Quelle classicalnatural.gi
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#############################################################################
##
## This file is part of recog, a package for the GAP computer algebra system
## which provides a collection of methods for the constructive recognition
## of groups.
##
## This files's authors include Max Neunhöffer, Ákos Seress.
##
## Copyright of recog belongs to its developers whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-3.0-or-later
##
##
## Constructive recognition of classical groups in their natural
## representation.
##
#############################################################################
InstallMethod( CharacteristicPolynomial, "for a memory element matrix",
[ IsMatrix and IsObjWithMemory ],
function(m)
return CharacteristicPolynomial(m!.el);
end );
InstallOtherMethod( \-, "for two memory elements",
[ IsMatrix and IsObjWithMemory, IsMatrix and IsObjWithMemory ],
function(m,n)
return m!.el - n!.el;
end );
InstallMethod( Eigenspaces, "for a field and a memory element matrix",
[ IsField, IsMatrix and IsObjWithMemory ],
function( f, m )
return Eigenspaces(f,m!.el);
end );
# Obsolete stuff?
# RECOG.RelativePrimeToqm1Part := function(q,n)
# local x,y;
# x := (q^n-1)/(q-1);
# repeat
# y := x/(q-1);
# x := NumeratorRat(y);
# until DenominatorRat(y) = q-1;
# return x;
# end;
#
# RECOG.SearchForElByCharPolFacts := function(g,f,degs,limit)
# local count,degrees,factors,pol,y;
# count := 0;
# while true do # will be left by return
# if InfoLevel(InfoRecog) >= 3 then Print(".\c"); fi;
# y:=PseudoRandom(g);
# pol:=CharacteristicPolynomial(f,f,StripMemory(y),1);
# factors:=Factors(PolynomialRing(f),pol);
# degrees:=List(factors,Degree);
# SortParallel(degrees,factors);
# if degrees = degs then
# if InfoLevel(InfoRecog) >= 3 then Print("\n"); fi;
# return rec( el := y, factors := factors, degrees := degrees );
# fi;
# count := count + 1;
# if count >= limit then return fail; fi;
# od;
# end;
#
# RECOG.SL_Even_godownone:=function(g,subspg,q,d)
# local n,y,yy,yyy,ready,order,es,null,subsph,z,x,a,b,c,h,r,divisors,cent,i,
# pol,factors,degrees;
#
# n:=DimensionOfMatrixGroup(g);
# #d:=Dimension(subspg);
# repeat
# ready:=false;
# y:=PseudoRandom(g);
# pol:=CharacteristicPolynomial(GF(q),GF(q),StripMemory(y),1);
# factors:=Factors(pol);
# degrees:=List(factors,Degree);
# if d-1 in degrees then
# order:=Order(y);
# yy:=y^(order/Gcd(order,q-1));
# if not IsOne(yy) then
# es:= Eigenspaces(GF(q),StripMemory(yy));
# es:=Filtered(es,x->Dimension(x)=d-1 and IsSubspace(subspg,x));
# if Length(es)>0 then
# subsph:=es[1];
# ready:=true;
# fi;
# yyy:=y^(Gcd(order,q-1));
# fi;
# fi;
# until ready;
#
# cent:=[yyy];
# for i in [1..4] do
# z:=PseudoRandom(g);
# x:=yy^z;
# a := x;
# b := x^yy;
# c := x^(yy^2);
# h := Group(a,b,c);
# ready:=false;
# repeat
# r:=PseudoRandom(h);
# r:=r^(q*(q+1));
# if not IsOne(r) and r*yy=yy*r then
# Add(cent,yyy^r);
# ready:=true;
# fi;
# until ready=true;
# od;
# return [Group(cent), subsph];
# end;
#
# RECOG.SL_FindSL2 := function(g,f)
# local V,a,bas,c,count,ev,gens,genss,genssm,gl4,h,i,j,n,ns,o,pos,pow,pr,q,r,
# res,sl2gens,sl3,slp,std,v,w,y,z,zz;
# q := Size(f);
# n := DimensionOfMatrixGroup(g);
# if q = 2 then
# # We look for a transvection:
# while true do # will be left by break
# r := RECOG.SearchForElByCharPolFacts(g,f,[1,1,n-2],3*n+20);
# if r = fail then return fail; fi;
# y := r.el^(q^(n-2)-1);
# if not IsOne(y) and IsOne(y^2) then break; fi;
# od;
# # Find a good random conjugate:
# repeat
# z := y^PseudoRandom(g);
# until Order(z*y) = 3;
# gens := [y,z];
# o := IdentityMat(n,f);
# w := [];
# for i in [1..2] do
# for j in [1..n] do
# w[i] := o[j]*gens[i]-o[j];
# if not IsZero(w[i]) then break; fi;
# od;
# od;
# return [Group(gens),VectorSpace(GF(q),w)];
# fi;
# if q = 3 and n = 3 then
# std := RECOG.MakeSL_StdGens(3,1,2,3);
# slp := RECOG.FindStdGensUsingBSGS(g,Concatenation(std.s,std.t),
# false,true);
# if slp = fail then return fail; fi;
# h := Group(ResultOfStraightLineProgram(slp,GeneratorsOfGroup(g)));
# o := IdentityMat(3,f);
# return [h,VectorSpace(f,o{[1..2]})];
# fi;
# if q = 3 and n = 4 then
# std := RECOG.MakeSL_StdGens(3,1,2,4);
# slp := RECOG.FindStdGensUsingBSGS(g,Concatenation(std.s,std.t),
# false,true);
# if slp = fail then return fail; fi;
# h := Group(ResultOfStraightLineProgram(slp,GeneratorsOfGroup(g)));
# o := IdentityMat(4,f);
# return [h,VectorSpace(f,o{[1..2]})];
# fi;
# if q = 3 then
# # We look for a transvection:
# while true do # will be left by break
# r := RECOG.SearchForElByCharPolFacts(g,f,[1,1,n-2],3*n+20);
# if r = fail then return fail; fi;
# y := r.el^(q^(n-2)-1);
# if not IsOne(y) and IsOne(y^3) then break; fi;
# od;
# # Find a two good random conjugates:
# while true do # will be left by return
# z := y^PseudoRandom(g);
# zz := y^PseudoRandom(g);
# gens := [y,z,zz];
# o := IdentityMat(n,f);
# ns := [];
# for j in [1..3] do
# for i in [1..n] do
# w := o[i]*gens[j]-o[i];
# if not IsZero(w) then break; fi;
# od;
# # Since y has order y at least one basis vector is moved.
# ns[j] := w;
# od;
# V := VectorSpace(f,ns);
# bas := Basis(V,ns);
# genss := List(StripMemory(gens),
# x->List(ns,i->Coefficients(bas,i*x)));
# genssm := GeneratorsWithMemory(genss);
# sl3 := Group(genssm);
# pr := ProductReplacer(genssm,rec( maxdepth := 400, scramble := 0,
# scramblefactor := 0 ) );
# sl3!.pseudorandomfunc := [rec(func := Next,args := [pr])];
# res := RECOG.SL_FindSL2(sl3,f);
# if res = fail then
# if InfoLevel(InfoRecog) >= 3 then Print("#\c"); fi;
# continue;
# fi;
# slp := SLPOfElms(GeneratorsOfGroup(res[1]));
# sl2gens := ResultOfStraightLineProgram(slp,gens);
# ns := BasisVectors(Basis(res[2])) * ns;
# ConvertToMatrixRep(ns,q);
# return [Group(sl2gens),VectorSpace(f,ns)];
# od;
# fi;
# if q = 4 and n = 3 then
# std := RECOG.MakeSL_StdGens(2,2,2,3);
# slp := RECOG.FindStdGensUsingBSGS(g,Concatenation(std.s,std.t),
# false,true);
# if slp = fail then return fail; fi;
# h := Group(ResultOfStraightLineProgram(slp,GeneratorsOfGroup(g)));
# o := IdentityMat(3,f);
# return [h,VectorSpace(f,o{[1..2]})];
# fi;
# if q = 5 and n = 4 then
# std := RECOG.MakeSL_StdGens(5,1,2,4);
# slp := RECOG.FindStdGensUsingBSGS(g,Concatenation(std.s,std.t),
# false,true);
