Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma


products/Sources/formale Sprachen/GAP/pkg/fining/doc/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 27.6.2023 mit Größe 172 kB

chap12_mj.html

Interaktion und
PortierbarkeitHTML

products/Sources/formale Sprachen/GAP/pkg/fining/doc/chap12_mj.html

<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<script type="text/javascript"
 src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (FinInG) - Chapter 12: Generalised Polygons</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap12" onload="jscontent()">

<div class="chlinktop">Goto Chapter: <a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chap7_mj.html">7</a> <a href="chap8_mj.html">8</a> <a href="chap9_mj.html">9</a> <a href="chap10_mj.html">10</a> <a href="chap11_mj.html">11</a> <a href="chap12_mj.html">12</a> <a href="chap13_mj.html">13</a> <a href="chap14_mj.html">14</a> <a href="chapA_mj.html">A</a> <a href="chapB_mj.html">B</a> <a href="chapC_mj.html">C</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap11_mj.html">[Previous Chapter]</a> <a href="chap13_mj.html">[Next Chapter]</a> </div>

<a href="chap12.html">[MathJax off]</a>
<a id="X7E1F10767D2A4D6A" name="X7E1F10767D2A4D6A"></a>
<div class="ChapSects"><a href="chap12_mj.html#X7E1F10767D2A4D6A">12 Generalised Polygons</a>
<div class="ContSect"> <a href="chap12_mj.html#X7CC6903E78F24167">12.1 Categories</a>

<div class="ContSSBlock">
 <a href="chap12_mj.html#X8141DFFF7B352E2D">12.1-1 IsGeneralisedPolygon</a>
 <a href="chap12_mj.html#X832E75AE7CCC5BB2">12.1-2 Subcategories in <code class="code">IsGeneralisedPolygon</code></a>

 <a href="chap12_mj.html#X86C124B3804C7569">12.1-3 IsWeakGeneralisedPolygon</a>
 <a href="chap12_mj.html#X8540C04887CF8824">12.1-4 Subcategories in <code class="code">IsProjectivePlaneCategory</code></a>

 <a href="chap12_mj.html#X7CF10DAE7847939D">12.1-5 Subcategories in <code class="code">IsGeneralisedQuadrangle</code></a>

 <a href="chap12_mj.html#X7A2ABB0B818E25C5">12.1-6 IsClassicalGeneralisedHexagon</a>
</div></div>
<div class="ContSect"> <a href="chap12_mj.html#X8614D6A779F9B1AA">12.2 Generic functions to create generalised polygons</a>

<div class="ContSSBlock">
 <a href="chap12_mj.html#X849A83597CC78B5F">12.2-1 GeneralisedPolygonByBlocks</a>
 <a href="chap12_mj.html#X8751B0FB8500D0B1">12.2-2 GeneralisedPolygonByIncidenceMatrix</a>
 <a href="chap12_mj.html#X7A6169127CE8ABC0">12.2-3 GeneralisedPolygonByElements</a>
</div></div>
<div class="ContSect"> <a href="chap12_mj.html#X864C966D8184A9C0">12.3 Attributes and operations for generalised polygons</a>

<div class="ContSSBlock">
 <a href="chap12_mj.html#X84F59A2687C62763">12.3-1 Order</a>
 <a href="chap12_mj.html#X836C15B57968A511">12.3-2 IncidenceGraphAttr</a>
 <a href="chap12_mj.html#X815BE6D57D623452">12.3-3 IncidenceGraph</a>
 <a href="chap12_mj.html#X81F531BB7DEA127D">12.3-4 IncidenceMatrixOfGeneralisedPolygon</a>
 <a href="chap12_mj.html#X83FF6FA0790D5747">12.3-5 CollineationGroup</a>
 <a href="chap12_mj.html#X8074EDB381D97984">12.3-6 CollineationAction</a>
 <a href="chap12_mj.html#X7D75521986B958DA">12.3-7 BlockDesignOfGeneralisedPolygon</a>
</div></div>
<div class="ContSect"> <a href="chap12_mj.html#X7A13D5EB82E01576">12.4 Elements of generalised polygons</a>

<div class="ContSSBlock">
 <a href="chap12_mj.html#X7E7607CA7D59D086">12.4-1 Collections of elements of generalised polygons</a>

