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Add RECOG.FixspaceMat and RECOG.EigenspaceMat #506

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Merged
fingolfin merged 1 commit into from
Jun 23, 2026
Merged

Add RECOG.FixspaceMat and RECOG.EigenspaceMat #506

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15 changes: 7 additions & 8 deletions gap/projective/sl.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -29,7 +29,7 @@ RECOG.FindStdGens_SL := function(sld)
# RECOG.SLn_UpStep often enough. Finally it returns an slp such
# that the SL(d,q) standard generators with respect to this basis are
# expressed by the slp in terms of the original generators of sld.
local V,b,bas,basi,basit,d,data,ext,fakegens,id,nu,nu2,p,q,resl2,sl2,sl2gens,
local V,b,bas,basi,basit,d,data,ext,fakegens,nu,nu2,p,q,resl2,sl2,sl2gens,
sl2gensf,sl2genss,sl2stdf,slp,slpsl2std,slptosl2,st,std,stdgens,i,ex,f;

# Some setup:
Expand Down Expand Up @@ -59,7 +59,7 @@ RECOG.FindStdGens_SL := function(sld)
stdgens := RECOG.MakeSL_StdGens(p,ext,4,4).all;
slpsl2std := RECOG.FindStdGensUsingBSGS(Group(sl2genss),stdgens,
false,false);
nu := List(sl2gens,x->NullspaceMat(x-One(x)));
nu := List(sl2gens, RECOG.FixspaceMat);
ex := SumIntersectionMat(nu[1],nu[2])[2];
for i in [3..Length(nu)] do
ex := SumIntersectionMat(nu[3],ex);
Expand All @@ -81,9 +81,8 @@ RECOG.FindStdGens_SL := function(sld)
StripMemory(GeneratorsOfGroup(sl2)));

# Extend basis by something invariant under SL2:
id := IdentityMat(d,f);
nu := NullspaceMat(StripMemory(st[1]-id));
nu2 := NullspaceMat(StripMemory(st[2]-id));
nu := RECOG.FixspaceMat(StripMemory(st[1]));
nu2 := RECOG.FixspaceMat(StripMemory(st[2]));
Append(bas,SumIntersectionMat(nu,nu2)[2]);
ConvertToMatrixRep(bas,q);
basi := bas^-1;
Expand Down Expand Up @@ -205,13 +204,13 @@ RECOG.SLn_constructsl4:=function(g,dim,q,r)
local s,h,count,readydim4,readydim3,ready,u,orderu,
nullr,nulls,nullspacer,nullspaces,int,intbasis,nullintbasis,
newu,newbasis,newbasisinv,newr,news,outputu,mat,i,shorts,shortr;
nullr:=NullspaceMat(StripMemory(r)-One(StripMemory(r)));
nullr:=RECOG.FixspaceMat(StripMemory(r));
nullspacer:=VectorSpace(GF(q),nullr);
mat:=One(r);
ready:=false;
repeat
s:=r^PseudoRandom(g);
nulls:=NullspaceMat(StripMemory(s)-One(StripMemory(s)));
nulls:=RECOG.FixspaceMat(StripMemory(s));
nullspaces:=VectorSpace(GF(q),nulls);
int:=Intersection(nullspacer,nullspaces);
intbasis:=Basis(int);
Expand Down Expand Up @@ -576,7 +575,7 @@ RECOG.SLn_UpStep := function(w)
Vnc := VectorSpace(w.f,c{[1..w.n]});
sum1 := ClosureLeftModule(Vn,Vnc);
if Dimension(sum1) = aimdim then
Fixc := VectorSpace(w.f,NullspaceMat(c-One(c)));
Fixc := VectorSpace(w.f,RECOG.FixspaceMat(c));
int1 := Intersection(Fixc,Vn);
for i in [1..Dimension(int1)] do
v := Basis(int1)[i];
Expand Down
17 changes: 17 additions & 0 deletions gap/utils.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -145,3 +145,20 @@ RECOG.CopySubVectorCompat := function(src, dst, from, to)
fi;
return dst;
end;

# compute the eigenspace of `mat` for the given eigenvalue lambda`
RECOG.EigenspaceMat := function(mat, lambda)
local i;
mat := MutableCopyMat( mat );
# since mat is a copy, we can efficiently "subtract an identity matrix"
for i in [1..NrRows(mat)] do
mat[i,i] := mat[i,i] - lambda;
od;
# since mat is a copy we can use NullspaceMatDestructive instead of NullspaceMat
return NullspaceMatDestructive(mat);
end;

# compute fixed space of mat, i.e. eigenspace for eigenvalue 1
RECOG.FixspaceMat := function(mat)
return RECOG.EigenspaceMat(mat, 1);
end;