Add OrbitalGraph for a group, base-pair, and posint

gap/OrbitalGraphs.gd

@@ -77,21 +77,42 @@ DeclareOperation("OrbitalGraphs", [IsPermGroup, IsInt]);
 #DeclareAttribute("OrbitalGraphsRepresentative", IsPermGroup);
 
 
-# TODO: Implement constructing the orbital graph with a given base pair
+# TODO Once orbital graphs can be defined on arbitrary vertices:
+# * Allow specifying a set of points rather than a pos int k.
+# * Implement a 2-arg version with MovedPoints(G) taken as the third arg.
 #! @BeginGroup orbital
 #! @Returns An orbital graph
-#! @Arguments G, u, v
+#! @Arguments G, basepair, k
 #! @Description
-#!   TODO.
-DeclareOperation("OrbitalGraph", [IsPermGroup, IsPosInt, IsPosInt]);
+#!   If <A>G</A> is a permutation group, <A>basepair</A> is a pair of
+#!   positive integers, and `k` is a positive integer such that `[1..<A>k</A>]`
+#!   is preserved by <A>G</A> and contains the entries of <A>basepair</A>,
+#!   then this function returns the orbital graph of <A>G</A> with the given
+#!   <A>basepair</A>, on the vertices `[1..<A>k</A>]`.
+#!
+#!   The resulting orbital graph will have <A>basepair</A> set as its
+#!   <Ref Attr="BasePair" Label="for IsOrbitalGraph"/> attribute.
 #! @EndGroup
 #! @Group orbital
-#! @Arguments G, basepair
 #! @BeginExampleSession
-#! gap> true;
-#! true
+#! gap> D8 := DihedralGroup(IsPermGroup, 8);
+#! Group([ (1,2,3,4), (2,4) ])
+#! gap> OrbitalGraph(D8, [1, 3], 4);
+#! <self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 4 vertices 
+#! with base-pair (1,3), 4 arcs>
+#! gap> OrbitalGraph(D8, [1, 3], 5);
+#! <self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 5 vertices 
+#! with base-pair (1,3), 4 arcs>
+#! gap> G := Group([ (1,2)(3,4) ]);;
+#! gap> OrbitalGraph(G, [1, 2], 2);
+#! <self-paired orbital graph of Group([ (1,2)(3,4) ]) on 2 vertices 
+#! with base-pair (1,2), 2 arcs>
 #! @EndExampleSession
-DeclareOperation("OrbitalGraph", [IsPermGroup, IsHomogeneousList]);
+DeclareOperation("OrbitalGraph", [IsPermGroup, IsHomogeneousList, IsPosInt]);
+#@Arguments G, basepair, points
+#DeclareOperation("OrbitalGraph", [IsPermGroup, IsHomogeneousList, IsHomogeneousList]);
+#@Arguments G, basepair
+#DeclareOperation("OrbitalGraph", [IsPermGroup, IsHomogeneousList]);
 
 
 #! @Section Information stored about orbital graphs at creation

gap/OrbitalGraphs.gi

@@ -114,6 +114,31 @@ function(G, points)
 end);
 
 
+# Individual orbital graphs
+
+InstallMethod(OrbitalGraph,
+"for a permutation group, homogeneous list, and pos int",
+[IsPermGroup, IsHomogeneousList, IsPosInt],
+function(G, basepair, k)
+  local D;
+    if not (Length(basepair) = 2 and ForAll(basepair, IsPosInt)) then
+        ErrorNoReturn("the second argument <basepair> must be a pair of ",
+                      "positive integers");
+    elif basepair[1] > k or basepair[2] > k or
+      ForAny(GeneratorsOfGroup(G), g -> ForAny([1 .. k], i -> i ^ g > k)) then
+        ErrorNoReturn("the third argument <k> must be such that [1..k] ",
+                      "contains the entries of <basepair> and is preserved ",
+                      "by G");
+    fi;
+
+    D := EdgeOrbitsDigraph(G, basepair, k);
+    SetFilterObj(D, IsOrbitalGraphOfGroup);
+    SetUnderlyingGroup(D, G);
+    SetBasePair(D, basepair);
+    return D;
+end);
+
+
 # Transformation semigroups
 
 InstallMethod(OrbitalGraphs, "for a transformation semigroup",

tst/orbitalgraphs01.tst

@@ -18,23 +18,33 @@ gap> OrbitalGraphs(D8);
     , <self-paired orbital graph of D8 on 4 vertices 
     with base-pair (1,2), 8 arcs> ]
 
-# doc/_Chapter_Orbital_graphs.xml:95-98
-gap> true;
-true
+# doc/_Chapter_Orbital_graphs.xml:101-114
+gap> D8 := DihedralGroup(IsPermGroup, 8);
+Group([ (1,2,3,4), (2,4) ])
+gap> OrbitalGraph(D8, [1, 3], 4);
+<self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 4 vertices 
+with base-pair (1,3), 4 arcs>
+gap> OrbitalGraph(D8, [1, 3], 5);
+<self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 5 vertices 
+with base-pair (1,3), 4 arcs>
+gap> G := Group([ (1,2)(3,4) ]);;
+gap> OrbitalGraph(G, [1, 2], 2);
+<self-paired orbital graph of Group([ (1,2)(3,4) ]) on 2 vertices 
+with base-pair (1,2), 2 arcs>
 