 <a href="chap12_mj.html#X858ADA3B7A684421">12.4-2 Size</a>
 <a href="chap12_mj.html#X7E9B2A217DBF2849">12.4-3 Creating elements from objects and retrieving objects from elements</a>

 <a href="chap12_mj.html#X83B0FA9E7AE3DF01">12.4-4 Incidence</a>

 <a href="chap12_mj.html#X875BE2957FAF6209">12.4-5 Span</a>
 <a href="chap12_mj.html#X8469B54180FE1E4C">12.4-6 Meet</a>
 <a href="chap12_mj.html#X8154BB13844AA0FD">12.4-7 Shadow elements</a>

 <a href="chap12_mj.html#X7807682E86D3BE10">12.4-8 DistanceBetweenElements</a>
</div></div>
<div class="ContSect"> <a href="chap12_mj.html#X7934EB788049B533">12.5 The classical generalised hexagons</a>

<div class="ContSSBlock">
 <a href="chap12_mj.html#X7BF1D2E57B7630CB">12.5-1 Trialities of the hyperbolic quadric and generalised hexagons</a>

 <a href="chap12_mj.html#X7C35384980AA9B77">12.5-2 IsLieGeometry</a>
 <a href="chap12_mj.html#X7A05FF0079F55291">12.5-3 SplitCayleyHexagon</a>
 <a href="chap12_mj.html#X7E17977384011587">12.5-4 TwistedTrialityHexagon</a>
 <a href="chap12_mj.html#X807A60A28264BD93">12.5-5 vector spaceToElement</a>
 <a href="chap12_mj.html#X7809B7C183FA7213">12.5-6 ObjectToElement</a>
 <a href="chap12_mj.html#X810D4D6D87069697">12.5-7 UnderlyingObject</a>
 <a href="chap12_mj.html#X87BDB89B7AAFE8AD"><code>12.5-8 \in</code></a>
 <a href="chap12_mj.html#X7B1380878358938C">12.5-9 Span and meet of elements</a>

 <a href="chap12_mj.html#X83FF6FA0790D5747">12.5-10 CollineationGroup</a>
</div></div>
<div class="ContSect"> <a href="chap12_mj.html#X7BA462527B2777BC">12.6 Elation generalised quadrangles</a>

<div class="ContSSBlock">
 <a href="chap12_mj.html#X86BD86C77BAAF887">12.6-1 Elation generalised quadrangles and Kantor families</a>

 <a href="chap12_mj.html#X7CC6903E78F24167">12.6-2 Categories</a>

 <a href="chap12_mj.html#X820A2D6A84A259FC">12.6-3 Kantor families</a>

 <a href="chap12_mj.html#X7B80E3AC7DEAF948">12.6-4 EGQByKantorFamily</a>
 <a href="chap12_mj.html#X80C93974807A342B">12.6-5 Representation of elements and underlying objects</a>

 <a href="chap12_mj.html#X7DCD7EAB839BD97F">12.6-6 Elation group and natural action on elements</a>

 <a href="chap12_mj.html#X8462F11584736E32">12.6-7 Kantor families, q-clans, and elation generalised quadrangles</a>

 <a href="chap12_mj.html#X806527387F1D5D42">12.6-8 qClan</a>
 <a href="chap12_mj.html#X858A1EA8843BEC13">12.6-9 Particular q-clans</a>

 <a href="chap12_mj.html#X7C59CF8079EA33D3">12.6-10 KantorFamilyByqClan</a>
 <a href="chap12_mj.html#X830222B084BF866D">12.6-11 EGQByqClan</a>
 <a href="chap12_mj.html#X7FAE48497B2F658A">12.6-12 BLT-sets, flocks, q-clans, and elation generalised quadrangles</a>

 <a href="chap12_mj.html#X7A8438537D2F1374">12.6-13 IsEGQByBLTSet</a>
 <a href="chap12_mj.html#X7ECC1871866AC286">12.6-14 BLTSetByqClan</a>
 <a href="chap12_mj.html#X84700DDF80B39332">12.6-15 EGQByBLTSet</a>
 <a href="chap12_mj.html#X79186E29799355BD">12.6-16 DefiningPlanesOfEGQByBLTSet</a>
 <a href="chap12_mj.html#X80C93974807A342B">12.6-17 Representation of elements and underlying objects</a>

