| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/fining/lib/ (GAP Algebra Version 4.15.1©) Datei vom 27.6.2023 mit Größe 38 kB |
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Quellcode-Bibliothek projectivespace.gi Sprache: Shell |
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############################################################################# ## ## projectivespace.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 projective spaces. ## ############################################################################# ############################################################################# # Low level help methods: ############################################################################# # CHECKED 6/09/11 jdb ############################################################################# #O Wrap( <geo>, <type>, <o> ) # This is an internal subroutine which is not expected to be used by the user; # they would be using VectorSpaceToElement. Recall that Wrap is declared in # geometry.gd. ## InstallMethod( [IsProjectiveSpace,IsObject]java.lang.StringIndexOutOfBoundsExcep "for a projective space and an object", [IsProjectiveSpace, IsPosInt function( geo, # If the object "v" to be unwrapped is a point of a vector space, then we do not want t local w;# w# "for a subspace of a projective # [ IsSubspaceOfProjectiveSpace# function( v, u ) java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 IsElementOfIncidenceStructureRep# CHECKED 6/09/11 jdb #O ProjectiveSpace( <d>,#O ProjectiveSpace( <d>, ite field. end) # CHECKED 6/09/11 jdb #jdb 30/10/15: This method is not used anymore after commenting out the Unwrapper stuff in geom #see the comment there. ############################################################################# #O \^( <v>, <u> ) # If the object "v" to be unwrapped is a point of a vector space, then we do not want to use # return v!.obj, but we want to return a list with one vector, i.e. [v!.obj] # e.g. if p is a point of a projective space # gap> p^_; # will return a list, with the coordinate vector of the point p ## #InstallMethod( \^, # "for a subspace of a projective space and an unwrapper", # [ IsSubspaceOfProjectiveSpace, IsUnwrapper ], # function( v, u ) # if v!.type = 1 then return [v!.obj]; # else return v!.obj; # fi; # end ); ############################################################################# # Constructor methods and some operations/attributes for projective spaces. ############################################################################# # CHECKED 6/09/11 jdb ############################################################################# #O ProjectiveSpace( <d>, <f> ) # returns PG(d,f), f a finite field. ## InstallMethod( IsProjectiveSpace andIsDesarguesianPlane; =NewType(GeometriesFamily IsProjectiveSpace and IsProjectiveSpaceRep ); local geo, ty; geo := rec( dimension : fi; : (f,d+1 ; if d = 2 then ty := NewType( GeometriesFamily, IsProjectiveSpace and IsProjectiveSpaceRep and IsDesarguesianPlane); else ty := NewType( GeometriesFamily, IsProjectiveSpaceif d=then fi; Objectify( ty, geo ); SetAmbientSpace(g; if d=2 then SetOrder(java.lang.StringIndexOutOfBoundsException: Range [0, 20) out of bounds fi; return geo; end## # CHECKED 6/09/11 jdb ############################################################################# #O ProjectiveSpace( <d>, <q> ) # returns PG(d,q). ## return (,GFq)); [ ,IsPosInt] function(####################################################################### ############################################################################# end; ############################################################################# # Display methods: ############################################################################# function p java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for l (p java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 1 Print((,!dimension" ,p.),""; end ,java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27 " projective "java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds fo [IsProjectiveSpace and IsProjectiveSpaceRep, ; InstallMethod( PrintObj,[and IsProjectiveSpaceRep ] nd ) PrintObj,[IsProjectiveSpaceand sProjectiveSpaceRep], function( p end) nt"ProjectiveSpace("p.imension""p.,))java.lang.StringIndexOutOfBoundsException: end ); InstallMethod( Display, [ IsProjectiveSpace and IsProjectiveSpaceRep ], #if HasDiagramOfGeometry( p ) then # Display #if HasDiagramOfGeometry( p ) then # Display( DiagramOfGeometry( p ) ); #fi; end );#O \=( <pg1>, <pg2> ) # CHECKED 6/09/11 jdb ############################################################################# #O \=( <pg1>, <pg2> ) ## InstallMethod( \=, "for two projective spaces", [IsProjectiveSpace, IsProjectiveSpace], function(pg1,pg2)[IsProjectiveSpace,IsProjectiveSpace] return (pg1 (pg2) end ); return UnderlyingVectorSpacep) =UnderlyingVectorSpace); # CHECKED 8/09/11 jdb ############################################################################# #O Rank( <ps> ) # returns the projective dimension of <ps> ## InstallMethod( Rank, "for a # returns the projective dimension of <ps> IsProjectiveSpace IsProjectiveSpaceRep, ps"or a projective space, ); ############################################################################# #O BaseField( <sub> ) # returns the basefield of an element of a projective space ## ps >ps!.imension foranelement space, [IsSubspaceOfProjectiveSpace] sub -> #O BaseField( <sub> ) # CHECKED 6/09/11 jdb ############################################################################# #A StandardFrame( <ps> ) # if the dimension of the projective space is n, then StandardFrame # makes a list of points with coordinates # (1,0,...0), (0,1,0,...,0), ..., (0,...,0,1) and (1,1,...,1) ## InstallMethod( StandardFrame, "ora projective space", [IsProjectiveSpace], function( pg ) local bas, frame, local bas, frame, unitpt], if not pg!.dimension > 0 then Error(Theargument needs to be a projective space of dimension at least 1!"; else bas:=#A StandardFrame( <ps> ) # if the dimension of the projective space is n, then# makes a list of points with coordinates unitpt## InstallMethod( StandardFrame, return frame; fi; end ); # CHECKED 11/09/11 jdb ############################################################################# #A RepresentativesOfElements( <ps> ) # Returns the canonical maximal flag for the projective space <ps> ## InstallMethod(RepresentativesOfElements "for a projective space", [IsProjectiveSpace], # returns the canonical maximal flag function( ps ) local dif not pg!dimension > 0then d =(ps); gf := id : IdentityMat(d+,gf); :ListB()v>ectorSpaceToElement,); return elts; end ); # CHECKED 18/4/2011 jdb ############################################################################# #O Hyperplanes( <ps> ) # returns Hyperplanes(ps,ps!.dimension), <ps> a projective space ## , "for a projective space", [ IsProjectiveSpace fijava.lang.StringIndexOutOfBoundsException: Index 5 out of bo function( ps# Returns the canonical maximal flag for the projective space <ps> (ps .); end) # CHECKED 11/09/11 jdb ############################################################################# #A TypesOfElementsOfIncidenceStructure( <ps> ) # returns the names of the types of the elements of the projective space <ps> # the is a helper operation. ## (TypesOfElementsOfIncidenceStructure "for a projective space", [IsProjectiveSpace], function( ps ) ocal d,,types pes: ""; d := ProjectiveDimension(ps); if d > elts : List(1.] - VectorSpaceToElement(s, id[.i]); if e) if # CHECKED 18/4################################################################## for i in [5..d] do Add(types,Concatenation("proj. ",String(i-1),"-space")); od; return types; end ); # CHECKED 11/09/11 jdb ############################################################################# #A TypesOfElementsOfIncidenceStructurePlural( <ps> ) # retunrs the plural of the names of the types of the elements of the # projective space <ps>. This is a helper operation. ## InstallMethod( TypesOfElementsOfIncidenceStructurePlural, "for a projective space", [IsProjectiveSpace], function( ps ) local d,i,types; types := ["points"]; d := ProjectiveDimension(ps); if d >= 2 then Add(types,"lines"); fi; if d >= 3 then Add(types,"planes"); fi; if d >= 4 Add(types,". "String(-1,-pace"); for i in [5..d] do Add(types,Concatenation("proj. ",String(i-1)," ; od; returntypes; end ); # CHECKED 11/09/11 jdb ############################################################################# #O ElementsOfIncidenceStructure( <ps>, <j> ) # returns the elements of the projective space <ps> of type <j> ## InstallMethod( ElementsOfIncidenceStructure, "for a projective space and an integer", [IsProjectiveSpace, IsPosInt], function( ps, j ) local r; r := Rank(ps); if j > r then Error("<ps> has no elements of type <j>"); else return Objectify( NewType( ElementsCollFamily, IsSubspacesOfProjectiveSpace and IsSubspacesOfProjectiv rec( geometry := ps, type := j, size := Size(Subspaces(ps!.vectorspace, j)) ) ); fi; end); # CHECKED 11/09/11 jdb ############################################################################# #O ElementsOfIncidenceStructure( <ps> ) # returns all the elements of the projective space <ps> ## InstallMethod( ElementsOfIncidenceStructure, " space, [IsProjectiveSpace] if > Add(types,"lines";fijava.lang.StringIndexOutOfBoundsExcepti java.lang.StringIndexOutOfBoundsException: Range [10, 9) out of bounds for lengt return Objectify( pe java.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for le rec( geometry#################################################################### type := "all# returns the elements of the projective space <ps> of java.lang.StringIndexOutOfB ); end)[, ] # CHECKED 14/09/11 jdb ############################################################################# #O \=( <x>, <y> ) # returns true if the collections <x> and <y> of all subspaces of a projective # space are the same. ## (\java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 1 "for set of all subspaces of java.lang.StringIndexOutOfBoundsException: Index 32 out o Space ElementsCollFamilyIsAllSubspacesOfProjectiveSpaceIsAllSubspacesOfProjective function(x,y) return ((x!.geometry! ) y!.geometryend; end ); # CHECKED 11/09/11 jdb ############################################################################# #O Size( <subs>) # returns the number of elements in the collection <subs>. ## java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for leng "for subspaces of a IsAllSubspacesOfProjectiveSpace ] [ and ], function(subsreturn (!g!.dimension=y.!dimension)and(!geometry. java.lang.StringInd return java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for leng end); ############################################################################# # Constructor methods and some operations/attributes for subspaces of # projective spaces. ############################################################################# ############################################################################# # VectorSpaceToElement methods ############################################################################# ## Things to check for (dodgy input) ## --------------------------------- ## - dimension ## - field ## - compress the matrix at the end ## - rank of matrix ## - an empty list ## Much of the following will need to change in the new ## version of GAP, with the new Row and Matrix types. ## Should we have methods for the new types given by the cvec package? ## Currently we don't load the cvec package. Since 18/3/14 we do. The next ## method also inserts a method for cmat. ## - rank of matrix ############################################################################# #O VectorSpaceToElement( <geom>, <v> ) returns the elements in <geom> determined # by the vectorspace <v>. Several checks are built in. # This method unpacks the cmat, and uses the other VectorSpaceToElement method for PlistRep. # this is maybe not too efficient, but row selection like x{list} (list is a list of positions) # seems to fail, although according to the documentation, it should work. # In the future this could be made better. ## InstallMethod( VectorSpaceToElement, "for a projective space and a # This method unpacks the cmat, and uses the other VectorSpaceToEl [IsProjectiveSpace# seems to fail, although according to the documentation, it should work. function( geom, v ) InstallMethod( , ) # CHECKED 20/09/11 # changed 19/01/16 (jdb): by a change of IsPlistRep, this method gets also # called when using a row vector, causing a problem with TriangulizeMat. # a solution was to add IsMatrix. ############################################################################# #O VectorSpaceToElement( <geom>, <v> ) returns the elements in <geom> determined # by the vectorspace <v>. Several checks are built in. ## InstallMethod( VectorSpaceToElement, "for a projective space############################################################ #O VectorSpaceToElement( <geom>, <v> ) # by the vectorspace <v>. java.lang.StringIndexOutOfBound function( geom, v ) x,n,,y;java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for len ## when v is empty... f (v "v does represent any "; fi; #x := EchelonMat(v).vectors; fi; java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 x;java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 2 ## dimension should be correct if Length( ifLength([1)< !dimension Error" "); fi; ## Remove zero rows. It is possible the the user ## has inputted a matrix which does not have full rank n := Length(x); # i := 0; orAllx-i] IsZero)do i := i+1; od; if i = n then return while i < nand ForAll([-],IsZero) do fi; x := x{[1..n-i]}; if Length( i := i+1; return geom; fi od; ## It is possible that (a) the user has entered a ## matrix with one row, or that (b) the user has matrix with rank1 (hus stage ## we will have a matrix with one row). # We must also compress our vector/matrix. y: (sCMatRep,!basefieldLengthx1)x; #NewMatrix is currently undocumented in cvec, but creates a CMat object from a list of lists, us ifLength()=1then ## matrix with one row, or that (b) the user has #x := x[1]; #ConvertToVectorRep(x, geom!.basefield); # the extra basefield is necessary. #return Wrap(geom, 1, x); #return Wrap(geom, 1, CVec(Unpack(x), geom!.basefield) ); # changed to cvec 18/3/2014. else #ConvertToMatrixRep(x, geom!.basefield); #return Wrap(geom, Length(x), x); Wrapgeom,Lengthy,y; fi; end ); # CHECKED 20/09/11 ############################################################################# #O VectorSpaceToElement( <geom>, <v> ) returns the elements in <geom> determined # by the vectorspace <v>. Several checks are built in. ## InstallMethod( VectorSpaceToElement, "for a projective space and a compressed GF(2)-matrix", [IsProjectiveSpace, IsGF2MatrixRep], function( geom, v ) local x, n, i, y; ## when v is empty... if IsEmpty(v) then Error("<v> does not represent any element"); fi; x := MutableCopyMat(v); TriangulizeMat(x); :=EchelonMat(().ectorsjava.lang.StringIndexOutOfBoundsException: Index 30 out of ## dimension should be correct if Length ConvertToVectorRep(x, geom!.basefield); # the extra basefield is necessary. Error("Dimensions are incompatible"); fi; #if Length(x) = 0 then # return EmptySubspace(geom); #fi; if Length(x)=ProjectiveDimension(geom)+1 then # return geom; #return Wrap(geom, Length(x), x); adoes n :=f; i := 0; whilei andxn-,)do i :=# CHECKED 20/09/11 od; if i = n then return EmptySubspace(geom); fi; x := x{[1..n-i]}; if Length(x)=ProjectiveDimension(geom) "for a projective a GF2-" eturn geom; fi; java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51 java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50 ## entered a matrix with rank 1 (thus at this stage ## we will have a matrix with one row). ## We must also compress our vector/matrix. InstallMethod VectorSpaceToElement, if Lengthy 1then Wrap(geom,1,y[]; #x := x[1]; #ConvertToVectorRep(x, geom!.basefield); # the extra basefield is necessary. #return Wrap(geom, 1, x); #return Wrap(geom, 1, CVec(Unpack(x), geom!.basefield) ); # changed to cvec 18/3/2014. else #ConvertToMatrixRep(x, geom!.basefield); return Wrap(geom, Length(y), y); fijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 ); # CHECKED 20/09/11 ############################################################################# #O VectorSpaceToElement( <geom>, <v> ) returns the elements in <geom> determined # by the vectorspace <v>. Several checks are built in. ## java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 2 " a compressed basis of a vector subspace" [IsProjectiveSpaceIs8BitMatrixRep], function geom,v) local x, n, i, y; ## when v is empty... if IsEmpty(v) then Error("<v> does not represent any java.lang.StringIndexOutOfBoundsException: Index 41 o fi; #x := EchelonMat(v).vectors; x : ## dimension should be correct if Length(v[1]) <> i = 0java.lang.StringIndexOutOfBoundsException: Index 9 out of bound Error(Dimensions areincompatible"; i #if Length(x) = 0 then # return EmptySubspace(geom); #fi; #if Length(x)=ProjectiveDimension(geom)+1 then # return geom; #fi; n := Length(x); i := 0; while i < n and ForAll(x[n fi; i := i+1 java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for lengt # if i = n then return EmptySubspace# fi; :=x[..-]} ()+ return geom; java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5 ## It is possible that (a) the user has entered a return Wrap(geom, 1, CVec(Unpack(x), geom!.basefield) ); # changed to cvec 18/3/2014. ## entered a matrix with rank 1 (thus at this stage # we will have a matrix with one row). ## We must also compress our vector/matrix. y ### The next mathod constructs an element using a cvec. return Wrap(geom, 1, y# ADDED 20/3/2014 jdb #x := x[1]; #ConvertToVectorRep(x, geom!.basefield); # the extra basefield is necessary.#O VectorSpa #return Wrap(geom, 1, x);## return Wrap(geom, 1, CVec(Unpack(x), geom!.basefield) ); # changed to cvec 18/3/2014. else #ConvertToMatrixRep(x, geom!.basefield); return Wrap(eom Lengthy,y; fi; end ); ### The next mathod constructs an element using a cvec. # ADDED 20/3/2014 jdb ############################################################################# #O VectorSpaceToElement( <geom>, <v> ) returns the elements in <geom> determined # by the rowvector <v>. Several checks are built in. ## InstallMethod(VectorSpaceToElement, "for a row vector", [IsProjectiveSpace, IsCVecRep], function(geom v ) ## di should be correct ## when v is empty... does this ever occur for a row vector? No. jdb 21/09/2011 #if IsEmpty(v) then not any element); #fi; x := ShallowCopy(v #java.lang.StringIndexOutOfBoundsException: Index 38 out of boun ## dimension should be correctcheckedthis with input .. if Length(v) <> geom!.dimension + return EmptySubspace(geom); Error("Dimensions are incompatible"); fi; ## We must also compress our vector. #ConvertToVectorRep(x, geom!.basefield); bad ,such as jdb,checked this with input zero vector... if IsZero(x) then eturn (geom); else MultVector(x,Inverse(java.lang.StringIndexOutOfBoundsException: Index 0 out of b return Wrap# CHECKED 21/09/2011 jdb fi; end#O VectorSpaceToElement( <geom>, <v> ) returns the elements in <geom> determined # CHECKED 11/04/15 jdb # CHANGED 19/9/2011 jdb + ml # CHECKED 21/09/2011 jdb ############################################################################# #O VectorSpaceToElement( <geom>, <v> ) returns the elements in <geom> determined # by the rowvector <v>. Several checks are built in. ## InstallMethod( VectorSpaceToElement, [sProjectiveSpace, IsRowVector], "for a row vector", [IsProjectiveSpace, IsRowVectorfunction(geom, v ) function( geom, v ) local x local x y; ## when v is empty... does this ever occur for a row vector? No. jdb 21/09/2011 #if IsEmpty(v) then # Error("<v> does not represent any element"); #fi; x := ShallowCopy(v); #fi; if: ShallowCopy(v); Error("Dimensions are incompatible"); fi; ## We must also compress our vector. ConvertToVectorRep(x, geom!.basefield); ## bad characters, such as jdb, checked this with input zero vector... if IsZero(x) then return EmptySubspace(geom); else MultVector(x,Inverse( x[PositionNonZero(java.lang.StringIndexOutOfBoundsException y : else #ConvertToVectorRep(x, geom!.basefield); return y := NewMatrix,eom.,(x,x)java.lang.StringIndexOutOfBoundsException: Index ; end ); # CHECKED 11/04/15 jdb # CHANGED 19/9/2011 jdb + ml ############################################################################# #O VectorSpaceToElement( <geom>, <v> ) returns the elements in <geom> determined # by the rowvector <v>. Several checks are built in. ## InstallMethod( VectorSpaceToElement, "for a java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for lengt ] function( geom,], x,n , ; ## when v is empty... , n, ; if IsEmpty(v) then return EmptySubspace(geom); fi; x := ShallowCopy(v); ## dimension should be correct if Length( Error if Lengthv >geom. +1 then fi; vectorjava.lang.StringIndexOutOfBoundsException: Range [38, 39) out of bounds fo #ConvertToVectorRep(x, geom!.basefield); ## bad characters, such as jdb, checked this with input zero vector... if IsZero(x) then return EmptySubspace(geom); else MultVector(x,Inverse( x[PositionNonZero(x)] )); y := NewMatrix(IsCMatRep, MultVector(,(x[x) )java.lang.StringIndexOutOfBoundsExcepti #ConvertToVectorRep(x, geom!.basefield); ConvertToVectorRep(x, geom!.basefield); fi; end ); ############################################################################# # attributes/operations for subspaces ############################################################################# # CHECKED 14/09/11 jdb ############################################################################# #O UnderlyingVectorSpace( <subspace> ) returns the underlying vectorspace of # <subspace>, i.e. the vectorspace determining <subspace> ## InstallMethod( UnderlyingVectorSpace, "for a subspace of a projective space", [IsSubspaceOfProjectiveSpace], function(subspace) local vspace,W; vspace:=UnderlyingVectorSpace(subspace!.geo); if subspace!.type = 1 then W:=SubspaceNC(vspace,[Unpack(subspace!.obj)]); #possibly unpack here to avoid bloody se else W:=SubspaceNC(vspace,Unpack(subspace!.obj)); fi; return W; end); # CHECKED 8/09/11 jdb ############################################################################# #O ProjectiveDimension( <v> ) returns the projective dimension of <v> ## InstallMethod( ProjectiveDimension, "for a subspace of a projective space", [ IsSubspaceOfProjectiveSpace ], function( v ) return v!.type - 1; end ); #InstallMethod( ProjectiveDimension, [ IsEmpty ], function(x) return -1;end ); # CHECKED 8/09/11 jdb ############################################################################# #O Dimension( <v> ) returns the projective dimension