#############################################################################
##
## affinegroup.gi FinInG package
## John Bamberg
## Anton Betten
## Jan De Beule
## Philippe Cara
## Michel Lavrauw
## Max Neunhoeffer
##
## Copyright 2018 Colorado State University
## Sabancı Üniversitesi
## Università degli Studi di Padova
## Universiteit Gent
## University of St. Andrews
## University of Western Australia
## Vrije Universiteit Brussel
##
##
## Implementation stuff for affine groups
##
#############################################################################
# **** The way it works ****
#
# An affine transformation of the form x+A
# can be written as a matrix...
# | 0|
# | A 0|
# | 0|
# | x 1|
# where the matrix A occurs in all but the last row and column of the matrix,
# x sits in all but the last element of the last row, and the rest is filled
# with 0's and a 1 in the last spot. We can then just use ProjElWithFrob
# wrapping in order to incoorporate the Frobenius map.
# CHECKED 25/3/14 jdb
#############################################################################
#O AffineGroup( <as> )
# returns AGL(d,q)
##
InstallMethod( AffineGroup,
"for an affine space",
[ IsAffineSpace ],
function( as )
## This operation returns the group commonly known as AGL(d,F)
local d, gf, vec, semi, gens, frob, newgens, agl, q;
d := as!.dimension;
gf := as!.basefield;
q := Size(gf);
vec := as!.vectorspace;
semi := SemidirectProduct(GL(d, gf), vec);
gens := GeneratorsOfGroup(semi);
frob := FrobeniusAutomorphism( gf );
newgens := List(gens, x -> [x, frob^0]);
newgens := ProjElsWithFrob(newgens);
agl := GroupWithGenerators(newgens);
SetName(agl, Concatenation("AGL(",String(d),",",String(q),")") );
SetSize(agl, Size(semi));
return agl;
end );
# CHECKED 27/3/2012 jdb
#############################################################################
#O CollineationGroup( <as> )
# returns AGammaL(d,q)
##
InstallMethod( CollineationGroup,
"for an affine space",
[ IsAffineSpace ],
function( as )
## This operation returns the group commonly known as AGammaL(d,F)
local d, gf, vec, semi, gens, frob, newgens, coll, q;
d := as!.dimension;
gf := as!.basefield;
q := Size(gf);
vec := as!.vectorspace;
semi := SemidirectProduct(GL(d, gf), vec);
gens := GeneratorsOfGroup(semi);
frob := FrobeniusAutomorphism( gf );
newgens := List(gens, x -> [x, frob^0]);
Add(newgens, [One(semi), frob]);
newgens := ProjElsWithFrob(newgens);
coll := GroupWithGenerators(newgens);
if LogInt(q, Characteristic(gf)) > 1 then
SetName( coll, Concatenation("AGammaL(",String(d),",",String(q),")") );
else
SetName( coll, Concatenation("AGL(",String(d),",",String(q),")") );
fi;
SetSize(coll, Size(semi) * Order(frob));
return coll;
end );
# CHECKED 27/3/2012 jdb
# cvec changed 25/3/14.
#############################################################################
#F OnAffinePoints( <y>, <el> )
# implements action of projective semilinear elements on affine points.
# <y> is subspace
# <el> is group element.
##
InstallGlobalFunction( OnAffinePoints,
function( y, el )
# Note that
# | 0|
# [y, 1] | A 0| = [yA + x, 1]
# | 0|
# | x 1|
local yobj, new, d, geo, bf;
geo := y!.geo;
bf := geo!.basefield;
d := geo!.dimension;
yobj := Unpack(y!.obj);
yobj := CVec(Concatenation(yobj, [ One(bf) ]),bf); #Concatenation will make this a real lis,
so CVec is applicable.
new := yobj * el!.mat;
new := new^el!.frob;
new := new{[1..d]};
return Wrap(geo, y!.type, new);
end );
# CHECKED 27/3/2012 jdb
# cvec/cmat change 25/3/14.
# unpacking three times, and using AffineSubspace seems to be the easiest solution.
#############################################################################
#F OnAffineNotPoints( <subspace>, <el> )
# implements action of projective semilinear elements on affine subspaces different
# from points.
##
# CHANGED 12/03/12 jdb
# new directions at infinity should be normalized, otherwise you get the orbits q-1 times.
##
InstallGlobalFunction( OnAffineNotPoints,
function( subspace, el )
local dir, v, vec, newv, ag, mat, newdir, d, frob, bf;
ag := subspace!.geo;
d := ag!.dimension;
vec := ag!.vectorspace;
v := Unpack(subspace!.obj[1]);
dir := Unpack(subspace!.obj[2]);
mat := Unpack(el!.mat);
frob := el!.frob;
newdir := dir * mat{[1..d]}{[1..d]};
newdir := newdir^frob;
TriangulizeMat(newdir);
newv := Concatenation(v, [ One(ag!.basefield) ]); #will be a plain list.
newv := newv * mat;
newv := newv^frob;
newv := newv{[1..d]};
newv := VectorSpaceTransversalElement(vec, newdir, newv);
return AffineSubspace(ag,newv,newdir);
#return Wrap(ag, subspace!.type, [newv, newdir]);
end );
# CHECKED 27/3/2012 jdb
#############################################################################
#F OnAffineSubspaces( <y>, <el> )
# This is the group action of an affine group on affine subspaces that
# the user should use.
##
InstallGlobalFunction( OnAffineSubspaces,
function( v, el )
if v!.type = 1 then
return OnAffinePoints(v, el);
else
return OnAffineNotPoints(v, el);
fi;
end );
# CHECKED 27/3/2012 jdb
#############################################################################
#F \^( <x>, <em> )
# shorcut to OnAffineSubspaces.
##
InstallOtherMethod( \^,
"for a subspace of an affine space and a projective element with frob",
[IsSubspaceOfAffineSpace, IsProjGrpElWithFrob],
function(x, em)
return OnAffineSubspaces(x,em);
end );