 <a href="chap12_mj.html#X7C57B0BD7A2E5877">12.6-18 CollineationSubgroup</a>
</div></div>
</div>

<h3>12 Generalised Polygons</h3>

A generalised n-gon is a point/line geometry whose incidence graph is bipartite of diameter <var class="Arg">n</var> and girth <var class="Arg">2n</var>. Although these rank 2 structures are very much a subdomain of GRAPE and Design, their significance in finite geometry warrants their inclusion in FinInG. By the famous theorem of Feit and Higman, a generalised n-gon which has at least three points on every line, must have \(n\) in \(\{2,3,4,6,8\}\). The case \(n=2\) concerns the complete multipartite graphs, which we disregard. The more interesting cases are accordingly projective planes (\(n=3\)), generalised quadrangles (\(n=4\)), generalised hexagons (\(n=6\)), and generalised octagons (\(n=8\)).

FinInG provides some basic functionality to deal with generalised polygons as incidence geometries. A lot of non-trivial interaction with the package GRAPE has been very useful and even necessary. Currently, generic functions to create generalised polygons, to create elements of generalised polygons, and to explore the elements are implemented. This generic functionality allows the user to construct generalised polygons through many different objects available in GAP and FinInG. Apart from these generic functions, some particular generalised polygons are available: the classical generalised hexagons and elation generalised quadrangles from different perspectives can be constructed.

<a id="X7CC6903E78F24167" name="X7CC6903E78F24167"></a>

<h4>12.1 Categories</h4>

<a id="X8141DFFF7B352E2D" name="X8141DFFF7B352E2D"></a>

<h5>12.1-1 IsGeneralisedPolygon</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralisedPolygon</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralisedPolygonRep</code></td><td class="tdright">( representation )</td></tr></table></div>
This category is a subcategory of <code class="code">IsIncidenceGeometry</code>, and contains all generalised polygons. Generalised polygons constructed through functions described in this chapter, all belong to <code class="code">IsGeneralisedPolygonRep</code>.

<a id="X832E75AE7CCC5BB2" name="X832E75AE7CCC5BB2"></a>

<h5>12.1-2 Subcategories in <code class="code">IsGeneralisedPolygon</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsProjectivePlaneCategory</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralisedQuadrangle</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralisedHexagon</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralisedOctagon</code></td><td class="tdright">( category )</td></tr></table></div>
All generalised polygons in FinInG belong to one of these four categories. It is not possible to construct generalised polygons of which the gonality is not known (or checked). Note that the classical generalised quadrangles (which are the classical polar spaces of rank 2) belong also to <code class="code">IsGeneralisedQuadrangle</code> and that the desarguesian projective planes (which are the projective spaces of dimension 2) also belong to <code class="code">IsProjectivePlaneCategory</code>, but both do not belong to <code class="code">IsGeneralisedPolygonRep</code>.

<a id="X86C124B3804C7569" name="X86C124B3804C7569"></a>

<h5>12.1-3 IsWeakGeneralisedPolygon</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWeakGeneralisedPolygon</code></td><td class="tdright">( category )</td></tr></table></div>
<code class="code">IsWeakGeneralisedPolygon</code> is the category for weak generalised polygons.

<a id="X8540C04887CF8824" name="X8540C04887CF8824"></a>

<h5>12.1-4 Subcategories in <code class="code">IsProjectivePlaneCategory</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDesarguesianPlane</code></td><td class="tdright">( category )</td></tr></table></div>
<code class="code">IsDesarguesianPlane</code> is declared as a subcategory of <code class="code">IsProjectivePlaneCategory</code> and <code class="code">IsProjecticeSpace</code>. Projective spaces of dimension 2 constructed using <code class="file">ProjectiveSpace</code> belong to <code class="code">IsDesarguesianPlane</code>.