of <v> ## InstallMethod( Dimension, "for a subspace of a projective space", [ IsSubspaceOfProjectiveSpace ], function( v ) return v!.type - 1; end ); #InstallMethod( Dimension, [ IsEmpty ], function(x) return -1;end ); # CHECKED 8/09/11 jdb # 31/5/2020: still ok, but undocumentend! ############################################################################# #O StandardFrame( <subspace> ) returns a standard frame for <subspace> ## InstallMethod( StandardFrame, "for a subspace of a projective space", [IsSubspaceOfProjectiveSpace], # if the dimension of the subspace is d (needs to be at least 1), then this returns d+2 # points of the subspace, the first d+1 are the points "basispoints" # the last point has as coordinates the sum of the basispoints. function( subspace ) local list,v; if not Dimension(subspace) > 0 then Error("The argument needs to be a projective space of dimension at least 1!"); else list:=ShallowCopy(subspace!.obj); Add(list,Sum(subspace!.obj)); return List(list,v->VectorSpaceToElement(subspace!.geo,v)); fi; end ); # CHECKED 8/09/11 jdb ############################################################################# #O Coordinates( <point> ) returns the coordinates of the projective point <point> ## InstallMethod( Coordinates, "for a point of a projective space", [IsSubspaceOfProjectiveSpace], function( point ) if not Dimension(point)=0 then Error("The argument is not a projective point"); else return ShallowCopy(Unpack(point!.obj)); fi; end ); # obsolete ############################################################################# #O CoordinatesOfHyperplane( <hyp> ) returns the coordinates of the hyperplane # <hyp> ## #InstallMethod( CoordinatesOfHyperplane, # "for a hyperplane of a projective space", # [IsSubspaceOfProjectiveSpace], # function(hyp) # local pg; # pg:=ShallowCopy(hyp!.geo); # if not hyp!.type=Dimension(pg) then # Error("The argument is not a hyperplane"); # else # #perp:=StandardDualityOfProjectiveSpace(pg); # return Coordinates(VectorSpaceToElement(pg,NullspaceMat(TransposedMat(hyp!.ob # fi; # end ); # came from varieties.gi ############################################################################# #O DualCoordinatesOfHyperplane( <hyp> ) # returns the dual coordinate of a hyperplane in a projective space. ## InstallMethod( DualCoordinatesOfHyperplane, "for a subspace of a projective space", [IsSubspaceOfProjectiveSpace], function(hyp) local mat,a,x; if not Dimension(hyp)=Dimension(hyp!.geo)-1 then Error("The argument is not a hyperplane"); else mat:=hyp!.obj; a:=NullspaceMat(TransposedMat(mat)); x := Unpack(a[1]); MultVector(x,Inverse( x[PositionNonZero(x)] )); return x; fi; end ); # came from varieties.gi ############################################################################# #O HyperplaneByDualCoordinates( <pg>,<vector> ) # returns the hyperplanes by given dual coordinates. ## InstallMethod( HyperplaneByDualCoordinates, "for a projective space and a list with coordinates", [IsProjectiveSpace,IsList], function(pg,a) local mat,list; if not Size(a)=Dimension(pg)+1 or not ForAll(a,x->x in pg!.basefield) then Error("The dual coordinates are not compatible with the projective space"); else mat:=[a]; list:=NullspaceMat(TransposedMat(mat)); return VectorSpaceToElement(pg,list); fi; end ); # CHECKED 8/09/11 jdb ############################################################################# #O EquationOfHyperplane( <hyp> ) returns the euqation of the hyperplane # <hyp> ## InstallMethod( EquationOfHyperplane, "for a hyperplane of a projective space", [IsSubspaceOfProjectiveSpace], function(hyp) local pg,r,v,indets; pg:=AmbientGeometry(hyp); r:=PolynomialRing(pg!.basefield,pg!.dimension + 1); indets:=IndeterminatesOfPolynomialRing(r); v:=DualCoordinatesOfHyperplane(hyp); return Sum(List([1..Size(indets)],i->v[i]*indets[i])); end ); # CHECKED 11/09/11 jdb # Commented out 28/11/11 jdb + pc, according to new regulations ############################################################################# #O AmbientSpace( <subspace> ) returns the ambient space of <subspace> ## #InstallMethod( AmbientSpace, [IsSubspaceOfProjectiveSpace], # function(subspace) # return subspace!.geo; # end ); ############################################################################# # Span/Meet for empty subspaces ############################################################################# # CHECKED 8/09/11 jdb ############################################################################# #O Span( <x>, <y> ) returns the span of <x> and <y> ## InstallMethod( Span, "for the empty subspace and a projective space", [ IsEmptySubspace, IsProjectiveSpace ], function( x, y ) if x!.geo!.vectorspace = y!.vectorspace then return y; else Error( "The subspace <x> has a different ambient space than <y>" ); fi; end ); # CHECKED 8/09/11 jdb ############################################################################# #O Span( <x>, <y> ) returns the span of <x> and <y> ## InstallMethod( Span, "for the empty subspace and a projective space", [ IsProjectiveSpace, IsEmptySubspace ], function( x, y ) if x!.vectorspace = y!.geo!.vectorspace then return x; else Error( "The subspace <x> has a different ambient space than <y>" ); fi; end ); # CHECKED 8/09/11 jdb ############################################################################# #O Meet( <x>, <y> ) returns the intersection of <x> and <y> ## InstallMethod( Meet, "for the empty subspace and a projective subspace", [ IsEmptySubspace, IsProjectiveSpace ], function( x, y ) if x!.geo!.vectorspace = y!.vectorspace then return x; else Error( "The subspace <x> java.lang.StringIndexOutOfBoundsException: Range [0, 31) out fi; end ); # CHECKED 8/09/11 jdb ############################################################################# #O Meet( <x>, <y> ) returns the intersection of <x> and <y> ## InstallMethod( Meet, "for the empty subspace and # The enumerator functionality is quite technical. We need many tin [ IsProjectiveSpace, IsEmptySubspace ], function( x, y# Very Low level help functions if x!.vectorspace = y!geo.then return("G_ffenumber" ); else Error( "The subspace "FG_unrank_GFQ" ); #never used Decl FG_log_alpha ; ; # CHECKED 8/09/11 jdb # 25/3/14 It turns out that there is a problem when we do not Unpack # cvec/cmat with Size(Subspaces(localfactorspace)). As there is never an action # on shadow of flag objects, it is not unreasonable to store the matrices as # GAP matrices rather than cvec/cmats. ############################################################################# #O ShadowOfElement(<ps>, <v>, <j> ). Recall that for every particular Lie # geometry a method for ShadowOfElement must be installed. ## InstallMethod ShadowOfElement, "or a projective space,an element, and integer", [IsProjectiveSpace, IsSubspaceOfProjectiveSpace, IsPosInt], # returns the shadow of an element v as a record containing the projective space (geometry), ; of elements () elementv(arentflag,and extra information shadows, e.g. iterator function( psjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for ,localouter localfactorspace; Error("<v> is not a subspace of <ps>"); fi; if j > ps!.dimension then Error("<ps> has no elements of type <j>"); elif j < v!.type then localinner := []; localouter := Unpack(v!.obj); elif j = v!.type then localinner := Unpack(v!.obj); localouter := localinner; else localinner := Unpack(v!.obj); localouter := BasisVectors(Basis(ps!.vectorspace)); fi; if IsVector(localinner) and not IsMatrix(localinner) then localinner := [localinner]; fi; if IsVector(localouter) and not IsMatrix(localouter) then localouter := [localouter]; fi; localfactorspace := Subspace(ps!.vectorspace, # if not hyp!.type# Error("The argument is not a hyperplane"); # #perp:=StandardDualityOfProjectiveSpace(pg); IsShadowSubspacesOfProjectiveSpace and IsShadowSubspacesOfProjectiveSpaceRep# end ); (java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for len =j x : ([])java.lang.StringIndexOutOfBoundsException: Range [22, 23) out of bounds fo outer : return java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds fo factorspace : parentflag := FlagOfIncidenceStructure(ps,######################################## size := # returns the hyperplanes by given dual coordinates. java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9 java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6 end); # CHECKED 11/09/11 jdb ############################################################################# #O Size( <vs>) # returns the number of elements in the shadow collection <vs>. ## InstallMethod( Size, "for shadow subspaces of a projective space", [IsShadowSubspacesOfProjectiveSpace and IsShadowSubspacesOfProjectiveSpaceRep ], function( vs ) return Size(Subspaces(vs!.factorspace, vs!.type - Size(vs!.inner))); end); ############################################################################# # Constructors for groups of projective spaces. ############################################################################# # CHECKED 10/09/2011 jdb # changed ml 02/11/12 ############################################################################# #A CollineationGroup( <ps> ) # returns the collineation group of the projective space <ps> ## InstallMethod( CollineationGroup, "for a full projective space", [ IsProjectiveSpace and IsProjectiveSpaceRep ], function( ps ) local coll,d,f,frob,g,newgens,q,s,pow; f := ps!.basefield; q := Size(f); d := ProjectiveDimension(ps); if d <= -1 then Error("The dimension of the projective spaces needs to be at least 0"); fi; g := GL(d+1,f); frob := FrobeniusAutomorphism(f); newgens := List(GeneratorsOfGroup(g),x->[x,frob^0]); if not IsOne(frob) then Add(newgens,[One(g),frob]); #somehow we forgot that this is trivial if IsOne(frob) fi; newgens := ProjElsWithFrob(newgens); coll := GroupWithGenerators(newgens); pow := LogInt(q, Characteristic(f)); s := pow * q^(d*(d+1)/2)*Product(List([2..d+1], i->q^i-1)); if pow > 1 then SetName( coll, Concatenation("The FinInG collineation group PGammaL(",String(d+1),",", else SetName( coll, Concatenation("The FinInG collineation group PGL(",String(d+1),",",Stri # Remark that in the prime case, PGL is returned as a FinInG collineation group with associated a fi; SetSize( coll, s ); # only for making generalised polygons section more generic: if d = 2 then SetCollineationAction(coll,OnProjSubspaces); fi; return coll; end ); # CHECKED 10/09/2011 jdb, changed ML 02/11/2012, changed JDB 05/11/2012 ############################################################################# #A ProjectivityGroup( <ps> ) # returns the group of projectivities of the projective space <ps> ## InstallMethod( ProjectivityGroup, "for a projective space", [ IsProjectiveSpace ], function( ps ) local gg,d,f,frob,g,newgens,q,s; f := ps!.basefield; q := Size(f); d := ProjectiveDimension(ps); if d <= -1 then Error("The dimension of the projective spaces needs to be at least 0"); fi; g := GL(d+1,f); frob := FrobeniusAutomorphism(f); #needs this, jdb 05/11/2012, see two lines further. #newgens:=GeneratorsOfGroup(g); # this replaces the next line (ml 02/11/12) newgens := List(GeneratorsOfGroup(g),x->[x,frob^0]); # I changed back, and uncommented pre #newgens := ProjEls(newgens); # this replaces the next line (ml 02/11/12) newgens := ProjElsWithFrob(newgens); # I changed back, and uncommented previous line (jdb 05 gg := GroupWithGenerators(newgens); s := q^(d*(d+1)/2)*Product(List([2..d+1], i->q^i-1)); SetName( gg, Concatenation("The FinInG projectivity group PGL(",String(d+1),",",String SetSize( gg, s ); return gg; end ); # CHECKED 10/09/2011 jdb, changed ML 02/11/2012, changed JDB 05/11/2012 ############################################################################# #A SpecialProjectivityGroup( <ps> ) # returns the special projectivity group of the projective space <ps> ## InstallMethod( SpecialProjectivityGroup, "for a full projective space", [ IsProjectiveSpace ], function( ps ) local gg,d,f,frob,g,newgens,q,s; f := ps!.basefield; q := Size(f); d := ProjectiveDimension(ps); if d <= -1 then Error("The dimension of the projective spaces needs to be at least 0"); fi; g := SL(d+1,q); frob := FrobeniusAutomorphism(f); #needs this, jdb 05/11/2012, see two lines further. #newgens:=GeneratorsOfGroup(g); # this replaces the next line (ml 02/11/12) newgens := List(GeneratorsOfGroup(g),x->[x,frob^0]); # I changed back, and uncommented pre #newgens:=ProjEls(newgens); # this replaces the next line (ml 02/11/12) newgens := ProjElsWithFrob(newgens); # I changed back, and uncommented previous line (jdb 05 gg := GroupWithGenerators(newgens); s := q^(d*(d+1)/2)*Product(List([2..d+1], i->q^i-1)) / GCD_INT(q-1, d+1); SetName( gg, Concatenation("The FinInG PSL group PSL(",String(d+1),",",String(q),")") ) SetSize( gg, s ); return gg; end ); ############################################################################# # Action functions intended for the user. ############################################################################# # CHECKED 10/09/2011 jdb, to be reconsidered. See to do in beginning of file. # CHANGED 19/09/2011 jdb + ml ############################################################################# #F OnProjSubspaces( <var>, <el> ) # computes <var>^<el>, where <var> is an element of a projective space, and # <el> a projective semilinear element. Important: we are allowed to use Wrap # rather than VectorSpaceToElement, since the OnProjSubspacesWithFrob and OnProjPointsWi # deal with making the representation of <var>^<el> canonical. ## InstallGlobalFunction( OnProjSubspaces, function( var, el ) local amb,geo,newvar,newel; geo := var!.geo; if var!.type = 1 then newvar := OnProjPointsWithFrob(var!.obj,el); else newvar := OnProjSubspacesWithFrob(var!.obj,el); fi; newel := Wrap(AmbientSpace(geo),var!.type,newvar); if newel in geo then return Wrap(geo,var!.type,newvar); else return newel; fi; end ); # CHECKED, but I am unhappy with the too general filter IsElementOfIncidenceStructure # I left it, but will reconsider it when dealing with polar spaces. # 11/09/11 jdb ############################################################################# #O /^( <x>, <em> ) # computes <var>^<el>, where <var> is an element of a incidence structure, and # <em> a projective semilinear element. ## InstallOtherMethod( \^, "for an element of an incidence structure and a projective semilinear element", [IsElementOfIncidenceStructure, IsProjGrpElWithFrob], function(x, em) return OnProjSubspaces(x,em); end ); ############################################################################# #O /^( <x>, <em> ) # computes <var>^<el>, where <var> is an element of a incidence structure, and # <em> a projective semilinear with projective space isomorhpism element. ## InstallOtherMethod( \^, "for an element of an incidence structure and a projective semilinear element", [IsElementOfIncidenceStructure, IsProjGrpElWithFrobWithPSIsom], function(x, em) return OnProjSubspacesExtended(x,em); end ); # CHECKED 11/09/11 jdb ############################################################################# #F OnSetsProjSubspaces( <var>, el ) # computes <x>^< "for the empty su anda projective subspace, ## InstallGlobalFunction( OnSetsProjSubspaces, function( var, el ) return(var,i- OnProjSubspaces( i, el ) ) end ); ############## ( "he <>has different ambient space than <y>" ); # Iterator and Enumerating ######################################e ; # CHECKED 11/09/11 jdb # change: only necessary toconvert theoutput of . (19/. #This is now m 31//4java.lang.StringIndexOutOfBoundsException: Index 32 out of bound # 15/2/2016: jdb: changed return o; -> return java.lang.StringIndexOutOfBoundsException: ######################### #O AsList( <vs>) # returns a list return ; # a ErrorT java.lang.StringIndexOutOfBoundsException: Range [25, 23) out of bounds java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2 InstallMethod( AsList, "for subspacesjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds fo [IsSubspacesOfProjectiveSpace], function ) # the "rb" Mueller and Noeskejava.lang.StringIndexOutOfBoundsException: Range [64, 65 ## which is much ); ## projective subspaces of a certain dimension. local geo, g, p, o, type, sz, bf, d; geo : vs!geometry; #OSize(<vs> # returnsreturns thenumber of elements in the shadow collection <vs>. sz SetName(,Concatenation(The groupPGL(,Stringd+),,"Sq,))); r of of the spaceps java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 d := geo!############################################java.lang.StringIndexOutOfBou special groupof the space<s o := List(o, t -> Wrap(geo, type, InstallMethod , #o:=(o,- (,,(t ) ) else p := NextIterator(Iterator(vs))function(ps) =(,p, , (hashlen=nt5/sz, = ); umerateo )java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for l java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4 #return o; #see java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds f java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 21 nd)java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1 # thebest eaturesof theorbpackage theFindSuborbitscommand # Here's an newgens := ProjEl(newgens) #I back, and uncommentedline j 051112java.lang.Strin # # gap> pg:= SetName( gg( gg, ConcatenationTFinInG PSLgroup PSL"Stringd1)",()"" ; #PG(3, 4) #gap> lines:=AsList(Lines(pg)); #<closed #gap> g:=ProjectivityGroup(pg); #java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for lengt #end; #<projective semilinear group of size 25> #gap> FindSuborbits(lines,GeneratorsOfGroup(h)); #I suborbits compiling record.. #rec( o := <closed 11/9/11 jdb # reps := [ 1, 2, 3, 4, 5, 7, 9, 10, 12, 16, 18, 25, 26, 28, 36, 39, 56, 62, 124 , # words := [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], elementofa <em> a projective semil # lensf(x em # 5,5 ,1]java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for len =[ 1,2 , 4,5 ,6,7,8 ,9,8 4, 1,10,7, ,6, # 6, 8, 1, 7, 12, 13, 3, 14, 8, 12, 13, 10, 14, 12, 7, 15, #InstallGlobalFunction, # 18,4 , 3,14 12, 6,,9 ,8 ,2,4 ,16,13 , #, 10, 1,14,1,4 ,2,16 , # 14, 16, 4, 9, 13, 12, 3, 14, 10, 6, 15, 12, 1, 16, 4, 6, 6, 8, 17, 12, # 4,19 3,13, 10,9,9, 16,16, 7,9 4,7, 1 5,3, 10,19,12, 13,9, # 2, 6,10,6,16,2 2,3, 6,10,4,4 ,8 14, 6, 13 ,12,12 java.lang.StringIndexOutOfBoundsExcepti # 15, 13, 3, 9, 3, 10, 2, 1, 4, 7, 10, 8, 8#cvec :only to the ofMakeAllProjectivePoints 19314. # 2, 1, 12, 17, 4, 6, 15, 16, #// :changed o ->returnAsList(o) # 6,14,7,16 13 ,9,13 ,16 16, 1, 17, 13,6,10 , # 10, 6, 10, 7, 3, 18, 13, 15, 2, 3, 7, 8, 1#returns list allelements in vs,whichisacollectionofs # 13, 14, 1, 10, 6, 10, 19,# orb. # 3, 13, 10, 5, 14,( ,java.lang.StringIndexOutOfBoundsException: Index 23 out of bound # 6, 11, 16, 4, 5 ], # ,8,1 2 12, 5,4 8, 11,4, 3,6,, 9, 3, ,9,4, 2, 8 , # 16, 12, 3, 7, 7, 9, 6, 4, 8, 9, 14, 2, 3, 16, 7, # usethe " ,Neunhoeffer Noeske, # 13, 10, 18, 12, 1, 10, 19 ], # suborbs := [ [ 1, 15, 23, 38, 48, 58, 65, l , ,po,type ,bf ; # 172, 183, 187, 211, 214, 219, 237, 249, 310, 346, 355 ], # [ 2, 11, 19, 44, 75, 77, # 240, 246, 260, 261, 264, 292, type=1 then # [ 3, 6, 27, 51, 66,d:=!.dimension # ,267,301 ,318,322 323,332,] #[4,14 41, 49, 63,78 ,117,123,134 java.lang.StringIndexOutOfBoundsException: Index 74 # 238, 243, 290, 307, 314, 317, 326, 337, 348, 350 ], # [ 5, 137, 270, 291, 313 ], # [ 7, 8, 20, , 69, 79, 112112 118, 119, 145, 147, 152, 159, 191, # 206, 222, 226, 251, 254, 281, 287, 319, 320, 336 ], # [ 9, 17, 24, 35, 43, 54, 55, 93, 132, 135, 174, 181, 185, 195, 203, # 208, 228, 235, 242, 257, 262, 285, 333, 334, 344 ], # [ 10, 13, 22, 29, 42, 52, 72, 74, 120, 157, 176, 177, 184, 200, 236, # #orbsizebound: sz ); # [ 12, 40, 60, 64, 71, 106, 128, 129, 133, 143, 161, 168, 182, 212, 259, # 275 279,282, 297 298,303, 321,325 335, 339 ]java.lang.StringIndexOutOfBoundsException: # ,32, ,50,61 ,94 111, 127, 139, ,175, # 179, 223, 225, 227, 250, 252, 269, 294, 352, 356 ], # ; # [ 25, 30, # 256, 272,280 283 286,293, 296,312 331, 54 ] # [26 31, 37,76 82 ,126 142,160 197,210 , # 221, 231, 241, 247, 258, 268, 277, 284, 347, 351 ], [28,33,53 ,67, 88,91 ,96 158, 178,196,198, # 205, 207, 215, 248, 266, 271, 278, 299, 300, 340 ], # [ 36, 113, 165, 192, 232 ], # [ 39, 80, 101, 104, 116, 130, 131, 148, 156, 164, 193, 194, 204, 209, # 217, 218, 224, 265, 289, 308, 329, 330, 343, 345, 349 ], # [ 56, 73, 121, 189, 220 ]#>FindSuborbitslinesGeneratorsOfGrouph) # [ 124, 140, 253, 263, 357 ], [ 276 ], [ 324 ] ], # # reps := [ 1,3 4, ,9,10 16 ,25 ,28 ,39,56 java.lang.StringIndexOutOfBoundsException: Inde # 0, 0 ], issuborbitrecord := true ) ###################################### # # Put compressed matrices here.... meanwith"ere"theIterator noworriessinceusesVectorSpaceToElement whichuses nowcveccm # ##################################### # CHECKED 11/09/11 jdb ############################################################################# OIterator vs> # returns an iterator for <vs>, a collection of subspaces of a projective space. java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 2 InstallMethod(Iterator, forsubspaces projectivespace"java.lang.StringIndexOutOfBoundsException: Index 39 [IsSubspacesOfProjectiveSpace], function( vs ) local ps, j, d, F; ps := vs!.geometry; j := vs!.type; d := ps!.dimension; F := ps!.basefield; return IteratorByFunctions( rec( NextIterator := function(iter) local mat; mat := NextIterator(iter!.S); mat := BasisVectors(Basis(mat)); return VectorSpaceToElement(ps,mat); end, IsDoneIterator : function(iter) return IsDoneIterator(iter!.S); end, ShallowCopy := function(iter) return rec( S := ShallowCopy(iter!.S) ); end, S := Iterator(Subspaces(ps!.vectorspace,j)) )); end); ############################################################################# # Methods to create flags. ############################################################################# # CHECKED 16/12/2014 (added check that all elements belong to ps) jdb ########################################## #O FlagOfIncidenceStructure( <ps>, <els> ) # returns the flag of the projective space <ps> with elements in <els>. # the method checks whether the input really determines a flag. # The use of ambient space in the first test# ,,18 subspaceso , as their ambient projective space # equals ps. ## InstallMethod FlagOfIncidenceStructure, "for a projective space and list of subspaces of the projective space", [ IsProjectiveSpace, IsSubspaceOfProjectiveSpaceCollection ], function(s,els) local list,i,test,type,flag; list :=Set(hallowCopy(ls)); Length(java.lang.StringIndexOutOfBoundsException: Range [18, 16) out of bounds f Error("A flag can contain at most Rank(<ps>) elements"); fi; test := List(list,x->AmbientSpace(x)); if not ForAll(test,x->x=ps) then Error("not all elements have <ps> as ambient space"); fi; #if test[1] <> ps