The GAP 4 package `UnitalSZ`

Algorithms and library of abstract unitals and their embeddings

Website: https://nagygp.github.io/UnitalSZ/
Repository: https://github.com/nagygp/UnitalSZ

Install

You can download the tar of the package from the homepage GAP Package UnitalSZ and you can install it for GAP the usual way. Check the homepage for the dependencies of the package UnitalSZ.

Example session in GAP

In the following example we define her to be the Hermitian unital of order 8 and bm_unital to be the orthogonal Buekenhout–Metz (BM) unital of order 8 as well. Note the “the”: for other qs there may be more non-isomorphic BM-unitals.

gap> LoadPackage( "UnitalSZ" );
gap> q := 2^3;;
gap> her := HermitianAbstractUnital( q );;
gap> KnownAttributesOfObject( her );
[ "Name", "Order" ]
gap> params := AllBuekenhoutMetzAbstractUnitalParameters( q );;
gap> Length( params );
1 
gap> bm_unital := OrthogonalBuekenhoutMetzAbstractUnital( q, params[1][1], params[1][2] );
OrthogonalBuekenhoutMetzAbstractUnital(8,Z(2^6)^62,Z(2^6)^60)
gap> KnownAttributesOfObject( bm_unital );
[ "Name", "Order", "PointNamesOfUnital" ]

Then we compute the automorphism group of bm_unital and the orbit lengths of this group. We also check whether the Hermitian unital her and the BM-unital bm_unital are isomorphic or not: the result fail indicates that they are not isomorphic (of course).

gap> ag := AutomorphismGroup( bm_unital );;
gap> OrbitLengths( ag );
[ 512 ]
gap> Isomorphism( her, bm_unital );
fail

As we have seen, there is some information stored about the unital objects her and bm_unital: we can access these with the command KnownAttributesOfObject. After computing the full points of bm_unital we obtain much more information about the unital. With SetInfoLevel we get some additional messages about the ongoing computation of the full points.

gap> SetInfoLevel( InfoUnitalSZ, 2);
gap> FullPointsOfUnitalRepresentatives( bm_unital );;
#I  1896 block pair(s) up to automorphisms computed
#I  1 full point(s) found
#I  2 full point(s) found
#I  3 full point(s) found
#I  4 full point(s) found
#I  5 full point(s) found
#I  6 full point(s) found
#I  7 full point(s) found
gap> KnownAttributesOfObject( bm_unital );
[ "Name", "Order", "AutomorphismGroup", "PointsOfUnital", "BlocksOfUnital",
"PointNamesOfUnital", "IncidenceDigraph", "FullPointsOfUnitalRepresentatives" ]

On unitals

The concept of unitals arises from unitary polarities over finite projective planes: the absolute points of such a polarity form a unital. The combinatorial properties of a unital obtained as above lead us to the definition of an abstract unital.

In design theory context we call a 2-(q³+1, q+1, 1) design an abstract unital. It means that we have q³+1 points and subsets with q+1 elements of the points called blocks and for every pair of points there is exactly one block containing these 2 points.

Currently there are 4 “named” unitals available in the package:

the Hermitian unital: HermitianAbstractUnital( q );
the orthogonal Buekenhout–Metz unital: OrthogonalBuekenhoutMetzAbstractUnital( q, alpha, beta );
the Buekenhout–Tits unital: BuekenhoutTitsAbstractUnital( q ) and
the cyclic unital due to Bagchi and Bagchi: BagchiBagchiCyclicUnital( n ).

Of course there are options for constructing your own unital:

by the list of its blocks: AbstractUnitalByDesignBlocks( blocklist );
by its incidence matrix (blocks × points): AbstractUnitalByIncidenceMatrix( incmat );
by a boolean matrix based on its incidence matrix: AbstractUnitalByBlistList( bmat ).

There is also access to different attributes of a unital:

points: PointsOfUnital( u );
names of the points: PointNamesOfUnital( u ). If there is any. It comes handy when the (abstract) unital can be represented in a more structured way (e.g., Buekenhout–Metz unital).
Blocks: BlocksOfUnital( u );
incidence (di)graph: IncidenceDigraph( u );
automorphism group: AutomorphismGroup( u ).

And as in the example above, one can check isomorphism between two unitals u1 and u2 with Isomorphism( u1, u2 ).

Available libraries of unitals

Apart from the unitals mentioned above, we’ve made 3 libraries of unitals available : they are results of different papers (see the section References in the documentation for more details). A useful command can be the following, which prints some essential informations about these libraries:

gap> DisplayUnitalLibraryInfo();

Full points and perspectivity groups

For the definition of a full point see the documentation. They can be seen as tools for obtaining non-embeddability results. The following command computes the full points of the first unital in the KNP library (up to automorphisms of the unital):

gap> FullPointsOfUnitalRepresentatives( KNPAbstractUnital(1) );
[ rec( block1 := [ 1, 2, 3, 4, 5 ], block2 := [ 6, 10, 27, 35, 41 ], fullpts :=
[ 39 ] ), rec( block1 := [ 1, 2, 3, 4, 5 ], block2 := [ 6, 36, 52, 58, 63 ],
fullpts := [ 9, 34, 50, 59, 64 ] ) ]

The group of perspectivities from one block to another is closely related to full points, and it serves as tool for testing embeddability. PerspectivityGroupOfUnitalsBlocks is the command for computing this group. For the exact specification of its arguments please refer to the documentation.

Acknowledgement

Support provided from the National Research, Development and Innovation Fund of Hungary, financed under the 2018-1.2.1-NKP funding scheme, within the SETIT Project (2018-1.2.1-NKP-2018-00004).
Partially supported by NKFIH-OTKA Grants 114614, 119687 and 115288.
D. Mezőfi has been supported by the UNKP-17-3 New National Excellence Program of the Ministry of Human Capacities.

The GAP 4 package UnitalSZ