<a id="X7CF10DAE7847939D" name="X7CF10DAE7847939D"></a>

<h5>12.1-5 Subcategories in <code class="code">IsGeneralisedQuadrangle</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsClassicalGQ</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsElationGQ</code></td><td class="tdright">( category )</td></tr></table></div>
<code class="code">IsClassicalGQ</code> is declared as a subcategory of <code class="code">IsGeneralisedQuadrangle</code> and <code class="code">IsClassicalPolarSpace</code>. All classical polar spaces of rank 2 belong to <code class="code">IsClassicalGQ</code>. <code class="code">IsElationGQ</code> is declared as subcategory of <code class="code">IsGeneralisedQuadrangle</code>. Elation GQs will be discussed in detail in Section <a href="chap12_mj.html#X7BA462527B2777BC">12.6</a>

<div class="example"><pre>
gap> gp := SymplecticSpace(3,2);
W(3, 2)
gap> IsGeneralisedPolygon(gp);
true
gap> IsGeneralisedQuadrangle(gp);
true
gap> IsClassicalGQ(gp);
true
gap> IsGeneralisedPolygonRep(gp);
false

</pre></div>

<a id="X7A2ABB0B818E25C5" name="X7A2ABB0B818E25C5"></a>

<h5>12.1-6 IsClassicalGeneralisedHexagon</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsClassicalGeneralisedHexagon</code></td><td class="tdright">( category )</td></tr></table></div>
<code class="code">IsClassicalGeneralisedHexagon</code> is declared as subcategory of <code class="code">IsGeneralisedHexagon</code> and <code class="code">IsLieGeometry</code>. The so called classical generalised hexagons are the hexagons that come from the triality of the hyperbolic quadric \(\mathrm{Q}^+(7,q)\). The Split Cayley hexagon is embedded in the parabolic quadric \(\mathrm{Q}(6,q)\), or \(\mathrm{W}(5,q)\) in even characteristic. The twisted triality hexagon is embedded in the hyperbolic quadric \(\mathrm{Q}^+(7,q)\). The construction of these hexagons in a subcategory of <code class="code">IsLieGeometry</code> means that the usual operations for Lie geometries become applicable. The classical generalised hexagons are in detail discussed in Section <a href="chap12_mj.html#X7934EB788049B533">12.5</a>

<div class="example"><pre>
gap> gp := SplitCayleyHexagon(3);
H(3)
gap> IsGeneralisedHexagon(gp);
true
gap> IsClassicalGeneralisedHexagon(gp);
true
gap> IsLieGeometry(gp);
true
gap> IsGeneralisedPolygonRep(gp);
true

</pre></div>

<a id="X8614D6A779F9B1AA" name="X8614D6A779F9B1AA"></a>

<h4>12.2 Generic functions to create generalised polygons</h4>

<a id="X849A83597CC78B5F" name="X849A83597CC78B5F"></a>

<h5>12.2-1 GeneralisedPolygonByBlocks</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralisedPolygonByBlocks</code>( <var class="Arg">l</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a generalised polygon

The argument <var class="Arg">l</var> is a finite homogeneous list consisting of ordered sets of a common size \(n+1\). This operation will assume that each element of <var class="Arg">l</var> represents a line of the generalised polygon. Its points are assumed to be the union of all elements of <var class="Arg">l</var>. The incidence is assumed to be symmetrised containment. From this information, an incidence graph is computed using GRAPE. If this graph has diameter \(d\) and girth \(2d\), a generalised polygon is returned. The thickness condition is not checked. If \(d \in \{3,4,6,8\}\), a projective plane, a generalised quadrangle, a generalised hexagon, a generalised octagon respectively, is returned. Note that for large input, this operation can be time consuming.

<div class="example"><pre>
gap> blocks := [ 
> [ 1, 2, 3, 4, 5 ], [ 1, 6, 7, 8, 9 ], [ 1, 10, 11, 12, 13 ],
> [ 1, 14, 15, 16, 17 ], [ 1, 18, 19, 20, 21 ], [ 2, 6, 10, 14, 18 ], 
> [ 2, 7, 11, 15, 19 ], [ 2, 8, 12, 16, 20 ], [ 2, 9, 13, 17, 21 ], 
> [ 3, 6, 11, 16, 21 ], [ 3, 7, 10, 17, 20 ], [ 3, 8, 13, 14, 19 ], 
> [ 3, 9, 12, 15, 18 ], [ 4, 6, 12, 17, 19 ], [ 4, 7, 13, 16, 18 ], 
> [ 4, 8, 10, 15, 21 ], [ 4, 9, 11, 14, 20 ], [ 5, 6, 13, 15, 20 ], 
> [ 5, 7, 12, 14, 21 ], [ 5, 8, 11, 17, 18 ], [ 5, 9, 10, 16, 19 ] ];;
gap> gp := GeneralisedPolygonByBlocks( blocks );
<projective plane order 4>

