add ConformalSymplecticGroup and some generalizations

init.g

@@ -14,3 +14,4 @@
 
 ReadPackage("forms","lib/forms.gd");
 ReadPackage("forms","lib/recognition.gd");
+ReadPackage("forms","lib/conformal.gd");

lib/conformal.gd

@@ -0,0 +1,152 @@
+#############################################################################
+##
+#A  InvariantBilinearFormUpToScalars( <matgrp> )
+##
+##  <#GAPDoc Label="InvariantBilinearFormUpToScalars">
+##  <ManSection>
+##  <Attr Name="InvariantBilinearFormUpToScalars" Arg='matgrp'/>
+##
+##  <Description>
+##  This attribute describes a bilinear form that is invariant up to scalars
+##  under the matrix group <A>matgrp</A>.
+##  The form is given by a record with the component <C>matrix</C>
+##  which is a matrix <M>J</M> such that for every generator <M>g</M> of
+##  <A>matgrp</A> the equation <M>g \cdot J \cdot g^{tr} = \lambda(g) J</M>
+##  holds, for <M>\lambda(g)</M> in the
+##  <Ref Attr="FieldOfMatrixGroup" BookName="ref"/> value of <A>matgrp</A>.
+##  </Description>
+##  </ManSection>
+##  <#/GAPDoc>
+##
+DeclareAttribute( "InvariantBilinearFormUpToScalars", IsMatrixGroup );
+
+
+#############################################################################
+##
+#P  IsFullSubgroupGLRespectingBilinearFormUpToScalars( <matgrp> )
+##
+##  <#GAPDoc Label="IsFullSubgroupGLRespectingBilinearFormUpToScalars">
+##  <ManSection>
+##  <Prop Name="IsFullSubgroupGLRespectingBilinearFormUpToScalars"
+##  Arg='matgrp'/>
+##
+##  <Description>
+##  This property tests whether the matrix group <A>matgrp</A> is the full
+##  subgroup of GL respecting, up to scalars, the form stored as the value of
+##  <Ref Attr="InvariantBilinearFormUpToScalars"/> for <A>matgrp</A>.
+##  </Description>
+##  </ManSection>
+##  <#/GAPDoc>
+##
+DeclareProperty( "IsFullSubgroupGLRespectingBilinearFormUpToScalars",
+    IsMatrixGroup );
+
+InstallTrueMethod( IsGroup,
+    IsFullSubgroupGLRespectingBilinearFormUpToScalars );
+
+
+#############################################################################
+##
+#O  ConformalSymplecticGroupCons( <filter>, <form> )
+#O  ConformalSymplecticGroupCons( <filter>, <matrix> )
+#O  ConformalSymplecticGroupCons( <filter>, <G> )
+#O  ConformalSymplecticGroupCons( <filter>, <d>, <R> )
+#O  ConformalSymplecticGroupCons( <filter>, <d>, <R>, <form> )
+#O  ConformalSymplecticGroupCons( <filter>, <d>, <R>, <matrix> )
+#O  ConformalSymplecticGroupCons( <filter>, <d>, <R>, <G> )
+#O  ConformalSymplecticGroupCons( <filter>, <d>, <q> )
+#O  ConformalSymplecticGroupCons( <filter>, <d>, <q>, <form> )
+#O  ConformalSymplecticGroupCons( <filter>, <d>, <q>, <matrix> )
+#O  ConformalSymplecticGroupCons( <filter>, <d>, <q>, <G> )
+##
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsBilinearForm ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsMatrixOrMatrixObj ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsGroup ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsRing ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsRing, IsBilinearForm ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsRing, IsMatrixOrMatrixObj ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsRing, IsGroup ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsPosInt ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsPosInt, IsBilinearForm ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsPosInt, IsMatrixOrMatrixObj ] );
