[GRAPE] 9 Partial Linear Spaces

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# 9 Partial Linear Spaces

### Sections

1. PartialLinearSpaces
2. A research application of PartialLinearSpaces

Let s and t be positive integers. A partial linear space (P,L), with parameters (s,t) consists of a set P of points, together with a set L of (s+1)-subsets of P called lines, such that every point is in exactly t+1 lines, and every pair of distinct points is contained in at most one line. The point graph of a partial linear space S having point-set P is the graph with vertex-set P and having [p,q] an edge if and only if p ¹ q and p,q are in a common line of S. Two partial linear spaces (P,L) and (P¢,L¢) (with parameters (s,t)) are said to be isomorphic if there is a bijection P® P¢ which induces a bijection L® L¢. An automorphism of a partial linear space is an isomorphism onto itself. The set of all automorphisms of a partial linear space S forms a group, called the automorphism group of S.

GRAPE contains a function `PartialLinearSpaces` to determine and classify partial linear spaces with given point graph and parameters. In this chapter we describe this function, and also give a research application of this function.

## 9.1 PartialLinearSpaces

- `PartialLinearSpaces(` ptgraph`,` s`,` t`)`
- `PartialLinearSpaces(` ptgraph`,` s`,` t`,` nspaces`)`
- `PartialLinearSpaces(` ptgraph`,` s`,` t`,` nspaces`,` printlevel`)`
- `PartialLinearSpaces(` ptgraph`,` s`,` t`,` nspaces`,` printlevel`,` cliques`)`

  This function classifies the partial linear spaces with given point graph ptgraph, and parameters (s,t). It calls functions making use of the nauty package within GRAPE, and so can only be used on UNIX systems on which GRAPE has been fully installed.

  The function `PartialLinearSpaces` returns a list of representatives of distinct isomorphism classes of partial linear spaces with (simple) point graph ptgraph, and parameters (s,t). The default is that representatives for all isomorphism classes are returned.

  The integer argument nspaces is optional, and has default value -1, which means that representatives for all isomorphism classes are returned. If nspaces is non-negative then exactly nspaces representatives are returned if there are at least nspaces isomorphism classes, otherwise representatives for all isomorphism classes are returned.

  In the output of this function, a partial linear space S is given by its incidence graph delta. The point-vertices of delta are 1,...,``ptgraph`.order`, with the name of point-vertex i being the name of vertex i of ptgraph. A line-vertex of delta is named by a list (not necessarily ordered) of the point-vertex names for the points on that line. We warn that this is a different naming convention to versions of GRAPE before 4.1. The group ``delta`.group` associated with the incidence graph delta is the automorphism group of S acting on point-vertices and line-vertices, and preserving both sets.

  If printlevel is bound then it controls the print-level (default 0). Permitted values for printlevel are 0,1,2.

  If cliques is bound then it is assumed to be a list (without repeats) of the (s +1)-cliques of ptgraph. If known, this can help the function to run faster.

      gap> K7:=CompleteGraph(SymmetricGroup(7));;
      gap> P:=PartialLinearSpaces(K7,2,2);
      [ rec( isGraph := true, order := 14,
            group := Group([ ( 1, 2)( 5, 6)( 9,11)(10,12),
                ( 1, 2, 3)( 5, 6, 7)( 9,11,13)(10,12,14),
                ( 1, 2, 3)( 4, 7, 6)( 9,12,14)(10,11,13),
                ( 1, 4, 7, 6, 2, 5, 3)( 8, 9,13,10,11,12,14) ]),
            schreierVector := [ -1, 1, 2, 4, 4, 1, 3, -2, 4, 1, 1, 3, 4, 2 ],
            adjacencies := [ [ 8, 9, 10 ], [ 1, 2, 3 ] ],
            representatives := [ 1, 8 ],
            names := [ 1, 2, 3, 4, 5, 6, 7, [ 1, 2, 3 ], [ 1, 4, 5 ], [ 1, 6, 7 ],
                [ 2, 4, 6 ], [ 2, 5, 7 ], [ 3, 4, 7 ], [ 3, 5, 6 ] ],
            isSimple := true ) ]
      gap> Size(P[1].group);
      168
      gap> T:=ComplementGraph(JohnsonGraph(10,2));;
      gap> P:=PartialLinearSpaces(T,4,6);;
      gap> List(P,x->Size(x.group));
      [ 216, 1512 ]

  ## 9.2 A research application of PartialLinearSpaces

  We now provide an extended example of the use of GRAPE which illustrates a research application of the `PartialLinearSpaces` function.

