|
############################################################################
##
## matmeths.gi IRREDSOL Burkhard Höfling
##
## Copyright © 2003–2016 Burkhard Höfling
##
############################################################################
##
#M Degree(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(Degree, "for matrix group", true, [IsMatrixGroup], 0,
DegreeOfMatrixGroup);
############################################################################
##
#M DegreeOfMatrixGroup(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(DegreeOfMatrixGroup, "for matrix group with dimension", true,
[IsMatrixGroup and HasDimension], 0,
Dimension);
############################################################################
##
#M IsIrreducible(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(IsIrreducible, "for matrix group", true, [IsMatrixGroup], 0,
function(G)
return IsIrreducibleMatrixGroup(G, FieldOfMatrixGroup(G));
end);
############################################################################
##
#M IsIrreducible(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(IsIrreducible, "for matrix group and field", IsMatGroupOverFieldFam,
[IsMatrixGroup, IsField], 0,
function(G, F)
return IsIrreducibleMatrixGroup(G, F);
end);
############################################################################
##
#M IsIrreducibleMatrixGroup(<G>)
##
##
InstallOtherMethod(IsIrreducibleMatrixGroup, "for matrix group", true, [IsMatrixGroup], 0,
function(G)
return IsIrreducibleMatrixGroup(G, FieldOfMatrixGroup(G));
end);
############################################################################
##
#M IsIrreducibleMatrixGroup(<G>, <F>)
##
InstallMethod(IsIrreducibleMatrixGroupOp, "for matrix group and finite field - use MeatAxe",
IsMatGroupOverFieldFam, [IsFFEMatrixGroup, IsField and IsFinite], 0,
function(G, F)
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
if DegreeOfMatrixGroup(G) = 1 then
return true;
elif IsTrivial(G) then
return false;
elif IsSubset(F, FieldOfMatrixGroup(G)) then
return MTX.IsIrreducible (GModuleByMats (GeneratorsOfGroup(G), F));
else
Error("G must be a matrix group over F");
fi;
end);
############################################################################
##
#M IsIrreducibleMatrixGroup(<G>, <F>)
##
InstallMethod(IsIrreducibleMatrixGroupOp, "for matrix group and finite field - test attr IsIrreducibleMatrixGroup",
IsMatGroupOverFieldFam, [IsFFEMatrixGroup and HasIsIrreducibleMatrixGroup,
IsField and IsFinite], 0,
function(G, F)
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
if IsIrreducibleMatrixGroup(G) then
if F = FieldOfMatrixGroup(G) then
return true;
fi;
elif IsSubset(F, FieldOfMatrixGroup(G)) then
return false;
fi;
TryNextMethod();
end);
############################################################################
##
#M IsIrreducibleMatrixGroup(<G>, <F>)
##
InstallMethod(IsIrreducibleMatrixGroupOp, "for matrix group and finite field - for absolutely irreducible matrix group",
IsMatGroupOverFieldFam, [IsFFEMatrixGroup and IsAbsolutelyIrreducibleMatrixGroup,
IsField and IsFinite], RankFilter(HasIsIrreducibleMatrixGroup),
function(G, F)
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
return true;
end);
############################################################################
##
#M IsAbsolutelyIrreducible(<G>)
##
InstallMethod(IsAbsolutelyIrreducible, "for matrix group", true, [IsMatrixGroup], 0,
function(G)
return IsAbsolutelyIrreducibleMatrixGroup(G);
end);
############################################################################
##
#M IsAbsolutelyIrreducibleMatrixGroup(<G>)
##
## we use `InstallOtherMethod' because otherwise there will be a warning
## generated by KeyDependentOperation
##
InstallOtherMethod(IsAbsolutelyIrreducibleMatrixGroup, "for mat group over finite field", true,
