1+u,s-1-u,-t-1];
return Matrix(F,4,4,[Basis(Eigenspace(B,th))[1]:th in theta]);
end function;

F<ss,tt>:=FunctionField(Rationals(),3);
s:=ss^2;
t:=tt^2;
u:=ss*tt;
P:=eigenmatrixOfGeneralizedHexagon(s,t,u);
n:=&+Eltseq(P[1]);
n eq (1+s)*(1+s*t+s^2*t^2);
Eltseq(P) eq [
1,s*(t+1),s^2*t*(t+1),s^3*t^2,
1,s-1+u,(s-1)*u-s,-s*u,
1,s-1-u,-(s-1)*u-s,s*u,
1,-t-1,t*(t+1),-t^2];
Q:=n*P^(-1);
&and{Q[i,j]/Q[1,j] eq P[j,i]/P[1,i]:i,j in {1..4}};

P:=eigenmatrixOfGeneralizedHexagon(2,2,2);
Eltseq(P) eq [
1,6,24,32,
1,3,0,-4,
1,-1,-4,4,
1,-3,6,-4];