# 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:= "A6 as 8 x 8 matrices over Z[b5].\n\ Representation 8b.\n\ Absolutely irreducible representation.\n\ Schur Index 1.\n\ \n\ SEED:\n\ Nonzero v fixed by = D8.\n\ v has 1 x 45 = 45 images under G; has 45 images under G.\n\ BASIS:\n\ All in v^G.\n\ \n\ Possible matrix entries are in {-1,0,1,b5,b5+1,-b5,-b5-1} (norms 0, 1 only).\n\ \n\ Average number of nonzero entries for any element of the group:\n\ 31 + 13/45 (about 31.289; 48.889%).\n\ \n\ Entry Av/Mat %Av/Mat\n\ 0 32.711 [32+32/45] 51.111 [51+1/9]\n\ nonzero 31.289 [31+13/45] 48.889 [48+8/9]\n\ ±1 15.644 [15+29/45] 24.444 [24+4/9]\n\ 1 7.822 [7+37/45] 12.222 [12+2/9]\n\ -1 7.822 [7+37/45] 12.222 [12+2/9]\n\ ±1±b5 12.644 [15+29/45] 24.444 [24+4/9]\n\ b5 3.911 [3+41/45] 6.111 [6+1/9]\n\ -b5 3.911 [3+41/45] 6.111 [6+1/9]\n\ 1+b5 3.911 [3+41/45] 6.111 [6+1/9]\n\ -1-b5 3.911 [3+41/45] 6.111 [6+1/9]\n\ "; result.symmetricforms:= []; result.antisymmetricforms:= []; result.hermitianforms:= []; result.centralizeralgebra:= []; result.generators:= List( [ [ 1,0,0,0,0,0,0,0, 1,-1,1,1,1,1,-1,1, 0,0,0,0,0,0,0,1, b5+1,0,0,b5,b5+1,b5,-b5-1,0, -b5-1,0,0,-b5,-b5-1,-b5,b5,0, 0,0,0,0,0,1,0,0, 0,0,0,-1,-1,0,0,0, 0,0,1,0,0,0,0,0] ,[ 1,0,1,0,0,1,0,1, 1,0,0,0,b5+1,0,-b5-1,0, 0,0,0,0,1,0,0,0, b5+1,0,0,0,1,0,-b5-1,0, -b5-1,0,0,b5,-1,-1,1,0, 0,-1,0,0,0,0,-1,0, 0,b5,0,0,-b5,0,0,1, 0,0,0,1,0,0,0,0] ], l -> List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); l:= [ 8,-3*b5-1,-1,-1,-3*b5-1,-4,3*b5+2,-1, -3*b5-1,8,3*b5+2,3*b5+2,-3*b5-1,3*b5+2,-4,-3*b5-1, -1,3*b5+2,8,-3*b5-1,-1,-1,-3*b5-1,-4, -1,3*b5+2,-3*b5-1,8,-4,3*b5+2,-3*b5-1,-1, -3*b5-1,-3*b5-1,-1,-4,8,-1,3*b5+2,3*b5+2, -4,3*b5+2,-1,3*b5+2,-1,8,-3*b5-1,-1, 3*b5+2,-4,-3*b5-1,-3*b5-1,3*b5+2,-3*b5-1,8,3*b5+2, -1,-3*b5-1,-4,-1,3*b5+2,-1,3*b5+2,8]; Add( result.symmetricforms, List( [ 0 .. 7 ], i -> l{ [ i*8+1 .. (i+1)*8 ] } ) ); Add( result.centralizeralgebra, IdentityMat(8) ); return result;