  First we give a definition. Let s and t be positive integers. A partial geometry is a partial linear space with parameters (s,t) for which there is an additional constant constant a > 0, such that, for every line l and every point p not on l, there are exactly a lines through p meeting l in some point.

  Our example shows that the Haemers partial geometry Haemers81 is uniquely determined (up to isomorphism) by its point graph, as is the dual of the Haemers geometry (where the role of points and lines are interchanged), and that each of these geoemetries has automorphism group isomorphic to A7.

  We first construct and study the Hoffman-Singleton graph, using the construction of Peter Cameron contained in Cam99. We then construct the point graph of the Haemers partial geometry Haemers81 (this partial geometry has (s,t)=(4,17) and a = 2). The vertices of this point graph are the edges of the Hoffman-Singleton graph, and two such vertices are adjacent in the point graph precisely when they are at distance 2 in the edge-graph of the Hoffman-Singleton graph (see Haemers81). We then construct and classify (up to isomorphism) all partial linear spaces with parameters (4,17) having point graph isomorphic to that of the Haemers partial geometry. We find that the Haemers partial geometry is the only possibility. It follows from basic theory of partial geometries that the Haemers partial geometry is uniquely determined up to isomorphism (as a partial geometry) by its point graph. We also show that the dual of the Haemers partial geometry is also uniquely determined by its point graph. Thus far, the only proof of these results is by GRAPE. Our example also shows that the Haemers partial geometry and its dual each has automorphism group isomorphic to A7.

  The total runtime (not including calls of nauty) was about 504 CPU-seconds on a Pentium II running at 350 MHz.

      gap> RequirePackage("grape");

      Loading  GRAPE 4.1  (GRaph Algorithms using PErmutation groups),
      by L.H.Soicher@qmul.ac.uk.

      true
      gap>
      gap> OnSetsRecursive:=function(x,g)
      > if not IsList(x) then
      >   return x^g;
      > else
      >   return Set(List(x, y->OnSetsRecursive(y,g)));
      > fi;
      > end;;
      gap>
      gap> HofSingAdjacency := function(x,y)
      > #
      > # This boolean function returns  true  iff  x  and  y  are
      > # adjacent in the Hoffman-Singleton graph, in Peter Cameron's
      > # construction.
      > #
      > if Size(x)=3 then                  # x is a 3-set
      >    if Size(y)=3 then               # y is a 3-set
      >       return Intersection(x,y)=[]; # join iff disjoint
      >    else                            # y is a projective plane
      >       return x in y;               # join iff x is a line of y
      >    fi;
      > else                               # x is a projective plane
      >    if Size(y)=3 then               # y is a 3-set
      >       return y in x;               # join iff y is a line of x
      >    else                            # y is a projective plane
      >       return false;                # don't join
      >    fi;
      > fi;
      > end;;
      gap>
      gap> projectiveplane:=
      >    Set([[1,2,4],[2,3,5],[3,4,6],[4,5,7],[1,5,6],[2,6,7],[1,3,7]]);;
      gap>
      gap> HofSingGraph:=Graph(AlternatingGroup(7),
      >                     [[1,2,3], projectiveplane], OnSetsRecursive,
      >                     HofSingAdjacency);;
      gap> GlobalParameters(HofSingGraph);
      [ [ 0, 0, 7 ], [ 1, 0, 6 ], [ 1, 6, 0 ] ]
      gap> autgrp := AutGroupGraph(HofSingGraph);;
      gap> Size(autgrp);
      252000
      gap> HofSingGraph := NewGroupGraph(autgrp,HofSingGraph);;
      gap> pointgraph:=DistanceGraph( EdgeGraph(HofSingGraph), 2);;
      gap> GlobalParameters(pointgraph);
      [ [ 0, 0, 72 ], [ 1, 20, 51 ], [ 36, 36, 0 ] ]
      gap> spaces:=PartialLinearSpaces(pointgraph,4,17);;
      gap> Length(spaces);
      1
      gap> haemers:=spaces[1];;
      gap> DisplayCompositionSeries(haemers.group);
      G (3 gens, size 2520)
       | A(7)
      1 (0 gens, size 1)
      gap> linegraph:=PointGraph(haemers, Adjacency(haemers,1)[1]);;
      gap> spaces:=PartialLinearSpaces(linegraph,17,4);;
      gap> Length(spaces);
      1
      gap> dualhaemers:=spaces[1];;
      gap> DisplayCompositionSeries(dualhaemers.group);
      G (4 gens, size 2520)
       | A(7)
      1 (0 gens, size 1)
      gap> quit;

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