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

SEED:
Nonzero v such that v.x = v and v.y^2*x*y^3*x*y^2 = i*v, where
<x,y^2*x*y^3*x*y^2> = 2 x 12.
Stab(<v>) = <x,y^2*x*y^3*x*y^2,(y*x*y^4*x*y)^3> = 2^{1+6}:3^{1+2}:4,
            where v*(y*x*y^4*x*y)^3 = -v.
Stab(v) = 2^{1+6}:3^{1+2} is normal in
               <x,y^2*x*y^3*x*y^2,y*x*y^4*x*y> = 2^{1+6}:3^{1+2}:2A4.
v has 4 x 990 = 3960 images under G; <v> has 990 images under G.
BASIS:
All in v^G.

Possible matrix entries (for any g\in G) are in
{0,±1,±i,±1±i,±2,±2*i,±2±2*i} (17 such).
Norms of entries are in {0,1,2,4,8} (5 such).

Average number of nonzero entries for any element of the group:
65 + 5/33  (about 65.152; 65.152%).

          Entry Norm  Av/Mat                %Av/Mat
              0    0  34.848  [34 + 28/33]   34.848  [34 + 28/33]
        nonzero   --  65.152  [65 +  5/33]   65.152  [65 +  5/33]
Each of   ±1,±i    1  12.929  [12 + 92/99]   12.929  [12 + 92/99]
Each of    ±1±i    2   3.232  [ 3 + 23/99]    3.232  [ 3 + 23/99]
Each of ±2,±2*i    4   0.101  [     10/99]    0.101  [     10/99]
Each of  ±2±2*i    8   0.025  [     5/198]    0.025  [     5/198]
         norm 1    1  51.717  [51 + 71/99]   51.717  [51 + 71/99]
         norm 2    2  12.929  [12 + 92/99]   12.929  [12 + 92/99]
         norm 4    4   0.404  [     40/99]    0.404  [     40/99]
         norm 8    8   0.101  [     10/99]    0.101  [     10/99]
*/


F<i>:=QuadraticField(-1);

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

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


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

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

// B2 (Hermitian form): Determinant 32 = 2^5.
B2:=MatrixAlgebra(F,10)![
4,-1-i,-1-i,2*i,-1-i,-2*i,1+i,-1+i,0,-2,
-1+i,4,-2,-1-i,-1-i,2*i,-1-i,-2*i,1+i,0,
-1+i,-2,4,0,1+i,1-i,0,-1+i,0,1-i,
-2*i,-1+i,0,4,-2,-1-i,-1-i,2*i,-1-i,1+i,
-1+i,-1+i,1-i,-2,4,0,1+i,1-i,0,0,
2*i,-2*i,1+i,-1+i,0,4,-2,-1-i,-1-i,-1-i,
1-i,-1+i,0,-1+i,1-i,-2,4,0,1+i,0,
-1-i,2*i,-1-i,-2*i,1+i,-1+i,0,4,-2,-1-i,
0,1-i,0,-1+i,0,-1+i,1-i,-2,4,1+i,
-2,0,1+i,1-i,0,-1+i,0,-1+i,1-i,4];

// B2a = B2/(1-i): Determinant i.
B2a:=MatrixAlgebra(F,10)![
2*(1+i),-i,-i,-1+i,-i,1-i,i,-1,0,-1-i,
-1,2*(1+i),-1-i,-i,-i,-1+i,-i,1-i,i,0,
-1,-1-i,2*(1+i),0,i,1,0,-1,0,1,
1-i,-1,0,2*(1+i),-1-i,-i,-i,-1+i,-i,i,
-1,-1,1,-1-i,2*(1+i),0,i,1,0,0,
-1+i,1-i,i,-1,0,2*(1+i),-1-i,-i,-i,-i,
1,-1,0,-1,1,-1-i,2*(1+i),0,i,0,
-i,-1+i,-i,1-i,i,-1,0,2*(1+i),-1-i,-i,
0,1,0,-1,0,-1,1,-1-i,2*(1+i),i,
-1-i,0,i,1,0,-1,0,-1,1,2*(1+i)];


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