# if slp = fail then return fail; fi;
# h := Group(ResultOfStraightLineProgram(slp,GeneratorsOfGroup(g)));
# o := IdentityMat(4,f);
# return [h,VectorSpace(f,o{[1..2]})];
# fi;
# if n mod (q-1) <> 0 and q <> 3 then # The generic case:
# # We look for an element with n-1 dimensional eigenspace:
# count := 0;
# while true do # will be left by break
# count := count + 1;
# if count > 20 then return fail; fi;
# r := RECOG.SearchForElByCharPolFacts(g,f,[1,n-1],3*n+20);
# if r = fail then return fail; fi;
# pow := RECOG.RelativePrimeToqm1Part(q,n-1);
# y := r.el^pow;
# o := Order(y);
# if o mod (q-1) = 0 then
# y := y^(o/(q-1));
# break;
# fi;
# od;
# # Now y has order q-1 and and n-1 dimensional eigenspace
# ev := -Value(r.factors[1],0*Z(q));
# ns := NullspaceMat(StripMemory(r.el)-ev*StripMemory(One(y)));
# # this is a 1xn matrix now
# ns := ns[1];
# pos := PositionNonZero(ns);
# ns := (ns[pos]^-1) * ns;
# count := 0;
# while true do # will be left by break
# count := count + 1;
# if count > 20 then return fail; fi;
# a := PseudoRandom(g);
# v := OnLines(ns,a);
# z := y^a;
# if OnLines(v,y) <> v and OnLines(ns,z) <> ns then
# # Now y and z most probably generate a GL(2,q), we need
# # the derived subgroup and then check:
# c := Comm(y,z);
# sl2gens := FastNormalClosure([y,z],[c],1);
# V := VectorSpace(f,[ns,v]);
# bas := Basis(V,[ns,v]);
# genss := List(sl2gens,x->List([ns,v],i->Coefficients(bas,i*x)));
# if RECOG.IsThisSL2Natural(genss,f) then break; fi;
# if InfoLevel(InfoRecog) >= 3 then Print("$\c"); fi;
# else
# if InfoLevel(InfoRecog) >= 3 then Print("-\c"); fi;
# fi;
# od;
# if InfoLevel(InfoRecog) >= 3 then Print("\n"); fi;
# return [Group(sl2gens),VectorSpace(f,[ns,v])];
# fi;
# # Now q-1 does divide n, we have to do something else:
# # We look for an element with n-2 dimensional eigenspace:
# while true do # will be left by break
# r := RECOG.SearchForElByCharPolFacts(g,f,[1,1,n-2],5*n+20);
# if r = fail then return fail; fi;
# pow := RECOG.RelativePrimeToqm1Part(q,n-2);
# y := r.el^pow;
# o := Order(y);
# if o mod (q-1) = 0 then
# y := y^(o/(q-1));
# if RECOG.IsScalarMat(y) = false then break; fi;
# fi;
# od;
# # Now y has order q-1 and n-2 dimensional eigenspace
# if r.factors[1] <> r.factors[2] then
# ev := -Value(r.factors[1],0*Z(q));
# ns := NullspaceMat(StripMemory(r.el)-ev*StripMemory(One(y)));
# if not IsMutable(ns) then ns := MutableCopyMat(ns); fi;
# # this is a 1xn matrix now
# ev := -Value(r.factors[2],0*Z(q));
# Append(ns,NullspaceMat(StripMemory(r.el)-ev*StripMemory(One(y))));
# # ns now is a 2xn matrix
# else
# ev := -Value(r.factors[1],0*Z(q));
# ns := NullspaceMat((StripMemory(r.el)
# -ev*StripMemory(One(y)))^2);
# if not IsMutable(ns) then ns := MutableCopyMat(ns); fi;
# fi;
#
# count := 0;
# while true do # will be left by break
# count := count + 1;
# if count > 20 then return fail; fi;
# if Length(ns) > 2 then ns := ns{[1..2]}; fi;
# a := PseudoRandom(g);
# Append(ns,ns * a);
# if RankMat(ns) < 4 then
# if InfoLevel(InfoRecog) >= 3 then Print("+\c"); fi;
# continue;
# fi;
# z := y^a;
# # Now y and z most probably generate a GL(4,q), we need
# # the derived subgroup and then check:
# V := VectorSpace(f,ns);
# bas := Basis(V,ns);
# genss := List([y,z],x->List(ns,i->Coefficients(bas,i*x)));
# genssm := GeneratorsWithMemory(genss);
# gl4 := Group(genssm);
# pr := ProductReplacer(genssm,rec( maxdepth := 400, scramble := 0,
# scramblefactor := 0 ) );
# gl4!.pseudorandomfunc := [rec(func := Next,args := [pr])];
# res := RECOG.SL_FindSL2(gl4,f);
# if res = fail then
# if InfoLevel(InfoRecog) >= 3 then Print("#\c"); fi;
# continue;
# fi;
# slp := SLPOfElms(GeneratorsOfGroup(res[1]));
# sl2gens := ResultOfStraightLineProgram(slp,[y,z]);
# ns := BasisVectors(Basis(res[2])) * ns;
# return [Group(sl2gens),VectorSpace(f,ns)];
# od;
# return fail;
# end;
#
#
# RECOG.SL_Even_constructdata:=function(g,q)
# local S,a,b,degrees,eva,factors,gens,h,i,n,ns,o,pol,pos,ready,ready2,
# ready3,subgplist,w,ww,y,yy,z;
#
# n:=DimensionOfMatrixGroup(g);
#
# if q=2 then
# repeat
# ready:=false;
# y:=PseudoRandom(g);
# pol:=CharacteristicPolynomial(GF(q),GF(q),StripMemory(y),1);
# factors:=Factors(pol);
# degrees:=List(factors,Degree);
# if SortedList(degrees)=[1,1,n-2] then
# yy := y^(q^(n-2)-1);
# if not IsOne(yy) and IsOne(yy^2) then ready := true; fi;
# fi;
# until ready;
# repeat
# z := yy^PseudoRandom(g);
# until Order(z*yy) = 3;
# o := OneMutable(z);
# i := 1;
# while i <= n do
# w := o[i]*yy-o[i];
# if not IsZero(w) then break; fi;
# i := i + 1;
# od;
# i := 1;
# while i <= n do
# ww := o[i]*z-o[i];
# if not IsZero(ww) then break; fi;
# i := i + 1;
# od;
# return [Group(z,yy),VectorSpace(GF(2),[w,ww])];
# else
# #case of q <> 2
# repeat
# ready:=false;
# y:=PseudoRandom(g);
# if InfoLevel(InfoRecog) >= 3 then Print(".\c"); fi;
# pol:=CharacteristicPolynomial(GF(q),GF(q),StripMemory(y),1);
# factors:=Factors(pol);
# degrees:=List(factors,Degree);
# if n-1 in degrees then
# yy := y^(RECOG.RelativePrimeToqm1Part(q,n-1));
# o := Order(yy);
# if o mod (q-1) = 0 then
# yy := yy^(o/(q-1));
# ready := true;
# fi;
# fi;
# until ready;
# if InfoLevel(InfoRecog) >= 3 then Print("\n"); fi;
#
# ready2:=false;
# ready3:=false;
# repeat
# gens:=[yy];
# a := PseudoRandom(g);
# b := PseudoRandom(g);
# Add(gens,yy^a);
# Add(gens,yy^b);
# h:=Group(gens);
# if q = 4 then
# S := StabilizerChain(h);
# if not Size(S) in [60480,3*60480] then continue; fi;
# pos := Position(degrees,1);
# eva := -Value(factors[pos],0*Z(q));
# ns := NullspaceMat(StripMemory(y)-eva*One(StripMemory(y)));
# return [h,
# VectorSpace(GF(q),[ns[1],ns[1]*StripMemory(a),ns[1]*StripMemory(b)])];
# fi;
#
# # Now check using ppd-elements:
# for i in [1..10] do
# z:=PseudoRandom(h);
# pol:=CharacteristicPolynomial(GF(q),GF(q),StripMemory(z),1);
# factors:=Factors(pol);
# degrees:=List(factors,Degree);
# if Maximum(degrees)=2 then
# ready2:=true;
# elif Maximum(degrees)=3 then
# ready3:=true;
# fi;
# if ready2 and ready3 then
# break;
# fi;
# od;
# if not (ready2 and ready3) then
# ready2:=false;
# ready3:=false;
# fi;
# until ready2 and ready3;
#
# subgplist:=RECOG.SL_Even_godownone(h,VectorSpace(GF(q),One(g)),q,3);
# fi;
#
# return subgplist;
# end;
RECOG.FindStdGensUsingBSGS := function(g,stdgens,projective,large)