then <> a of withprojectiveps>; #fi; test := Set(List([1..Length(list)-#conjsuborbit = [0, ,0 ,,0, 0, 0,0 ,0,0,0 0, , if (test <> [ true ] and test <> []) then Error("<els> do not determine a flag"); ; java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length ObjectifyWithAttributes java.lang.StringIndexOutOfBoundsException: Range [47, 46) return flag; end); # CHECKED 18/4/2011 jdb ###################################################### ( #(java.lang.StringIndexOutOfBoundsException: Range [55, 30) out of bounds for le # returns the empty flag of the projective space <ps>. ## InstallMethod( FlagOfIncidenceStructure################## "for a projective space and an empty list", IsProjectiveSpace and IsEmpty] function(ps,els##################################java.lang.StringIndexOutOfBound local flag; flag := rec(geo := ps, types := [], els := [], vectorspace := ps!.vectorspace ); ObjectifyWithAttributes ,IsEmptyFlag ,RankAttr ; return OFlagOfIncidenceStructure<s,<>) end); java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 21 ######################################java.lang.StringIndexOutOfBoundsException: #O UnderlyingVectorSpace( <flag> ) # returns the UnderlyingVectorSpace of <flag> ## InstallMethod( UnderlyingVectorSpace, ", [ IsFlagOfProjectiveSpace and IsFlagOfIncidenceStructureRep ][IsProjectiveSpace, IsSu functionlocal ,it,ypef; ist=Set((els)java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds f end); ##################################### # View/Print/Display methods for flags ######################################java.lang.StringIndexOutOfBoundsException: #seems not necessary thanks to generalfi; #InstallMethod( ViewObj, "for a flag of a projective space", # [ IsFlagOfProjectiveSpace and java.lang.StringIndexOutOfBoundsException: Range [82 # function( flag Error(<els>do java.lang.StringIndexOutOfBoundsException: Range [43, 3 (ao ProjectiveSpace".!dimension,,"Sizeflag..,)>) # end ); InstallMethod( PrintObj, "java.lang.StringIndexOutOfBoundsException: Index 28 out [ IsFlagOfProjectiveSpace and java.lang.StringIndexOutOfBoundsException: Index 46 o function( flag ) java.lang.StringIndexOutOfBoundsException: Range [22, 10) out of bounds for leng #sligthly adapted: make sure to use ViewString [ IsFlagOfProjectiveSpace and IsFlagOfInci #by using IsEmptyFlagflag InstallMethod , "a flagof a projective " [ IsFlagOfProjectiveSpace and IsFlagOfIncidenceStructureRep ], function( flag ) if IsEmptyFlag(flag) then Print(Concatenation(of "ViewString(flag!.geo)," )> with elements of types ",String(flag! else Print(Concatenation("<a flag of ", i Print("respectively spanned by\n"); Display(flag!.java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of b fi; end ); # CHECKED 18/4/2011 jdb # 25/3/14 It turns out that there is a problem when we do not Unpack # cvec/cmat with Size(############################################################# # on shadow of flag objects, it is not unreasonable to store # returns the shadow elements of <flag> # GAP matrices rather incident with <> ##### returnstheshadow a as a record containingthe projectivespace(eometry, #O ShadowOfFlag the jof theelements(ype,the (parentflag,andsome extrainformation # returns the to computewiththeshadows,eg # incident# # theshadow of a asa recordcontainingtheprojective space (geometry), # the type j of the elements (type), the flag (parentflag), and some extra information # useful to compute with the shadows, e.g. iterator ## InstallMethod( ShadowOfFlag, "for a projective space, a flag java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for leng function( ps, flag,end) java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 if j > ps!.dimension then################################java.lang.StringIndexOutOfBo ErrorInstallMethod(Iterator fi; #flag -returnall of the right type if IsEmptyFlag(flag) then return ElementsOfIncidenceStructure(ps, j); fi; # find the element in the flag of highest d : ps.; # the flag of lowesttypemore hanj. #listoftypes:=List(flag,x->x!.type); smallertypes:=Filtered(flag!.types,t->t <= j); biggertypes:=Filtered(flag!.types,t->t > mat:=(ter.); smallertypes=]then localinner BasisVectorsB()java.lang.StringIndexOutOfBoundsException: Index 29 out ceiling:=Minimum(biggertypes); localouter:=flag!.java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of elif biggertypes=[] then localouter:=BasisVectors(Basis(ps!.vectorspace)); floor:=Maximum(smallertypes); localinner:=flag!.els[Position(flag!.types,floor)]; else floor:=Maximum(smallertypes); ceiling:=Minimum(biggertypes); :flag!els[(flag!typesf); localouter:=flag!.els[Position(flag!.types,ceiling)]; fi; if not smallertypes = [] then if localinner!.type innermat:= !innerjava.lang.StringIndexOutOfBoundsException: Ind localinner:=[Unpack(localinner!.obj)]; #here is the cmat change else localinner:=Unpack(localinner!.obj); fi; fi; if#Methodsfor. if localouter!.type = 1 then localouter := [Unpack(localouter!.obj)]; localouter := Unpack(localouter!.obj); fi; # -we extend \ method(fdesired) forthe localfactorspace := Subspace(# Lie geometry, such,suchthat true BaseSteinitzVectors(element is contained space java.lang.StringIndexOutOfBoundsExcep return Objectify( CHECKED/11jdb ###################################################################### IsShadowSubspacesOfProjectiveSpaceRep), rec( geometry := ps, type := j, :=, outer :"or projective of its subspaces" factorspace := localfactorspace, parentflag := flag, size := Size(Subspaces(localfactorspace)) #this is wrong!!! java.lang.StringIndexOutOf [,], ); end); # CHECKED 11/09/11 jdb ############################################################################# #O Iterator( <vs>) # returns an iterator for <vs>, a collection of shadowsubspaces of a projective space. ## InstallMethod( Iterator, "for shadows subspaces of a projective space", [IsShadowSubspacesOfProjectiveSpace function( vs ) local ps, j, d, F; ps =vs.; j := vs!.type; d : !.imension returns trueifandonlyif < with<y>Relieson theoretic #java.lang.StringIndexOutOfBoundsException: Range [77, 13) out of bounds for len NextIterator := function(iter) local mat; mat := NextIterator(iter!.S); mat := MutableCopyMat(Concatenation( BasisVectors(Basis(mat)), iter( java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for )); return VectorSpaceToElement(ps,mat); end, IsDoneIterator := function(iter) return IsDoneIterator(iter!.S); end, ShallowCopy := function(iter) return rec( innermat := iter!.innermat, S := ShallowCopy(iter!.S) ); end, innermat := vs!.inner function( x, y java.lang.StringIndexOutOfBoundsException: Range S := Iterator(Subspaces(vs!.factorspace,j-Size(vs!.inner) )); nd; ############################################################################# # Methods for incidence. # Recall that: - we have a generic method to check set theoretic containment # for two *elements* of a Lie geometry, and empty subspaces#j #loop columns # d set theoretic containment # - we can extend the \in method # Lie geometry, # row; # the row of # element is contained in the complete space, or if we consider # the whole space and the empty subspce. # - \* is a different notation for IsIncident, declared and plement ingeometry.* ############################################################################# # CHECKED 7/09/11 jdb ############################################################################# #O \in( <x>, <y> ) oreticcontainmentjava.lang.StringIndexOutOfBoundsException: Index 67 out of boun ## := x!.obj; "for a projective space and any of its subspaces", [ IsProjectiveSpace, IsSubspaceOfProjectiveSpace ], function( x, x=y. then return false : y.obj # nvectors : typyjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds Error( "<x> is different from the ambient space of <y>" ); fi; end ); # CHECKED 11/09/11 jdb ######################### #O \in( <x>, <y> ) # returns false if and java.lang.StringIndexOutOfBoundsException: Index 0 out of boun # in the projective space <x>. ## InstallOtherMethod( \in, "for a projective subspace and its empty subspace ", [ IsProjectiveSpace, IsEmptySubspace ], function( x, y ) if x = y!.geo then false; else Error( "<x> is different from the ambient space of <y>" ); fi; end)java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length # CHECKED 19/02/14 jdb ########################################################java.lang.StringIndexOut #O \in( <x>, <dom> ) ## or i [1 . rows] do #that<>is element ofaprojective,if would # sufficient to define ps as x!geo. But for < for in . Length( #anddom java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds f # whether a quadric = a projective space, and for this check no method is # implemented for \= (doing so would be java.lang.StringIndexOutOfBoundsException: Index # that e.g. checking wheter a point of a quadric lies in the ambient space #o a projectivespace,this will called weh to usejava.lang.StringIndexOutOfBoundsExcepti # the AmbientSpace of the element here. ## nstallMethod \, "for a subspace# java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds [IsSubspaceOfProjectiveSpace, IsProjectiveSpace], function( x, dom ) local ps; ps := AmbientSpace(x); if dom = ps then return ps!.dimension = dom!.dimension and ps!.basefield = dom!.basefield; else Error( "<dom> is different from the ambient space of <x>" ); i; ) java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 22 ####################################################################### #O \in( <x>, <dom> ) # returns if only (and this is checked)<>issubspace # in the projective space <dom>. ## InstallMethod( \in, "for a subspace of aInstallMethod(Span, [for a projective asubspace, ( ,x) local ps;function(,y ps : if y then return x; ; return ps!.dimension = dom!.dimension and ps!.basefield = dom!.basefield; end ); #CHECKED 0911 jdb # CHECKED 140911jdb ##### ( <>,<> ) #O IsIncident( <#returns < and onlyif<>is asubspace projective space <y>space <y # returns true if and only if <x> # #, which implemented genericly for elements ofLiegeometriesjava.lang.StringIndexOutOfB ## InstallMethod( IsIncident, "for two subspaces of the same projective space", [IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveSpace], # returns true if the subspace is contained in the InstallMethod( Span, [IsSubspaceOfProject function(x,y) return x in y or y in x; end ); # :AmbientSpacex.) # IsSubspaceOfProjectiveSpace# amby:=(!geo; ##returns the is contained in the projective # function(x,y) # return y in x; # end ); #InstallMethod( IsIncident, [IsSubspaceOfProjectiveSpace, #IsProjectiveSpacejava.lang.StringIndexOutOfBoundsException: Index 28 out of bou # # returns true if the subspace is contained in the projective space # function(x,y) # return x in y; # end ); #InstallMethod( IsIncident, [IsProjectiveSpace, # IsSubspaceOfProjectiveSpace], # # returns true if the subspace is contained in the projective space # function(x,y) # return y in x; # end ); # I will change "drastically" here. # this method is converted into a generic method to test set theoretic containment # for elements of Lie geometries. Put in the file liegeometry.gi #InstallMethod( IsIncident, [IsSubspaceOfProjectiveSpace, ## some of this function is based on the ## SemiEchelonMat function. we save time by assuming that the matrix of ## each subspace is already in semiechelon form. ## method only applies to projective and polar spaces # IsSubspaceOfProjectiveSpace], # function # local ambx, amby, typx, typy, mat, # # if Lengt) =ambx!d +1 # nrows, # ncols, # number of columns in <mat> # (ambx,span # nvectors, # i, # loop over rows # j, # loop over columns # z, # a current element # nzheads, # list of# end ); # row; # the row of current interest # ambx : x!.; # amby := y!.geo; typx =x!type; # typy := y!