/*
U4(2) as 5 x 5 matrices over Z[w].
Representation 5a.
Absolutely irreducible representation.
Schur Index 1.

MONOMIAL GROUP:
M := <y> = 5.

SEED:
Nonzero v fixed by x.
Stab_G(<v>) is <x,y*x*y^-1*x*y,y^2*x*y^-2*x*y^2> = 2.(A4xA4).2
with v.y*x*y^-1*x*y = w*v and v.y^2*x*y^-2*x*y^2 = -w^2*v.
v has 6 x 45 = 270 images under G; <v> has 45 images under G.
BASIS:
v, v*y, v*y^2, v*y^3, v*y^4.

Possible matrix entries are in {0,1,-1,w,-w,w^2,-w^2}
(norms 0, 1 only).

Average number of nonzero entries for any element of the group:
15  (exactly 60%).

  Entry   Av/Mat             %Av/Mat
      0   10              40
nonzero   15              60
      1    2.5  [2+1/2]   10
     -1    2.5  [2+1/2]   10
      w    2.5  [2+1/2]   10
     -w    2.5  [2+1/2]   10
    w^2    2.5  [2+1/2]   10
   -w^2    2.5  [2+1/2]   10


p-MODULAR REDUCTIONS:
p = 2: 1.4a (uniserial).
other: absolutely irreducible.
*/


F<w>:=CyclotomicField(3);

G<x,y>:=MatrixGroup<5,F|[
1,0,0,0,0,
-w,-1,0,0,0,
w^2,0,-1,0,0,
w,0,0,-1,0,
-w^2,0,0,0,-1]
,\[
0,1,0,0,0,
0,0,1,0,0,
0,0,0,1,0,
0,0,0,0,1,
1,0,0,0,0]
>;
a:=x;b:=y;

xc:=GL(5,F)![Conjugate(u,2):u in Eltseq(x)];
yc:=GL(5,F)![Conjugate(u,2):u in Eltseq(y)];


// Forms: B1 (Hermitian).
// B1 (Hermitian form): Determinant 2.
B1:=MatrixAlgebra(F,5)![
2,-w^2,w,w^2,-w,
-w,2,-w^2,w,w^2,
w^2,-w,2,-w^2,w,
w,w^2,-w,2,-w^2,
-w^2,w,w^2,-w,2];

// x*B1*Transpose(xc) eq B1;
// y*B1*Transpose(yc) eq B1;


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,5)![
1,0,0,0,0,
0,1,0,0,0,
0,0,1,0,0,
0,0,0,1,0,
0,0,0,0,1];