############################################################################## ## ## Green functions for ## E_8(q), q power of good prime ## ## http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/Green/ ## Frank.Luebeck@Math.RWTH-Aachen.de ## ## This file can be read with GAP 3 or GAP 4. # defining q if not bound and a helper function if not IsBound(SetName) then FamilyObj:=0;LaurentPolynomialByCoefficients:=0; fi; if not IsBound(q) then q := Indeterminate(Cyclotomics); if not IsBound(SetName) then q.name := "q"; else SetName(q,"q"); fi; fi; if q = Indeterminate(Cyclotomics) then if not IsBound(SetName) then tmpf:=function(cv) return Polynomial(Cyclotomics,cv[1],cv[2]);end; else tmpf := function(cv) return LaurentPolynomialByCoefficients( FamilyObj(1),cv[1],cv[2]); end; fi; else if not IsBound(SetName) then tmpf:=function(cv) return Value(Polynomial(Cyclotomics,cv[1],cv[2]),q);end; else tmpf := function(cv) return Value(LaurentPolynomialByCoefficients( FamilyObj(1),cv[1],cv[2]),q); end; fi; fi; tmpff := function(t) t.GFOrder := tmpf(t.GFOrder); t.classLengths := List(t.classLengths,tmpf); t.table := List(t.table,r-> List(r,tmpf)); end; greenE8good1mod3 := rec( classLengths := [ [ [ 1 ], 0 ], [ [ -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 ], 0 ], [ [ 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, -3, 0, -3, 0, -2, 0, -5, 0, -3, 0, -3, 0, -5, 0, -2, 0, -3, 0, -3, 0, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1 ], 6 ], [ [ -1, 0, -1, 0, -1, 0, -2, 0, -2, 0, -2, 0, -3, 0, -2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 5, 0, 6, 0, 5, 0, 7, 0, 6, 0, 5, 0, 6, 0, 3, 0, 2, 0, 2, 0, -2, 0, -2, 0, -3, 0, -6, 0, -5, 0, -6, 0, -7, 0, -5, 0, -6, 0, -5, 0, -3, 0, -3, 0, -1, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1 ], 14 ], [ [ 1/2, 0, 0, 0, 0, 1/2, 1/2, 0, 0, 1/2, 1/2, 1/2, 1/2, 0, 0, 1, 1/2, 1/2, 1/2, 0, 0, 1, 1/2, 0, 0, 0, 0, 1, 0, -1, -1/2, 0, 0, 0, -1/2, -3/2, -1/2, 0, -1/2, -1, -1, -3/2, -1/2, 0, -1/2, -3/2, -1, -1, -1/2, 0, -1/2, -3/2, -1/2, 0, 0, 0, -1/2, -1, 0, 1, 0, 0, 0, 0, 1/2, 1, 0, 0, 1/2, 1/2, 1/2, 1, 0, 0, 1/2, 1/2, 1/2, 1/2, 0, 0, 1/2, 1/2, 0, 0, 0, 0, 1/2 ], 28 ], [ [ 1/2, 0, 0, 0, 0, -1/2, 1/2, 0, 0, -1/2, 1/2, -1/2, 1/2, 0, 0, -1, 1/2, -1/2, 1/2, 0, 0, -1, 1/2, 0, 0, 0, 0, -1, 0, 1, -1/2, 0, 0, 0, -1/2, 3/2, -1/2, 0, -1/2, 1, -1, 3/2, -1/2, 0, -1/2, 3/2, -1, 1, -1/2, 0, -1/2, 3/2, -1/2, 0, 0, 0, -1/2, 1, 0, -1, 0, 0, 0, 0, 1/2, -1, 0, 0, 1/2, -1/2, 1/2, -1, 0, 0, 1/2, -1/2, 1/2, -1/2, 0, 0, 1/2, -1/2, 0, 0, 0, 0, 1/2 ], 