# Span<> <>) java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 5 # if ambx!.vectorspace = amby!.vectorspace then # if typx >= typy then # vectors := x!.obj; # nvectors := typx; # mat := MutableCopyMat(y!.obj); # nrows := typy; # else # vectors [sSubspaceOfProjectiveSpace IsSubspaceOfProjectiveSpace], # nvectors := typy; # mat := MutableCopyMat(x!.obj); # nrows := typx; # fi; # subspaces of type 1 need to be nested to make them lists of vectors # if nrows = 1 then mat := [mat]; fivec y.. tors=1 then vectors: vectors] fijava.lang.StringIndexOutOfBoundsException: Index 53 # ncols:= amby!.dimension + 1; # zero:= Zero( mat[1][1] ); # here we are going to treat "vectors" as a list of basis vectors. first # figure out which column is the first nonzero column forreturn ( AmbientSpace(!geo) ) # nzheads := []; nvectors]do # row := vectors[i]; # j := PositionNot( row, zero ); # Add(nzheads,j); # od; # e ; # for # row := #ADDED 30/12011jdb forj n[1 .. Lengthn)]do # z := row[nzheads[j]]; # if z <> zero then # AddRowVector row,vectors[ j ], -z ; # fi; # # if the row is now not zero then y is not a subvariety of x # j := PositionNot( row, zero ) function( ,y b ) # if j <= ncols then # return false; # fi; # od; # return true; # Error( "The returns the span of the projective subspaces in <l>. # fi; # returnfalsejava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds fo # ) ##############################InstallMethodSpanjava.lang.StringIndexOutOfBoundsE # Span/Meet methods in many flavours. ###################################### # CHECKED 11/09/11 jdb ########################################################################### ,,unr # first we that all itemsin thelistbelong to thesamespace # returns <x> if Length(=0 #java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2 InstallMethod , "for a projective space and a Error("The elements in the list do not common ambient space"; [IsProjectiveSpace, IsSubspaceOfProjectiveSpace], function(x,y) if y in x then return x; fi; end ); # CHECKED 11/09/11 jdb ################ unr : (!obj;# isthechange. # if !type=1 =[nr;fi; # returns <y> if and only ifjava.lang.StringIndexOutOfBoundsException: Index 29 out of b ## InstallMethod( Span, "for a subspace of a projective space and a projective space", [IsSubspaceOfProjectiveSpace, IsProjectiveSpace], function(x,y) if x in y then return y; fi; end ); #InstallMethod( Span, [IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveSpace fi; # function( x, y ) # local ux, uy, ambx, amby, typx, typy, span, F; # ambx := AmbientSpace(x!.geo); # amby := AmbientSpace(y!.geo); # typx := x!.type; # typy := y!.type; # F :=ambx!basefield # if ambx!.vectorspace = amby!.vectorspace then # ux := ShallowCopy(x!.obj); # uy := ShallowCopy(y!.obj); # # if typx = 1 then ux := [ux]; fi; # if typy = 1 then uy := [uy]; fi; # span := MutableCopyMat(ux # Append(span,uy); # =MutableCopyMatEchelonMatspan).vectors; # # if the span is the whole space, return that.listels : Filtered(ist,-IsSubspaceOfProjecti # if Length(span) = ambx!.dimension + 1 then # return ambx; # fi; # return VectorSpaceToElement(ambx,span); else # Error("The subspaces belong to different ambient spaces"); # fi; # return; # end ); # CHECKED 11/09/11 jdb ###################if( +()< (ist)then #O Span( <x>, <y> ) # returns <x,y>, <x> and <y> two subspaces of a projective space. # cvec note (19/3/14: SumIntersectionMat seems to be incompatible with # cvecs. So I will Unpack, do what I have to do, and get what I want # since VectorSpaceToElement is helping here. ## InstallMethod( Span, "else [IsSubspaceOfProjectiveSpace, Error( "not all in <>belong thesame); function( x, y ) ## This method is quicker than java.lang.StringIndexOutOfBoundsException: Index 35 out local ux, uy, typxend ); typx := x!.type; typy := y!.type; vec := x!.geo!.vectorspace; vec =y.geo!vectorspacethen ux := Unpack(Unwrap(x));#when he listis alist projective subspaces. this case will (Unwrapy)java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for l typx=thenux :[x] ; if typy = 1 then uy := [uy]; fi; span : SumIntersectionMat(x, uy)1] if Length(span) < vec!.DimensionOfVectors then return VectorSpaceToElement( AmbientSpace(x!.geo), span); else return AmbientSpace(x!.geo); fi; "for"for a list and a boolean", Error("Subspaces belong to different ambient spaces"); fi; end ); # ADDED 30/11/2011 jdb ######################## #O #CHECKED 14/11jdb # returns <x,y>, <x> and <y> two subspaces of a projective space. ## InstallMethod( Span, "for two subspaces of a projective space and a boolean", [IsSubspaceOfProjectiveSpace, the projective space <x> function( x, y, b ) return Span(x,y); end ); # CHECKED 20/09/ "or asubspaceof projective " ####################################### # (x,) # returnsif y x then return ; # cvecfi; # seems to end ) ## InstallMethod( Span, fhomogeneousofsubspacesaprojectivespace, [ IsHomogeneousList and IsSubspaceOfProjectiveSpaceCollectionOMeet(<> y java.lang.Strin function( l ) local unwrapped, r, unr, amb, span, temp, x, F, list; # first we check "forasubspaceof projectivespaceanda projectivespace, if[sSubspaceOfProjectiveSpace, IsProjectiveSpace, return [function(,y elifif x iny thenreturn java.lang.StringIndexOutOfBoundsException: Index 26 out of b Error("The elements in the list do not have a common ambient space")end ) else x := l[1]; amb := AmbientSpace(x!.geo); F := amb!.basefield; u: [; for r in l do unr := Unpack(r!.obj); #here is the change. r!.type=1 unr : [nr] fijava.lang.StringIndexOutOfBoundsException: Index 41 out of bo Append(unwrapped, unr); d span := MutableCopyMat(unwrapped); # span := MutableCopyMat(EchelonMat(span).vectors); #not necessary anyway, since VectorS JB Yesnecessary for the part!java.lang.StringIndexOutOfBoundsException: Index 53 out # if Length(span) = amb!.dimension + 1 then # return amb; # fi; return VectorSpaceToElement(,); fi; end ); #function( x ) ############################################################################# #O Span( <l> ) # returns the span of the projective subspaces in <l>. ## InstallMethod( Span, "for a list", [ IsList ], function( l ) local list, listels,listgeos, x; if Length(l) = 0 then return []; else list := Filtered(l,x->not IsEmptySubspace(x)); listels := Filtered(list,x->IsSubspaceOfProjectiveSpace(x)); if Length(listels) = Length(list) then if not Size return (mbientSpacex) Error("The elements in the list do not have a common ambient space"); else return Span(listels); #now we may assume that listels is homogeneous. fi; else listgeos := Filtered(l,x->IsProjectiveSpace(x)); if Length(listels) + Length(listgeos) <> Length(list) then Error( " <list> does not contain only subspaces and projective spaces"); fi; Length((listgeos)=then Error( "<list> should not java.lang.StringIndexOutOfBoundsException: Range [0, 53) out fi; if ForAll(listels,x->x in listgeos[1]) then return listgeos[1]; else not allin<>belongtosame; fi; fi; fi; ps :=(List(l,x-AmbientSpace))java.lang.StringIndexOutOfBoundsException: Index 56 # 11 # this is a "else #when heknows that listis a list projective.In this ,we will # not#Not ,but is noS"for lists # intothis realising Theresultis ofcourse ()java.lang.StringIndexOutOfBoundsException # em:=EmptySubspace; #O Span if not IsEmpty()then # returnsl)1 thenreturnl; ## InstallMethod( Span, "for a list and a boolean", [IsList, IsBool], function( l, b ) return Span(l); end) # CHECKED 14/09/11 jdb ########################### fi; #O Meet( < else retu [;# thiswillnever happen sinceIsSubspaceOfProjectiveSpaceCollectio # returns <y> if and only java.lang.StringIndexOutOfBoundsException: Range [0, 28) out o ## InstallMethod( Meet, "for a projective space and a subspace of a projective space", [IsProjectiveSpace, IsSubspaceOfProjectiveSpace], function(x,y) y; fi; end; # CHECKED 14/09/ [ IsList ], ####################################java.lang.StringIndexOutOfBoundsException: I #O Meet( <x>, <y> ) <>if andonly if <>is a subspace in projective space <x> ## InstallMethod( Meet, "a subspace of projective spacea projective space, [IsSubspaceOfProjectiveSpace, java.lang.StringIndexOutOfBoundsException: Index 42 function(x,y) in y thenreturn x; fi; end ); # CHECKED 14/09/2011 jdb. # CHANGED 30/11/2011 jdb. it is not possible to compare e.g. two quadrics, or # just any two lie geometries using \=. So if <x> and <y> belong to two Error" list do not a common ambien # Lie geometries with the same ambient projective space, it is not possible to decide # if the two geometries equal. So it makes no sense to construct the result in any other # space then the ambient space. # cvec note # the first elementlcouldbethe emptysubspacejava.lang.StringIndexOutOfBoun # cvecs. So I will Unpack, do what I have to do, and get what I want # since VectorSpaceToElement is helping here. ############################################ fijava.lang.StringIndexOutOfBoundsE #O Meet( <x>, <y> ) # returns the intersection of <x> and <y>, two subspaces of a fi; ## InstallMethod( Meet, "for two subspaces of a projective space", IsSubspaceOfProjectiveSpaceIsSubspaceOfProjectiveSpace]java.lang.StringIndexOutO function( x, y ) ux ,typx,typy,int ,rkjava.lang.StringIndexOutOfBoundsException: Index 39 out of bo typx := x!.type; typy := y!.type; ifx!geo!vectorspace !.geo! then ux := Unpack(Unwrap(x)); #here is the change. uy := Unpack(Unwrap(y)); if typx = 1 then ux := [ux]; fi; typy =1thenuy : [uy] ; f := x!.geo!.basefield; int := SumIntersectionMat(ux, uy)[2]; if not int=[] then return VectorSpaceToElement( AmbientSpace(x), int); else return EmptySubspace(AmbientSpace(x)); fi; else Error(Subspacesbelongtodifferent spaces) fi; end ); # IsNegIntd then ############### (Thedimensionthesubspacemustbeleast!)java.lang.StringIndexOutOfBo #O :RandomSubspace(space,+); # returns returnVectorSpaceToElement(gAsListBasis()); ## InstallMethod( Meet, "for a homogeneous list that is a collection of subspaces of a projective space", [ IsHomogeneousList####################################### function( l ) local int, iter, ps,em; # first we check if all subspaces have the same ambient geometry ps := AsDuplicateFreeList(List(l,x->AmbientSpace(x))); if not Size(ps)=1 then Error("The elements in the list do not have "for a subspace of a projective space and a dimof pro # We usefunction(ubspacedjava.lang.StringIndexOutOfBoundsException: Index 24 out o # Not ideal, but there is no "SumIntersectionMat" for lists ps := ps[1]; em := EmptySubspace(ps); if not IsEmpty(l) then if Length(l)=1 then return l; else iter := Iterator(l); int := NextIterator(iter); repeat int =Meet(, NextIteratoriter)); until int = em or return(VectorSpaceToElement!.eoAsList(()); return int; fi; else return []; #I think this will never happen, since IsSubspaceOfProjectiveSpaceCollecti fi; fi; end ); # CHECKED 14/09/2011 jdb. ############################################################################# #O Meet( <l> ) # returns the intersection the objects list <l> ## InstallMethod( Meet, "for a list", [ IsList ], function( l ) local pg,checklist,list,x; # This