# F:=Rationals(); local result, l; result:= rec(); result.comment:= "A6 as 5 x 5 matrices over Z.\n\ Representation 5b.\n\ Absolutely irreducible representation.\n\ Schur Index 1.\n\ \n\ SEED:\n\ Nonzero v fixed by = A5 [L2(5)-type].\n\ v has 1 x 6 = 6 images under G; has 6 images under G.\n\ BASIS:\n\ NSB([x,y,y^2]) with above v.\n\ \n\ Possible matrix entries are in {-1,0,1}.\n\ \n\ Average number of nonzero entries for any element of the group:\n\ 8 + 1/3 (about 8.333; 33.333%).\n\ \n\ Entry Av/Mat %Av/Mat\n\ 0 16.667 [16+2/3] 66.667 [66+2/3]\n\ ±1 8.333 [8+1/3] 33.333 [33+1/3]\n\ 1 4.167 [4+1/6] 16.667 [16+2/3]\n\ -1 4.167 [4+1/6] 16.667 [16+2/3]\n\ "; result.symmetricforms:= []; result.antisymmetricforms:= []; result.hermitianforms:= []; result.centralizeralgebra:= []; result.generators:= List( [ [ 1,0,0,0,0, 0,0,0,1,0, 0,0,0,0,1, 0,1,0,0,0, 0,0,1,0,0] ,[ 0,1,0,0,0, 0,0,1,0,0, 0,0,0,0,1, -1,-1,-1,-1,-1, 1,0,0,0,0] ], l -> List( [ 0 .. 4 ], i -> l{ [ i*5+1 .. (i+1)*5 ] } ) ); l:= [ 5,-1,-1,-1,-1, -1,5,-1,-1,-1, -1,-1,5,-1,-1, -1,-1,-1,5,-1, -1,-1,-1,-1,5]; Add( result.symmetricforms, List( [ 0 .. 4 ], i -> l{ [ i*5+1 .. (i+1)*5 ] } ) ); Add( result.centralizeralgebra, IdentityMat(5) ); return result;