# stdgens generators for the matrix group g
# returns an SLP expressing stdgens in the generators of g
# set projective to true for projective mode
# set large to true if we should not bother finding nice base points!
local S,dim,gens,gm,i,l,strong;
dim := DimensionOfMatrixGroup(g);
if IsObjWithMemory(GeneratorsOfGroup(g)[1]) then
gm := GroupWithMemory(StripMemory(GeneratorsOfGroup(g)));
else
gm := GroupWithMemory(g);
fi;
if HasSize(g) then SetSize(gm,Size(g)); fi;
if large then
S := StabilizerChain(gm,rec( Projective := projective,
Cand := rec( points := One(g),
ops := ListWithIdenticalEntries(dim, OnLines) ) ) );
else
S := StabilizerChain(gm,rec( Projective := projective ) );
fi;
strong := ShallowCopy(StrongGenerators(S));
ForgetMemory(S);
l := List(stdgens,x->SiftGroupElementSLP(S,x));
gens := EmptyPlist(Length(stdgens));
for i in [1..Length(stdgens)] do
if not l[i].isone then
return fail;
fi;
Add(gens,ResultOfStraightLineProgram(l[i].slp,strong));
od;
return SLPOfElms(gens);
end;
RECOG.ResetSLstd := function(r)
r.left := One(r.a);
r.right := One(r.a);
if not IsBound(r.cache) then
r.cache := [EmptyPlist(100),EmptyPlist(100),
List([1..r.ext],i->[]), # rowopcache
List([1..r.ext],i->[])]; # colopcache
fi;
return r;
end;
# TODO: document the parameters
RECOG.InitSLstd := function(f,d,s,t,a,b)
local r;
r := rec( f := f, p := Characteristic(f), ext := DegreeOverPrimeField(f),
q := Size(f), d := d, all := Concatenation(s,t,[a],[b]),
one := One(f), One := One(s[1]), s := s, t := t, a := a, b := b );
return RECOG.ResetSLstd(r);
end;
RECOG.FindFFCoeffs := function(r,lambda)
return IntVecFFE(Coefficients(CanonicalBasis(r.f),lambda));
end;
# TODO: document this; what does "fake" mean????
# Theory: the fake gens are only used for their memory. Since we are only
# interested in the memory (to produce slps), we use trivial permutations for
# the underlying group elements, so that the multiplication is cheap.
# Verify and then document this.
RECOG.InitSLfake := function(f,d)
local ext,l;
ext := DegreeOverPrimeField(f);
l := ListWithIdenticalEntries(2*ext+2,());
l := GeneratorsWithMemory(l);
return RECOG.InitSLstd(f,d,l{[1..ext]},l{[ext+1..2*ext]},
l[2*ext+1],l[2*ext+2]);
end;
RECOG.DoRowOp_SL := function(m,i,j,lambda,std)
# add lambda times j-th row to i-th row, i<>j
# by left-multiplying with an expression in the standard generators:
# a : e_n -> e_{n-1} -> ... -> e_1 -> (-1)^(n+1) e_n
# b : e_n -> e_{n-1} -> ... -> e_2 -> (-1)^n e_n and e_1 -> e_1
# s : e_1 -> e_1+ * e_2, e_i -> e_i for i > 1
# t : e_2 -> e_1+ * e_2, e_i -> e_i for i <> 2
# s and t are lists of length ext to span over GF(p) all the scalars
# in *.
# Note that V_i = <e_1,...,e_i>.
# So <s,t> is an SL_2 in the upper left corner, a is an n-cycle
# b is an n-1 cycle with garbage fixing the first vector
# This only modifies the record std collecting a straight line program.
local Getai,Getbj,coeffs,k,new,newnew;
Getai := function(l)
local pos;
if l < 0 then pos := std.d - l;
else pos := l;
fi;
if not IsBound(std.cache[1][pos]) then
std.cache[1][pos] := std.a^l;
fi;
return std.cache[1][pos];
end;
Getbj := function(l)
local pos;
if l < 0 then pos := std.d - l;
else pos := l;
fi;
if not IsBound(std.cache[2][pos]) then
std.cache[2][pos] := std.b^l;
fi;
return std.cache[2][pos];
end;
newnew := std.One;
coeffs := RECOG.FindFFCoeffs(std,lambda);
for k in [1..std.ext] do
if not IsZero(coeffs[k]) then
if IsBound(std.cache[3][k][i]) and
IsBound(std.cache[3][k][i][j]) then
new := std.cache[3][k][i][j];
else;
new := std.One;
if i < j then
# We need to multiply from the left with the element
# a^(i-1) * b^(j-i-1) * s_k * b^-(j-i-1) * a^-(i-1)
# from the left.
if i > 1 then new := Getai(-(i-1)) * new; fi;
if j > i+1 then new := Getbj(-(j-i-1)) * new; fi;
new := std.s[k] * new;
if j > i+1 then new := Getbj(j-i-1) * new; fi;
if i > 1 then new := Getai(i-1) * new; fi;
elif i > j then
# We need to multiply from the left with the element
# a^(j-1) * b^(i-j-1) * t_k * b^-(i-j-1) * a^-(j-1)
# from the left.
if j > 1 then new := Getai(-(j-1)) * new; fi;
if i > j+1 then new := Getbj(-(i-j-1)) * new; fi;
new := std.t[k] * new;
if i > j+1 then new := Getbj(i-j-1) * new; fi;
if j > 1 then new := Getai(j-1) * new; fi;
fi;
if not IsBound(std.cache[3][k][i]) then
std.cache[3][k][i] := [];
fi;
std.cache[3][k][i][j] := new;
fi;
std.left := new^coeffs[k] * std.left;
newnew := new^coeffs[k] * newnew;
fi;
od;
if m <> false and not IsZero(lambda) then m[i] := m[i] + m[j] * lambda; fi;
return newnew;
end;
RECOG.DoColOp_SL := function(m,i,j,lambda,std)
# add lambda times i-th column to j-th column, i<>j
# by left-multiplying with an expression in the standard generators:
# a : e_n -> e_{n-1} -> ... -> e_1 -> (-1)^(n+1) e_n
# b : e_n -> e_{n-1} -> ... -> e_2 -> (-1)^n e_n and e_1 -> e_1
# s : e_1 -> e_1+ * e_2, e_i -> e_i for i > 1
# t : e_2 -> e_1+ * e_2, e_i -> e_i for i <> 2
# s and t are lists of length ext to span over GF(p) all the scalars
# in *.
# Note that V_i = <e_1,...,e_i>.
# So <s,t> is an SL_2 in the upper left corner, a is an n-cycle
# b is an n-1 cycle with garbage fixing the first vector
# This only modifies the record std collecting a straight line program.
local Getai,Getbj,coeffs,k,new,newnew;
Getai := function(l)
local pos;
if l < 0 then pos := std.d - l;
else pos := l;
fi;
if not IsBound(std.cache[1][pos]) then
std.cache[1][pos] := std.a^l;
fi;
return std.cache[1][pos];
end;
Getbj := function(l)
local pos;
if l < 0 then pos := std.d - l;
else pos := l;
fi;
if not IsBound(std.cache[2][pos]) then
std.cache[2][pos] := std.b^l;
fi;
return std.cache[2][pos];
end;
newnew := std.One;
coeffs := RECOG.FindFFCoeffs(std,lambda);
for k in [1..std.ext] do
if not IsZero(coeffs[k]) then
if IsBound(std.cache[4][k][i]) and
IsBound(std.cache[4][k][i][j]) then
new := std.cache[4][k][i][j];
else;
new := std.One;
if i < j then
# We need to multiply from the right with the element
# a^(i-1) * b^(j-i-1) * s_k * b^-(j-i-1) * a^-(i-1)
# from the right.
if i > 1 then new := Getai(-(i-1)) * new; fi;
if j > i+1 then new := Getbj(-(j-i-1)) * new; fi;
new := std.s[k] * new;
if j > i+1 then new := Getbj(j-i-1) * new; fi;
if i > 1 then new := Getai(i-1) * new; fi;