[IsFFEMatrixGroup], 0,
function(G)
local M;
if DegreeOfMatrixGroup(G) = 1 then
return true;
elif IsTrivial(G) then
return false;
else
M := GModuleByMats (GeneratorsOfGroup(G), DefaultFieldOfMatrixGroup(G));
return MTX.IsIrreducible (M) and MTX.IsAbsolutelyIrreducible (M);
fi;
end);
############################################################################
##
#M IsPrimitive(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(IsPrimitive, "for matrix group", true, [IsMatrixGroup], 0,
function(G)
return IsPrimitiveMatrixGroup(G, FieldOfMatrixGroup(G));
end);
############################################################################
##
#M IsPrimitiveMatrixGroup(<G>)
##
## we need an OtherMethod to avoid the warning generated by
## KeyDependentOperation
##
InstallOtherMethod(IsPrimitiveMatrixGroup, "for matrix group", true, [IsMatrixGroup], 0,
function(G)
return IsPrimitiveMatrixGroup(G, FieldOfMatrixGroup(G));
end);
############################################################################
##
#M IsPrimitive(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(IsPrimitive, "for matrix group over field",
IsMatGroupOverFieldFam, [IsMatrixGroup, IsField], 0,
function(G, F)
return IsPrimitiveMatrixGroup(G, F);
end);
############################################################################
##
#M IsPrimitiveMatrixGroupOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(IsPrimitiveMatrixGroupOp, "for matrix group over finite field, construct IsomorphismPcGroup",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and IsSolvableGroup, IsField and IsFinite], 0,
function(G, F)
local iso, inv;
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
iso := IsomorphismPcGroup(G);
inv := InverseGeneralMapping (iso);
SetIsBijective (inv, true);
SetIsGroupHomomorphism (inv, true);
return SmallBlockDimensionOfRepresentation (ImagesSource (iso), inv, F, DegreeOfMatrixGroup(G)) = DegreeOfMatrixGroup(G);
end);
############################################################################
##
#M IsPrimitiveMatrixGroupOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(IsPrimitiveMatrixGroupOp, "for matrix group over finite field, use RepresentationIsomorphism",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and HasRepresentationIsomorphism, IsField and IsFinite],
RankFilter (IsHandledByNiceMonomorphism) + 1, # rank higher than the nice mono. method
function(G, F)
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
return SmallBlockDimensionOfRepresentation (
Source (RepresentationIsomorphism (G)), RepresentationIsomorphism (G), F, DegreeOfMatrixGroup(G)) = DegreeOfMatrixGroup(G);
end);
############################################################################
##
#M IsPrimitiveMatrixGroupOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(IsPrimitiveMatrixGroupOp, "for matrix group over finite field, use nice monomorphism",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and IsHandledByNiceMonomorphism, IsField and IsFinite],
0,
function(G, F)
local iso, inv;
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
iso := NiceMonomorphism (G);
inv := GroupHomomorphismByFunction(NiceObject (G), G,
h -> PreImagesRepresentative(iso, h),
g -> ImageElm(iso, g));
SetIsBijective (inv, true);
return SmallBlockDimensionOfRepresentation (NiceObject (G), inv, F, DegreeOfMatrixGroup(G)) = DegreeOfMatrixGroup(G);
end);
############################################################################
##
#M IsPrimitiveMatrixGroupOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(IsPrimitiveMatrixGroupOp, "for matrix group over finite field, try if IsPrimitive is set",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and HasIsPrimitive, IsField and IsFinite],
