# R:=PolynomialRing(Rationals());F:=NumberField(X^2+X-1);r5:=2*b5+1; local result, l, r5, b5; r5:= Sqrt(5); b5:=(-1+r5)/2; result:= rec(); result.comment:= "2a\"M20 = 2^5:A5 as 12 x 12 matrices over Z[b5].\n\ Representation 12a.\n\ Absolutely irreducible representation.\n\ Schur Index 1.\n\ \n\ SEED:\n\ Nonzero v fixed by x^2 and y*x*y^-1 where = [2^4].\n\ v has 2 x 30 = 60 images under G; has 30 images under G.\n\ BASIS:\n\ All in v^G (induced representation).\n\ \n\ Possible matrix entries are in {-1,0,1,b5,-b5} (norms 0, 1 only).\n\ \n\ Average number of nonzero entries for any element of the group:\n\ 19 + 1/5 (19.2; about 13.333%).\n\ \n\ Entry Av/Mat %Av/Mat\n\ 0 124.8 [124+4/5] 86.667 [86+2/3]\n\ nonzero 19.2 [ 19+1/5] 13.333 [13+1/3]\n\ ±1 9.6 [ 9+3/5] 6.667 [ 6+2/3]\n\ ±b5 9.6 [ 9+3/5] 6.667 [ 6+2/3]\n\ 1 4.8 [ 4+4/5] 3.333 [ 3+1/3]\n\ -1 4.8 [ 4+4/5] 3.333 [ 3+1/3]\n\ b5 4.8 [ 4+4/5] 3.333 [ 3+1/3]\n\ -b5 4.8 [ 4+4/5] 3.333 [ 3+1/3]\n\ "; result.symmetricforms:= []; result.antisymmetricforms:= []; result.hermitianforms:= []; result.centralizeralgebra:= []; result.generators:= List( [ [ 0,1,0,0,0,0,0,0,0,0,0,0, 1,0,0,0,0,0,0,0,0,0,0,0, 0,0,1,0,0,0,0,0,0,0,0,0, 0,0,b5,-1,0,0,0,0,0,0,0,0, 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,-1,0,0,0,0,0,0,0, 0,0,0,0,0,-1,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,-1, 0,0,0,0,0,0,0,0,0,0,1,-b5, 0,0,0,0,0,0,0,0,b5,-1,0,0, 0,0,0,0,0,0,0,0,1,0,0,0] ,[ 0,0,1,0,0,0,0,0,0,0,0,0, 0,0,0,1,0,0,0,0,0,0,0,0, 0,0,0,0,1,0,0,0,0,0,0,0, 0,0,0,0,0,1,0,0,0,0,0,0, 1,0,0,0,0,0,0,0,0,0,0,0, 0,1,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,1,0,0,0, 0,0,0,0,0,0,0,0,0,1,0,0, 0,0,0,0,0,0,0,0,0,0,1,0, 0,0,0,0,0,0,0,0,0,0,0,1, 0,0,0,0,0,0,1,0,0,0,0,0, 0,0,0,0,0,0,0,1,0,0,0,0] ], l -> List( [ 0 .. 11 ], i -> l{ [ i*12+1 .. (i+1)*12 ] } ) ); l:= [ 2,b5,0,0,0,0,0,0,0,0,0,0, b5,2,0,0,0,0,0,0,0,0,0,0, 0,0,2,b5,0,0,0,0,0,0,0,0, 0,0,b5,2,0,0,0,0,0,0,0,0, 0,0,0,0,2,b5,0,0,0,0,0,0, 0,0,0,0,b5,2,0,0,0,0,0,0, 0,0,0,0,0,0,2,b5,0,0,0,0, 0,0,0,0,0,0,b5,2,0,0,0,0, 0,0,0,0,0,0,0,0,2,b5,0,0, 0,0,0,0,0,0,0,0,b5,2,0,0, 0,0,0,0,0,0,0,0,0,0,2,b5, 0,0,0,0,0,0,0,0,0,0,b5,2]; Add( result.symmetricforms, List( [ 0 .. 11 ], i -> l{ [ i*12+1 .. (i+1)*12 ] } ) ); Add( result.centralizeralgebra, IdentityMat(12) ); return result;