elif i > j then
# We need to multiply from the right with the element
# a^(j-1) * b^(i-j-1) * t_k * b^-(i-j-1) * a^-(j-1)
# from the left.
if j > 1 then new := Getai(-(j-1)) * new; fi;
if i > j+1 then new := Getbj(-(i-j-1)) * new; fi;
new := std.t[k] * new;
if i > j+1 then new := Getbj(i-j-1) * new; fi;
if j > 1 then new := Getai(j-1) * new; fi;
fi;
if not IsBound(std.cache[4][k][i]) then
std.cache[4][k][i] := [];
fi;
std.cache[4][k][i][j] := new;
fi;
std.right := std.right * new^coeffs[k];
newnew := newnew * new^coeffs[k];
fi;
od;
if m <> false and not IsZero(lambda) then
for k in [1..Length(m)] do
m[k][j] := m[k][j] + m[k][i] * lambda;
od;
fi;
return newnew;
end;
RECOG.MakeSL_StdGens := function(p,ext,n,d)
local a,b,f,i,q,s,t,x,res;
f := GF(p,ext);
q := Size(f);
a := IdentityMat(d,f);
a := a{Concatenation([n],[1..n-1],[n+1..d])};
ConvertToMatrixRep(a,q);
b := IdentityMat(d,f);
b := b{Concatenation([1,n],[2..n-1],[n+1..d])};
ConvertToMatrixRep(b,q);
if IsEvenInt(n) then
a[1] := -a[1];
else
b[2] := -b[2];
fi;
s := [];
t := [];
for i in [0..ext-1] do
x := IdentityMat(d,f);
x[1,2] := Z(p,ext)^i;
Add(s,x);
x := IdentityMat(d,f);
x[2,1] := Z(p,ext)^i;
Add(t,x);
od;
res := rec( s := s, t := t, a := a, b := b, f := f, q := q, p := p,
ext := ext, One := IdentityMat(d,f), one := One(f),
d := d );
res.all := Concatenation( res.s, res.t, [res.a], [res.b] );
return res;
end;
RECOG.ExpressInStd_SL2 := function(m,std)
local mi;
if IsObjWithMemory(m) then
mi := InverseMutable(StripMemory(m));
else
mi := InverseMutable(m);
fi;
std.left := std.One;
if not IsOne(mi[1,1]) then
if IsZero(mi[2,1]) then
RECOG.DoRowOp_SL(mi,2,1,std.one,std);
# Now mi[2,1] is non-zero
fi;
RECOG.DoRowOp_SL(mi,1,2,(std.one-mi[1,1])/mi[2,1],std);
fi;
# Now mi[1,1] is equal to one
if not IsZero(mi[2,1]) then
RECOG.DoRowOp_SL(mi,2,1,-mi[2,1],std);
fi;
# Now mi[2,1] is equal to zero and thus mi[2,2] equal to one
if not IsZero(mi[1,2]) then
RECOG.DoRowOp_SL(mi,1,2,-mi[1,2],std);
fi;
# Now mi is the identity matrix, the element collected in std
# is the one to multiply on the left hand side to transform mi to the
# identity. Thus it is equal to m.
return SLPOfElm(std.left);
end;
RECOG.ExpressInStd_SL := function(m,std)
# m a matrix, std a fake standard generator record with trivial
# generators with memory
local d,i,j,mi,pos;
if IsObjWithMemory(m) then
mi := InverseMutable(StripMemory(m));
else
mi := InverseMutable(m);
fi;
std.left := std.One;
d := Length(m);
for i in [1..d] do
if not IsOne(mi[i,i]) then
pos := First([i+1..d],k->not IsZero(mi[k,i]));
if pos = fail then
pos := i+1;
RECOG.DoRowOp_SL(mi,pos,i,std.one,std);
fi;
RECOG.DoRowOp_SL(mi,i,pos,(std.one-mi[i,i])/mi[pos,i],std);
fi;
# Now mi[i,i] is equal to one
for j in Concatenation([1..i-1],[i+1..d]) do
if not IsZero(mi[j,i]) then
RECOG.DoRowOp_SL(mi,j,i,-mi[j,i],std);
fi;
od;
# Now mi[i,i] is the only non-zero entry in the column
od;
# Now mi is the identity matrix, the element collected in std
# is the one to multiply on the left hand side to transform mi to the
# identity. Thus it is equal to m.
return SLPOfElm(std.left);
end;
# BindGlobal("FunnyProductObjsFamily",NewFamily("FunnyProductObjsFamily"));
# DeclareCategory("IsFunnyProductObject",
# IsPositionalObjectRep and IsMultiplicativeElement and
# IsMultiplicativeElementWithInverse );
# BindGlobal("FunnyProductObjsType",
# NewType(FunnyProductObjsFamily,IsFunnyProductObject));
# DeclareOperation("FunnyProductObj",[IsObject,IsObject]);
#
#
# InstallOtherMethod( \*, "for two funny product objects",
# [ IsFunnyProductObject, IsFunnyProductObject ],
# function(a,b)
# return Objectify(FunnyProductObjsType,[a![1]+a![2]*b![1],a![2]*b![2]]);
# end );
#
# InstallOtherMethod( InverseSameMutability, "for a funny product object",
# [ IsFunnyProductObject ],
# function(a)
# local i;
# i := a![2]^-1;
# return Objectify(FunnyProductObjsType,[-i*a![1],i]);
# end );
#
# InstallOtherMethod( OneMutable, "for a funny product object",
# [ IsFunnyProductObject ],
# function(a)
# return Objectify(FunnyProductObjsType,[Zero(a![1]),OneMutable(a![2])]);
# end );
#
# InstallMethod( FunnyProductObj, "for two arbitrary objects",
# [ IsObject, IsObject ],
# function(a,b)
# return Objectify(FunnyProductObjsType,[a,b]);
# end );
#
# FIXME: unused? but see misc/work/DOWORK.
# Perhaps this was / is meant as a replacement for RECOG.FindStdGens_SL
# in even characteristic.
# But in a quick test based on misc/work/DOWORK, the code there
# seems to be faster.
# RECOG.FindStdGens_SL_EvenChar := function(sld,f)
# # gens of sld must be gens for SL(d,q) in its natural rep with memory
# # This function calls RECOG.SL_Even_constructdata and then extends
# # the basis to a basis of the full row space and 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,diffv,diffw,el,ext,fakegens,gens,i,id,