28 ], [ [ 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, -1, 0, -1, 0, -3, 0, -2, 0, -3, 0, -4, 0, -2, 0, -3, 0, -3, 0, 0, 0, -2, 0, 1, 0, 2, 0, 1, 0, 4, 0, 4, 0, 3, 0, 6, 0, 3, 0, 4, 0, 4, 0, 1, 0, 2, 0, 1, 0, -2, 0, 0, 0, -3, 0, -3, 0, -2, 0, -4, 0, -3, 0, -2, 0, -3, 0, -1, 0, -1, 0, -1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1 ], 20 ], [ [ -1/2, 0, 0, -1/2, -1/2, -1/2, -1, -1/2, -1/2, -3/2, -3/2, -1, -3/2, -3/2, -1, -5/2, -2, -1, -3/2, -2, -1, -2, -3/2, 0, 0, -3/2, 0, 1/2, 1/2, 2, 2, 1/2, 2, 4, 3, 7/2, 4, 3, 7/2, 13/2, 9/2, 7/2, 9/2, 9/2, 7/2, 6, 4, 3/2, 3, 4, 3/2, 5/2, 3/2, -3/2, 0, 3/2, -3/2, -5/2, -3/2, -4, -3, -3/2, -4, -6, -7/2, -9/2, -9/2, -7/2, -9/2, -13/2, -7/2, -3, -4, -7/2, -3, -4, -2, -1/2, -2, -2, -1/2, -1/2, 0, 3/2, 0, 0, 3/2, 2, 1, 2, 3/2, 1, 2, 5/2, 1, 3/2, 3/2, 1, 3/2, 3/2, 1/2, 1/2, 1, 1/2, 1/2, 1/2, 0, 0, 1/2 ], 28 ], [ [ -1/2, 0, 0, 1/2, -1/2, 1/2, -1, 1/2, -1/2, 3/2, -3/2, 1, -3/2, 3/2, -1, 5/2, -2, 1, -3/2, 2, -1, 2, -3/2, 0, 0, 3/2, 0, -1/2, 1/2, -2, 2, -1/2, 2, -4, 3, -7/2, 4, -3, 7/2, -13/2, 9/2, -7/2, 9/2, -9/2, 7/2, -6, 4, -3/2, 3, -4, 3/2, -5/2, 3/2, 3/2, 0, -3/2, -3/2, 5/2, -3/2, 4, -3, 3/2, -4, 6, -7/2, 9/2, -9/2, 7/2, -9/2, 13/2, -7/2, 3, -4, 7/2, -3, 4, -2, 1/2, -2, 2, -1/2, 1/2, 0, -3/2, 0, 0, 3/2, -2, 1, -2, 3/2, -1, 2, -5/2, 1, -3/2, 3/2, -1, 3/2, -3/2, 1/2, -1/2, 1, -1/2, 1/2, -1/2, 0, 0, 1/2 ], 28 ], [ [ 1, 0, 1, 0, 2, 0, 3, 0, 3, 0, 4, 0, 4, 0, 3, 0, 3, 0, 1, 0, -1, 0, -2, 0, -6, 0, -7, 0, -9, 0, -12, 0, -11, 0, -12, 0, -12, 0, -8, 0, -8, 0, -4, 0, 1, 0, 2, 0, 8, 0, 11, 0, 12, 0, 17, 0, 16, 0, 16, 0, 17, 0, 12, 0, 11, 0, 8, 0, 2, 0, 1, 0, -4, 0, -8, 0, -8, 0, -12, 0, -12, 0, -11, 0, -12, 0, -9, 0, -7, 0, -6, 0, -2, 0, -1, 0, 1, 0, 3, 0, 3, 0, 4, 0, 4, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1 ] , 32 ], [ [ -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -2, 0, 0, 0, -1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 3, 0, 2, 0, 5, 0, 3, 0, 3, 0, 5, 0, 1, 0, 3, 0, 2, 0, -1, 0, 1, 0, -2, 0, -3, 0, -1, 0, -5, 0, -3, 0, -3, 0, -5, 0, -2, 0, -3, 0, -3, 0, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1 ], 50 ], [ [ -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 3, 0, 5, 0, 5, 0, 3, 0, 5, 0, 2, 0, 0, 0, 1, 0, -4, 0, -4, 0, -4, 0, -9, 0, -6, 0, -7, 0, -9, 0, -4, 0, -6, 0, -4, 0, 1, 0, -1, 0, 4, 0, 6, 0, 4, 0, 9, 0, 7, 0, 6, 0, 9, 0, 4, 0, 4, 0, 4, 0, -1, 0, 0, 0, -2, 0, -5, 0, -3, 0, -5, 0, -5, 0, -3, 0, -4, 0, -2, 0, -1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1 ], 36 ], [ [ 1/2, 