RankFilter (IsHandledByNiceMonomorphism) + 3, # rank higher than the nice mono. method
function(G, F)
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
if IsPrimitive(G) then
if FieldOfMatrixGroup(G) = F then
return true;
fi;
else
return false;
fi;
TryNextMethod();
end);
############################################################################
##
#F SmallBlockDimensionOfRepresentation(G, hom, F, limit)
##
## hom must be a homomorphism G -> GL(n, F), where G is a group and F a finite
## field such that Image(hom, G) is irreducible over F. limit is an integer
## The function returns an integer k such that Im hom has a block system
## of block dimension k, where k < limit, or k >= limit and G has no
## block system of block dimension < limit
##
InstallGlobalFunction(SmallBlockDimensionOfRepresentation, function(G, hom, F, limit)
# computes a block dimension smaller than limit, if it exists,
# or the smallest block dimension otherwise
local max, min, dim, M, m, cf, i;
max := AttributeValueNotSet(MaximalSubgroupClassReps, G);
min := DegreeOfMatrixGroup(Range (hom));
for M in max do
if not IsTrivial(M) then
m := GModuleByMats (List(GeneratorsOfGroup(M), x -> ImageElm(hom, x)), F);
if not MTX.IsIrreducible (m) then
cf := First (MTX.CompositionFactors(m),
cf -> MTX.Dimension (cf) * IndexNC(G, M) = DegreeOfMatrixGroup(Range (hom))
and Length(MTX.Homomorphisms (cf, m)) > 0);
if cf <> fail then
dim := SmallBlockDimensionOfRepresentation (M,
GroupHomomorphismByImagesNC(M, GL(MTX.Dimension (cf), Size(F)),
GeneratorsOfGroup(M), MTX.Generators (cf)),
F, limit);
if dim < min then
min := dim;
if min < limit then
return min;
fi;
fi;
fi;
fi;
fi;
od;
return min;
end);
############################################################################
##
#M MinimalBlockDimension(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(MinimalBlockDimension, "for matrix group", true, [IsMatrixGroup], 0,
function(G)
return MinimalBlockDimensionOfMatrixGroup(G);
end);
############################################################################
##
#M MinimalBlockDimension(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(MinimalBlockDimension, "for matrix group and field",
IsMatGroupOverFieldFam, [IsMatrixGroup, IsField], 0,
function(G, F)
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
return MinimalBlockDimensionOfMatrixGroup(G, F);
end);
############################################################################
##
#M MinimalBlockDimensionOfMatrixGroup(<G>)
##
## see IRREDSOL documentation
##
InstallOtherMethod(MinimalBlockDimensionOfMatrixGroup, "for matrix group", true,
[IsMatrixGroup], 0,
function(G)
return MinimalBlockDimensionOfMatrixGroup(G, FieldOfMatrixGroup(G));
end);
############################################################################
##
#M MinimalBlockDimensionOfMatrixGroupOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(MinimalBlockDimensionOfMatrixGroupOp, "for matrix group over finite field",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and IsSolvableGroup, IsField and IsFinite], 0,
function(G, F)
local iso, inv;
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
if not IsIrreducibleMatrixGroup(G, F) then
TryNextMethod();
fi;
iso := IsomorphismPcGroup(G);
inv := InverseGeneralMapping (iso);
SetIsBijective (inv, true);
SetIsGroupHomomorphism (inv, true);
return SmallBlockDimensionOfRepresentation (ImagesSource (iso), inv, F, 2);
end);
############################################################################
##
#M MinimalBlockDimensionOfMatrixGroupOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(MinimalBlockDimensionOfMatrixGroupOp,