# lambda,mu,n,notinv,nu,nu2,oldyf,oldyy,p,pos,q,resl2,sl2,sl2gens,
# sl2gensf,sl2genss,sl2stdf,slp,slpsl2std,slptosl2,st,std,stdsl2,
# w,x,xf,y,y2f,y3f,yf,yy,yy2,yy3,yyy,yyy2,yyy3,z,zf,zzz,goodguy;
#
# # Some setup:
# p := Characteristic(f);
# q := Size(f);
# ext := DegreeOverPrimeField(f);
# d := DimensionOfMatrixGroup(sld);
# if not IsObjWithMemory(GeneratorsOfGroup(sld)[1]) then
# sld := GroupWithMemory(sld);
# fi;
#
# # First find an SL2 with the space it acts on:
# Info(InfoRecog,2,"Finding an SL2...");
# #data := RECOG.SL_Even_constructdata(sld,q);
# repeat
# data := RECOG.SL_FindSL2(sld,f);
# until data <> fail;
# bas := ShallowCopy(BasisVectors(Basis(data[2])));
# sl2 := data[1];
# slptosl2 := SLPOfElms(GeneratorsOfGroup(sl2));
#
# # Now compute the natural SL2 action and run constructive recognition:
# sl2gens := StripMemory(GeneratorsOfGroup(sl2));
# V := VectorSpace(f,bas);
# b := Basis(V,bas);
# sl2genss := List(sl2gens,x->List(BasisVectors(b),v->Coefficients(b,v*x)));
# for i in sl2genss do
# ConvertToMatrixRep(i,q);
# od;
# Info(InfoRecog,2,
# "Recognising this SL2 constructively in 2 dimensions...");
# sl2genss := GeneratorsWithMemory(sl2genss);
# if IsEvenInt(q) then
# resl2 := RECOG.RecogniseSL2NaturalEvenChar(Group(sl2genss),f,false);
# else
# resl2 := RECOG.RecogniseSL2NaturalOddCharUsingBSGS(Group(sl2genss),f);
# fi;
# slpsl2std := SLPOfElms(resl2.all);
# bas := resl2.bas * bas;
# # We need the actual transvections:
# slp := SLPOfElms([resl2.s[1],resl2.t[1]]);
# st := ResultOfStraightLineProgram(slp,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));
# Append(bas,SumIntersectionMat(nu,nu2)[2]);
# ConvertToMatrixRep(bas,q);
# basi := bas^-1;
# basit := TransposedMatMutable(basi);
#
# # Now set up fake generators for keeping track what we do:
# fakegens := ListWithIdenticalEntries(Length(GeneratorsOfGroup(sld)),());
# fakegens := GeneratorsWithMemory(fakegens);
# sl2gensf := ResultOfStraightLineProgram(slptosl2,fakegens);
# sl2stdf := ResultOfStraightLineProgram(slpsl2std,sl2gensf);
# std := RECOG.InitSLstd(f,d,sl2stdf{[1..ext]},sl2stdf{[ext+1..2*ext]},
# sl2stdf[2*ext+1],sl2stdf[2*ext+2]);
#
# # workrec := rec( n := 2, slnstdf := sl2stdf, bas := bas, basi := basi,
# # std := std, sld := sld, sldf := fakegens, f := f );
# #
# #Error("... now go on with alternative going up...");
#
# Info(InfoRecog,2,"Going up to SL_d again...");
# for n in [Dimension(data[2])..d-1] do
# if InfoLevel(InfoRecog) >= 3 then Print(n," \c"); fi;
# while true do # will be left by break at the end
# x := PseudoRandom(sld);
# slp := SLPOfElm(x);
# xf := ResultOfStraightLineProgram(slp,fakegens);
# # From now on plain matrices, we have to keep track with the
# # fake ones!
# x := StripMemory(x);
#
# # Find a new basis vector:
# y := st[1]^x;
# notinv := First([1..n],i->bas[i]*y<>bas[i]);
# if notinv = fail then continue; fi; # try next x
# w := bas[notinv]*y-bas[notinv];
# if ForAll(basit{[n+1..d]},v->IsZero(ScalarProduct(v,w))) then
# continue; # try next x
# fi;
# # Now make it so that w is invariant under SL_n by modifying
# # it by something in the span of bas{[1..n]}:
# for i in [1..n] do
# w := w - bas[i] * ScalarProduct(w,basit[i]);
# od;
# if w*y=w then
# if InfoLevel(InfoRecog) >= 3 then Print("!\c"); fi;
# continue;
# fi;
#
# # w is supposed to become the next basis vector number n+1.
# # So we need to throw away one of bas{[n+1..d]}:
# i := First([n+1..d],i->not IsZero(ScalarProduct(w,basit[i])));
# Remove(bas,i);
# Add(bas,w,n+1);
# # However, we want that the rest of them bas{[n+2..d]} is invariant
# # under y which we can achieve by adding a multiple of w:
# diffw := w*y-w;
# pos := PositionNonZero(diffw);
# for i in [n+2..d] do
# diffv := bas[i]*y-bas[i];
# if not IsZero(diffv) then
# bas[i] := bas[i] - (diffv[pos]/diffw[pos]) * w;
# fi;
# od;
# basi := bas^-1;
# basit := TransposedMat(basi);
#
# # Compute the action of y-One(y) on Span(bas{[1..n+1]})
# yy := EmptyPlist(n+1);
# for i in [1..n+1] do
# Add(yy,(bas[i]*y-bas[i])*basi);
# yy[i] := yy[i]{[1..n+1]};
# od;
# if q > 2 and IsOne(yy[n+1,n+1]) then
# if InfoLevel(InfoRecog) >= 3 then Print("#\c"); fi;
# continue;
# fi;
# ConvertToMatrixRep(yy,q);
# break;
# od;
# yf := xf^-1*std.s[1]*xf;
#
# # make sure that rows n-1 and n are non-zero:
# std.left := std.One;
# std.right := std.One;
# if IsZero(yy[n-1]) then
# RECOG.DoRowOp_SL(yy,n-1,notinv,std.one,std);
# RECOG.DoColOp_SL(yy,n-1,notinv,-std.one,std);
# fi;
# if IsZero(yy[n]) then
# RECOG.DoRowOp_SL(yy,n,notinv,std.one,std);
# RECOG.DoColOp_SL(yy,n,notinv,-std.one,std);
# fi;
# yf := std.left * yf * std.right;
#
# oldyy := MutableCopyMat(yy);
# oldyf := yf;
#
# if q = 2 then
# # In this case y is already good after cleaning out!
# # (remember that y+One(y) has rank 1 and does not fix bas[notinv])
# std.left := std.One;
# std.right := std.One;
# for i in [1..n-1] do
# lambda := -yy[i,n+1]/yy[n,n+1];
# RECOG.DoRowOp_SL(yy,i,n,lambda,std);
# RECOG.DoColOp_SL(yy,i,n,-lambda,std);
# od;
# yf := std.left * yf * std.right;
# z := yy+One(yy);
# zf := yf;
# if not IsZero(z[n,n]) or not IsOne(z[n,n+1]) or
# not IsZero(z[n+1,n+1]) or not IsOne(z[n+1,n]) then
# ErrorNoReturn("How on earth could this happen???");
# fi;
# else # q > 2
# while true do # will be left by break when we had success!
# # Note that by construction yy[n,n+1] is not zero!
# yy2 := MutableCopyMat(yy);
# std.left := std.One;
# std.right := std.One;
# # We want to be careful not to kill row n:
# repeat
# lambda := PrimitiveRoot(f)^Random(0,q-1);
# until lambda <> -yy2[n,n+1]/yy2[n-1,n+1];
# RECOG.DoRowOp_SL(yy2,n,n-1,lambda,std);
# RECOG.DoColOp_SL(yy2,n,n-1,-lambda,std);
# mu := lambda;
# y2f := std.left * yf * std.right;
#
# yy3 := MutableCopyMat(yy);
# std.left := std.One;
# std.right := std.One;
# # We want to be careful not to kill row n:
# repeat
# lambda := PrimitiveRoot(f)^Random(0,q-1);
# until (lambda <> -yy3[n,n+1]/yy3[n-1,n+1]) and
# (lambda <> mu or q = 3);
# # in GF(3) there are not enough values!
# RECOG.DoRowOp_SL(yy3,n,n-1,lambda,std);
# RECOG.DoColOp_SL(yy3,n,n-1,-lambda,std);
# y3f := std.left * yf * std.right;
#
# # We now perform conjugations such that the ys leave
# # bas{[1..n-1]} fixed:
#
# # (remember that y-One(y) has rank 1 and does not fix bas[notinv])
# std.left := std.One;
# std.right := std.One;
# for i in [1..n-1] do
# lambda := -yy[i,n+1]/yy[n,n+1];
# RECOG.DoRowOp_SL(yy,i,n,lambda,std);
# RECOG.DoColOp_SL(yy,i,n,-lambda,std);
# od;
# yf := std.left * yf * std.right;
#
# std.left := std.One;
# std.right := std.One;
# for i in [1..n-1] do
# lambda := -yy2[i,n+1]/yy2[n,n+1];
# RECOG.DoRowOp_SL(yy2,i,n,lambda,std);
# RECOG.DoColOp_SL(yy2,i,n,-lambda,std);
# od;
# y2f := std.left * y2f * std.right;
#
# std.left := std.One;
# std.right := std.One;
# for i in [1..n-1] do
# lambda := -yy3[i,n+1]/yy3[n,n+1];
# RECOG.DoRowOp_SL(yy3,i,n,lambda,std);
# RECOG.DoColOp_SL(yy3,i,n,-lambda,std);
# od;
# y3f := std.left * y3f * std.right;
#
# gens :=[ExtractSubMatrix(yy,[n,n+1],[n,n+1])+IdentityMat(2,f),
# ExtractSubMatrix(yy2,[n,n+1],[n,n+1])+IdentityMat(2,f),
# ExtractSubMatrix(yy3,[n,n+1],[n,n+1])+IdentityMat(2,f)];
# if RECOG.IsThisSL2Natural(gens,f) = true then break; fi;
# if InfoLevel(InfoRecog) >= 3 then Print("$\c"); fi;
# yy := MutableCopyMat(oldyy);
# yf := oldyf;
# od;
#
