/usr/share/gap/pkg/Polycyclic/gap/cohom/twocohom.gi is in gap-polycyclic 2.11-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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##
#F CollectedTwoCR( A, u, v )
##
InstallGlobalFunction( CollectedTwoCR, function( A, u, v )
local n, word, tail, rels, wstack, tstack, p, c, l, g, e, mat, s, t, r, i;
# set up and push u into result
n := Length( A.mats );
word := ShallowCopy( u.word );
tail := ShallowCopy( u.tail );
rels := RelativeOrdersOfPcp( A.factor );
# catch a trivial case
if v.word = 0 * v.word then
AddTailVectorsCR( tail, v.tail );
return rec( word := word, tail := tail );
fi;
# create stacks and put v onto stack
wstack := [WordOfVectorCR( v.word )];
tstack := [v.tail];
p := [1];
c := [1];
# run until stacks are empty
l := 1;
while l > 0 do
# take a generator and its exponent
g := wstack[l][p[l]][1];
e := wstack[l][p[l]][2];
# take operation mat
if e < 0 then
mat := A.invs[g];
else
mat := A.mats[g];
fi;
# push g through module tail
for i in [1..Length(tail)] do
if IsBound( tail[i] ) then
tail[i] := tail[i] * mat;
fi;
od;
# correct the stacks
c[l] := c[l] + 1;
if AbsInt(e) < c[l] then # exponent overflow
c[l] := 1;
p[l] := p[l] + 1;
if Length(wstack[l]) < p[l] then # word overflow - add tail
AddTailVectorsCR( tail, tstack[l] );
tstack[l] := 0;
l := l - 1;
fi;
fi;
# push g through word
for i in [ n, n-1 .. g+1 ] do
if word[i] <> 0 then
# get relator and tail
t := [];
if e > 0 then
s := Position( A.enumrels, [i, g] );
r := PowerWord( A, A.relators[i][g], word[i] );
t[s] := PowerTail( A, A.relators[i][g], word[i] );
elif e < 0 then
s := Position( A.enumrels, [i, i+g] );
r := PowerWord( A, A.relators[i][i+g], word[i] );
t[s] := PowerTail( A, A.relators[i][i+g], word[i] );
fi;
# add to stacks
AddTailVectorsCR( tail, t );
l := l+1;
wstack[l] := r;
tstack[l] := tail;
tail := [];
c[l] := 1;
p[l] := 1;
fi;
# reset
word[i] := 0;
od;
# increase exponent
if e < 0 then
word[g] := word[g] - 1;
else
word[g] := word[g] + 1;
fi;
# insert power relators if exponent has reached rel order
if rels[g] > 0 and word[g] = rels[g] then
word[g] := 0;
r := A.relators[g][g];
s := Position( A.enumrels, [g, g] );
for i in [1..Length(r)] do
word[r[i][1]] := r[i][2];
od;
t := []; t[s] := A.one;
AddTailVectorsCR( tail, t );
# insert power relators if exponent is negative
elif rels[g] > 0 and word[g] < 0 then
word[g] := rels[g] + word[g];
if Length(A.relators[g][g]) <= 1 then
r := A.relators[g][g];
for i in [1..Length(r)] do
word[r[i][1]] := -r[i][2];
od;
s := Position( A.enumrels, [g, g] );
t := []; t[s] := - MappedWordCR( r, A.mats, A.invs );
AddTailVectorsCR( tail, t );
else
r := InvertWord( A.relators[g][g] );
s := Position( A.enumrels, [g, g] );
t := []; t[s] := - MappedWordCR( r, A.mats, A.invs );
AddTailVectorsCR( tail, t );
l := l+1;
wstack[l] := r;
tstack[l] := tail;
tail := [];
c[l] := 1;
p[l] := 1;
fi;
fi;
od;
return rec( word := word, tail := tail );
end );
#############################################################################
##
#F TwoCocyclesCR( A )
##
InstallGlobalFunction( TwoCocyclesCR, function( A )
local C, n, e, id, l, gn, gp, gi, eq, pairs, i, j, k, w1, w2, d, sys, h;