+DeclareConstructor( "ConformalSymplecticGroupCons", [ IsGroup, IsPosInt, IsPosInt, IsGroup ] );
+
+
+#############################################################################
+##
+#F  ConformalSymplecticGroup( [<filt>, ]<d>, <q>[, <form>] ) conf. sympl. gp.
+#F  ConformalSymplecticGroup( [<filt>, ]<d>, <R>[, <form>] ) conf. sympl. gp.
+#F  ConformalSymplecticGroup( [<filt>, ]<form> )   conformal symplectic group
+#F  CSp( [<filt>, ]<d>, <q>[, <form>] )            conformal symplectic group
+#F  CSp( [<filt>, ]<d>, <R>[, <form>] )            conformal symplectic group
+#F  CSp( [<filt>, ]<form> )                        conformal symplectic group
+##
+##  <#GAPDoc Label="ConformalSymplecticGroup">
+##  <ManSection>
+##  <Heading>ConformalSymplecticGroup</Heading>
+##  <Func Name="ConformalSymplecticGroup" Arg='[filt, ]d, R[, form]'
+##   Label="for dimension and a ring"/>
+##  <Func Name="ConformalSymplecticGroup" Arg='[filt, ]d, q[, form]'
+##   Label="for dimension and field size"/>
+##  <Func Name="ConformalSymplecticGroup" Arg='[filt, ]form'
+##   Label="for form"/>
+##  <Func Name="CSp" Arg='[filt, ]d, R[, form]'
+##   Label="for dimension and a ring"/>
+##  <Func Name="CSp" Arg='[filt, ]d, q[, form]'
+##   Label="for dimension and field size"/>
+##  <Func Name="CSp" Arg='[filt, ]form'
+##   Label="for form"/>
+##
+##  <Description>
+##  constructs a group isomorphic to the conformal symplectic group
+##  CSp( <A>d</A>, <A>R</A> ) of those <M><A>d</A> \times <A>d</A></M>
+##  matrices over the ring <A>R</A> or the field with <A>q</A> elements,
+##  respectively,
+##  that respect a fixed nondegenerate symplectic form up to scalars,
+##  in the category given by the filter <A>filt</A>.
+##  <P/>
+##  Currently only finite fields <A>R</A> are supported.
+##  <P/>
+##  If <A>filt</A> is not given it defaults to
+##  <Ref Filt="IsMatrixGroup" BookName="ref"/>,
+##  and the returned group is the conformal symplectic group itself.
+##  Another supported value for <A>filt</A> is
+##  <Ref Filt="IsPermGroup" BookName="ref"/>;
+##  in this case, the argument <A>form</A> is not supported.
+##  <P/>
+##  If the arguments describe a matrix group over a finite field then
+##  the desired bilinear form can be specified via <A>form</A>,
+##  which can be either a matrix
+##  or a form object in <Ref Filt="IsBilinearForm"/>
+##  or a group with stored
+##  <Ref Attr="InvariantBilinearForm" BookName="ref"/> or
+##  <Ref Attr="InvariantBilinearFormUpToScalars"/> value
+##  (and then this form is taken).
+##  <P/>
+##  A given <A>form</A> determines and <A>d</A>, and also <A>R</A>
+##  except if <A>form</A> is a matrix that does not store its
+##  <Ref Attr="BaseDomain" Label="for a matrix object" BookName="ref"/> value.
+##  These parameters can be entered, and an error is signalled if they do
+##  not fit to the given <A>form</A>.
+##  <P/>
+##  If <A>form</A> is not given then a default is chosen as described in the
+##  introduction to Section <Ref Sect="Classical Groups" BookName="ref"/>.
+##  <P/>
+##  <Example><![CDATA[
+##  gap> g:= ConformalSymplecticGroup( 4, 2 );
+##  CSp(4,2)
+##  gap> Size( g );
+##  720
+##  gap> StructureDescription( g );
+##  "S6"
+##  gap> ConformalSymplecticGroup( IsPermGroup, 4, 2 );
+##  Perm_CSp(4,2)
+##  ]]></Example>
+##  </Description>
+##  </ManSection>
+##  <#/GAPDoc>
+##
+DeclareGlobalFunction( "ConformalSymplecticGroup" );
+
+DeclareSynonym( "CSp", ConformalSymplecticGroup );
+