</pre></div>

<a id="X8751B0FB8500D0B1" name="X8751B0FB8500D0B1"></a>

<h5>12.2-2 GeneralisedPolygonByIncidenceMatrix</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralisedPolygonByIncidenceMatrix</code>( <var class="Arg">incmat</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a generalised polygon

The argument \(incmat\) is a matrix representing the incidence matrix of a point line geometry. The points are represented by the columns, the rows represent the lines. From <var class="Arg">incmat</var> a homogeneous list of sets of column entries is derived, which is then passed to <code class="file">GeneralisedPolygonByBlocks</code>. When <var class="Arg">incmat</var> indeed represents a generalised polygon, it is returned. The checks are performed by <code class="file">GeneralisedPolygonByBlocks</code>.

<div class="example"><pre>
gap> incmat := [ 
> [ 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], 
> [ 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], 
> [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], 
> [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0 ], 
> [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1 ], 
> [ 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0 ], 
> [ 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0 ], 
> [ 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0 ], 
> [ 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 ], 
> [ 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 ], 
> [ 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0 ], 
> [ 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0 ], 
> [ 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0 ], 
> [ 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0 ], 
> [ 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0 ], 
> [ 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 ], 
> [ 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0 ], 
> [ 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0 ], 
> [ 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1 ], 
> [ 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0 ], 
> [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0 ] ];;
gap> pp := GeneralisedPolygonByIncidenceMatrix( incmat );
<projective plane order 4>

</pre></div>

<a id="X7A6169127CE8ABC0" name="X7A6169127CE8ABC0"></a>

<h5>12.2-3 GeneralisedPolygonByElements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralisedPolygonByElements</code>( <var class="Arg">pts</var>, <var class="Arg">lns</var>, <var class="Arg">inc</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneralisedPolygonByElements</code>( <var class="Arg">pts</var>, <var class="Arg">lns</var>, <var class="Arg">inc</var>, <var class="Arg">grp</var>, <var class="Arg">act</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a generalised polygon

The argument <var class="Arg">pts</var>, <var class="Arg">lns</var> and <var class="Arg">inc</var> are respectively a set of objects, a set of objects and a function. The function <var class="Arg">inc</var> must represent an incidence relation between objects of <var class="Arg">pts</var> and <var class="Arg">lns</var>. The first version of <code class="file">GeneralisedPolygonByElements</code> will construct an incidence graph, and if this graph has diameter \(d\) and girth \(2d\), a generalised polygon is returned. The thickness condition is not checked. If \(d \in \{3,4,6,8\}\), a projective plane, a generalised quadrangle, a generalised hexagon, a generalised octagon respectively, is returned. The argument <var class="Arg">grp</var> is a group, and <var class="Arg">act</var> a function, representing an action of the elements of <var class="Arg">grp</var> on the objects in the lists <var class="Arg">pts</var> and <var class="Arg">lns</var>, preserving the incidence. The second version of <code class="file">GeneralisedPolygonByElements</code> acts as the first version, but uses <var class="Arg">grp</var> and <var class="Arg">act</var> to construct the incidence graph in a more efficient way, so if <var class="Arg">grp</var> is a non trivial group, the construction of the graph will be faster. This operation can typically be used to construct generalised polygons from objects that are available in FinInG. This difference in time is shown in the first two examples. The third examples shows the construction of the generalised quadrangle \(T_2(O)\).