# # Now perform a constructive recognition in the SL2 in the lower
# # right corner:
# gens := GeneratorsWithMemory(gens);
# if IsEvenInt(q) then
# resl2 := RECOG.RecogniseSL2NaturalEvenChar(Group(gens),f,gens[1]);
# else
# resl2 := RECOG.RecogniseSL2NaturalOddCharUsingBSGS(Group(gens),f);
# fi;
# stdsl2 := RECOG.InitSLfake(f,2);
# goodguy := Reversed(IdentityMat(2,f));
# goodguy[1,2] := - goodguy[1,2];
# slp := RECOG.ExpressInStd_SL2(resl2.bas*goodguy*resl2.basi,stdsl2);
# el := ResultOfStraightLineProgram(slp,resl2.all);
# slp := SLPOfElm(el);
#
# yy := yy+One(yy);
# yy2 := yy2+One(yy2);
# yy3 := yy3+One(yy3);
# yyy := FunnyProductObj(ExtractSubMatrix(yy,[n,n+1],[1..n-1]),
# ExtractSubMatrix(yy,[n,n+1],[n,n+1]));
# yyy2 := FunnyProductObj(ExtractSubMatrix(yy2,[n,n+1],[1..n-1]),
# ExtractSubMatrix(yy2,[n,n+1],[n,n+1]));
# yyy3 := FunnyProductObj(ExtractSubMatrix(yy3,[n,n+1],[1..n-1]),
# ExtractSubMatrix(yy3,[n,n+1],[n,n+1]));
# zzz := ResultOfStraightLineProgram(slp,[yyy,yyy2,yyy3]);
# z := OneMutable(yy);
# CopySubMatrix(zzz![1],z,[1..2],[n,n+1],[1..n-1],[1..n-1]);
# CopySubMatrix(zzz![2],z,[1..2],[n,n+1],[1..2],[n,n+1]);
# zf := ResultOfStraightLineProgram(slp,[yf,y2f,y3f]);
# fi;
#
# std.left := std.One;
# std.right := std.One;
# # Now we clean out the last row of z:
# for i in [1..n-1] do
# if not IsZero(z[n+1,i]) then
# RECOG.DoColOp_SL(z,n,i,-z[n+1,i],std);
# fi;
# od;
# # Now we clean out the second last row of z:
# for i in [1..n-1] do
# if not IsZero(z[n,i]) then
# RECOG.DoRowOp_SL(z,n,i,-z[n,i],std);
# fi;
# od;
# zf := std.left * zf * std.right;
#
# # Now change the standard generators in the fakes:
# std.a := std.a * zf;
# std.b := std.b * zf;
# std.all[std.ext*2+1] := std.a;
# std.all[std.ext*2+2] := std.b;
# RECOG.ResetSLstd(std);
#
# od;
# if InfoLevel(InfoRecog) >= 3 then Print(".\n"); fi;
# return rec( slpstd := SLPOfElms(std.all), bas := bas, basi := basi );
# end;
# TODO: which algorithm is this? reference?
RECOG.FindStdGens_SL := function(sld,f)
# gens of sld must be gens for SL(d,q) in its natural rep with memory
# This function calls RECOG.SLn_constructsl2 and then extends
# the basis to a basis of the full row space and calls
# 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,
sl2gensf,sl2genss,sl2stdf,slp,slpsl2std,slptosl2,st,std,stdgens,i,ex;
# Some setup:
p := Characteristic(f);
q := Size(f);
ext := DegreeOverPrimeField(f);
d := DimensionOfMatrixGroup(sld);
if not IsObjWithMemory(GeneratorsOfGroup(sld)[1]) then
sld := GroupWithMemory(sld);
fi;
# First find an SL2 with the space it acts on:
Info(InfoRecog,2,"Finding an SL2...");
data := RECOG.SLn_constructsl2(sld,d,q);
bas := ShallowCopy(BasisVectors(Basis(data[2])));
sl2 := data[1];
slptosl2 := SLPOfElms(GeneratorsOfGroup(sl2));
sl2gens := StripMemory(GeneratorsOfGroup(sl2));
V := data[2];
b := Basis(V,bas);
sl2genss := List(sl2gens,x->RECOG.LinearAction(b,f,x));
if q in [2,3,4,5,9] then
Info(InfoRecog,2,"In fact found an SL4...");
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)));
ex := SumIntersectionMat(nu[1],nu[2])[2];
for i in [3..Length(nu)] do
ex := SumIntersectionMat(nu[3],ex);
od;
Append(bas,ex);
ConvertToMatrixRep(bas,q);
basi := bas^-1;
else
# Now compute the natural SL2 action and run constructive recognition:
Info(InfoRecog,2,
"Recognising this SL2 constructively in 2 dimensions...");
sl2genss := GeneratorsWithMemory(sl2genss);
if IsEvenInt(q) then
resl2 := RECOG.RecogniseSL2NaturalEvenChar(Group(sl2genss),f,false);
else
resl2 := RECOG.RecogniseSL2NaturalOddCharUsingBSGS(Group(sl2genss),f);
fi;
slpsl2std := SLPOfElms(resl2.all);
bas := resl2.bas * bas;
# We need the actual transvections:
slp := SLPOfElms([resl2.s[1],resl2.t[1]]);
st := ResultOfStraightLineProgram(slp,
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));
Append(bas,SumIntersectionMat(nu,nu2)[2]);
ConvertToMatrixRep(bas,q);
basi := bas^-1;
fi;
# Now set up fake generators for keeping track what we do:
fakegens := ListWithIdenticalEntries(Length(GeneratorsOfGroup(sld)),1);
fakegens := GeneratorsWithMemory(fakegens);
sl2gensf := ResultOfStraightLineProgram(slptosl2,fakegens);
sl2stdf := ResultOfStraightLineProgram(slpsl2std,sl2gensf);
std := rec( f := f, d := d, n := 2, bas := bas, basi := basi,
sld := sld, sldf := fakegens, slnstdf := sl2stdf,
p := p, ext := ext );
Info(InfoRecog,2,"Going up to SL_d again...");
while std.n < std.d do
RECOG.SLn_UpStep(std);
od;
return rec( slpstd := SLPOfElms(std.slnstdf),
bas := std.bas, basi := std.basi );
end;
RECOG.RecogniseSL2NaturalOddCharUsingBSGS := function(g,f)
local ext,p,q,res,slp,std;
p := Characteristic(f);
ext := DegreeOverPrimeField(f);
q := Size(f);
std := RECOG.MakeSL_StdGens(p,ext,2,2);
slp := RECOG.FindStdGensUsingBSGS(g,std.all,false,true);
if slp = fail then
return fail;
fi;
res := rec( g := g, one := One(f), One := One(g), f := f, q := q,
p := p, ext := ext, d := 2, bas := IdentityMat(2,f),
basi := IdentityMat(2,f) );
res.all := ResultOfStraightLineProgram(slp,GeneratorsOfGroup(g));
res.s := res.all{[1..ext]};
res.t := res.all{[ext+1..2*ext]};
res.a := res.all[2*ext+1];
res.b := res.all[2*ext+2];
return res;
end;
RECOG.RecogniseSL2NaturalEvenChar := function(g,f,torig)
# f a finite field, g equal to SL(2,Size(f)), t either an involution
# or false.
# Returns a set of standard generators for SL_2 and the base change
# to expose it. Works with memory. Uses PseudoRandom.
local a,actpos,am,b,bas,bm,c,can,ch,cm,co,co2,el,ev,eva,evb,evbi,ext,gens,
i,j,k,kk,mas,masi,mat,mati,mb,o,one,os,pos,q,res,s,ss,ssm,t,tb,tm,
tt,ttm,u,v,x,xb,xm;
q := Size(f);
gens := GeneratorsOfGroup(g);
if torig = false then
for a in gens do
if not IsOne(a) and IsOne(a^2) then
torig := a;
break;
fi;
od;
fi;
if torig = false then
# if no involution t has been given, compute one, using Proposition 4 from
# [KK15].
repeat
am:=PseudoRandom(g);
until not IsOneProjective(am);
k := Order(am);
if IsEvenInt(k) then
tm := am^(k/2);
else
# find a conjugate of a which does not commute with a.
repeat
bm := am^PseudoRandom(g);
cm := am*bm;
tm := bm*am;
until cm<>tm;
tm := tm^-1 * cm;
if not IsOneProjective(StripMemory(tm)^2) then
tm := cm^((q^2-2)/2) * am;
fi;
fi;
else
tm := torig;
fi;
t := StripMemory(tm);
Assert(1, IsOne(t^2));
ch := Factors(CharacteristicPolynomial(f,f,t,1));
if Length(ch) <> 2 or ch[1] <> ch[2] then
ErrorNoReturn("matrix is not triagonalizable - this should never happen!");
fi;
one := OneMutable(t);
bas := MutableCopyMat(NullspaceMat(Value(ch[1],t)));
Add(bas,one[1]);
if RankMat(bas) < 2 then
bas[2] := one[2];
fi;
tb := bas*t*bas^-1;
can := CanonicalBasis(f);
tt := [t];
ttm := [tm];
mat := [Coefficients(can,tb[2,1])];
mb := MutableBasis(GF(2),mat);
o := [gens[1]];
os := [gens[1]];
actpos := 1;
j := 1;
ext := DegreeOverPrimeField(f);
while Length(tt) < ext do
repeat
repeat
while j > Length(o) do
for k in gens do
kk := o[actpos]*k;
pos := PositionSorted(os,kk);
if pos > Length(os) or os[pos] <> kk then
Add(o,kk);
Add(os,kk,pos);
fi;
od;
actpos := actpos + 1;
od;
xm := o[j];
j := j + 1;
c := Comm(tm,xm);
until not IsOne(c^2);
xm := xm * c^(((q-1)*(q+1)-1)/2);
x := StripMemory(xm);
xb := bas*x*bas^-1;
co := Coefficients(can,xb[2,1]);
until not IsContainedInSpan(mb,co);
CloseMutableBasis(mb,co);
Add(tt,x);
Add(ttm,xm);
Add(mat,co);
od;
ConvertToMatrixRep(mat,2);
mati := mat^-1;