# set up system of length d
n := Length( A.mats );
e := RelativeOrdersOfPcp( A.factor );
l := Length( A.enumrels );
if IsBound(A.endosys) then
sys := List( A.endosys, x -> CRSystem( x[2], l, 0 ) );
for i in [1..Length(sys)] do sys[i].full := true; od;
else
sys := CRSystem( A.dim, l, A.char );
fi;
# set up for equations
id := IdentityMat(n);
gn := List( id, x -> rec( word := x, tail := [] ) );
# precompute (ij) for i > j
pairs := List( [1..n], x -> [] );
for i in [1..n] do
if e[i] > 0 then
h := rec( word := (e[i] - 1) * id[i], tail := [] );
pairs[i][i] := CollectedTwoCR( A, h, gn[i] );
fi;
for j in [1..i-1] do
pairs[i][j] := CollectedTwoCR( A, gn[i], gn[j] );
od;
od;
# consistency 1: k(ji) = (kj)i
for i in [ n, n-1 .. 1 ] do
for j in [ n, n-1 .. i+1 ] do
for k in [ n, n-1 .. j+1 ] do
w1 := CollectedTwoCR( A, gn[k], pairs[j][i] );
w2 := CollectedTwoCR( A, pairs[k][j], gn[i] );
if w1.word <> w2.word then
Error( "k(ji) <> (kj)i" );
else
AddEquationsCR( sys, w1.tail, w2.tail, true );
fi;
od;
od;
od;
# consistency 2: j^(p-1) (ji) = j^p i
for i in [n,n-1..1] do
for j in [n,n-1..i+1] do
if e[j] > 0 then
h := rec( word := (e[j] - 1) * id[j], tail := [] );
w1 := CollectedTwoCR( A, h, pairs[j][i]);
w2 := CollectedTwoCR( A, pairs[j][j], gn[i]);
if w1.word <> w2.word then
Error( "j^(p-1) (ji) <> j^p i" );
else
AddEquationsCR( sys, w1.tail, w2.tail, true );
fi;
fi;
od;
od;
# consistency 3: k (i i^(p-1)) = (ki) i^p-1
for i in [n,n-1..1] do
if e[i] > 0 then
h := rec( word := (e[i] - 1) * id[i], tail := [] );
l := CollectedTwoCR( A, gn[i], h );
for k in [n,n-1..i+1] do
w1 := CollectedTwoCR( A, gn[k], l );
w2 := CollectedTwoCR( A, pairs[k][i], h );
if w1.word <> w2.word then
Error( "k i^p <> (ki) i^(p-1)" );
else
AddEquationsCR( sys, w1.tail, w2.tail, true );
fi;
od;
fi;
od;
# consistency 4: (i i^(p-1)) i = i (i^(p-1) i)
for i in [ n, n-1 .. 1 ] do
if e[i] > 0 then
h := rec( word := (e[i] - 1) * id[i], tail := [] );
l := CollectedTwoCR( A, gn[i], h );
w1 := CollectedTwoCR( A, l, gn[i] );
w2 := CollectedTwoCR( A, gn[i], pairs[i][i] );
if w1.word <> w2.word then
Error( "i i^p-1 <> i^p" );
else
AddEquationsCR( sys, w1.tail, w2.tail, true );
fi;
fi;
od;
# consistency 5: j = (j -i) i
gi := List( id, x -> rec( word := -x, tail := [] ) );
for i in [n,n-1..1] do
for j in [n,n-1..i+1] do
if e[i] = 0 then
w1 := CollectedTwoCR( A, gn[j], gi[i] );
w2 := CollectedTwoCR( A, w1, gn[i] );
if w2.word <> id[j] then
Error( "j <> (j -i) i" );
else
AddEquationsCR( sys, w2.tail, [], true );
fi;
fi;
od;
od;
# consistency 6: i = -j (j i)
for i in [n,n-1..1] do
for j in [n,n-1..i+1] do
if e[j] = 0 then
w1 := CollectedTwoCR( A, gi[j], pairs[j][i] );
if w1.word <> id[i] then
Error( "i <> -j (j i)" );
else
AddEquationsCR( sys, w1.tail, [], true );
fi;
fi;
od;
od;
# consistency 7: -i = -j (j -i)
for i in [n,n-1..1] do
for j in [n,n-1..i+1] do
if e[i] = 0 and e[j] = 0 then
w1 := CollectedTwoCR( A, gn[j], gi[i] );
w1 := CollectedTwoCR( A, gi[j], w1 );
if w1.word <> -id[i] then
Error( "-i <> -j (j -i)" );
else
AddEquationsCR( sys, w1.tail, [], true );
fi;
fi;
od;
od;
# add a check ((j ^ i) ^-i ) = j
for i in [1..n] do
for j in [1..i-1] do
w1 := CollectedTwoCR( A, gi[j], pairs[i][j] );
w1 := CollectedTwoCR( A, gn[j], w1 );
