local result, l, i6; i6:= Sqrt(-6); result:= rec(); result.comment:= "2\"A5 as 4 x 4 matrices over Z[i6].\n\ Absolutely irreducible representation.\n\ Schur Index 2.\n\ \n\ SEED:\n\ Nonzero v s.t. v.(1+i6*x+y+y^2) = 0. v is fixed by = 3.\n\ v has 2 x 20 = 40 images under G; has 20 images under G.\n\ BASIS:\n\ v, v.x, v.y, v.y^2*x.\n\ \n\ Possible matrix entries for any element of the group:\n\ The 31 elements of {0,±1,±2,±4,±i6,±i6±1,±i6±2,±i6±3,±i6±5,±2i6,±2i6±4} only.\n\ The possible norms are in {0,1,4,6,7,10,15,16,24,31,40} only.\n\ \n\ Average number of nonzero entries for any element of the group:\n\ 13 + 1/5 (13.2; 82.5% exactly).\n\ \n\ Entry Av/Mat %Av/Mat\n\ 0 2.8 17.5\n\ ±1 2.0 ea 12.5 each\n\ ±2 0.4 ea 2.5 each\n\ ±4 0.6 ea 3.75 each\n\ ±i6 0.4 ea 2.5 each\n\ ±i6±1 0.6 ea 3.75 each\n\ ±i6±2 0.2 ea 1.25 each\n\ ±i6±3 0.2 ea 1.25 each\n\ ±i6±5 0.2 ea 1.25 each\n\ ±2i6 0.4 ea 2.5 each\n\ ±2i6±4 0.2 ea 1.25 each\n\ Norm 0 [0] 2.8 17.5\n\ Norm 1 [±1] 4.0 25\n\ Norm 4 [±2] 0.8 5\n\ Norm 6 [±i6] 0.8 5\n\ Norm 7 [±i6±1] 2.4 15\n\ Norm 10 [±i6±2] 0.8 5\n\ Norm 15 [±i6±3] 0.8 5\n\ Norm 16 [±4] 1.2 7.5\n\ Norm 24 [±2i6] 0.8 5\n\ Norm 31 [±i6±5] 0.8 5\n\ Norm 40 [±2i6±4] 0.8 5\n\ nonzero 13.2 82.5\n\ "; result.symmetricforms:= []; result.antisymmetricforms:= []; result.hermitianforms:= []; result.centralizeralgebra:= []; result.generators:= List( [ [ 0,1,0,0, -1,0,0,0, i6,-1,0,-1, 1,i6,1,0] ,[ 0,0,1,0, 0,1,0,0, -1,-i6,-1,0, -1,-2,i6+1,1] ], l -> List( [ 0 .. 3 ], i -> l{ [ i*4+1 .. (i+1)*4 ] } ) ); l:= [ 0,0,1,0, 0,0,0,-1, -1,0,0,-1, 0,1,1,0]; Add( result.antisymmetricforms, List( [ 0 .. 3 ], i -> l{ [ i*4+1 .. (i+1)*4 ] } ) ); l:= [ 12,-4*i6,i6+6,-4*i6, 4*i6,12,4*i6,-i6+6, -i6+6,-4*i6,12,-i6-6, 4*i6,i6+6,i6-6,12]; Add( result.hermitianforms, List( [ 0 .. 3 ], i -> l{ [ i*4+1 .. (i+1)*4 ] } ) ); Add( result.centralizeralgebra, IdentityMat(4) ); return result;