-# doc/_Chapter_Orbital_graphs.xml:129-132
+# doc/_Chapter_Orbital_graphs.xml:145-148
 gap> true;
 true
 
-# doc/_Chapter_Orbital_graphs.xml:148-151
+# doc/_Chapter_Orbital_graphs.xml:164-167
 gap> true;
 true
 
-# doc/_Chapter_Orbital_graphs.xml:167-170
+# doc/_Chapter_Orbital_graphs.xml:183-186
 gap> true;
 true
 
-# doc/_Chapter_Orbital_graphs.xml:199-211
+# doc/_Chapter_Orbital_graphs.xml:215-227
 gap> OrbitalClosure(PSL(2,5)) = SymmetricGroup(6);
 true
 gap> C6 := CyclicGroup(IsPermGroup, 6);;
@@ -47,7 +57,7 @@ gap> IsConjugate(SymmetricGroup(6),
 >                closure, WreathProduct(Group([(1,2)]), Group([(1,2,3)])));
 true
 
-# doc/_Chapter_Orbital_graphs.xml:227-236
+# doc/_Chapter_Orbital_graphs.xml:243-252
 gap> OrbitalIndex(PSL(2,5));
 12
 gap> OrbitalIndex(PGL(2,5));
@@ -57,23 +67,23 @@ gap> OrbitalIndex(AlternatingGroup(6));
 gap> OrbitalIndex(DihedralGroup(IsPermGroup, 6));
 1
 
-# doc/_Chapter_Orbital_graphs.xml:263-270
+# doc/_Chapter_Orbital_graphs.xml:279-286
 gap> IsOrbitalGraphRecognisable(QuaternionGroup(IsPermGroup, 8));
 true
 gap> IsOGR(AlternatingGroup(8));
 false
 gap> IsOGR(TrivialGroup(IsPermGroup));
 true
 
-# doc/_Chapter_Orbital_graphs.xml:294-301
+# doc/_Chapter_Orbital_graphs.xml:310-317
 gap> IsStronglyOrbitalGraphRecognisable(CyclicGroup(IsPermGroup, 8));
 true
 gap> IsStronglyOGR(QuaternionGroup(IsPermGroup, 8));
 false
 gap> IsStronglyOGR(TrivialGroup(IsPermGroup));
 true
 
-# doc/_Chapter_Orbital_graphs.xml:325-332
+# doc/_Chapter_Orbital_graphs.xml:341-348
 gap> IsAbsolutelyOrbitalGraphRecognisable(DihedralGroup(IsPermGroup, 8));
 true
 gap> IsAbsolutelyOGR(CyclicGroup(IsPermGroup, 8));

tst/orbitals.tst

@@ -1,4 +1,4 @@
-#@local S50product, orbitals, G, trivial
+#@local S50product, orbitals, G, trivial, x
 gap> START_TEST("OrbitalGraphs package: orbitalgraphs.tst");
 gap> LoadPackage("orbitalgraphs", false);;
 
@@ -74,5 +74,36 @@ gap> OrbitalGraphs(SymmetricGroup(3), [2, 1]);
 Error, the second argument <points> must be fixed setwise by the first argumen\
 t <G>
 
+# OrbitalGraph: Error checking
+gap> G := DihedralGroup(IsPermGroup, 8);;
+gap> OrbitalGraph(G, [0, 1], 4);
+Error, the second argument <basepair> must be a pair of positive integers
+gap> OrbitalGraph(G, [1, 1], 3);
+Error, the third argument <k> must be such that [1..k] contains the entries of\
+ <basepair> and is preserved by G
+gap> x := OrbitalGraph(G, [1, 5], 4);
+Error, the third argument <k> must be such that [1..k] contains the entries of\
+ <basepair> and is preserved by G
+gap> x := OrbitalGraph(G, [1, 1], 4);
+<self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 4 vertices 
+with base-pair (1,1), 4 arcs>
+gap> DigraphEdges(x);
+[ [ 1, 1 ], [ 2, 2 ], [ 3, 3 ], [ 4, 4 ] ]
+gap> x := OrbitalGraph(G, [2, 1], 4);
+<self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 4 vertices 
+with base-pair (2,1), 8 arcs>
+gap> DigraphEdges(x);
+[ [ 1, 2 ], [ 1, 4 ], [ 2, 1 ], [ 2, 3 ], [ 3, 2 ], [ 3, 4 ], [ 4, 1 ], 
+  [ 4, 3 ] ]
+gap> x := OrbitalGraph(G, [3, 1], 5);
+<self-paired orbital graph of Group([ (1,2,3,4), (2,4) ]) on 5 vertices 
+with base-pair (3,1), 4 arcs>
+gap> DigraphEdges(x);
+[ [ 1, 3 ], [ 2, 4 ], [ 3, 1 ], [ 4, 2 ] ]
+gap> OrbitalGraphs(G) =
+>   List(OrbitalGraphs(G),
+>        x -> OrbitalGraph(G, BasePair(x), LargestMovedPoint(G)));
+true
+
 #
 gap> STOP_TEST("OrbitalGraphs package: orbitalgraphs.tst", 0);