[GAP Forum] semisimple algebra invariant subspace decomposition

[mim_ at op.pl]

Hi R.N.,

I do not know why your code returns error. I have rewritten it in following way and it returned:

[ 3, 13, 15, 17 ] A*V=V 8
[ 4, 6, 8, 21 ] A*V=V 8
[ 7, 9, 11, 20 ] A*V=V 8

I have marked my changes with #MM

Regards,
Marek

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 Test2:=function()local E8,bE8,e,f,F4,D4,bF4,bD4,vD4,T,TT,V,M,t;
 # define E8, DefaultField([ER(2),ER(3)]) necessary to find an F4
 E8:=SimpleLieAlgebra("E",8,DefaultField([ER(2),ER(3)]));
 F:=DefaultField([ER(2),ER(3)]);  # MM
 bE8:=ChevalleyBasis(E8);
 e:=bE8[1]{[8,7,6,5,4,3,1,2]};
 f:=bE8[2]{[8,7,6,5,4,3,1,2]};
 # define F4 subalgebra of E8
 F4:=Subalgebra(E8,[e[8],e[5],(e[4]+e[6])/ER(2),(e[3]+e[7])/ER(2),f[8],f[5],(f[4]+f[6])/ER(2),(f[3]+f[7])/ER(2)]);
 Print(SemiSimpleType(F4),"\n");
 bF4:=ChevalleyBasis(F4);
 # D4 subalgebra in F4 generated by long roots
 T:=[1,2,5,10,12,14,16,18,19,22,23,24];
 vD4:=Concatenation(bF4[1]{T},bF4[2]{T},bF4[3]);
 D4:=Subalgebra(F4,vD4);
 Print(SemiSimpleType(D4),"\n");
 #short roots
 TT:=Difference([1..24],T);
 #look for 8 dimensional subspaces spanned by short roots, heuristic
 for t in Combinations(TT,4) do
  V:=Concatenation(bF4[1]{t},bF4[2]{t});
  bb:=MutableBasis(F, V);   # MM
  M:=List(vD4,x->List(V,y->IsContainedInSpan(bb,x*y)));   #MM
  if(not(false in Flat(M)))then Print(t," A*V=V ",Length(V),"\n");fi; #MM
 od;
end;