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grahamknockillareegrahamknockillaree
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Merge pull request #150 from giuliao27/extra-methods
Additional methods for constructing congruence subgroups of SL2 and SL3
2 parents 6fdb585 + 4f74499 commit 169ec61

6 files changed

Lines changed: 177 additions & 4 deletions

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‎lib/Orru/decsOrru.gd‎

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Original file line numberDiff line numberDiff line change
@@ -3,6 +3,7 @@ IsHapRightTransversalSL3ZSubgroup:=NewFilter("IsHapRightTransversalSL3ZSubgroup"
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DeclareGlobalFunction("HAP_GenericCongruenceSubgroup");
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DeclareOperation("FiniteProjectiveLine",[IsInt]);
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DeclareOperation("FiniteProjectivePlane",[IsInt]);
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DeclareOperation("FiniteProjectiveLine_alt",[IsInt]);
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DeclareGlobalFunction("HAP_FiniteProjectiveLineIntegers");
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DeclareGlobalFunction("HAP_FiniteProjectiveLineIntegers_alt");
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DeclareGlobalFunction("HAP_FiniteProjectivePlaneIntegers");

‎lib/Orru/finiteProjectiveSpaces.gi‎

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Original file line numberDiff line numberDiff line change
@@ -20,6 +20,12 @@ return HAP_FiniteProjectivePlaneIntegers(n);
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end);
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########################################################
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########################################################
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InstallMethod(FiniteProjectiveLine_alt,
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"Finite projective line for the ring Z/nZ",
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[IsInt],
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function(n)
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return HAP_FiniteProjectiveLineIntegers_alt(n);
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end);
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‎lib/Orru/initOrru.gi‎

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Original file line numberDiff line numberDiff line change
@@ -2,3 +2,5 @@ ReadPackageHap( "lib/Orru/subgroups.gi");
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ReadPackageHap( "lib/Orru/finiteProjectiveSpaces.gi");
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ReadPackageHap( "lib/Orru/HAP_SL3ZSubgroupTree_fast.gi");
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ReadPackageHap( "lib/Orru/SL3ZTopHomology.gi");
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ReadPackageHap( "lib/Orru/sl2methodsExtra.gi");
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ReadPackageHap( "lib/Orru/sl3methods.gi");

‎lib/Orru/sl2methodsExtra.gi‎

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Original file line numberDiff line numberDiff line change
@@ -0,0 +1,79 @@
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InstallMethod(CosetPosFunction,
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"Returns cosetPos(g) function for the congruence subgroup G",
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[ IsIntegerMatrixGroup and IsHAPCongruenceSubgroupGamma0 ],
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function(G)
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local cosetPos, canonicalRep, n, ProjLine;
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if DimensionOfMatrixGroup(G) <> 2 then
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TryNextMethod();
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fi;
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n := LevelOfCongruenceSubgroup(G);
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ProjLine := FiniteProjectiveLine(n);
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canonicalRep := function(g)
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local v, vv, U, d, dd, x, y;
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v := [g[1][1], g[2][1]];
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vv := List(v, x -> x mod n);
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U := Units(Integers mod n);
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if vv[1] mod n = 0 then
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return [0,1];
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elif ZmodnZObj(vv[1],n) in U then
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return [1,(Inverse(vv[1]) mod n)*vv[2] mod n];
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else
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d := Gcd(vv[1],n);
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dd := n/d;
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x := vv[1]/d;
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y := vv[2]/x mod dd;
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while not Gcd(d,y) = 1 do
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y := y + dd;
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od;
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return [d, y];
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fi;
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end;
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cosetPos := function(g)
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local w;
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w := canonicalRep(g);
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return Position(ProjLine,w);
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end;
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return cosetPos;
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end);
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InstallMethod(CosetRepFunction,
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"Returns cosetPos(g) function for the congruence subgroup G",
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[ IsIntegerMatrixGroup and IsHAPCongruenceSubgroupGamma0 ],
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function(G)
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local cosetOfInt, cosetRep, n, ProjLine, cosetPos;
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if DimensionOfMatrixGroup(G) <> 2 then
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TryNextMethod();
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fi;
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n := LevelOfCongruenceSubgroup(G);
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ProjLine := FiniteProjectiveLine(n);
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cosetOfInt := function(i)
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local a, c, b, d, gg;
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a := ProjLine[i][1];
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c := ProjLine[i][2];
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if a = 0 then
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return [[0,-1],[1,0]];
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fi;
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gg := Gcdex(a,c);
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b := -gg.coeff2;
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d := gg.coeff1;
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return [[a,b],[c,d]];
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end;
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cosetPos := CosetPosFunction(G);
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cosetRep:=function(g);
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return cosetOfInt(cosetPos(g));
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end;
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return cosetRep;
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end);