# Now we can add an arbitrary multiple of the first row to the
# second and an arbitrary multiple of the second column to the first.
# Therefore we quickly find other complimentary transvections:
ss := [];
ssm := [];
mas := [];
mb := MutableBasis(GF(2),mas,ZeroMutable(mat[1]));
j := 1;
while Length(ss) < ext do
while true do # will be left by break
repeat
while j > Length(o) do
for k in gens do
kk := o[actpos]*k;
pos := PositionSorted(os,kk);
if pos > Length(os) or os[pos] <> kk then
Add(o,kk);
Add(os,kk,pos);
fi;
od;
actpos := actpos + 1;
od;
xm := o[j];
j := j + 1;
x := MutableCopyMat(bas*StripMemory(xm)*bas^-1);
until not IsZero(x[1,2]);
if not IsOne(x[2,2]) then
el := (One(f)-x[2,2])/x[1,2];
co := Coefficients(can,el) * mati;
for i in [1..Length(co)] do
if not IsZero(co[i]) then
xm := ttm[i] * xm;
fi;
od;
x[2] := x[2] + x[1] * el;
if x <> bas*StripMemory(xm)*bas^-1 then
# FIXME: sometimes triggered by RecognizeGroup(GL(2,16));
ErrorNoReturn("!!!");
fi;
fi;
# now x[2,2] is equal to One(f)
# we postpone the actual computation of the final x until we
# know it is needed:
co := Coefficients(can,x[1,2]);
if IsContainedInSpan(mb,co) then continue; fi;
# OK, we need it, so let's make it:
el := x[2,1];
co2 := Coefficients(can,el) * mati;
for i in [1..Length(co2)] do
if not IsZero(co2[i]) then
xm := xm * ttm[i];
fi;
od;
# TODO: add sanity check here, too???
x := StripMemory(xm);
# now x[2,1] is equal to Zero(f) and thus x[1,1] is One(f) as well
break;
od;
CloseMutableBasis(mb,co);
Add(ss,x);
Add(ssm,xm);
Add(mas,co);
od;
ConvertToMatrixRep(mas,2);
masi := mas^-1;
# Now we replace all the s and the t by some products to get rid
# of the base changes:
s := EmptyPlist(ext);
t := EmptyPlist(ext);
for i in [1..ext] do
co := Positions(masi[i],Z(2));
Add(s,Product(ssm{co}));
co := Positions(mati[i],Z(2));
Add(t,Product(ttm{co}));
od;
res := rec( g := g, t := t, s := s, bas := bas, basi := bas^-1,
one := One(f), a := s[1]*t[1]*s[1], b := One(s[1]),
One := One(s[1]), f := f, q := q, p := 2, ext := ext,
d := 2 );
res.all := Concatenation(res.s,res.t,[res.a],[res.b]);
return res;
end;
# RECOG.GuessSL2ElmOrder := function(x,f)
# local facts,i,j,o,p,q,r,s,y,z;
# p := Characteristic(f);
# q := Size(f);
# if IsOne(x) then return 1;
# elif IsOne(x^2) then return 2;
# fi;
# if p > 2 then
# y := x^p;
# if IsOne(y) then return p;
# elif IsOddInt(p) and IsOne(y^2) then
# return 2*p;
# fi;
# fi;
# if IsOne(x^(q-1)) then
# facts := Collected(FactInt(q-1:cheap)[1]);
# s := Product(facts,x->x[1]^x[2]);
# r := (q-1)/s;
# else
# facts := Collected(FactInt(q+1:cheap)[1]);
# s := Product(facts,x->x[1]^x[2]);
# r := (q+1)/s;
# fi;
# y := x^r;
# o := r;
# for i in [1..Length(facts)] do
# p := facts[i];
# j := p[2]-1;
# while j >= 0 do
# z := y^(s/p[1]^(p[2]-j));
# if not IsOne(z) then break; fi;
# j := j - 1;
# od;
# o := o * p[1]^(j+1);
# od;
# return o;
# end;
RECOG.GuessProjSL2ElmOrder := function(x,f)
local facts,i,j,o,p,q,r,s,y,z;
p := Characteristic(f);
q := Size(f);
if IsOneProjective(x) then return 1;
elif IsEvenInt(p) and IsOneProjective(x^2) then return 2;
fi;
if p > 2 then
y := x^p;
if IsOneProjective(y) then
return p;
fi;
fi;
if IsOneProjective(x^(q-1)) then
facts := Collected(FactInt(q-1:cheap)[1]);
s := Product(facts,x->x[1]^x[2]);
r := (q-1)/s;
else
facts := Collected(FactInt(q+1:cheap)[1]);
s := Product(facts,x->x[1]^x[2]);
r := (q+1)/s;
fi;
y := x^r;
o := r;
for i in [1..Length(facts)] do
p := facts[i];
j := p[2]-1;
while j >= 0 do
z := y^(s/p[1]^(p[2]-j));
if not IsOneProjective(z) then break; fi;
j := j - 1;
od;
o := o * p[1]^(j+1);
od;
return o;
end;
RECOG.IsThisSL2Natural := function(gens,f)
# Checks quickly whether or not this is SL(2,f).
# The answer is not guaranteed to be correct, this is Las Vegas.
local CheckElm,a,b,clos,coms,i,isabelian,j,l,notA5,p,q,S,seenqm1,seenqp1,x;
# The following method does not work for q <= 11, as then
# the projective orders are either q+1, or else less than 5.
# Hence seenqm1 never gets set.
CheckElm := function(x)
local o;
o := RECOG.GuessProjSL2ElmOrder(x,f);
if o in [1,2] then
return false;
fi;
if o > 5 then
if notA5 = false then Info(InfoRecog,4,"SL2: Group is not A5"); fi;
notA5 := true;
if seenqp1 and seenqm1 then
return true;
fi;
fi;
if o = p or o <= 5 then
return false;
fi;
if (q+1) mod o = 0 then
if not seenqp1 then
Info(InfoRecog,4,"SL2: Found element of order dividing q+1.");
seenqp1 := true;
if seenqm1 and notA5 then
return true;
fi;
fi;
else
if not seenqm1 then
Info(InfoRecog,4,"SL2: Found element of order dividing q-1.");
seenqm1 := true;
if seenqp1 and notA5 then
return true;
fi;
fi;
fi;
return false;
end;
if Length(gens) <= 1 then
Info(InfoRecog,4,"SL2: Group cyclic");
return false;
fi;
q := Size(f);
p := Characteristic(f);
# For small q, comput the order of the group via a stabilizer chain.
# Note that at this point we are usually working projective, and thus
# scalars are factored out "implicitly". Thus the generators we are
# looking at may generate a group which only contains SL2 as a subgroup.
if q <= 11 then # this could be increased if needed
Info(InfoRecog,4,"SL2: Computing stabiliser chain.");
S := StabilizerChain(Group(gens));
Info(InfoRecog,4,"SL2: size is ",Size(S));
return Size(S) mod (q*(q-1)*(q+1)) = 0;
fi;
seenqp1 := false;
seenqm1 := false;
notA5 := false;
for i in [1..Length(gens)] do
if CheckElm(gens[i]) then
return true;
fi;
od;
CheckElm(gens[1]*gens[2]);
if Length(gens) >= 3 then
CheckElm(gens[1]*gens[3]);
CheckElm(gens[2]*gens[3]);
fi;
# First we check the derived group:
coms := EmptyPlist(20);
l := Length(gens);
if l <= 4 then
Info(InfoRecog,4,"SL2: Computing commutators of gens...");
for i in [1..l-1] do
for j in [i+1..l] do
x := Comm(gens[i],gens[j]);
if CheckElm(x) then
return true;
fi;
Add(coms,x);
od;
od;
else
Info(InfoRecog,4,"SL2: Computing 6 random commutators...");
for i in [1..6] do
a := RECOG.RandomSubproduct(gens,rec());
b := RECOG.RandomSubproduct(gens,rec());
x := Comm(a,b);
if CheckElm(x) then
return true;
fi;
Add(coms,x);
od;
fi;
if ForAll(coms, IsDiagonalMat) then
Info(InfoRecog,4,"SL2: Group is soluble, commutators are central");
return false;
fi;
Info(InfoRecog,4,"SL2: Computing normal closure...");
clos := FastNormalClosure(gens,coms,5);
for i in [Length(coms)+1..Length(clos)] do
if CheckElm(clos[i]) then
return true;
fi;
od;
if ForAll(clos{[Length(coms)+1..Length(clos)]}, IsDiagonalMat) then
Info(InfoRecog,4,"SL2: Group is soluble, derived subgroup central");
return false;
fi;
Info(InfoRecog,4,"SL2: Computing 6 random commutators...");
isabelian := true;
for i in [1..6] do
a := RECOG.RandomSubproduct(clos,rec());
b := RECOG.RandomSubproduct(clos,rec());
x := Comm(a,b);
if RECOG.IsScalarMat(x) = false then isabelian := false; break; fi;
od;
if isabelian then
Info(InfoRecog,4,
"SL2: Group is soluble, derived subgroup abelian mod scalars");
return false;
fi;