0, 1/2, 0, 1/2, 0, 3/2, 0, 1, 0, 1, 0, 2, 0, 1/2, 0, 1/2, 0, 1, 0, -3/2, 0, -1, 0, -3/2, 0, -4, 0, -5/2, 0, -4, 0, -5, 0, -5/2, 0, -9/2, 0, -3, 0, -1/2, 0, -5/2, 0, 3/2, 0, 5/2, 0, 1, 0, 6, 0, 9/2, 0, 4, 0, 8, 0, 4, 0, 9/2, 0, 6, 0, 1, 0, 5/2, 0, 3/2, 0, -5/2, 0, -1/2, 0, -3, 0, -9/2, 0, -5/2, 0, -5, 0, -4, 0, -5/2, 0, -4, 0, -3/2, 0, -1, 0, -3/2, 0, 1, 0, 1/2, 0, 1/2, 0, 2, 0, 1, 0, 1, 0, 3/2, 0, 1/2, 0, 1/2, 0, 1/2 ], 44 ], [ [ 1/2, 0, -1/2, 0, 1/2, 0, -1/2, 0, 0, 0, 0, 0, 0, 0, -1/2, 0, 1/2, 0, -1, 0, 1/2, 0, 0, 0, -1/2, 0, 1, 0, -1/2, 0, 0, 0, 1, 0, -1/2, 0, 1/2, 0, 1, 0, -3/2, 0, 3/2, 0, -1/2, 0, -1/2, 0, 1, 0, -1, 0, -1/2, 0, 1, 0, -2, 0, 1, 0, -1/2, 0, -1, 0, 1, 0, -1/2, 0, -1/2, 0, 3/2, 0, -3/2, 0, 1, 0, 1/2, 0, -1/2, 0, 1, 0, 0, 0, -1/2, 0, 1, 0, -1/2, 0, 0, 0, 1/2, 0, -1, 0, 1/2, 0, -1/2, 0, 0, 0, 0, 0, 0, 0, -1/2, 0, 1/2, 0, -1/2, 0, 1/2 ], 44 ], [ [ -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 3, 0, 5, 0, 5, 0, 3, 0, 5, 0, 2, 0, 0, 0, 1, 0, -4, 0, -4, 0, -4, 0, -9, 0, -6, 0, -7, 0, -9, 0, -4, 0, -6, 0, -4, 0, 1, 0, -1, 0, 4, 0, 6, 0, 4, 0, 9, 0, 7, 0, 6, 0, 9, 0, 4, 0, 4, 0, 4, 0, -1, 0, 0, 0, -2, 0, -5, 0, -3, 0, -5, 0, -5, 0, -3, 0, -4, 0, -2, 0, -1, 0, -1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1 ], 44 ], [ [ 1, 0, 1, 0, 2, 0, 3, 0, 3, 0, 4, 0, 4, 0, 3, 0, 3, 0, 1, 0, -1, 0, -2, 0, -6, 0, -7, 0, -9, 0, -12, 0, -11, 0, -12, 0, -12, 0, -8, 0, -8, 0, -4, 0, 1, 0, 2, 0, 8, 0, 11, 0, 12, 0, 17, 0, 16, 0, 16, 0, 17, 0, 12, 0, 11, 0, 8, 0, 2, 0, 1, 0, -4, 0, -8, 0, -8, 0, -12, 0, -12, 0, -11, 0, -12, 0, -9, 0, -7, 0, -6, 0, -2, 0, -1, 0, 1, 0, 3, 0, 3, 0, 4, 0, 4, 0, 3, 0, 3, 0, 2, 0, 1, 0, 1 ] , 50 ], [ [ 1/6, 0, 0, 0, 1/3, 0, 1/6, 0, 1/3, 0, 1/3, 0, 1/3, 0, 1/6, 0, 1/3, 0, -1/6, 0, 0, 0, -1/3, 0, -2/3, 0, -1/2, 0, -1, 0, -1, 0, -5/6, 0, -7/6, 0, -5/6, 0, -1/2, 0, -5/6, 0, 1/6, 0, 0, 0, 1/3, 0, 1, 0, 5/6, 0, 7/6, 0, 5/3, 0, 1, 0, 5/3, 0, 7/6, 0, 5/6, 0, 1, 0, 1/3, 0, 0, 0, 1/6, 0, -5/6, 0, -1/2, 0, -5/6, 0, -7/6, 0, -5/6, 0, -1, 0, -1, 0, -1/2, 0, -2/3, 0, -1/3, 0, 0, 0, -1/6, 0, 1/3, 0, 1/6, 0, 1/3, 0, 1/3, 0, 1/3, 0, 1/6, 0, 1/3, 0, 0, 0, 1/6 ], 54 ], [ [ -1/2, 0, 0, 0, 0, 0, -1/2, 0, 0, 0, 0, 0, 0, 0, 1/2, 0, 0, 0, 1/2, 0, 1, 0, 0, 0, 1, 0, 1/2, 0, 0, 0, 1, 0, -1/2, 0, -1/2, 0, 1/2, 0, -3/2, 0, -1/2, 0, -1/2, 0, -2, 0, 0, 0, -1, 0, -3/2, 0, 1/2, 0, -1, 0, 0, 0, 1, 0, -1/2, 0, 3/2, 0, 1, 0, 0, 0, 2, 0, 1/2, 0, 1/2, 0, 3/2, 0, -1/2, 0, 1/2, 0, 1/2, 0, -1, 0, 0, 0, -1/2, 0, -1, 0, 0, 0, -1, 0, -1/2, 0, 0, 0, -1/2, 0, 0, 0, 0, 0, 0, 0, 1/2, 0, 0, 0, 