/*
U5(2) as 10 x 10 matrices over Z[w].
Absolutely irreducible representation with character values in Q.
Schur index 2.

SEED:
Nonzero v such that v.x = -v and v.y*x*y^4*x*y^2 = -w*v, where
<x,y*x*y^4*x*y^2> = 2^{1+6}:3^{1+2}:6.
Stab(v) = 2^{1+6}:3^{1+2} is normal in a copy of 2^{1+6}:3^{1+2}:2A4.
v has 6 x 660 = 3960 images under G; <v> has 660 images under G.
BASIS:
All in v^G.

Possible matrix entries (for any g\in G) are in
{0,±1,±w,±w^2,±2,±2*w,±2*w^2} (13 such).
Norms of entries are in {0,1,4} (3 such).

Average number of nonzero entries for any element of the group:
58 + 7/11  (about 58.636; 58.636%).

                 Entry Norm  Av/Mat                %Av/Mat
                     0    0  41.364  [41 + 4/11]   41.364  [41 + 4/11]
               nonzero   --  58.636  [58 + 7/11]   58.636  [58 + 7/11]
Each of     ±1,±w,±w^2    1   9.697  [ 9 +23/33]    9.697  [ 9 +23/33]
Each of ±2,±2*w,±2*w^2    4   0.076  [     5/66]    0.076  [     5/66]
                norm 1    1  58.182  [58 + 2/11]   58.182  [58 + 2/11]
                norm 4    4   0.455  [     5/11]    0.455  [     5/11]
*/


F<w>:=CyclotomicField(3);
W:=w^2;i3:=2*w+1;

G<x,y>:=MatrixGroup<10,F|[
-1,0,0,0,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,0,0,
0,0,-1,0,0,0,0,0,0,0,
0,1,0,0,0,0,0,0,0,0,
0,W,0,W,-1,w,0,-1,0,0,
0,0,0,0,0,0,0,-W,0,0,
0,w,0,w,0,0,-1,0,0,0,
0,0,0,0,0,-w,0,0,0,0,
0,0,0,0,0,0,0,0,-1,0,
0,0,0,0,0,-1,0,W,0,-1]
,[
0,1,0,0,0,0,0,0,0,0,
0,0,0,0,1,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,1,0,0,0,
0,0,0,0,0,1,0,0,0,0,
0,0,1,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,0,
0,0,0,1,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,1,
0,0,0,0,0,0,0,1,0,0]
>;
a:=x;b:=y;

xc:=GL(10,F)![Conjugate(u,2):u in Eltseq(x)];
yc:=GL(10,F)![Conjugate(u,2):u in Eltseq(y)];


// Forms: B1 (Antisymmetric), B2 (Hermitian).

// B1 (Antisymmetric form): Determinant 1.
B1:=MatrixAlgebra(F,10)![
0,0,0,0,0,0,0,0,w,W,
0,0,0,0,0,0,0,W,0,w,
0,0,0,0,0,0,w,0,W,0,
0,0,0,0,-W,-w,0,0,0,0,
0,0,0,W,0,0,0,w,0,0,
0,0,0,w,0,0,W,0,0,0,
0,0,-w,0,0,-W,0,0,0,0,
0,-W,0,0,-w,0,0,0,0,0,
-w,0,-W,0,0,0,0,0,0,0,
-W,-w,0,0,0,0,0,0,0,0];

// B2 (Hermitian form): Determinant 7776 = 6^5.
B2:=MatrixAlgebra(F,10)![
6,0,0,0,-3*W,-3*w,3*W,-3,-i3*W,-i3,
0,6,-3*w,-3,0,-3*W,0,-i3,3*W,-i3*W,
0,-3*W,6,3*W,-3*w,0,-i3*W,0,-i3,-3,
0,-3,3*w,6,i3,i3*w,3*W,3*w,-3*W,-3*w,
-3*w,0,-3*W,-i3,6,0,-3,-i3*W,0,3*W,
-3*W,-3*w,0,-i3*W,0,6,-i3,3*W,-3,0,
3*w,0,i3*w,3*w,-3,i3,6,-3*w,3*W,-3*W,
-3,i3,0,3*W,i3*w,3*w,-3*W,6,-3*w,3*w,
i3*w,3*w,i3,-3*w,0,-3,3*w,-3*W,6,3*W,
i3,i3*w,-3,-3*W,3*w,0,-3*w,3*W,3*w,6];

// B2a = -B2/i3: Determinant -32.
B2a:=MatrixAlgebra(F,10)![
2*i3,0,0,0,-i3*W,-i3*w,i3*W,-i3,W,1,
0,2*i3,-i3*w,-i3,0,-i3*W,0,1,i3*W,W,
0,-i3*W,2*i3,i3*W,-i3*w,0,W,0,1,-i3,
0,-i3,i3*w,2*i3,-1,-w,i3*W,i3*w,-i3*W,-i3*w,
-i3*w,0,-i3*W,1,2*i3,0,-i3,W,0,i3*W,
-i3*W,-i3*w,0,W,0,2*i3,1,i3*W,-i3,0,
i3*w,0,-w,i3*w,-i3,-1,2*i3,-i3*w,i3*W,-i3*W,
-i3,-1,0,i3*W,-w,i3*w,-i3*W,2*i3,-i3*w,i3*w,
-w,i3*w,-1,-i3*w,0,-i3,i3*w,-i3*W,2*i3,i3*W,
-1,-w,-i3,-i3*W,i3*w,0,-i3*w,i3*W,i3*w,2*i3];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,10)!1;