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ThomasBreuerThomasBreuer
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add ConformalSymplecticGroup and some generalizations
Add `ConformalSymplecticGroup`, essentially analogous to the implementation of `SymplecticGroup`: - introduce a plain function `ConformalSymplecticGroup` and a constructor `ConformalSymplecticGroupCons`, - install various methods for `ConformalSymplecticGroupCons`, - introduce a property `IsFullSubgroupGLRespectingBilinearFormUpToScalars` that is set in the groups returned by `ConformalSymplecticGroup`, - install a `\in` method for a matrix and a group in `IsFullSubgroupGLRespectingBilinearFormUpToScalars`, - add tests for the new code. The proposed implementation differs from that of `SymplecticGroup` and is intended as a pattern for changing the implementation of `SymplecticGroup` and the other classical groups, and for adding the other types of conformal groups. These are the differences: - Do *not* delegate from the variants where a finite field is given to the variants with given size of the field, but the other way round. This way, it will be possible to use elements from `StandardFiniteField( p, n )` instead of `GF( p, n )` in the matrices. - We might recommend to choose an `IsMatrixObj` representation for the matrices if the entries are from `StandardFiniteField( p, n )`. For that (and also in general), an "experimental" global option `ConstructingFilter` is supported that prescribes the internal format of the matrices. Currently one can use for example `IsPlistRepMatrix` as the value for this option, such that one can make experiments with matrix groups whose elements are in `IsMatrixObj`. More useful `IsMatrixObj` representations will hopefully become available. (Note: We can view `IsPlistRep` as the default for the `ConstructingFilter` option, but then the matrices need not be in `IsPlistRep` because the `ImmutableMatrix` calls in the code convert the generators to `IsGF2MatrixRep` or `Is8BitMatrixRep` if possible.) Once we decide about the setup that we want for `ConformalSymplecticGroup`, the implementation of `SymplecticGroup` etc. in the GAP library and the relevant methods in the Forms package can be rewritten accordingly.
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‎init.g‎

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ReadPackage("forms","lib/forms.gd");
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ReadPackage("forms","lib/recognition.gd");
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ReadPackage("forms","lib/conformal.gd");

