/*
U5(2) as 11 x 11 matrices over Z[w].
Representation 11a, absolutely irreducible.
Schur index 1.

SEED:
Nonzero v s.t. v*y = v and v*x*y^2*x*y^3*x*y*x*y*x*y*x*y^4*x = w*v,
where <y,x*y^2*x*y^3*x*y*x*y*x*y*x*y^4*x> = 2^{4+4}:(3 x A5).
v has 3 x 297 = 891 images under G; <v> has 297 images under G.
BASIS:
All in v^G.

Possible matrix entries (for any g\in G) are in {0,1,w,w^2,-1,-w,-w^2}.

Average number of nonzero entries for any element of the group:
66  (66; about 54.545%).

  Entry  Av/Mat   %Av/Mat
      0    55      45.455 [45+5/11]
nonzero    66      54.545 [54+6/11]
      1    11       9.091  [9+1/11]
     -1    11       9.091  [9+1/11]
      w    11       9.091  [9+1/11]
     -w    11       9.091  [9+1/11]
    w^2    11       9.091  [9+1/11]
   -w^2    11       9.091  [9+1/11]
*/


F<w>:=CyclotomicField(3);
W:=w^2;

G<x,y>:=MatrixGroup<11,F|[
0,1,0,0,0,0,0,0,0,0,0,
1,0,0,0,0,0,0,0,0,0,0,
0,0,0,1,0,0,0,0,0,0,0,
0,0,1,0,0,0,0,0,0,0,0,
-1,-1,0,0,-1,0,1,0,0,0,0,
0,0,W,W,0,-1,0,0,0,0,0,
0,0,0,0,0,0,1,0,0,0,0,
0,0,0,0,0,0,-W,-1,0,0,0,
0,0,-1,-1,0,0,W,0,-1,0,0,
0,0,1,1,0,0,0,0,0,-1,0,
1,1,0,0,0,0,0,0,0,0,-1]
,[
1,0,0,0,0,0,0,0,0,0,0,
0,0,1,0,0,0,0,0,0,0,0,
0,0,0,0,1,0,0,0,0,0,0,
0,0,0,0,0,1,0,0,0,0,0,
0,0,0,0,0,0,0,w,-1,w,0,
0,0,0,0,0,0,0,0,1,0,0,
1,0,1,0,-W,-W,0,0,0,w,0,
0,0,-W,w,-1,-W,0,1,0,0,-1,
0,0,-1,-1,0,0,W,1,0,1,0,
-W,w,0,0,1,0,0,-1,0,-1,-w,
0,0,-W,w,0,-W,0,-w,1,-w,0]
>;
a:=x;b:=y;

xc:=GL(11,F)![Conjugate(u,2):u in Eltseq(x)];
yc:=GL(11,F)![Conjugate(u,2):u in Eltseq(y)];


// Forms: B1 (Hermitian).
// B1 (Hermitian form): Determinant 486 = 3^5*2.
B1:=MatrixAlgebra(F,11)![
4,-2,-2,1,-2,1,1,W,1,1,1,
-2,4,1,-2,1,W,1,1,w,-2,1,
-2,1,4,-2,1,-2,W,w,W,1,w,
1,-2,-2,4,W,-2*W,W,W,w,1,W,
-2,1,1,w,4,-2,1,1,-2,W,-2,
1,w,-2,-2*w,-2,4,w,w,-2*W,W,1,
1,1,w,w,1,W,4,-2*w,w,w,1,
w,1,W,w,1,W,-2*W,4,W,-2,1,
1,W,w,W,-2,-2*w,W,w,4,w,1,
1,-2,1,1,w,w,W,-2,W,4,W,
1,1,W,w,-2,1,1,1,1,w,4];


// Centralising algebra: Scalars only.
C1:=MatrixAlgebra(F,11)!1;