/*
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 negated by x, y*x*y^-1, y^2*x*y^-2, y^3*x*y^-3 where.
<x,y*x*y^-1,y^2*x*y^-2,y^3*x*y^-3> = 3^3:S4.
v has 2 x 40 = 80 images under G; <v> has 40 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: 4a.1 (uniserial).
other: absolutely irreducible.
*/




F<w>:=CyclotomicField(3);
th:=2*w+1; // This is \theta [= i3] in the ATLAS.

G<x,y>:=MatrixGroup<5,F|[
-1,0,0,0,0,
0,-1,0,0,0,
0,0,-1,0,0,
0,0,0,-1,0,
-w^2,w,w^2,-w,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 16 = 2^4.
B1:=MatrixAlgebra(F,5)![
3,-1,-th,th,-1,
-1,3,-1,-th,th,
th,-1,3,-1,-th,
-th,th,-1,3,-1,
-1,-th,th,-1,3];

// 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];