‎lib/conformal.gd‎

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#############################################################################
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##
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#A InvariantBilinearFormUpToScalars( <matgrp> )
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##
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## <#GAPDoc Label="InvariantBilinearFormUpToScalars">
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## <ManSection>
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## <Attr Name="InvariantBilinearFormUpToScalars" Arg='matgrp'/>
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##
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## <Description>
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## This attribute describes a bilinear form that is invariant up to scalars
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## under the matrix group <A>matgrp</A>.
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## The form is given by a record with the component <C>matrix</C>
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## which is a matrix <M>J</M> such that for every generator <M>g</M> of
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## <A>matgrp</A> the equation <M>g \cdot J \cdot g^{tr} = \lambda(g) J</M>
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## holds, for <M>\lambda(g)</M> in the
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## <Ref Attr="FieldOfMatrixGroup" BookName="ref"/> value of <A>matgrp</A>.
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## </Description>
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## </ManSection>
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## <#/GAPDoc>
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##
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DeclareAttribute( "InvariantBilinearFormUpToScalars", IsMatrixGroup );
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#############################################################################
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##
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#P IsFullSubgroupGLRespectingBilinearFormUpToScalars( <matgrp> )
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##
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## <#GAPDoc Label="IsFullSubgroupGLRespectingBilinearFormUpToScalars">
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## <ManSection>
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## <Prop Name="IsFullSubgroupGLRespectingBilinearFormUpToScalars"
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## Arg='matgrp'/>
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##
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## <Description>
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## This property tests whether the matrix group <A>matgrp</A> is the full
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## subgroup of GL respecting, up to scalars, the form stored as the value of
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## <Ref Attr="InvariantBilinearFormUpToScalars"/> for <A>matgrp</A>.
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## </Description>
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## </ManSection>
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## <#/GAPDoc>
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##
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DeclareProperty( "IsFullSubgroupGLRespectingBilinearFormUpToScalars",
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IsMatrixGroup );
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InstallTrueMethod( IsGroup,
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IsFullSubgroupGLRespectingBilinearFormUpToScalars );
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#############################################################################
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##
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#O ConformalSymplecticGroupCons( <filter>, <form> )
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#O ConformalSymplecticGroupCons( <filter>, <matrix> )
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#O ConformalSymplecticGroupCons( <filter>, <G> )
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#O ConformalSymplecticGroupCons( <filter>, <d>, <R> )
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#O ConformalSymplecticGroupCons( <filter>, <d>, <R>, <form> )
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#O ConformalSymplecticGroupCons( <filter>, <d>, <R>, <matrix> )
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#O ConformalSymplecticGroupCons( <filter>, <d>, <R>, <G> )
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#O ConformalSymplecticGroupCons( <filter>, <d>, <q> )
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#O ConformalSymplecticGroupCons( <filter>, <d>, <q>, <form> )
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#O ConformalSymplecticGroupCons( <filter>, <d>, <q>, <matrix> )
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#O ConformalSymplecticGroupCons( <filter>, <d>, <q>, <G> )
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##
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsBilinearForm ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsMatrixOrMatrixObj ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsGroup ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsRing ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsRing, IsBilinearForm ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsRing, IsMatrixOrMatrixObj ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsRing, IsGroup ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsPosInt ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsPosInt, IsBilinearForm ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsPosInt, IsMatrixOrMatrixObj ] );
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DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsPosInt, IsGroup ] );
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#############################################################################
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##
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#F ConformalSymplecticGroup( [<filt>, ]<d>, <q>[, <form>] ) conf. sympl. gp.
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#F ConformalSymplecticGroup( [<filt>, ]<d>, <R>[, <form>] ) conf. sympl. gp.
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#F ConformalSymplecticGroup( [<filt>, ]<form> ) conformal symplectic group
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#F CSp( [<filt>, ]<d>, <q>[, <form>] ) conformal symplectic group
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#F CSp( [<filt>, ]<d>, <R>[, <form>] ) conformal symplectic group
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#F CSp( [<filt>, ]<form> ) conformal symplectic group
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##
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## <#GAPDoc Label="ConformalSymplecticGroup">
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## <ManSection>
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## <Heading>ConformalSymplecticGroup</Heading>
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## <Func Name="ConformalSymplecticGroup" Arg='[filt, ]d, R[, form]'
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## Label="for dimension and a ring"/>
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## <Func Name="ConformalSymplecticGroup" Arg='[filt, ]d, q[, form]'
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## Label="for dimension and field size"/>
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## <Func Name="ConformalSymplecticGroup" Arg='[filt, ]form'
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## Label="for form"/>
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## <Func Name="CSp" Arg='[filt, ]d, R[, form]'
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## Label="for dimension and a ring"/>
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## <Func Name="CSp" Arg='[filt, ]d, q[, form]'
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## Label="for dimension and field size"/>
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## <Func Name="CSp" Arg='[filt, ]form'
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## Label="for form"/>
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##
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## <Description>
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## constructs a group isomorphic to the conformal symplectic group
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## CSp( <A>d</A>, <A>R</A> ) of those <M><A>d</A> \times <A>d</A></M>
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## matrices over the ring <A>R</A> or the field with <A>q</A> elements,
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## respectively,
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## that respect a fixed nondegenerate symplectic form up to scalars,
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## in the category given by the filter <A>filt</A>.
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## <P/>
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## Currently only finite fields <A>R</A> are supported.
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## <P/>
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## If <A>filt</A> is not given it defaults to
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## <Ref Filt="IsMatrixGroup" BookName="ref"/>,
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## and the returned group is the conformal symplectic group itself.
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## Another supported value for <A>filt</A> is
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## <Ref Filt="IsPermGroup" BookName="ref"/>;
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## in this case, the argument <A>form</A> is not supported.
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## <P/>
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## If the arguments describe a matrix group over a finite field then
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## the desired bilinear form can be specified via <A>form</A>,
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## which can be either a matrix
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## or a form object in <Ref Filt="IsBilinearForm"/>
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## or a group with stored
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## <Ref Attr="InvariantBilinearForm" BookName="ref"/> or
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## <Ref Attr="InvariantBilinearFormUpToScalars"/> value
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## (and then this form is taken).
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## <P/>
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## A given <A>form</A> determines and <A>d</A>, and also <A>R</A>
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## except if <A>form</A> is a matrix that does not store its
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## <Ref Attr="BaseDomain" Label="for a matrix object" BookName="ref"/> value.
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## These parameters can be entered, and an error is signalled if they do
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## not fit to the given <A>form</A>.
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## <P/>
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## If <A>form</A> is not given then a default is chosen as described in the
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## introduction to Section <Ref Sect="Classical Groups" BookName="ref"/>.
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## <P/>
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## <Example><![CDATA[
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## gap> g:= ConformalSymplecticGroup( 4, 2 );
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## CSp(4,2)
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## gap> Size( g );
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## 720
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## gap> StructureDescription( g );
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## "S6"
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## gap> ConformalSymplecticGroup( IsPermGroup, 4, 2 );
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## Perm_CSp(4,2)
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## ]]></Example>
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## </Description>
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## </ManSection>
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## <#/GAPDoc>
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##
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DeclareGlobalFunction( "ConformalSymplecticGroup" );
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DeclareSynonym( "CSp", ConformalSymplecticGroup );
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