0, 0, 1/2 ], 54 ], [ [ 1/3, 0, 0, 0, -1/3, 0, 1/3, 0, -1/3, 0, -1/3, 0, 2/3, 0, -2/3, 0, -1/3, 0, 2/3, 0, -1, 0, 1/3, 0, 2/3, 0, -1, 0, 1, 0, 0, 0, -2/3, 0, 5/3, 0, -2/3, 0, 0, 0, 4/3, 0, -5/3, 0, 1, 0, 2/3, 0, -2, 0, 5/3, 0, -2/3, 0, -5/3, 0, 2, 0, -5/3, 0, -2/3, 0, 5/3, 0, -2, 0, 2/3, 0, 1, 0, -5/3, 0, 4/3, 0, 0, 0, -2/3, 0, 5/3, 0, -2/3, 0, 0, 0, 1, 0, -1, 0, 2/3, 0, 1/3, 0, -1, 0, 2/3, 0, -1/3, 0, -2/3, 0, 2/3, 0, -1/3, 0, -1/3, 0, 1/3, 0, -1/3, 0, 0, 0, 1/3 ], 54 ], [ [ 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -2, 0, 1, 0, -1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 3, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, -2, 0, 0, 0, -1, 0, -4, 0, -1, 0, -4, 0, -4, 0, -1, 0, -4, 0, -1, 0, 0, 0, -2, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 3, 0, 3, 0, 1, 0, 2, 0, 1, 0, -1, 0, 1, 0, -2, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ], 46 ], [ [ 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, -2, 0, 0, 0, -1, 0, -2, 0, 0, 0, -2, 0, -1, 0, 1, 0, -2, 0, 1, 0, 1, 0, -1, 0, 3, 0, 1, 0, 0, 0, 4, 0, 0, 0, 1, 0, 3, 0, -1, 0, 1, 0, 1, 0, -2, 0, 1, 0, -1, 0, -2, 0, 0, 0, -2, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 ], 68 ], [ [ -1, 0, -1, 0, -2, 0, -2, 0, -2, 0, -2, 0, -1, 0, 0, 0, 1, 0, 3, 0, 4, 0, 5, 0, 7, 0, 6, 0, 7, 0, 6, 0, 4, 0, 3, 0, 0, 0, -3, 0, -4, 0, -8, 0, -9, 0, -10, 0, -12, 0, -10, 0, -10, 0, -9, 0, -5, 0, -4, 0, 0, 0, 4, 0, 5, 0, 9, 0, 10, 0, 10, 0, 12, 0, 10, 0, 9, 0, 8, 0, 4, 0, 3, 0, 0, 0, -3, 0, -4, 0, -6, 0, -7, 0, -6, 0, -7, 0, -5, 0, -4, 0, -3, 0, -1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 0, 1 ], 52 ], [ [ -1/6, 0, -1/3, 0, -1/2, 0, -2/3, 0, -2/3, 0, -2/3, 0, -1/2, 0, -1/6, 0, 1/6, 0, 2/3, 0, 7/6, 0, 3/2, 0, 2, 0, 13/6, 0, 13/6, 0, 13/6, 0, 5/3, 0, 7/6, 0, 1/2, 0, -1/2, 0, -7/6, 0, -2, 0, -17/6, 0, -19/6, 0, -11/3, 0, -11/3, 0, -10/3, 0, -19/6, 0, -7/3, 0, -3/2, 0, -2/3, 0, 2/3, 0, 3/2, 0, 7/3, 0, 19/6, 0, 10/3, 0, 11/3, 0, 11/3, 0, 19/6, 0, 17/6, 0, 2, 0, 7/6, 0, 1/2, 0, -1/2, 0, -7/6, 0, -5/3, 0, -13/6, 0, -13/6, 0, -13/6, 0, -2, 0, -3/2, 0, -7/6, 0, -2/3, 0, -1/6, 0, 1/6, 0, 1/2, 0, 2/3, 0, 2/3, 0, 2/3, 0, 1/2, 0, 1/3, 0, 1/6 ], 54 ], [ [ 1/2, 0, 0, 0, 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"E_8(a_1)", 1 ], [ "E_8", 1 ] ], charInfo := [ [ "L=A_0", "A_0" ], [ "L=A_0", "8A_1" ], [ "L=A_0", "4A_1'" ], [ "L=A_0", "2A_1" ], [ "L=A_0", "6A_1" ], [ "L=A_0", "2D_4(a_1)" ], [ "L=A_0", "4A_1''" ], [ "L=A_0", "A_2" ], [ "L=A_0", "D_4 + 4A_1" ], [ "L=A_0", "4A_2" ], [ "L=A_0", "E_8(a_8)" ], [ "L=A_0", "3A_2" ], [ "L=A_0", "E_7(a_4) + A_1" ], [ "L=A_0", "2A_2" ], [ "L=A_0", "2D_4" ], [ "L=A_0", "D_4(a_1)" ], [ "L=A_0", "2A_3 + 2A_1" ], [ "L=A_0", "2A_3'" ], [ "L=A_0", "A_3 + A_1" ], [ "L=A_0", "A_3 + 3A_1" ], [ "L=A_0", "D_8(a_3)" ], [ "L=A_0", "2A_3''" ], [ "L=A_0", "A_4" ], [ "L=A_0", "D_6 + 2A_1" ], [ "L=A_0", "2A_4" ], [ "L=A_0", "E_8(a_6)" ], [ "L=A_0", "A_2 + 4A_1" ], [ "L=A_0", "D_4" ], [ "L=A_0", "E_6(a_2) + A_2" ], [ "L=A_0", "A_2 + 2A_1" ], [ "L=A_0", "D_4 + 2A_1" ], [ "L=A_0", "E_8(a_3)" ], [ "L=A_0", "E_6(a_2)" ], [ "L=A_0", "A_5 + A_2 + A_1" ], [ "L=A_0", "A_5 + A_1'" ], [ "L=A_0", "D_4 + A_2" ], [ "L=A_0", "2A_2 + 2A_1" ], [ "L=A_0", "D_6(a_2)" ], [ "L=A_0", "D_8(a_1)" ], [ "L=A_0", "A_5 + A_1''" ], [ "L=A_0", "A_6" ], [ "L=A_0", "D_8" ], [ "L=A_0", "D_5 + A_1" ], [ "L=A_0", "D_6(a_1)" ], [ "L=A_0", "A_7 + A_1" ], [ "L=A_0", "E_6(a_1)" ], [ "L=A_0", "E_7 + A_1" ], [ "L=A_0", "A_8" ], [ "L=A_0", "E_8(a_4)" ], [ "L=A_0", "A_4 + 2A_1" ], [ "L=A_0", "D_6" ], [ "L=A_0", "E_8(a_2)" ], [ "L=A_0", "D_4(a_1) + A_2" ], [ "L=A_0", "D_5(a_1) + A_3" ], [ "L=A_0", "E_6 + A_2" ], [ "L=A_0", "E_8(a_7)" ], [ "L=A_0", "E_6" ], [ "L=A_0", "E_7(a_2) + A_1" ], [ "L=A_0", "A_3 + A_2 + A_1" ], [ "L=A_0", "D_5(a_1) + A_1" ], [ "L=A_0", "E_8(a_1)" ], [ "L=A_0", "A_4 + A_2" ], [ "L=A_0", "D_8(a_2)" ], [ "L=A_0", "E_8(a_5)" ], [ "L=A_0", "E_8" ], [ "L=A_0", "A_1" ], [ "L=A_0", "7A_1" ], [ "L=A_0", "3A_1" ], [ "L=A_0", "5A_1" ], [ "L=A_0", "A_3" ], [ "L=A_0", "A_3 + 4A_1" ], [ "L=A_0", "A_3 + 2A_1'" ], [ "L=A_0", "D_4(a_1) + A_1" ], [ "L=A_0", "2A_3 + A_1" ], [ "L=A_0", "D_4(a_1) + A_3" ], [ "L=A_0", "A_3 + 2A_1''" ], [ "L=A_0", "A_2 + A_1" ], [ "L=A_0", "D_4 + 3A_1" ], [ "L=A_0", "3A_2 + A_1" ], [ "L=A_0", "E_7(a_4)" ], [ "L=A_0", "A_2 + 3A_1" ], [ "L=A_0", "D_4 + A_1" ], [ "L=A_0", "2A_2 + A_1" ], [ "L=A_0", "D_6(a_2) + A_1" ], [ "L=A_0", "A_5 + A_2" ], [ "L=A_0", "E_6(a_2) + A_1" ], [ "L=A_0", "A_5" ], [ "L=A_0", "A_5 + 2A_1" ], [ "L=A_0", "D_5" ], [ "L=A_0", "D_5 + 2A_1" ], [ "L=A_0", "A_7'" ], [ "L=A_0", "A_7''" ], [ "L=A_0", "A_4 + A_1" ], [ "L=A_0", "D_6 + A_1" ], [ "L=A_0", "A_3 + A_2 + 2A_1" ], [ "L=A_0", "D_5(a_1)" ], [ "L=A_0", "A_3 + A_2" ], [ "L=A_0", "D_4 + A_3" ], [ "L=A_0", "D_5(a_1) + A_2" ], [ "L=A_0", "D_7" ], [ "L=A_0", "E_6 + A_1" ], [ "L=A_0", "E_7(a_2)" ], [ "L=A_0", "A_6 + A_1" ], [ "L=A_0", "E_7(a_1)" ], [ "L=A_0", "E_6(a_1) + A_1" ], [ "L=A_0", "E_7" ], [ "L=A_0", "A_4 + A_3" ], [ "L=A_0", "D_7(a_1)" ], [ "L=A_0", "D_5 + A_2" ], [ "L=A_0", "D_7(a_2)" ], [ "L=A_0", "A_4 + A_2 + A_1" ], [ "L=A_0", "E_7(a_3)" ], [ "L=E_8", [ 1 ] ] ], GFOrder := [ [ 1, 0, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 0, -2, 0, 3, 0, -1, 0, -1, 0, 3, 0, -3, 0, 1, 0, 1, 0, -4, 0, 3, 0, -1, 0, -3, 0, 4, 0, -4, 0, 1, 0, 3, 0, -5, 0, 4, 0, 0, 0, -3, 0, 6, 0, -3, 0, 0, 0, 4, 0, -5, 0, 3, 0, 1, 0, -4, 0, 4, 0, -3, 0, -1, 0, 3, 0, -4, 0, 1, 0, 1, 0, -3, 0, 3, 0, -1, 0, -1, 0, 3, 0, -2, 0, 1, 0, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, -1, 0, 1 ], 120 ] ); tmpff(greenE8good1mod3); # almost the same, we save memory by mainly pointing to data for 1 mod 3 greenE8good2mod3 := rec(name := "Green( E_8(q) ), 29 mod 30"); greenE8good2mod3.GFOrder := greenE8good1mod3.GFOrder;; greenE8good2mod3.classInfo := greenE8good1mod3.classInfo;; greenE8good2mod3.charInfo := greenE8good1mod3.charInfo;; greenE8good2mod3.classLengths := greenE8good1mod3.classLengths;; greenE8good2mod3.table := List(greenE8good1mod3.table, ShallowCopy);; greenE8good2mod3.diffs := [ [ -1680*q^10 - 1300*q^9 - 320*q^8, 1680*q^10 + 1300*q^9 + 320*q^8, -1680*q^10 - 1300*q^9 - 320*q^8 ], [ 1680*q^10 - 1300*q^9 + 320*q^8, -1680*q^10 + 1300*q^9 - 320*q^8, 1680*q^10 - 1300*q^9 + 320*q^8 ], [ -20*q^9, 20*q^9, -20*q^9 ], [ -8*q^10 - 20*q^9 - 32*q^8, 8*q^10 + 20*q^9 + 32*q^8, -8*q^10 - 20*q^9 - 32*q^8 ], [ 8*q^10 - 20*q^9 + 32*q^8, -8*q^10 + 20*q^9 - 32*q^8, 8*q^10 - 20*q^9 + 32*q^8 ], [ -20*q^9, 20*q^9, -20*q^9 ], [ -20*q^9, 20*q^9, -20*q^9 ], [ -42*q^10 - 130*q^9 - 68*q^8, 42*q^10 + 130*q^9 + 68*q^8, -42*q^10 - 130*q^9 - 68*q^8 ], [ 42*q^10 - 130*q^9 + 68*q^8, -42*q^10 + 130*q^9 - 68*q^8, 42*q^10 - 130*q^9 + 68*q^8 ], [ 120*q^10 - 130*q^9 + 40*q^8, -120*q^10 + 130*q^9 - 40*q^8, 120*q^10 - 130*q^9 + 40*q^8 ], [ -120*q^10 - 130*q^9 - 40*q^8, 120*q^10 + 130*q^9 + 40*q^8, -120*q^10 - 130*q^9 - 40*q^8 ], [ -6*q^10 - 4*q^9 + 4*q^8, 6*q^10 + 4*q^9 - 4*q^8, -6*q^10 - 4*q^9 + 4*q^8 ] , [ 6*q^10 - 4*q^9 - 4*q^8, -6*q^10 + 4*q^9 + 4*q^8, 6*q^10 - 4*q^9 - 4*q^8 ], [ 12*q^10 + 14*q^9 + 4*q^8, -12*q^10 - 14*q^9 - 4*q^8, 12*q^10 + 14*q^9 + 4*q^8 ], [ -12*q^10 + 14*q^9 - 4*q^8, 12*q^10 - 14*q^9 + 4*q^8, -12*q^10 + 14*q^9 - 4*q^8 ], [ -40*q^10 - 68*q^9 - 32*q^8, 40*q^10 + 68*q^9 + 32*q^8, -40*q^10 - 68*q^9 - 32*q^8 ], [ 40*q^10 - 68*q^9 + 32*q^8, -40*q^10 + 68*q^9 - 32*q^8, 40*q^10 - 68*q^9 + 32*q^8 ], [ -4*q^9, 4*q^9, -4*q^9 ], [ -4*q^10 + 4*q^9, 4*q^10 - 4*q^9, -4*q^10 + 4*q^9 ], [ 4*q^10 + 4*q^9, -4*q^10 - 4*q^9, 4*q^10 + 4*q^9 ], [ 4*q^9, -4*q^9, 4*q^9 ], [ -4*q^9, 4*q^9, -4*q^9 ], [ 10*q^10 - 