Add ConformalSymplecticGroup

doc/manualbib.xml

@@ -4901,6 +4901,15 @@
   <url>https://arxiv.org/abs/2201.09155</url>
 </article></entry>
 
+<entry id="Tay21"><misc>
+  <author>
+    <name><first>D. E.</first><last>Taylor</last></name>
+  </author>
+  <title>Conjugacy Classes in Finite Conformal Symplectic Groups</title>
+  <year>2021</year>
+  <howpublished>https://www.maths.usyd.edu.au/u/don/code/Magma/CSpConjugacy.pdf</howpublished>
+</misc></entry>
+
 <entry id="Theissen93"><mastersthesis>
   <author>
     <name><first>Heiko</first><last>Theißen</last></name>

doc/ref/grpmat.xml

@@ -91,6 +91,8 @@ that represent a faithful action on vectors.
 
 <#Include Label="InvariantBilinearForm">
 <#Include Label="IsFullSubgroupGLorSLRespectingBilinearForm">
+<#Include Label="InvariantBilinearFormUpToScalars">
+<#Include Label="IsFullSubgroupGLRespectingBilinearFormUpToScalars">
 <#Include Label="InvariantSesquilinearForm">
 <#Include Label="IsFullSubgroupGLorSLRespectingSesquilinearForm">
 <#Include Label="InvariantQuadraticForm">

doc/ref/makedocreldata.g

@@ -13,6 +13,7 @@ GAPInfo.ManualDataRef:= rec(
     "../../src/sysfiles.c",
     "../../grp/basic.gd",
     "../../grp/classic.gd",
+    "../../grp/conformal.gd",
     "../../grp/perf.gd",
     "../../grp/ree.gd",
     "../../grp/suzuki.gd",

grp/basicmat.gi

@@ -139,23 +139,29 @@ InstallMethod( GeneralLinearGroupCons,
       IsInt and IsPosRat,
       IsField and IsFinite ],
 function( filter, n, f )
-    local   q,  z,  o,  mat1,  mat2,  i,  g;
+    local   q,  filt,  z,  o,  mat1,  mat2,  i,  g;
 
     q:= Size( f );
 
+    # Decide about the internal representation of group generators.
+    filt:= ValueOption( "ConstructingFilter" );
+    if filt = fail then
+      filt:= IsPlistRep;
+    fi;
+
     # small cases
     if q = 2 and 1 < n  then
 #T why 1 < n?
-        return SL( n, 2 );
+        return SL( n, f );
     fi;
 
     # construct the generators
     z := PrimitiveRoot( f );
     o := One( f );
 
-    mat1 := IdentityMat( n, f );
+    mat1 := IdentityMatrix( filt, f, n );
     mat1[1,1] := z;
-    mat2 := List( Zero(o) * mat1, ShallowCopy );
+    mat2 := ZeroMatrix( filt, f, n, n );
     mat2[1,1] := -o;
     mat2[1,n] := o;
     for i  in [ 2 .. n ]  do mat2[i,i-1]:= -o;  od;
@@ -189,22 +195,28 @@ InstallMethod( SpecialLinearGroupCons,
       IsField and IsFinite ],
 
 function( filter, n, f )
-    local   q,  g,  o,  z,  mat1,  mat2,  i,  size,  qi;
+    local   q,  filt,  g,  o,  z,  mat1,  mat2,  i,  size,  qi;
 
     q:= Size( f );
 
+    # Decide about the internal representation of group generators.
+    filt:= ValueOption( "ConstructingFilter" );
+    if filt = fail then
+      filt:= IsPlistRep;
+    fi;
+
     # handle the trivial case first
     if n = 1 then
-        g := GroupByGenerators( [ ImmutableMatrix( f, [[One(f)]],true ) ] );
+        g := GroupByGenerators( [ ImmutableMatrix( f, IdentityMatrix( filt, f, 1 ), true ) ] );
 