"for matrix group over finite field with representation homomorphism",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and HasRepresentationIsomorphism, IsField and IsFinite],
RankFilter (IsHandledByNiceMonomorphism) + 1, # rank higher than the nice mono. method
function(G, F)
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
if not IsIrreducibleMatrixGroup(G, F) then
TryNextMethod();
fi;
return SmallBlockDimensionOfRepresentation (
Source (RepresentationIsomorphism (G)), RepresentationIsomorphism (G), F, 2) ;
end);
############################################################################
##
#M MinimalBlockDimensionOfMatrixGroupOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(MinimalBlockDimensionOfMatrixGroupOp, "for matrix group over finite field, use NiceMonomorphism",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and IsHandledByNiceMonomorphism, IsField and IsFinite],
0,
function(G, F)
local iso, inv;
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
if not IsIrreducibleMatrixGroup(G, F) then
TryNextMethod();
fi;
iso := NiceMonomorphism (G);
inv := GroupHomomorphismByFunction(NiceObject (G), G,
h -> PreImagesRepresentative(iso, h),
g -> ImageElm(iso, g));
SetIsBijective (inv, true);
return SmallBlockDimensionOfRepresentation (NiceObject (G), inv, F, 2);
end);
############################################################################
##
#M MinimalBlockDimensionOfMatrixGroupOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(MinimalBlockDimensionOfMatrixGroupOp,
"for matrix group over finite field which has MinimalBlockDimension",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and HasMinimalBlockDimension, IsField and IsFinite],
RankFilter (IsHandledByNiceMonomorphism) + 1, # rank higher than the nice mono. method
function(G, F)
if not IsSubset(F, FieldOfMatrixGroup(G)) then
Error("G must be a matrix group over F");
fi;
if F = FieldOfMatrixGroup(G) then
return MinimalBlockDimension (G);
elif MinimalBlockDimension (G) = 1 then
return 1;
fi;
TryNextMethod();
end);
############################################################################
##
#M CharacteristicOfField(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(CharacteristicOfField, "for matrix group", true, [IsMatrixGroup], 0,
M -> Characteristic (DefaultFieldOfMatrixGroup(M)));
############################################################################
##
#M Characteristic(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(Characteristic, "for matrix group", true, [IsMatrixGroup], 0,
CharacteristicOfField);
############################################################################
##
#M RepresentationIsomorphism(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(RepresentationIsomorphism, "for mat group handled by nice mono.", true,
[IsMatrixGroup and IsHandledByNiceMonomorphism], 0,
function(G)
local nice, H;
nice := NiceMonomorphism (G);
H := NiceObject(G);
if IsSolvableGroup(H) then
nice := IsomorphismPcGroup(G);
H := Range (nice);
fi;
return GroupHomomorphismByFunction(H, G,
x -> PreImagesRepresentative(nice, x),
x -> ImageElm(nice, x));
end);
############################################################################
##
#M RepresentationIsomorphism(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(RepresentationIsomorphism, "soluble group: inverse of IsomorphismPcGroup", true,
[IsMatrixGroup], 0,
function(G)
local nice;
nice := IsomorphismPcGroup(G);
return GroupHomomorphismByFunction(Range(nice), G,
x -> PreImagesRepresentative(nice, x),
x -> ImageElm(nice, x));
end);
############################################################################
##
#F ImprimitivitySystemsForRepresentation(G, rep, F, limit)