10*q^8, -10*q^10 + 10*q^8, 10*q^10 - 10*q^8 ], [ -10*q^10 + 10*q^8, 10*q^10 - 10*q^8, -10*q^10 + 10*q^8 ], [ 0*q^0, 0*q^0, 0*q^0 ], [ 0*q^0, 0*q^0, 0*q^0 ], [ 6*q^10 - 2*q^9 + 12*q^8, -6*q^10 + 2*q^9 - 12*q^8, 6*q^10 - 2*q^9 + 12*q^8 ], [ -6*q^10 - 2*q^9 - 12*q^8, 6*q^10 + 2*q^9 + 12*q^8, -6*q^10 - 2*q^9 - 12*q^8 ], [ -2*q^9, 2*q^9, -2*q^9 ], [ -2*q^10 - 2*q^9 + 4*q^8, 2*q^10 + 2*q^9 - 4*q^8, -2*q^10 - 2*q^9 + 4*q^8 ] , [ 2*q^10 - 2*q^9 - 4*q^8, -2*q^10 + 2*q^9 + 4*q^8, 2*q^10 - 2*q^9 - 4*q^8 ], [ -2*q^9, 2*q^9, -2*q^9 ], [ -6*q^10 - 20*q^9 - 12*q^8, 6*q^10 + 20*q^9 + 12*q^8, -6*q^10 - 20*q^9 - 12*q^8 ], [ 6*q^10 - 20*q^9 + 12*q^8, -6*q^10 + 20*q^9 - 12*q^8, 6*q^10 - 20*q^9 + 12*q^8 ], [ -2*q^9, 2*q^9, -2*q^9 ], [ -2*q^9, 2*q^9, -2*q^9 ], [ 4*q^10 - 2*q^9 + 4*q^8, -4*q^10 + 2*q^9 - 4*q^8, 4*q^10 - 2*q^9 + 4*q^8 ], [ -4*q^10 - 2*q^9 - 4*q^8, 4*q^10 + 2*q^9 + 4*q^8, -4*q^10 - 2*q^9 - 4*q^8 ] , [ -2*q^9, 2*q^9, -2*q^9 ], [ -2*q^9, 2*q^9, -2*q^9 ], [ 2*q^9 + 2*q^8, -2*q^9 - 2*q^8, 2*q^9 + 2*q^8 ], [ 2*q^9 - 2*q^8, -2*q^9 + 2*q^8, 2*q^9 - 2*q^8 ], [ -4*q^9, 4*q^9, -4*q^9 ], [ 4*q^10 + 4*q^9, -4*q^10 - 4*q^9, 4*q^10 + 4*q^9 ], [ -4*q^10 + 4*q^9, 4*q^10 - 4*q^9, -4*q^10 + 4*q^9 ], [ 2*q^9 - 2*q^8, -2*q^9 + 2*q^8, 2*q^9 - 2*q^8 ], [ 2*q^9 + 2*q^8, -2*q^9 - 2*q^8, 2*q^9 + 2*q^8 ], [ 2*q^9 - 2*q^8, -2*q^9 + 2*q^8, 2*q^9 - 2*q^8 ], [ 2*q^9 + 2*q^8, -2*q^9 - 2*q^8, 2*q^9 + 2*q^8 ], [ 2*q^10 - 2*q^8, -2*q^10 + 2*q^8, 2*q^10 - 2*q^8 ], [ -2*q^10 + 2*q^8, 2*q^10 - 2*q^8, -2*q^10 + 2*q^8 ], [ 0*q^0, 0*q^0, 0*q^0 ], [ -10*q^10 - 2*q^9 + 4*q^8, 10*q^10 + 2*q^9 - 4*q^8, -10*q^10 - 2*q^9 + 4*q^8 ], [ 10*q^10 - 2*q^9 - 4*q^8, -10*q^10 + 2*q^9 + 4*q^8, 10*q^10 - 2*q^9 - 4*q^8 ], [ -4*q^10 - 2*q^9 + 4*q^8, 4*q^10 + 2*q^9 - 4*q^8, -4*q^10 - 2*q^9 + 4*q^8 ] , [ 4*q^10 - 2*q^9 - 4*q^8, -4*q^10 + 2*q^9 + 4*q^8, 4*q^10 - 2*q^9 - 4*q^8 ], [ 2*q^10 + 4*q^9 + 4*q^8, -2*q^10 - 4*q^9 - 4*q^8, 2*q^10 + 4*q^9 + 4*q^8 ], [ -2*q^10 + 4*q^9 - 4*q^8, 2*q^10 - 4*q^9 + 4*q^8, -2*q^10 + 4*q^9 - 4*q^8 ] , [ 2*q^10 - 2*q^9, -2*q^10 + 2*q^9, 2*q^10 - 2*q^9 ], [ -2*q^10 - 2*q^9, 2*q^10 + 2*q^9, -2*q^10 - 2*q^9 ], [ -2*q^9, 2*q^9, -2*q^9 ], [ -2*q^10 + 2*q^8, 2*q^10 - 2*q^8, -2*q^10 + 2*q^8 ], [ 2*q^10 - 2*q^8, -2*q^10 + 2*q^8, 2*q^10 - 2*q^8 ], [ 0*q^0, 0*q^0, 0*q^0 ], [ 0*q^0, 0*q^0, 0*q^0 ], [ -252*q^10 - 320*q^9 - 128*q^8, 252*q^10 + 320*q^9 + 128*q^8, -252*q^10 - 320*q^9 - 128*q^8 ], [ 252*q^10 - 320*q^9 + 128*q^8, -252*q^10 + 320*q^9 - 128*q^8, 252*q^10 - 320*q^9 + 128*q^8 ], [ -20*q^10, 20*q^10, -20*q^10 ], [ 20*q^10, -20*q^10, 