     # now the general case
     else
 
         # construct the generators
         o := One(f);
         z := PrimitiveRoot(f);
-        mat1 := IdentityMat( n, f );
-        mat2 := List( Zero(o) * mat1, ShallowCopy );
+        mat1 := IdentityMatrix( filt, f, n );
+        mat2 := ZeroMatrix( filt, f, n, n );
         mat2[1,n] := o;
         for i  in [ 2 .. n ]  do mat2[i,i-1]:= -o;  od;
 

grp/classic.gd

@@ -18,20 +18,28 @@
 ##  <#GAPDoc Label="[1]{classic}">
 ##  The following functions return classical groups.
 ##  <P/>
+##  <E>Generators</E>
+##  <P/>
+##  <List>
+##  <Item>
 ##  For the linear, symplectic, and unitary groups (the latter in dimension
 ##  at least <M>3</M>),
 ##  the generators are taken from&nbsp;<Cite Key="Tay87"/>.
+##  </Item>
+##  <Item>
 ##  For the unitary groups in dimension <M>2</M>, the isomorphism of
 ##  SU<M>(2,q)</M> and SL<M>(2,q)</M> is used,
 ##  see for example&nbsp;<Cite Key="Hup67"/>.
-##  <P/>
+##  </Item>
+##  <Item>
 ##  The generators of the general and special orthogonal groups are taken
 ##  from&nbsp;<Cite Key="IshibashiEarnest94"/> and
 ##  <Cite Key="KleidmanLiebeck90"/>,
 ##  except that the generators of the groups in odd dimension in even
 ##  characteristic are constructed via the isomorphism to a symplectic group,
 ##  see for example&nbsp;<Cite Key="Car72a"/>.
-##  <P/>
+##  </Item>
+##  <Item>
 ##  The generators of the groups <M>\Omega^\epsilon(d, q)</M> are taken
 ##  from&nbsp;<Cite Key="RylandsTalor98"/>,
 ##  except that in odd dimension and even characteristic,
@@ -45,23 +53,45 @@
 ##  except in the case <M>(d,q) = (5,2)</M>,
 ##  where the matrices from&nbsp;<Cite Key="RylandsTalor98"/> generate
 ##  the simple group <M>S_4(2)'</M> and not the group <M>S_4(2)</M>.
-##  <P/>
+##  </Item>
+##  <Item>
 ##  The generators for the semilinear groups are constructed from the
 ##  generators of the corresponding linear groups plus one additional
 ##  generator that describes the action of the group of field automorphisms;
 ##  for prime integers <M>p</M> and positive integers <M>f</M>,
 ##  this yields the matrix groups <M>Gamma</M>L<M>(d, p^f)</M> and
 ##  <M>Sigma</M>L<M>(d, p^f)</M> as groups of <M>d f \times df</M> matrices
 ##  over the field with <M>p</M> elements.
+##  </Item>
+##  <Item>
+##  The generators for the conformal symplectic groups are taken
+##  from&nbsp;<Cite Key="Tay21"/>.
+##  </Item>
+##  </List>
+##  <P/>
+##  <E>Invariant forms</E>
 ##  <P/>
+##  <List>
+##  <Item>
 ##  For symplectic and orthogonal matrix groups returned by the functions
 ##  described below, the invariant bilinear form is stored as the value of
 ##  the attribute <Ref Attr="InvariantBilinearForm"/>.
-##  Analogously, the invariant sesquilinear form defining the unitary groups
+##  </Item>
+##  <Item>
+##  The invariant sesquilinear form defining the unitary groups
 ##  is stored as the value of the attribute
 ##  <Ref Attr="InvariantSesquilinearForm"/>).
+##  </Item>
+##  <Item>
 ##  The defining quadratic form of orthogonal groups is stored as the value
 ##  of the attribute <Ref Attr="InvariantQuadraticForm"/>.
+##  </Item>
+##  <Item>
+##  The bilinear form that is invariant up to scalars under the
+##  conformal symplectic group is stored as the value of the attribute
+##  <Ref Attr="InvariantBilinearFormUpToScalars"/>.
+##  </Item>
+##  </List>
 ##  <P/>
 ##  Note that due to the different sources for the generators,
 ##  the invariant forms for the groups <M>\Omega(e,d,q)</M> are in general
@@ -707,45 +737,51 @@ DeclareConstructor( "SymplecticGroupCons", [ IsGroup, IsPosInt, IsRing ] );
 