##
## G is a group, F a finite field, rep: G -> GL(n, F)
##
## If G has no block system with block dimension <= limit, the function
## computes a list of all imprimitivity systems of Im rep as a
## subgroup of GL(n, F). Otherwise, the function computes systems of imprimitivity,
## one of which will have block dimension <= limit.
##
## Each imprimitivity system is represented by a record with the following entries:
## bases: a list of lists of vectors, each list of vectors being a basis of a block
## in the imprimitivity system
## stab1: the stabilizer in G of the first block (i. e., the block with basis bases[1])
## min: true if the block system is a minimal block system amongst the systems returned
##
InstallGlobalFunction(ImprimitivitySystemsForRepresentation, function(G, rep, F, limit)
local systems, max, M, gens, m, c, cf, hom, subsys, sys, newsys, homBasis, homSpace, bas, bas2, newbasis, pos, orb;
if DegreeOfMatrixGroup(Range (rep))< limit then
return [rec(bases := [IdentityMat (DegreeOfMatrixGroup(Range (rep)), F)], stab1 := G, min := true)];
fi;
systems := [];
max := AttributeValueNotSet(MaximalSubgroupClassReps, G);
for M in max do
if not IsTrivial(M) then # inducing up from the trivial rep gives a reducible representation
gens := List(GeneratorsOfGroup(M), x -> ImageElm(rep, x));
m := GModuleByMats (gens, F);
if not MTX.IsIrreducible (m) then
for c in MTX.CollectedFactors(m) do
cf := c[1];
if MTX.Dimension (cf) * IndexNC(G, M) = Degree (Range (rep)) then
homBasis := MTX.Homomorphisms (cf, m);
if Length(homBasis) > 0 then # submodule isomorphic with cf
# get imprimitivity systems for cf
hom := GroupHomomorphismByImagesNC(M, GL(MTX.Dimension (cf), Size(F)),
GeneratorsOfGroup(M), MTX.Generators (cf));
subsys := ImprimitivitySystemsForRepresentation (M, hom, F, limit);
Add(subsys, rec(bases := [IdentityMat (MTX.Dimension (cf), F)], stab1 := M,
min := Length(subsys) = 0));
# translate result back
homSpace := VectorSpace(F, homBasis{[2..Length(homBasis)]}, 0*homBasis[1], "basis");
for bas2 in Enumerator(homSpace) do
bas := homBasis[1] + bas2;
for sys in subsys do
newbasis := List(sys.bases[1]*bas, ShallowCopy);
TriangulizeMat (newbasis);
if ForAll (systems, sys -> not newbasis in sys.bases) then
orb := Orbit (ImagesSet(rep, G), newbasis, OnSubspacesByCanonicalBasis);
Assert(1, Length(orb) * Length(newbasis) = Degree (Range (rep)));
Assert(1, Length(orb) = IndexNC(G, sys.stab1));
if orb[1] <> newbasis then
pos := Position (orb, newbasis);
orb{[1, pos]} := orb{[pos, 1]};
fi;
Add(systems, rec(bases := orb, stab1 := sys.stab1, min := sys.min));
fi;
od;
od;
fi;
fi;
od;
fi;
fi;
od;
# add trivial system
Add(systems, rec(bases := [IdentityMat (Degree (Range (rep)), F)], stab1 := G, min := Length(systems) = 0));
return systems;
end);
############################################################################
##
#A ImprimitivitySystemsOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(ImprimitivitySystemsOp, "for matrix group handled by nice mono. and finite field",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and IsHandledByNiceMonomorphism, IsField and IsFinite], 0,
function(G, F)
local rep;
rep := RepresentationIsomorphism (G);
return ImprimitivitySystemsForRepresentation (Source (rep), rep, F, 0);
end);
############################################################################
##
#M ImprimitivitySystemsOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(ImprimitivitySystemsOp, "for matrix group handled by nice mono. and finite field",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and IsHandledByNiceMonomorphism, IsField and IsFinite], 0,
function(G, F)
local rep;
rep := RepresentationIsomorphism (G);
return ImprimitivitySystemsForRepresentation (Source (rep), rep, F, 0);
end);
############################################################################
##
#M ImprimitivitySystems(<G>)
##
## see IRREDSOL documentation
##
InstallOtherMethod(ImprimitivitySystems, "for matrix group: use FieldOfMatrixGroup", true,
[IsFFEMatrixGroup], 0,
function(G)
return ImprimitivitySystems (G, FieldOfMatrixGroup(G));
end);