<div class="example"><pre>
gap> pg := PG(2,25);
ProjectiveSpace(2, 25)
gap> pts := Set(Points(pg));;
gap> lns := Set(Lines(pg));;
gap> inc := \*;
<Operation "*">
gap> gp := GeneralisedPolygonByElements(pts,lns,inc);
<projective plane order 25>
gap> time;
26427
gap> grp := CollineationGroup(pg);
The FinInG collineation group PGammaL(3,25)
gap> act := OnProjSubspaces;
function( var, el ) ... end
gap> gp := GeneralisedPolygonByElements(pts,lns,inc,grp,act);
<projective plane order 25>
gap> time;
127
gap> q := 4;
4
gap> conic := Set(Points(ParabolicQuadric(2,q)));
[ <a point in Q(2, 4)>, <a point in Q(2, 4)>, <a point in Q(2, 4)>,
 <a point in Q(2, 4)>, <a point in Q(2, 4)> ]
gap> pg := PG(3,q);
ProjectiveSpace(3, 4)
gap> hyp := HyperplaneByDualCoordinates(pg,[1,0,0,0]*Z(q)^0);
<a plane in ProjectiveSpace(3, 4)>
gap> em := NaturalEmbeddingBySubspace(PG(2,q),pg,hyp);
<geometry morphism from <All elements of ProjectiveSpace(2,
4)> to <All elements of ProjectiveSpace(3, 4)>>
gap> O := List(conic,x->x^em);;
gap> group := CollineationGroup(pg);
The FinInG collineation group PGammaL(4,4)
gap> stab := FiningSetwiseStabiliser(group,O);
#I Computing adjusted stabilizer chain...
<projective collineation group with 6 generators>
gap> points1 := Set(Filtered(Points(pg),x->not x in hyp));;
gap> tangents := List(conic,x->TangentSpace(x)^em);
[ <a line in ProjectiveSpace(3, 4)>, <a line in ProjectiveSpace(3, 4)>,
 <a line in ProjectiveSpace(3, 4)>, <a line in ProjectiveSpace(3, 4)>,
 <a line in ProjectiveSpace(3, 4)> ]
gap> planes := List(tangents,x->Filtered(Planes(x),y->not y in hyp));;
gap> points2 := Union(planes);;
gap> points3 := [hyp];
[ <a plane in ProjectiveSpace(3, 4)> ]
gap> linesa := Union(List(O,x->Filtered(Lines(x),y->not y in hyp)));;
gap> linesb := Set(O);;
gap> pts := Union(points1,points2,points3);;
gap> lns := Union(linesa,linesb);;
gap> inc := \*;
<Operation "*">
gap> gp := GeneralisedPolygonByElements(pts,lns,inc,stab,\^);
<generalised quadrangle of order [ 4, 4 ]>
gap> time;
50

</pre></div>

<a id="X864C966D8184A9C0" name="X864C966D8184A9C0"></a>

<h4>12.3 Attributes and operations for generalised polygons</h4>

All operations described in this section are applicable on objects in the category <code class="code">IsGeneralisedPolygon</code>.

<a id="X84F59A2687C62763" name="X84F59A2687C62763"></a>

<h5>12.3-1 Order</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Order</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a pair of positive integers

This method returns the parameters \((s,t)\) of the generalised polygon <var class="Arg">gp</var>. That is, \(s+1\) is the number of points on any line of <var class="Arg">gp</var>, and \(t+1\) is the number of lines incident with any point of <var class="Arg">gp</var>.

<div class="example"><pre>
gap> gp := TwistedTrialityHexagon(2^3);
T(8, 2)
gap> Order(gp);
[ 8, 2 ]
gap> gp := HermitianPolarSpace(4,25);
H(4, 5^2)
gap> Order(gp);
[ 25, 125 ]
gap> gp := EGQByqClan(LinearqClan(3));
#I Computed Kantor family. Now computing EGQ...
<EGQ of order [ 9, 3 ] and basepoint 0>
gap> Order(gp);
[ 9, 3 ]

</pre></div>

<a id="X836C15B57968A511" name="X836C15B57968A511"></a>

<h5>12.3-2 IncidenceGraphAttr</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IncidenceGraphAttr</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
This attribute is declared for objects in <code class="code">IsGeneralisedPolygon</code>. It is a mutable attribute and can be accessed by the operation <code class="file">IncidenceGraph</code>.

