# F:=Rationals(); local result, l; result:= rec(); result.comment:= "2a\"M20 = 2^5:A5 as 20 x 20 matrices over Z.\n\ Representation 20[a].\n\ Absolutely irreducible representation.\n\ Schur Index 1.\n\ \n\ SEED:\n\ Nonzero v fixed by x^2 and x^(yxyxy^-1xy^-1x) and negated by yxyxy^-1xy^-1\n\ where = [2^5].\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}.\n\ \n\ Average number of nonzero entries for any element of the group:\n\ 26 + 2/3 (about 26.667; 6.667%).\n\ \n\ Entry Av/Mat %Av/Mat\n\ 0 373.333 [373+1/3] 93.333 [93+1/3]\n\ ±1 26.667 [ 26+2/3] 6.667 [ 6+2/3]\n\ 1 13.333 [ 13+1/3] 3.333 [ 3+1/3]\n\ -1 13.333 [ 13+1/3] 3.333 [ 3+1/3]\n\ "; result.symmetricforms:= []; result.antisymmetricforms:= []; result.hermitianforms:= []; result.centralizeralgebra:= []; result.generators:= List( [ [ 0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0, 0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-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, 0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0] ,[ 0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0, 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1] ], l -> List( [ 0 .. 19 ], i -> l{ [ i*20+1 .. (i+1)*20 ] } ) ); l:= [ 2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, -1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,2,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,-1,2,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,-1,2,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-1,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,2,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-1,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,2,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,-1, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,2]; Add( result.symmetricforms, List( [ 0 .. 19 ], i -> l{ [ i*20+1 .. (i+1)*20 ] } ) ); Add( result.centralizeralgebra, IdentityMat(20) ); return result;