############################################################################
##
#M MinimalBlockDimensionOfMatrixGroupOp(<G>, <F>)
##
## see IRREDSOL documentation
##
InstallMethod(MinimalBlockDimensionOfMatrixGroupOp, "for matrix group having imprimitivity systems",
IsMatGroupOverFieldFam,
[IsFFEMatrixGroup and HasComputedImprimitivitySystemss, IsField],
0,
function(G, F)
local known, i, sys, d, l;
known := ComputedImprimitivitySystemss (G);
for i in [1,3..Length(known)-1] do
if known[i] = F then
d := DegreeOfMatrixGroup(G);
for sys in known[i+1] do
l := Length(sys.bases[1]);
if l < d then
d := l;
fi;
od;
return d;
fi;
od;
TryNextMethod();
end);
############################################################################
##
#M TraceField(<G>)
##
## see IRREDSOL documentation
##
InstallMethod(TraceField, "for irreducible matrix group over finite field", true,
[IsFFEMatrixGroup and IsFinite], 1,
function(G)
local ext, gens, q, q0, module, module2, gens2, c, i, p;
q := Size(FieldOfMatrixGroup(G));
Info(InfoIrredsol, 3, "TraceField: matrix group is over GF(",q,")");
if IsTrivial(G) then
Info(InfoIrredsol, 4, "TraceField: trivial group case");
return FieldOfMatrixGroup(G);
fi;
# guess a field contained in the smallest field over which module can be realised
q0 := Size( Field (List([1..LogInt(Size(G), 2) + 10], i -> TraceMat(PseudoRandom(G)))));
if q = q0 then
Info(InfoIrredsol, 3, "TraceField: trace field is GF(", q0, ")");
return FieldOfMatrixGroup(G);
fi;
Info(InfoIrredsol, 3, "TraceField: trace field contains GF(",q0, ")");
module := GModuleByMats (GeneratorsOfGroup(G), FieldOfMatrixGroup(G));
if not MTX.IsIrreducible (module) then
Info(InfoIrredsol, 3, "TraceField: trace field contains GF(",q0, ")");
TryNextMethod();
fi;
for c in Collected (Factors(LogInt(q, q0))) do
p := c[1];
for i in [1..c[2]] do
Info(InfoIrredsol, 1, "TraceField: trying if trace field is GF(",q0, ")");
# compute Galois conjugate of module
gens2 := List(MTX.Generators (module), g -> List(g, row -> List(row, a -> a^q0)));
module2 := GModuleByMats (gens2, MTX.Field (module));
# If module1 and module2 are conjugate, their traces are the same, and thus
# invariant under the Galois automorphism. Therefore the traces must belong to GF(q0).
# Thus by a theorem of Brauer, module1 (and module2) can be written over GF(q0).
# Conversely, if module1^x is over GF(q0) for some matrix x,
# then module1^x = (module1^x)^q0 = (module1^q0)^(x^q0) = module2^(x^q0)
# which shows that module1 and module2 are conjugate.
# see also S. P. Glasby, R. B. Howlett, Writing representations over mnimal fields,
# Comm. Alg. 25 (1997), 1703--1711
if MTX.Isomorphism (module, module2) <> fail then
Info(InfoIrredsol, 1, "TraceField: trace field is GF(",q0, ")");
return GF(q0);
fi;
q0 := q0^p; # size of minimal superfield of GF(q0)
od;
od;
Info(InfoIrredsol, 1, "TraceField: not rewritable over subfield: trace field is GF(",q0, ")");
return GF(q0);
end);
############################################################################
##
#M TraceField(<G>)
##
InstallMethod(TraceField, "generic method for finite matrix groups via conjugacy classes", true,
[IsMatrixGroup and IsFinite], 0,
function(G)
local F, rep;
F := FieldOfMatrixGroup(G);
if IsPrimeField (F) then
return F;
elif F = Field (List(GeneratorsOfGroup(G), TraceMat)) then
return F;
else
rep := RepresentationIsomorphism (G);
return Field (List(ConjugacyClasses (Source (rep)), cl -> TraceMat(ImageElm(rep, Representative(cl)))));
fi;
end);
RedispatchOnCondition (TraceField, true, [IsMatrixGroup], [IsFinite], 0);
############################################################################
##
#M SplittingField(<G>)
##
InstallMethod(SplittingField, "use MeatAxe", true,
[IsFFEMatrixGroup], 0,
function(G)
local F, module;
F := FieldOfMatrixGroup(G);
module := GModuleByMats (GeneratorsOfGroup(G), F);
if MTX.IsIrreducible (module) then
return GF(Characteristic (F)^MTX.DegreeSplittingField (module));
else
Error("G must be irreducible over FieldOfMatrixGroup(G)");
fi;
end);
############################################################################
##
#M ConjugatingMatTraceField(<G>)
