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############################################################################
#
# this file contains a few preliminary functions
#
#
#################################################
#Input: nilpotent matrix
#Output: its Jordan NF and base change
#################################################
## jb := function (n)
## local m, i;
## m := 0*IdentityMat(n);
## for i in [1..n-1] do m[i][i+1] := 1; od;
## return m;
## end;
## mydbm := function ( mats )
## local n, M, m, d;
## n := Sum( mats, function ( m )
## return Length( m[1] );
## end );
## M := NullMat( n, n );
## n := 0;
## for m in mats do
## d := Length( m );
## M{[ 1 .. d ] + n}{[ 1 .. d ] + n} := m;
## n := n + d;
## od;
## return M;
## end;
## rjb := function(n)
## local part, gr, k,i;
## part := Random(Partitions(n));
## Print("took these part ",part,"\n");
## gr := mydbm(List(part,x->jb(x)));
## repeat k := RandomMat(n,n); until not Determinant(k)=0;
## return k*gr*k^-1;
## end;
## could use that for the two adjoint matrices of nilpotent elts
## (that is, two nilp matrices in sln). If the JNF has only even
## blocks and the determinant of base change matrices are positive
## in one case, negative in the other, then the two elts are not
## conjugate
corelg.JNFofNilpotent2 := function(m)
local F, V, kern, rws, i, new, elts, exps, preim, toExp, bas, mm;
Print("new version\n");
F := DefaultFieldOfMatrix(m);
V := VectorSpace(F,m^0);
kern := Subspace(V,NullspaceMat(m));
rws := [V];
i := 0;
repeat
i := i+1;
new := Subspace(V,m^i);
if not Dimension(new)=0 then Add(rws,new); fi;
until Dimension(new)=0;
rws := Reversed(rws);
elts := List(BasisVectors(Basis(rws[1])),x->[x]);;
exps := ListWithIdenticalEntries(Length(elts),0);
if not ForAll(elts,x->x[1]*m=0*x[1]) then Error("1"); fi;
for i in [2..Length(rws)] do
mm := List(BasisVectors(Basis(rws[i])),x->x*m);
preim := List(elts,x->SolutionMat(mm,x[Length(x)])); #need preim in rws[i]
preim := List(preim,x-> x*BasisVectors(Basis(rws[i])));
toExp := List(elts,x->x[1]);
new := BasisVectors(Basis(Intersection(kern,rws[i])));
new := List(BaseSteinitzVectors(new,toExp).factorspace,x->[x]);
for i in [1..Length(elts)] do Add(elts[i],preim[i]); od;
elts := Concatenation(elts,new);
exps := Concatenation(exps+1,ListWithIdenticalEntries(Length(new),0));
if not ForAll(elts,x->x[1]*m=0*x[1]) then Error("1"); fi;
od;
SortParallel(exps,elts);
bas := Concatenation(elts);
return [bas*m*bas^-1,Determinant(bas)];
#return [bas,bas*m*bas^-1,exps];
end;
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