/* U5(2) as 11 x 11 matrices over Z[w]. Representation 11b, 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^2*v, where = 2^{4+4}:(3 x A5). v has 3 x 297 = 891 images under G; 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:=CyclotomicField(3); W:=w^2; G:=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;