##
## returns a matrix x such that the matrix entries of G^x lie in the
## trace field of G.
##
## The absolutely irreducible case is an impelemntation of an algorithm by
## S. P. Glasby, R. B. Howlett, Writing representations over mnimal fields,
## Comm. Alg. 25 (1997), 1703--1711
##
InstallMethod(ConjugatingMatTraceField, "for irreducible FFE matrix group",
true,
[IsFFEMatrixGroup], 0,
function(G)
local ext, q, q1, t, C, CC, D, Y, A, i, j, mu, nu,
basis, moduleG, module, module2, module3, absirred;
if Length(GeneratorsOfGroup(G)) = 0 or TraceField(G) = FieldOfMatrixGroup(G) then
return One(G);
fi;
q := Size(TraceField (G));
moduleG := GModuleByMats (GeneratorsOfGroup(G), FieldOfMatrixGroup(G));
# reduce to the absolutely irreducible case
if not MTX.IsIrreducible (moduleG) then
TryNextMethod();
fi;
absirred := MTX.IsAbsolutelyIrreducible (moduleG);
if absirred then
module := moduleG;
ext := FieldOfMatrixGroup(G);
else
ext := GF(CharacteristicOfField (G)^MTX.DegreeSplittingField (moduleG));
module := GModuleByMats (GeneratorsOfGroup(G), ext);
repeat
basis := MTX.ProperSubmoduleBasis (module);
module := MTX.InducedActionSubmodule (module, basis);
until MTX.IsIrreducible (module);
Assert(1, MTX.IsAbsolutelyIrreducible (module));
fi;
# moduleG can be rewritten over TraceField(G) but over no proper subfield
# let GF(q1) be the trace field of module, then over GF(q1),
# moduleG has block diagonal matrices which are conjugate via
# a Galois automorphism of GF(q1) of order Deg(G)/Dim(module)
# so GF(q1) has dimension Dim(moduleG)/Dim(module) over TraceField(G).
# Thus, taking q1-th powers is a generator of Gal (MTX.Field(module)/GF(q1))
q1 := q^(MTX.Dimension (moduleG)/MTX.Dimension(module));
t := LogInt(Size(ext), q1);
module2 := GModuleByMats (List(MTX.Generators (module),
A -> List(A, row -> List(row, x -> x^q1))), ext);
C := MTX.Isomorphism (module, module2);
if C = fail then
Error("panic: cannot rewrite matrix group over trace field!");
fi;
CC := C;
D := C;
for i in [1..t-1] do
CC := List(CC, row -> List(row, x -> x^q1));
D := D*CC;
od;
mu := D[1][1];
repeat
nu := Random(ext);
until Norm(ext, GF(q1), nu) = mu;
C := nu^-1 * C;
repeat
Y := RandomMat(MTX.Dimension(module), MTX.Dimension(module), ext);
A := Y;
for i in [2..t] do
A := Y + C * List(A, row -> List(row, x -> x^q1));
od;
until Length(NullspaceMat(A)) = 0;
Assert(1, ForAll(MTX.Generators(module), g -> IsSubset(GF(q1), Field(Flat(g^A)))));
# now we have a solution for the absolutely irreducible case
if absirred then
MakeImmutable(A);
ConvertToMatrixRep(A, ext);
return A;
fi;
basis := CanonicalBasis(AsVectorSpace(GF(q), GF(q1)));
module3 := GModuleByMats(List(MTX.Generators(module), x -> BlownUpMat(basis, x^A)), MTX.Field(moduleG));
A := Immutable(MTX.Isomorphism(moduleG, module3));
if A = fail then
Error("could not find conjugating matrix");
fi;
Assert(1, ForAll(GeneratorsOfGroup(G), g -> IsSubset(GF(q), Field(Flat(g^A)))));
ConvertToMatrixRep(A, FieldOfMatrixGroup(G));
return A;
end);
############################################################################
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#E
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