# Now we know that the group is not dihedral!
return false;
end;
# The going down method:
#Version 1.2
# finds first element of a list that is relative prime to all others
# input: list=[SL(d,q), d, q, SL(n,q)] acting as a subgroup of some big SL(n,q)
# output: list=[rr, dd] for a ppd(2*dd;q)-element rr
RECOG.SLn_godown:=function(list)
local d, first, q, g, gg, i, r, pol, factors, degrees, newdim, power, rr, ss,
newgroup, colldegrees, exp, count;
first:=function(list)
local i;
for i in [1..Length(list)] do
if list[i]>1 and Gcd(list[i],Product(list)/list[i])=1 then
return list[i];
fi;
od;
return fail;
end;
g:=list[1];
d:=list[2];
q:=list[3];
gg:=list[4];
Info(InfoRecog,2,"Dimension: ",d);
#find an element with irreducible action of relative prime dimension to
#all other invariant subspaces
#count is just safety, if things go very bad
count:=0;
repeat
count:=count+1;
if InfoLevel(InfoRecog) >= 3 then Print(".\c"); fi;
r:=PseudoRandom(g);
pol:=CharacteristicPolynomial(r);
factors:=Factors(pol);
degrees:=AsSortedList(List(factors,Degree));
newdim:=first(degrees);
until (count>10) or (newdim <> fail and newdim<=Maximum(2,d/4));
if count>10 then
return fail;
fi;
# raise r to a power so that acting trivially outside one invariant subspace
degrees:=Filtered(degrees, x->x<>newdim);
colldegrees:=Collected(degrees);
power:=Lcm(List(degrees, x->q^x-1))*q;
# power further to cancel q-part of element order
if degrees[1]=1 then
exp:=colldegrees[1][2]-(DimensionOfMatrixGroup(gg)-d);
if exp>0 then
power:=power*q^exp;
fi;
fi;
rr:=r^power;
#conjugate rr to hopefully get a smaller dimensional SL
#ss:=rr^PseudoRandom(gg);
#newgroup:=Group(rr,ss);
return [rr,newdim];
end;
# input is (group,dimension,q)
# output is a group element acting irreducibly in two dimensions, and fixing
# a (dimension-2)-dimensional subspace
RECOG.SLn_constructppd2:=function(g,dim,q)
local out, list ;
list:=[g,dim,q,g];
repeat
out:=RECOG.SLn_godown(list);
if out=fail or out[1]*out[1]=One(out[1]) then
if InfoLevel(InfoRecog) >= 3 then Print("B\c"); fi;
list:=[g,dim,q,g];
out:=fail;
else
if out[2]>2 then
list:=[Group(out[1],out[1]^PseudoRandom(g)),2*out[2],q,g];
fi;
fi;
until out<>fail and out[2]=2;
return out[1];
end;
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(r-One(r));
nullspacer:=VectorSpace(GF(q),nullr);
mat:=One(r);
ready:=false;
repeat
s:=r^PseudoRandom(g);
nulls:=NullspaceMat(s-One(s));
nullspaces:=VectorSpace(GF(q),nulls);
int:=Intersection(nullspacer,nullspaces);
intbasis:=Basis(int);
newbasis:=[];
for i in [1..Length(intbasis)] do
Add(newbasis,intbasis[i]);
od;
i:=0;
repeat
i:=i+1;
if not mat[i] in int then
Add(newbasis,mat[i]);
int:=VectorSpace(GF(q),newbasis);
fi;
until Dimension(int)=dim;
ConvertToMatrixRep(newbasis);
newbasisinv:=newbasis^(-1);
newr:=newbasis*r*newbasisinv;
news:=newbasis*s*newbasisinv;
#shortr, shorts do not need memory
#we shall throw away the computations in h
#check that we have SL(4,q), by non-constructive recognition
shortr:=newr{[dim-3..dim]}{[dim-3..dim]};
shorts:=news{[dim-3..dim]}{[dim-3..dim]};
h:=Group(shortr,shorts);
count:=0;
readydim4:=false;
readydim3:=false;
repeat
u:=PseudoRandom(h);
orderu:=Order(u);
if orderu mod ((q^4-1)/(q-1)) = 0 then
readydim4:=true;
elif Gcd(orderu,(q^2+q+1)/Gcd(3,q-1))>1 then
readydim3:=true;
fi;
if readydim4 = true and readydim3 = true then
ready:=true;
break;
fi;
count:=count+1;
until count=30;
until ready=true;
return Group(r,s);
end;
#g=SL(d,q), given as a subgroup of SL(dim,q)
#output: [SL(2,q), and a basis for the 2-dimensional subspace where it acts
RECOG.SLn_godownfromd:=function(g,q,d,dim)
local y,yy,ready,order,es,dims,subsp,z,x,a,b,c,h,vec,vec2,
pol,factors,degrees,comm1,comm2,comm3,image,basis,action,vs,readyqpl1,
readyqm1,count,u,orderu;
repeat
ready:=false;
y:=PseudoRandom(g);
pol:=CharacteristicPolynomial(y);
factors:=Factors(pol);
degrees:=List(factors,Degree);
if d-1 in degrees then
order:=Order(y);
if order mod (q-1)=0 then
yy:=y^(order/(q-1));
else
yy:=One(y);
fi;
if not IsOne(yy) then
es:= Eigenspaces(GF(q),yy);
dims:=List(es,Dimension);
if IsSubset(Set([1,d-1,dim-d]),Set(dims)) and
(1 in Set(dims)) then
es:=Filtered(es,x->Dimension(x)=1);
vec:=Basis(es[1])[1];
if vec*yy=vec then
vec:=Basis(es[2])[1];
fi;
repeat
z:=PseudoRandom(g);
x:=yy^z;
a:=Comm(x,yy);
b:=a^yy;
c:=a^x;
comm1:= Comm(a,c);
comm2:=Comm(a,b);
comm3:=Comm(b,c);
if comm1<>One(a) and comm2<>One(a) and
comm3<>One(a) and Comm(comm1,comm2)<>One(a) then
vec2:=vec*z;
vs:=VectorSpace(GF(q),[vec,vec2]);
basis:=Basis(vs);
#check that the action in 2 dimensions is SL(2,q)
#by non-constructive recognition, finding elements of
#order (q-1) and (q+1)
#we do not need memory in the group image
action:=List([a,b,c],x->RECOG.LinearAction(basis,q,x));
image:=Group(action);
count:=0;
readyqpl1:=false;
readyqm1:=false;
repeat
u:=PseudoRandom(image);
orderu:=Order(u);
if orderu = q-1 then
readyqm1:=true;
elif orderu = q+1 then
readyqpl1:=true;
fi;
if readyqm1 = true and readyqpl1 = true then
ready:=true;
break;
fi;
count:=count+1;
until count=20;
fi;
until ready=true;
fi;
fi;
fi;
until ready;
h:=Group(a,b,c);
subsp:=VectorSpace(GF(q),[vec,vec2]);
return [h,subsp];
end;
#going down from 4 to 2 dimensions, when q=2,3,4,5,9
#just construct the 4-dimensional invariant space and generators
#for the group acting on it
RECOG.SLn_exceptionalgodown:=function(h,q,dim)
local basis, v, vs, i, gen;
vs:=VectorSpace(GF(q),One(h));
basis:=[];
repeat
if InfoLevel(InfoRecog) >= 3 then Print("C"); fi;
for i in [1..4] do
v:=PseudoRandom(vs);
for gen in GeneratorsOfGroup(h) do
Add(basis,v*gen-v);
od;
od;
basis:=ShallowCopy(SemiEchelonMat(basis).vectors);
until Length(basis)=4;
return [h,VectorSpace(GF(q),basis)];
end;
RECOG.SLn_constructsl2:=function(g,d,q)
local r,h;
r:=RECOG.SLn_constructppd2(g,d,q);
h:=RECOG.SLn_constructsl4(g,d,q,r);
if not (q in [2,3,4,5,9]) then
return RECOG.SLn_godownfromd(h,q,4,d);
else
return RECOG.SLn_exceptionalgodown(h,q,d);
# return ["sorry only SL(4,q)",h];
fi;
end;
# Now the going up code:
RECOG.LinearAction := function(bas,field,el)
local mat,vecs;
if IsGroup(el) then
return Group(List(GeneratorsOfGroup(el),
x->RECOG.LinearAction(bas,field,x)));
fi;
if IsBasis(bas) then
vecs := BasisVectors(bas);
else
vecs := bas;
bas := Basis(VectorSpace(field,bas),bas);
fi;
mat := List(vecs,v->Coefficients(bas,v*el));
ConvertToMatrixRep(mat,field);
return mat;
end;
RECOG.SLn_UpStep := function(w)
# w has components:
# d : size of big SL
# n : size of small SL
# slnstdf : fakegens for SL_n standard generators
# bas : current base change, first n vectors are where SL_n acts
# rest of vecs are invariant under SL_n
# basi : current inverse of bas
# sld : original group with memory generators, PseudoRandom
# delivers random elements
# sldf : fake generators to keep track of what we are doing
# f : field
# The following are filled in automatically if not already there:
# p : characteristic
# ext : q=p^ext
# One : One(slnstdf[1])
# can : CanonicalBasis(f)
# canb : BasisVectors(can)
# transh : fakegens for the "horizontal" transvections n,i for 1<=i<=n-1
# entries can be unbound in which case they are made from slnstdf
# transv : fakegens for the "vertical" transvections i,n for 1<=i<=n-1
--> --------------------
--> maximum size reached
--> --------------------
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2026-04-02
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