 #############################################################################
 ##
+#F  SymplecticGroup( [<filt>, ]<d>, <R>[, <form>] ) . . . .  symplectic group
 #F  SymplecticGroup( [<filt>, ]<d>, <q>[, <form>] ) . . . .  symplectic group
 #F  SymplecticGroup( [<filt>, ]<form> ) . . . . . . . . . .  symplectic group
+#F  Sp( [<filt>, ]<d>, <R>[, <form>] )
 #F  Sp( [<filt>, ]<d>, <q>[, <form>] )
 #F  Sp( [<filt>, ]<form> )
+#F  SP( [<filt>, ]<d>, <R>[, <form>] )
 #F  SP( [<filt>, ]<d>, <q>[, <form>] )
 #F  SP( [<filt>, ]<form> )
 ##
 ##  <#GAPDoc Label="SymplecticGroup">
 ##  <ManSection>
 ##  <Heading>SymplecticGroup</Heading>
+##  <Func Name="SymplecticGroup" Arg='[filt, ]d, R[, form]'
+##   Label="for dimension and a ring"/>
 ##  <Func Name="SymplecticGroup" Arg='[filt, ]d, q[, form]'
 ##   Label="for dimension and field size"/>
-##  <Func Name="SymplecticGroup" Arg='[filt, ]d, ring[, form]'
-##   Label="for dimension and a ring"/>
 ##  <Func Name="SymplecticGroup" Arg='[filt, ]form'
 ##   Label="for form"/>
+##  <Func Name="Sp" Arg='[filt, ]d, R[, form]'
+##   Label="for dimension and a ring"/>
 ##  <Func Name="Sp" Arg='[filt, ]d, q[, form]'
 ##   Label="for dimension and field size"/>
-##  <Func Name="Sp" Arg='[filt, ]d, ring[, form]'
-##   Label="for dimension and a ring"/>
 ##  <Func Name="Sp" Arg='[filt, ]form'
 ##   Label="for form"/>
+##  <Func Name="SP" Arg='[filt, ]d, R[, form]'
+##   Label="for dimension and a ring"/>
 ##  <Func Name="SP" Arg='[filt, ]d, q[, form]'
 ##   Label="for dimension and field size"/>
-##  <Func Name="SP" Arg='[filt, ]d, ring[, form]'
-##   Label="for dimension and a ring"/>
 ##  <Func Name="SP" Arg='[filt, ]form'
 ##   Label="for form"/>
 ##
 ##  <Description>
 ##  constructs a group isomorphic to the symplectic group
-##  Sp( <A>d</A>, <A>q</A> ) of those <M><A>d</A> \times <A>d</A></M>
-##  matrices over the field with <A>q</A> elements (respectively the ring
-##  <A>ring</A>)
+##  Sp( <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,
 ##  in the category given by the filter <A>filt</A>.
 ##  <P/>
 ##  If <A>filt</A> is not given it defaults to <Ref Filt="IsMatrixGroup"/>,
 ##  and the returned group is the symplectic group itself.
+##  Another supported value for <A>filt</A> is
+##  <Ref Filt="IsPermGroup"/>;
+##  in this case, the argument <A>form</A> is not supported.
 ##  <P/>
 ##  At the moment finite fields or residue class rings
 ##  <C>Integers mod <A>q</A></C>, with <A>q</A> an odd prime power, are
@@ -759,7 +795,7 @@ DeclareConstructor( "SymplecticGroupCons", [ IsGroup, IsPosInt, IsRing ] );
 ##  or a group with stored <Ref Attr="InvariantBilinearForm"/> value
 ##  (and then this form is taken).
 ##  <P/>
-##  A given <A>form</A> determines and <A>d</A>, and also <A>q</A>
+##  A given <A>form</A> determines <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"/> value.
 ##  These parameters can be entered, and an error is signalled if they do