w1 := CollectedTwoCR( A, w1, gi[j] );
if w1.word <> id[i] then
Error("in rel check ");
elif not IsZeroTail( w2.tail ) then
# Error("relations bug");
AddEquationsCR( sys, w1.tail, [], true );
fi;
od;
od;
# and return solution
return KernelCR( A, sys );
end );
#############################################################################
##
#F TwoCoboundariesCR( A )
##
InstallGlobalFunction( TwoCoboundariesCR, function( A )
local n, e, l, sys, R, c, tail, i, t, j;
# set up system of length d
n := Length( A.mats );
e := RelativeOrdersOfPcp( A.factor );
l := Length( A.enumrels );
if IsBound(A.endosys) then
sys := List( A.endosys, x -> CRSystem( x[2], l, 0 ) );
for i in [1..Length(sys)] do sys[i].full := true; od;
else
sys := CRSystem( A.dim, l, A.char );
fi;
# loop over relators
R := [];
for c in A.enumrels do
tail := CollectedRelatorCR( A, c[1], c[2] );
SubtractTailVectors( tail[1], tail[2] );
Add( R, tail[1] );
od;
# shift into system
for i in [1..n] do
t := [];
for j in [1..l] do
if IsBound(R[j][i]) then t[j] := TransposedMat(R[j][i]); fi;
od;
if IsList(sys) then
AddEquationsCREndo( sys, t );
else
AddEquationsCRNorm( sys, t, true );
fi;
od;
# return
return ImageCR( A, sys ).basis;
end );
#############################################################################
##
#F TwoCohomologyCR( A )
##
InstallGlobalFunction( TwoCohomologyCR, function( A )
local cc, cb, exp, l, B, b, Q, U, V, i;
cc := TwoCocyclesCR( A );
cb := TwoCoboundariesCR( A );
if not IsBound(A.endosys) then
return rec( gcc := cc, gcb := cb,
factor := AdditiveFactorPcp( cc, cb, A.char ));
fi;
Q := [];
for i in [1..Length(cc)] do
if Length(cc[i]) = 0 then Add( Q, AbelianPcpGroup([])); fi;
exp := A.mats[1][i]!.exp;
l := Length(cc[i][1])/Length(exp);
B := AbelianPcpGroup( Concatenation(List([1..l], x -> exp)) );
b := Igs(B);
U := Subgroup( B, List(cc[i], x -> MappedVector(x,b)));
V := Subgroup( B, List(cb[i], x -> MappedVector(x,b)));
Add(Q, U/V);
od;
return Q;
end );
#############################################################################
##
#F TwoCohomologyTrivialModule( G, d[, p] )
##
TwoCohomologyTrivialModule := function(arg)
local G, d, m, C, c;
# catch arguments
G := arg[1];
d := arg[2];
if Length(arg)=2 then
m := List(Igs(G), x -> IdentityMat(d));
elif Length(arg)=3 then
m := List(Igs(G), x -> IdentityMat(d,arg[3]));
fi;
# construct H^2
C := CRRecordByMats(G, m);
c := TwoCohomologyCR(C);
return c.factor.rels;
end;
#############################################################################
##
#F CheckTrivialCohom( G )
##
CheckTrivialCohom := function(G)
local mats, C, cb, cc, c, E;
# compute cohom
Print("compute cohomology \n");
mats := List( Pcp(G), x -> IdentityMat( 1 ) );
C := CRRecordByMats( G, mats );
cb := TwoCoboundariesCR( C );
Print("cb has length ", Length(cb)," \n");
cc := TwoCocyclesCR( C );
Print("cc has length ", Length(cc)," \n");
# first check
Print("check cb in cc \n");
c := First( cb, x -> IsBool( SolutionMat( cc,x ) ) );
if not IsBool( c ) then
Print(" coboundary is not contained in cc \n");
return c;
fi;
# second check
Print("check cc \n");
for c in cc do
E := ExtensionCR( C, c );
od;
end;
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