20*q^10 ], [ -8*q^10 - 40*q^9 - 32*q^8, 8*q^10 + 40*q^9 + 32*q^8, -8*q^10 - 40*q^9 - 32*q^8 ], [ 8*q^10 - 40*q^9 + 32*q^8, -8*q^10 + 40*q^9 - 32*q^8, 8*q^10 - 40*q^9 + 32*q^8 ], [ -8*q^9, 8*q^9, -8*q^9 ], [ -4*q^10, 4*q^10, -4*q^10 ], [ 4*q^10, -4*q^10, 4*q^10 ], [ -8*q^9, 8*q^9, -8*q^9 ], [ -8*q^9, 8*q^9, -8*q^9 ], [ 18*q^10 + 10*q^9 - 8*q^8, -18*q^10 - 10*q^9 + 8*q^8, 18*q^10 + 10*q^9 - 8*q^8 ], [ -18*q^10 + 10*q^9 + 8*q^8, 18*q^10 - 10*q^9 - 8*q^8, -18*q^10 + 10*q^9 + 8*q^8 ], [ 18*q^10 - 32*q^9 + 16*q^8, -18*q^10 + 32*q^9 - 16*q^8, 18*q^10 - 32*q^9 + 16*q^8 ], [ -18*q^10 - 32*q^9 - 16*q^8, 18*q^10 + 32*q^9 + 16*q^8, -18*q^10 - 32*q^9 - 16*q^8 ], [ 10*q^10 - 6*q^9, -10*q^10 + 6*q^9, 10*q^10 - 6*q^9 ], [ -10*q^10 - 6*q^9, 10*q^10 + 6*q^9, -10*q^10 - 6*q^9 ], [ -2*q^9 + 4*q^8, 2*q^9 - 4*q^8, -2*q^9 + 4*q^8 ], [ -2*q^9 - 4*q^8, 2*q^9 + 4*q^8, -2*q^9 - 4*q^8 ], [ -2*q^10, 2*q^10, -2*q^10 ], [ 2*q^10, -2*q^10, 2*q^10 ], [ 4*q^10 + 6*q^9, -4*q^10 - 6*q^9, 4*q^10 + 6*q^9 ], [ -4*q^10 + 6*q^9, 4*q^10 - 6*q^9, -4*q^10 + 6*q^9 ], [ 4*q^10 + 8*q^9, -4*q^10 - 8*q^9, 4*q^10 + 8*q^9 ], [ -4*q^10 + 8*q^9, 4*q^10 - 8*q^9, -4*q^10 + 8*q^9 ], [ 0*q^0, 0*q^0, 0*q^0 ], [ 0*q^0, 0*q^0, 0*q^0 ], [ -2*q^10 + 2*q^8, 2*q^10 - 2*q^8, -2*q^10 + 2*q^8 ], [ 2*q^10 - 2*q^8, -2*q^10 + 2*q^8, 2*q^10 - 2*q^8 ], [ 6*q^10 - 14*q^9 + 12*q^8, -6*q^10 + 14*q^9 - 12*q^8, 6*q^10 - 14*q^9 + 12*q^8 ], [ -6*q^10 - 14*q^9 - 12*q^8, 6*q^10 + 14*q^9 + 12*q^8, -6*q^10 - 14*q^9 - 12*q^8 ], [ -2*q^10 + 2*q^9 + 4*q^8, 2*q^10 - 2*q^9 - 4*q^8, -2*q^10 + 2*q^9 + 4*q^8 ] , [ 2*q^10 + 2*q^9 - 4*q^8, -2*q^10 - 2*q^9 + 4*q^8, 2*q^10 + 2*q^9 - 4*q^8 ], [ -2*q^9, 2*q^9, -2*q^9 ], [ -2*q^9, 2*q^9, -2*q^9 ], [ 2*q^10, -2*q^10, 2*q^10 ], [ -2*q^10, 2*q^10, -2*q^10 ], [ 2*q^9 - 2*q^8, -2*q^9 + 2*q^8, 2*q^9 - 2*q^8 ], [ 2*q^9 + 2*q^8, -2*q^9 - 2*q^8, 2*q^9 + 2*q^8 ], [ -2*q^9 - 2*q^8, 2*q^9 + 2*q^8, -2*q^9 - 2*q^8 ], [ -2*q^9 + 2*q^8, 2*q^9 - 2*q^8, -2*q^9 + 2*q^8 ], [ 2*q^10 - 2*q^8, -2*q^10 + 2*q^8, 2*q^10 - 2*q^8 ], [ -2*q^10 + 2*q^8, 2*q^10 - 2*q^8, -2*q^10 + 2*q^8 ], [ -2*q^10 + 2*q^9, 2*q^10 - 2*q^9, -2*q^10 + 2*q^9 ], [ 2*q^10 + 2*q^9, -2*q^10 - 2*q^9, 2*q^10 + 2*q^9 ], [ -2*q^10 + 2*q^8, 2*q^10 - 2*q^8, -2*q^10 + 2*q^8 ], [ 2*q^10 - 2*q^8, -2*q^10 + 2*q^8, 2*q^10 - 2*q^8 ] ]; for tmpvari in [1..112] do for tmpvarj in [1..3] do greenE8good2mod3.table[tmpvari][86+tmpvarj] := greenE8good1mod3.table[tmpvari][86+tmpvarj] + greenE8good2mod3.diffs[tmpvari][tmpvarj]; od; od; Unbind(tmpvari); Unbind(tmpvarj);Unbind(tmpf);Unbind(tmpff);