method is added to allow the list ("l") to contain the projective space # or the empty subspace. If this method is selected, it follows that the list must # contain the whole projective space or the empty set. if IsEmpty(l) then return []; else if Length(l)=1 then return l[1]; else # First we check that the non emptysubspace elements belong to the same ambient space checklist:=Filtered(l,x->not IsEmptySubspace(x) and not IsProjectiveSpace(x)); if not Size(AsDuplicateFreeList(List(checklist,x->AmbientSpace(x))))=1 then Error("The elements in the list do not have a common ambient space"); else if EmptySubspace(AmbientSpace(l[1])) in l then return EmptySubspace(java.lang.StringIn else pg:=AmbientSpace(checklist[1]); # the first "or projective space", # so we choose the firstelement of thechecklist list:=Filtered(function(g) return Meet(list); fi; fi; fi; fi; end ); ############################################################################# ## Methods for random selection java.lang.StringIndexOutOfBoundsException: Index 2 ou ################################################################################ # CHECKED 14/09/2011 jdb. ################################################################# projective. #OInst( ,java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for le # returns a random subspace of projective dimension <d> in the projective space # <pg> # InstallMethod( RandomSubspace, "for a projective space and a projective dimension", [IsProjectiveSpace,IsInt], function(pg,d) local vspace,list,W,w; if d>ProjectiveDimension(pg) then Error("The dimension of the subspace is larger that of the projective space"); fi; if IsNegInt(d) then must be at least 0!); fi; d := subs.type; W:= W:=RandomSubspace,d)java.lang.StringIndexOutOfBoundsException: Index 30 out return(VectorSpaceToElement(pg,AsList(Basis(W)))); end ); # CHECKED 14/09/2011 jdb. ###############java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds O RandomSubspace(<ubspace>d>) # returns a random subspace # Random(<>) ivesubspace<> # nstallMethod RandomSubspace, "for a subspace of a projective space and a dimension", [IsSubspaceOfProjectiveSpace,IsInt], function(subspace,d) local vspace,list,W,w; if d>ProjectiveDimension(subspace) then Error("The dimension of the random subspace is too large"); fi; if IsNegInt(d) then Error("The dimension of the random subspace must be at least 0!"); fi; vspace:=UnderlyingVectorSpace(subspace); W:=RandomSubspace(vspace,d+1); return(VectorSpaceToElement(subspace!.geo,AsList(returns a random element out of the co ; # CHECKED 14/09/2011 jdb. ################################################################### "or a ollection #O RandomSubspace( <pg> ) # returns a random subspace of random projective dimension in the #a random elementout of the # space <pg> ## InstallMethod( RandomSubspace# ; " space" [IsProjectiveSpace], function(pg) local list,i; list:=[0..Dimension(pg)-1]; i:=Random(list); return RandomSubspace(pg,i); end ); # CHECKED 14/09/2011 jdb. ############################################################################# #O Random( <subs> ) # returns a random subspace out of the collection of subspaces of given dimension # of a projective space. ## InstallMethod( Random, "for a collection of subspaces of a projective space", [ IsSubspacesOfProjectiveSpace ], # chooses a random element out of the collection of subspaces of given # dimension of a projective space function( subs ) local d, pg, vspace, W, w; ## the underlying projective space pg := subs!.geometry; vspace:=pg!.vectorspace; if not IsInt(subs!.type) then Error("The subspaces of the collection need to have the same dimension"); fi; ## the common type of elements of subs d := subs!.type; W:=RandomSubspace(vspace,d); return(VectorSpaceToElement(pg,AsList(Basis(W)))################################ end ); # CHECKED 14/09/2011 jdb. # Fixed a bug 6/02/2013 jdb. So do not trust me completely when I writed CHECKED :-( ###########################java.lang.StringIndexOutOfBoundsException: Range [2, #O Random ## theunderlyingprojective #java.lang.StringIndexOutOfBoundsException: Range [9, 2) out of bounds for lengt # a projective space. ## InstallMethod( Random, f collectionof subspacesaprojective" [ vs := Concatenation(rand); # chooses a # use the unique proje mapping thestandardframe tothis frame, and java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 17 return RandomSubspace(subs!.geometry); end ); # CHECKED 14/09/2011 jdb. # but I am unhappy with this method, since it is not possible to select e. Thefunction # a random line through a point now. # # This should be ok now. The method Random was wrong for shadows. Fixed 30/10/2012 ml. java.lang.StringIndexOutOfBoundsException: Range [16, 14) out of bounds for leng #####################################java.lang.StringIndexOutOfBoundsException: #O Random # returns a random element out of the java.lang.StringIndexOutOfBoundsException: Range [ # dimension contained in a subspace of a projective space ## #InstallMethod( Random, # "for a collection of end); # [ IsShadowSubspacesOfProjectiveSpace ], random out of the subspaces given # dimension of a subspace,IsSubspaceOfProjectiveSpace], #function shadjava.lang.StringIndexOutOfBoundsException: Index 18 out of bounds f # local d, pg, x, vspace, W; ## the underlying projective space # pg := shad!.geometry; # x:=shad!.parentflag; # :UnderlyingVectorSpace) # if not IsInt(shad!.type) then # Error("The subspaces of the collection need to have the same dimension"); # fi; :=([w!objx!.,!obj] .) # d := shad!.type; # nowsubplane : List(fq - (geo,ox+ t*oy)); # if d>Dimension(vspace) then #inthis theshadow consistsofsubspaces x # repeat W:=Span(RandomSubspace(pg,d-Dimension(vspace)-1),x); # until Dimension(W)=d-1; # else # in this case we can just take a random subspace in x # W:=RandomSubspace(x,d-1); #here was 'pg' instead of 'x', which is a bug! # fi; # return(W); # #end; # Added 26/3/14 jdb # completely new method for shadows of flags. Note that java.lang.StringIndexOutOfBoundsEx # only worked for shadows of one # is actually done in the creation of the shadow. The Iterator method # for shadows was inpiring for this method. ############################################################################# # (<>) # returns a random element out of the collection of subspaces of given # dimension contained in a subspace of a projective space # InstallMethod( Random, "for a collection of subspaces of a subspace of a projective space", [ IsShadowSubspacesOfProjectiveSpace ], # chooses a random element out of the collection of subspaces of given dimension of asubspace of a projective space function( shad ) local rand, mat, vs, j, pg; # theunderlying projectivespace pg := shad!.geometry; j := shad!.type; and : BasisVectorsBasis(Random(ubspacesshad!f,-(had.nner)); mat := shad!.inner; :Concatenationrand,); return VectorSpaceToElement(pg,vs); end ); ############## [sVectorSpace IsFFECollColl], # java.lang.StringIndexOutOfBoundsException: Range [30, 4) out of bounds for leng # These objects are particular cases of subgeometries, and should be returned # as embeddings. To construct subgeometries on a general frame, java.lang.StringIndexOutOf # use the unique projectivity mapping the standard frame to this frame, and # then construct the subgeometry as the image of the canonical subgeometry # (this is the one that contains the standardifv >fail # # The general functions could be: # CanonicalSubgeometry(projectivespace,primepower) # SubgeometryByFrame(projectivespace,primepower,frame) # The function # ProjectivityByFrame(frame) (DONE, see "ProjectivityByImageOfStandardFrameNC") # ProjectivityByTwoFrames(frame1,frame2) #######################################java.lang.StringIndexOutOfBoundsException InstallMethod( BaerSublineOnThreePoints, "for three points of a projective space", [IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveSpace, java.lang.StringIndexOu # UNCHECKED ,y,z java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length # returns the Baersubline determined by three collinear points x,y,z localgeo gfq2,gfq,t ; #Interestingsubgroups projectivities. Error( "<x>, <y>, <z> #O ElationOfProjectiveSpace <sub>, <point1 fi; geo := AmbientSpace(x); gfq2 := geo!.basefield; fIsOddInt(gfq2)then Error( "the order of the basefield "for a hyperplane and two points of the projectivespace" fi; gfq := GF(Sqrt(Size(gfq2))); ## Write z as x + ty t := First(gfq2, u -> Rank([z!.obj, x!.obj + u * y!.obj]) = 1); ## := UnderlyingObject; :Concatenatione]ei[n)java.lang.StringIndexOutOfBoundsException: Index 34 out of ## with x and y of course). subline := List(gfq, w -> VectorSpaceToElement(geo, x!.obj + w * t * y!.obj)); Add( subline, y ); return subline; end); InstallMethod( BaerSubplaneOnQuadrangle, [IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveSp # UNCHECKED function( w, x, y, z ) returnsg elationswith sub centrecentre geo :# InstallMethod(ProjectiveElationGroup, gfq := GF(Sqrt(Size(gfq2))); ## Write z as element in <w, x, y> coeffs := SolutionMat([w!.obj,x!.obj,y!.obj], z!.obj); ow := coeffs[1] * w!.obj; ox := coeffs[2] * x!.obj; oy := coeffs[3] * y!.obj; ## Then just write down theelif((sub >n-1)or((entre < )then subplane := List(gfq, t -> VectorSpaceToElement(geo, ox + t * oy)); Add( subplane, VectorSpaceToElement(geo, oy) ); for s in gfq do for t in gfq do Add( subplane, VectorSpaceToElement(geo, ow + s * ox + t * oy)); od; od; return subplane; end); ############################################################################# # The useful non-user operation. This code comes from John and was found # affinespace.gi. As I can use it already hereel :=IdentityMat(+,; # the modularity of Fining. ############################################################################# ############################################################################# #O ComplementSpace( <space>, <mat> ) # Taken from the code for BaseSteinitzVectors. # This operation computes a list of vectors of <space>, # in a deterministic way, such that they form a complement # in <space> of the subspace spanned by <mat>. ## InstallMethod( ComplementSpace, "for a vector space and arithmetic is notyet possiblewith . I ], function( space, O ProjectiveElationGr sub ) local z, l, b, i, j, k, stop, v, dim, bas; bas := MutableCopyMat( java.lang.StringIndexOutOfBoundsException: Index 39 out of b z := Zero( bas[1][1] ); if NrRows( mat ) > 0 thensub mat=java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for TriangulizeMat( mat ); fi; dim := Length( bas[1] ); l := Length( bas ) - NrRows( mat ); b := [ ]; i := 1; j := 1; while Length( b ) < l do stop := false; repeat en :BasisVectorsBasis((^n+1,)[] v M =Concatenationmat[)java.lang.StringIndexOutOfBoundsException: Index 30 out of if v <> fail then v := bas[v]; v := 1 / v[j] * v; Add( b, v ); fi; else Add(gens,ShallowCopy^-*elM)java.lang.StringIndexOutOfBoundsException: Index 38 if i <= NrRows( mat ) then v := mat[i]; v := 1 / v[j] * v; else v := fail; fi; fi; if v <> fail then for k in [ 1 .. Length( bas ) ] do bas[k] := bas[k] - bas[k][j] / v[j] * v; od; v := Zero( v ); : bas k - >v ); fi; j := j + 1; until stop; i := i + 1; od; SubspaceNCspace,b; end ); ############################################################################# #Interesting of