‎lib/Orru/sl3methods.gi‎

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@@ -0,0 +1,80 @@
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##########################################################################
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##
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## Methods for 3x3 congruence subgroups of SL3
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##########################################################################
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##
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## CosetPosFunction( <G> )
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##
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## Returns a function cosetPos(g) giving the position of the coset gG in
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## the ambient group.
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InstallMethod(CosetPosFunction,
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"Returns cosetPos(g) function for the congruence subgroup G",
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[ IsIntegerMatrixGroup and IsHAPCongruenceSubgroupGamma0 ],
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function(G)
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local cosetPos, n, ProjPlane;
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if DimensionOfMatrixGroup(G) <> 3 then
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TryNextMethod();
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fi;
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n := LevelOfCongruenceSubgroup(G);
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ProjPlane := FiniteProjectivePlane(n);
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cosetPos := function(g)
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local v, vv, U, u, w;
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v := [g[1][1], g[2][1], g[3][1]];
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vv := List(v, x -> x mod n);
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U := Units(Integers mod n);
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for u in U do
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w := List(vv, x -> (Int(u)*x) mod n);
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if w in ProjPlane.Reps then
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return Position(ProjPlane.Reps,w);
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fi;
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od;
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end;
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return cosetPos;
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end);
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##########################################################################
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##
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## CosetRepFunction( <G> )
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##
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## Returns a function cosetPos(g) giving a canonical rpresentative of the
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## coset gG in the ambient group.
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InstallMethod(CosetRepFunction,
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"Returns cosetRep(g) function for the congruence subgroup G",
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[ IsIntegerMatrixGroup and IsHAPCongruenceSubgroupGamma0 ],
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function(G)
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local MatrixInSL3_Hermite, cosetOfInt, cosetRep, n, ProjPlane, cosetPos;
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if DimensionOfMatrixGroup(G) <> 3 then
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TryNextMethod();
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fi;
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n := LevelOfCongruenceSubgroup(G);
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MatrixInSL3_Hermite := function(v)
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local Herm;
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Herm := HermiteNormalFormIntegerMatTransform([[v[1]],[v[2]],[v[3]]]);
59+
return Inverse(Herm!.rowtrans);
60+
end;
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ProjPlane := FiniteProjectivePlane(n);
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cosetOfInt:=function(i)
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local x,y,z;
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x := ProjPlane.Reps[i][1];
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y := ProjPlane.Reps[i][2];
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z := ProjPlane.Reps[i][3];
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return MatrixInSL3_Hermite([x,y,z]);
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end;
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cosetPos := CosetPosFunction(G);
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cosetRep:=function(g);
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return cosetOfInt(cosetPos(g));
77+
end;
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return cosetRep;
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end);

‎lib/Orru/subgroups.gi‎

Lines changed: 9 additions & 4 deletions
Original file line numberDiff line numberDiff line change
@@ -124,7 +124,7 @@ function(n,m)
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G!.index := m*Product(List(SSortedList(Factors(m)), p->1+1/p));
125125
fi;
126126

127-
ProjLine := FiniteProjectiveLine(m);
127+
ProjLine := FiniteProjectiveLine_alt(m);
128128
CosetPos := function(g)
129129
local v, vv, U, u, w;
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v := [g[1][1], g[2][1]];
@@ -225,7 +225,7 @@ function(n,m)
225225
ProjLine := FiniteProjectiveLine(m);
226226

227227
CanonicalRep := function(g)
228-
local v, vv, U, d, dd;
228+
local v, vv, U, d, dd, x, y;
229229
v := [g[1][1], g[2][1]];
230230
vv := List(v, x -> x mod m);
231231
U := Units(Integers mod m);
@@ -236,7 +236,12 @@ function(n,m)
236236
else
237237
d := Gcd(vv[1],m);
238238
dd := m/d;
239-
return [vv[1], vv[2] mod dd];
239+
x := vv[1]/d;
240+
y := vv[2]/x mod dd;
241+
while not Gcd(d,y) = 1 do
242+
y := y + dd;
243+
od;
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return [d, y];
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fi;
241246
end;
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@@ -306,7 +311,7 @@ function(G,H)
306311
TryNextMethod();
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fi;
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309-
HAPCongruenceSubgroupTree(H);
314+
#HAPCongruenceSubgroupTree(H);
310315

311316
return HAP_TransversalCongruenceSubgroupInAmbientGroup(G,H);
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end);

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