<a id="X815BE6D57D623452" name="X815BE6D57D623452"></a>

<h5>12.3-3 IncidenceGraph</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IncidenceGraph</code>( <var class="Arg">gp</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a graph

The argument <var class="Arg">gp</var> is a generalised polygon. This operation returns the incidence graph of <var class="Arg">gp</var>. If <var class="Arg">gp</var> is constructed using <code class="file">GeneralisedPolygonByBlocks</code>, <code class="file">GeneralisedPolygonByElements</code> or <code class="file">GeneralisedPolygonByIncidenceMatrix</code>, an incidence graph is computed to check the input, and is stored as an attribute. For the particular generalised polygons available in FinInG, there is no precomputed incidence graph. Note that computing an incidence graph may require some time, especially when the <var class="Arg">gp</var> has no collineation group computed. Therefore, this operation will return an error when <var class="Arg">gp</var> has no collineation group computed. As <code class="file">CollineationGroup</code> is an attribute for objects in <code class="code">IsGeneralisedPolygon</code>, the user should compute the collineation group and then reissue the command to compute the incidence graph.

We should also point out that this method returns a mutable attribute of <var class="Arg">gp</var>, so that acquired information about the incidence graph can be added. For example, the automorphism group of the incidence graph may be computed and stored as a record component after the incidence graph is stored as an attribute of <var class="Arg">gp</var>. Normally, attributes of GAP objects are immutable.

Note that the factor 2 as difference in the order of the collineation group of \(Q(4,4)\) and the order of the automorphism group of its incidence graph is easily explained by the fact that the \(Q(4,4)\) is self dual.

<div class="example"><pre>
gap> blocks := [ 
> [ 1, 2, 3, 4, 5 ], [ 1, 6, 7, 8, 9 ], [ 1, 10, 11, 12, 13 ],
> [ 1, 14, 15, 16, 17 ], [ 1, 18, 19, 20, 21 ], [ 2, 6, 10, 14, 18 ], 
> [ 2, 7, 11, 15, 19 ], [ 2, 8, 12, 16, 20 ], [ 2, 9, 13, 17, 21 ], 
> [ 3, 6, 11, 16, 21 ], [ 3, 7, 10, 17, 20 ], [ 3, 8, 13, 14, 19 ], 
> [ 3, 9, 12, 15, 18 ], [ 4, 6, 12, 17, 19 ], [ 4, 7, 13, 16, 18 ], 
> [ 4, 8, 10, 15, 21 ], [ 4, 9, 11, 14, 20 ], [ 5, 6, 13, 15, 20 ], 
> [ 5, 7, 12, 14, 21 ], [ 5, 8, 11, 17, 18 ], [ 5, 9, 10, 16, 19 ] ];;
gap> gp := GeneralisedPolygonByBlocks( blocks );
<projective plane order 4>
gap> incgraph := IncidenceGraph( gp );;
gap> Diameter( incgraph );
3
gap> Girth( incgraph );
6
gap> VertexDegrees( incgraph );
[ 5 ]
gap> aut := AutGroupGraph( incgraph );
<permutation group with 9 generators>
gap> DisplayCompositionSeries(aut);
G (9 gens, size 241920)
| Z(2)
S (5 gens, size 120960)
| Z(2)
S (5 gens, size 60480)
| Z(3)
S (4 gens, size 20160)
| A(2,4) = L(3,4)
1 (0 gens, size 1)
gap> gp := ParabolicQuadric(4,4);
Q(4, 4)
gap> incgraph := IncidenceGraph( gp );;
Error, No collineation group computed. Please compute collineation group before compu
ting incidence graph,n called from
<function "unknown">( <arguments> )
called from read-eval loop at line 24 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> CollineationGroup(gp);
PGammaO(5,4)
gap> Order(last);
1958400
gap> incgraph := IncidenceGraph( gp );;
#I Computing incidence graph of generalised polygon...
gap> aut := AutGroupGraph( incgraph );
<permutation group with 10 generators>
gap> Order(aut);
3916800

</pre></div>

<a id="X81F531BB7DEA127D" name="X81F531BB7DEA127D"></a>

<h5>12.3-4 IncidenceMatrixOfGeneralisedPolygon</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IncidenceMatrixOfGeneralisedPolygon</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a matrix

This method returns the incidence matrix of the generalised polygon via the operation <code class="code">CollapsedAdjacencyMat</code> in the GRAPE package. The rows of the matrix correspond to the points of <var class="Arg">gp</var>, and the columns correspond to the lines. Note that since this operation relies on <code class="file">IncidenceGraph</code>, for some generalised polygons, it is necessary to compute a collineation group first.