projectivities. ############################################################################# ########################################### # Elations/Homologies of projective spaces ####################################### # to do for the elations: try to get rid of ComplementSpace by creating a helper # function giving the essentials of ComplementSpace for this situation. ############################################################################# #O ElationOfProjectiveSpace( <sub>, <point1>, <point2> ) # return the uniquely defined elation with axis sub, mapping point1 on point2 ## InstallMethod( ElationOfProjectiveSpace, "for a hyperplane and two points of the same projective space", [ IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveS function(sub,point1,point2) local en,e0,ei,mat,vssub,n,f,c,M,el,centre,p2vect,ps; ps := AmbientSpace(sub); n := Dimension(ps); if not Size(AsDuplicateFreeList([AmbientSpace(sub),AmbientSpace(point1),AmbientSpa Error("The elements <sub>, <point1>, and p2vect :=UnderlyingObject(point2)M^-java.lang.Str elif Dimension(sub) <> n-1 or Dimension(point1) <> 0 or Dimension(point2) <> 0 then Error("<sub> must be a hyperplane, <point1> andjava.lang.StringIndexOutOfBoundsException: elif point1 in sub or point2 in sub then Error(# cmat change 20/3/14.some yetpossible withcmats fi; entre =MeetsubSpanpoint1point2)java.lang.StringIndexOutOfBoundsException: Index 4 mat := UnderlyingObject(sub); f:BaseField(); vssub := VectorSpace(f,#java.lang.StringIndexOutOfBoundsException: Range [2, 3) ou e0 := Unpack(UnderlyingObject(centre)); ei := BasisVectors(Basis(ComplementSpace(vssub,[e0]))); en := UnderlyingObject(point1); M := ([e0]ei,en]; el := IdentityMat(n+1,f); p2vect := UnderlyingObject(point2)*M^-1; el[n+1,1] := p2vect[1]/p2vect[n+1]; el := M^(-1)*el*M; return CollineationOfProjectiveSpace(el,f); end) ############################################################################# #O ProjectiveElationGroup( <sub>, <centre> ) # returns group of elations with axis sub and centre centre ## InstallMethod( ProjectiveElationGroup, "for a hyperplane and a point of a projective space",M : Concatenation(npackUnderlyingObjec [ IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveSpace ], function(sub,centre) local en,e0,ei,mat,vssub,n,f,c,M,el,ps,gens,fbas,x,group; ps := AmbientSpace(sub); n := Dimension(ps); if not (ps=AmbientSpace(centre)) then Error("The elements <sub> and <centre> do not have a common ambient space"); elif (Dimension(sub) <> n-1) or (Dimension(centre) <> 0) then Error("<sub> must be a hyperplane, <centre> must be a point"); elif not centre in sub then Error("The point <centre> and <sub> must be incident"); fi; mat := UnderlyingObject(sub); f := BaseField(sub); :=VectorSpacef,mat)java.lang.StringIndexOutOfBoundsException: Index 29 out of bo e0 := Unpack(UnderlyingObject(centre)); : BasisVectorsB(omplementSpacevssub,]); en : #companion matrix M := mat := List((basis,t - Coefficients(basis *omega)java.lang.StringIndexOutOfBoundsE el := IdentityMat(n+1,f); gens := []; fbas := BasisVectors(Basis(f)); for x in fbas do el[n+1,1] := x; Add(gens,ShallowCopy(M^(-1)*el*M)); od; #group := SubgroupNC(ProjectivityGroup(ps),List(gens,x->CollineationOfProjectiveSpa group := Group(List(gens,x->CollineationOfProjectiveSpace(x,f))); SetOrder(group,Size(f)); return group; end ); .some arithmetic is not yetpossible cmats #### SingerCycleCollineation <>,q> ) <sub>) # returns group of elations with axis sub ## InstallMethod(ProjectiveElationGroup "for a hyperplane of java.lang.StringIndexOutOfBoundsException: Range [0, 23) out of [ IsSubspaceOfProjectiveSpace ], function(sub) local en,mat,vssub,n,f,c,M,el,ps,gens,fbas,x,group,i; ps := AmbientSpace(sub); n := Dimension(ps); if (Dimension(sub) <> n-1) then Error("<sub> must be a hyperplane"); mat := Unpack(UnderlyingObject(sub)); f := BaseField(sub); vssub := VectorSpace(InstallMethod( IncidenceGraph, #e0 := UnderlyingObject(centre); #ei := BasisVectors(Basis(ComplementSpace(vssub,[e0]))); en := BasisVectors(Basis(ComplementSpace(f^(n+1),mat)))[1]; M := Concatenation(mat,[en]); el := ShallowCopy(IdentityMat(n+1,f)); gens := []; fbas := BasisVectors(Basis(f)); for x in fbas do for i [..n] do el[n+1,i] := x elements : Concatenation(List..Rankps),i ->List(AsList(ElementsOfInciden Add(gens,ShallowCopy(M^(-1)*el*M)); od; ; #group := SubgroupNC(return IsIncident,y)java.lang.StringIndexOutOfBoundsException: group := Group(List(gens,x->CollineationOfProjectiveSpace(x,f))); SetOrder(group,Size(f)^n); return group; end ); ##################### Setter(IncidenceGraphAttr )( ps, graph ); #O HomologyOfProjectiveSpace( <sub>, <centre>, <point1>, <point2> ) # return the uniquely defined homology java.lang.StringIndexOutOfBoundsException: Inde ## InstallMethod( HomologyOfProjectiveSpace, foraandthreepointsof thesprojectivespace, [ IsSubspaceOfProjectiveSpaceOEvaluateForm(<>, <el1>, <el2> ) function(sub,centre,point1,point2) local en,e0,ei,mat,vssub,n,f,c,M,el,p1vect,p2vect,ps; ps := AmbientSpace(sub); n := Dimension(ps); ub),AmbientSpace(centre),mbientSpacepoint1)())= Error"elements sub> centre> point1>, and point2 ave a "; elif Dimension(sub) <> n-1 or Dimension(centre) <> 0 or Dimension(point1) <> 0 or Dimension(point2) <> Error("<sub> must be a hyperplane, <centre>, <point1> and <java.lang.StringIndexOutOfBoundsExce elif centre in sub or point1 in sub or point2 in sub then Error("The points <centre>, <point1> and <point2> must not be incident with <sub>"); elif centre=point1 or centre=point2 then Error("<centre> is fixed and must be different from <point1> and <point2>"); elif Dimension(Span([centre,point1,point2])) <> 1 then Error("<centre>, <point1>, and java.lang.StringIndexOutOfBoundsException: Range [15, 16 mat := UnderlyingObject(sub); f := BaseField(sub); vssub := VectorSpace(f,mat); e0 := UnderlyingObject(Meet(Span(centre,point1),sub)); ei := BasisVectors(Basis(ComplementSpace(vssub,[e0]))); en := UnderlyingObject(centre); M := Concatenation([e0],ei,[en]); el := IdentityMat(n+1,f); java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length l[,1 :(p2vectn+]p1vect[]/pvect[]*1n]); el := M^(-1)*el*M; eturnCollineationOfProjectiveSpaceel,) end ); ## cmat change 20/3/14. some arithmetic is not yet possible with cmats. ###### " quadratic two of a geometry" #O ProjectiveHomologyGroupIsQuadraticFormIsElementOfLieGeometryjava.lang.StringIn # returns group of homologies with axis sub and centre centre ## ProjectiveHomologyGroup "for a hyperplane and a point of a projective space fi; [ IsSubspaceOfProjectiveSpace, IsSubspaceOfProjectiveSpace ], function(sub,centre) local n,M,el,ps,x,group,f,q; ps := AmbientSpace(sub); n := Dimension(ps); not (=AmbientSpace() Error("The elements <sub> and <centre> do not have a common ambient space")#\(<>,< java.lang.String elif (Dimension(sub) <> n-1) or (Dimension(centre) <> 0) then Error("<sub> must be a hyperplane, <centre> must be a point"); elif centre in sub then Error("The point <centre> and <sub> must not be "for an element of a Lie geometry and a quadratic fi; f(l1 form f := BaseField(sub); q := Size(f); el := IdentityMat(n+1,f); el[n+1,n+1] := Z(q); return Group(CollineationOfProjectiveSpace(M^(-1)################################ end ); ############################################################################# #O SingerCycleCollineation( <dim>, <q> ) # returns a matrix which is a Singer cycle, of GF(q)^dim ## InstallMethod( SingerCycleMat, [ IsInt, IsInt], function(n, q) local basis, omega, mat; basis := Basis(AsVectorSpace(GF(q),GF(q^(n+1)))); omega := Z(q^(n+1)); # companion matrix mat := List(BasisVectors(basis), t -> Coefficients(basis, t*omega)); return mat; end); ######################java.lang.StringIndexOutOfBoundsException: Index 77 out of #O SingerCycleCollineation( <dim>, <q> ) # returns a Singer cycle of PG(<dim>,<q>) ## InstallMethod( SingerCycleCollineation, [IsInt, IsInt], function(n, q) rCollineationOfProjectiveSpaceS(nq, GF(); end); ############################################################################# #O IncidenceGraph( <gp> ) # Note that computing the collineation group of a projective space is zero # computation time. So useless to print the warning here if the group is not # yet computed. ### InstallMethod( IncidenceGraph, "for a projective space", [ IsProjectiveSpace ], function( ps ) local elements, graph, adj, coll, sz; if IsBound(ps!.IncidenceGraphAttr) then return ps!.IncidenceGraphAttr; fi; coll := CollineationGroup(ps); elements := Concatenation(List([1..Rank(ps)], i -> List(AsList(ElementsOfIncidenceStru adj := function(x,y) if x!.type <> y!.type then return IsIncident(x,y); else return false; fi; end; graph := Graph(coll,elements,OnProjSubspaces,adj,true); Setter( IncidenceGraphAttr )( ps, graph ); return graph; end ); # Added 18/09/18 jdb. ############################################################################# #O EvaluateForm( <form>, <el1>, <el2> ) # returns the "action" of <form> (sesquilinear) on the subspaces <el1> and <el2>. # All it does is unpack the underlyingobject and call the existing method # for EvaluateForm and matrices/vectors (and counts on checking of input # of the existing method. # InstallMethod( EvaluateForm, "for a sesquilinear form and two elements of a Lie geometry", [IsSesquilinearForm, IsElementOfLieGeometry, IsElementOfLieGeometry], function(form, el1, el2) local list; list := Set([AmbientSpace(el1)!.vectorspace,AmbientSpace(el2)!.vectorspace]); if Size(list) <> 1 or not form!.vectorspace in list then Error("underlying vectorspaces of <form>, <el1> and <el2> do not match"); fi; return EvaluateForm(form,Unpack(UnderlyingObject(el1)),Unpack(UnderlyingObject(el end ); # Added 18/09/18 jdb. ############################################################################# #O EvaluateForm( <form>, <el1> ) # returns the "action" of <form> (quadratic) on the subspace <el1>. # All it does is unpack the underlyingobject and call the existing method # for EvaluateForm and matrices/vectors (and counts on checking of input # of the existing method. # InstallMethod( EvaluateForm, "for a quadratic form and two elements of a Lie geometry", [IsQuadraticForm, IsElementOfLieGeometry ], function(form, el1 ) if not form!.vectorspace = AmbientSpace(el1)!.vectorspace then Error("underlying vectorspaces of <form> and <el1> do not match"); fi; return EvaluateForm(form,Unpack(UnderlyingObject(el1))); end ); # Added 18/09/18 jdb. ############################################################################# #O \^( <el>, <form> ) # returns the "action" of <form> (quadratic) on the subspace <el1>. # All it does is call EvaluateForm # InstallMethod( \^, "for an element of a Lie geometry and a quadratic form", [IsElementOfLieGeometry, IsQuadraticForm ], function(el1, form) return EvaluateForm(form,el1); end ); # Added 18/09/18 jdb. ############################################################################# #O \^( <pair>, <form> ) # returns the "action" of <form> (quadratic) on the subspace <el1>. # All it does is call EvaluateForm # InstallMethod( \^, "for two elements of a Lie geometry and a sesquilinear form", [IsSubspaceOfProjectiveSpaceCollection, IsSesquilinearForm ], function(pair, form) if Size(pair) <> 2 then Error("The first argument must be a pair of subspaces of a projective space"); fi; return EvaluateForm(form,pair[1],pair[2]); end );
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