<div class="example"><pre>
gap> gp := SymplecticSpace(3,2);
W(3, 2)
gap> CollineationGroup(gp);
PGammaSp(4,2)
gap> mat := IncidenceMatrixOfGeneralisedPolygon(gp);
#I Computing incidence graph of generalised polygon...
[ [ 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
 [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
 [ 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
 [ 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
 [ 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0 ],
 [ 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0 ],
 [ 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
 [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0 ],
 [ 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
 [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0 ],
 [ 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0 ],
 [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1 ],
 [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1 ],
 [ 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0 ],
 [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1 ] ]

</pre></div>

<a id="X83FF6FA0790D5747" name="X83FF6FA0790D5747"></a>

<h5>12.3-5 CollineationGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CollineationGroup</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a group

This attribute returns the full collineation group of the generalised polygon <var class="Arg">gp</var>. For some particular generalised polygons, a (subgroup) of the full collineation group can be computed efficiently without computing the incidence graph of <var class="Arg">gp</var>: the full collineation group of classical generalised quadrangles and classical generalised hexagons; and an elation group with relation to a base-point of an elation generalised quadrangle. For generalised polygons constructed by the operations <code class="file">GeneralisedPolygonByBlocks</code>, <code class="file">GeneralisedPolygonByElements</code> or <code class="file">GeneralisedPolygonByIncidenceMatrix</code>, the full collineation group is computed using the full automorphism group of the underlying incidence graph, the latter being computed by the package GRAPE.

The collineation groups computed for classical generalised quadrangles and classical generalised hexagons are collineation groups in the sense of FinInG, and come equipped with a NiceMonomorphism. The collineation groups computed in all other cases are permutations groups, acting on the vertices of the underlying incidence graph.

Note that the computation of the automorphism group of the underlying graph can be time consuming, also if the complete collineation group of the generalised polygon has been used as an argument in e.g. <code class="file">GeneralisedPolygonByElements</code>.

The first example illustrates that <code class="file">CollineationGroup</code> is naturally applicable to all classical generalised Polygons.

<div class="example"><pre>
gap> gp := PG(2,2);
ProjectiveSpace(2, 2)
gap> CollineationGroup(gp);
The FinInG collineation group PGL(3,2)
gap> gp := EllipticQuadric(5,4);
Q-(5, 4)
gap> CollineationGroup(gp);
PGammaO-(6,4)
gap> gp := TwistedTrialityHexagon(3^3);
T(27, 3)
gap> CollineationGroup(gp);
#I Computing nice monomorphism...
#I Found permutation domain...
3D_4(27)
gap> time;
40691

</pre></div>

The second example illustrates the computation of collineation groups of generalised polygons constructed using different objects.

<div class="example"><pre>
gap> mat := [ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ],
> [ 1, 0, 1, 0, 0, 0, 1 ], [ 0, 1, 1, 1, 0, 0, 0 ],
> [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],
> [ 0, 0, 1, 0, 1, 1, 0 ] ];
[ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ], [ 1, 0, 1, 0, 0, 0, 1 ],
 [ 0, 1, 1, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],
 [ 0, 0, 1, 0, 1, 1, 0 ] ]
gap> gp := GeneralisedPolygonByIncidenceMatrix(mat);
<projective plane order 2>
gap> group := CollineationGroup(gp);
Group([ (3,4)(5,7)(9,10)(13,14), (3,7)(4,5)(11,12)(13,14), (2,3)(6,7)(8,9)
(12,13), (2,6)(4,5)(11,13)(12,14), (1,2)(4,7)(9,11)(10,12) ])
gap> gp := EGQByqClan(FisherqClan(3));
#I Computed Kantor family. Now computing EGQ...
<EGQ of order [ 9, 3 ] and basepoint 0>
gap> group := CollineationGroup(gp);
#I Computing incidence graph of generalised polygon...
#I Using elation of the collineation group...
<permutation group of size 26127360 with 8 generators>
gap> Order(group);
26127360
gap> Random(group);
(1,75,27,191,96,50,9,110,88,53,63,154,115,213,229,19,236,226,49,143,16,266,58,
245,11,270,57,44)(2,181,116,225,262,223,17)(3,33,187,149,108,120,177,164,167,
261,198,26,196,276,52,73,94,222,101,176,32,39,43,89,31,280,65,71)(4,250,173,
112,246,38,142,138,54,208,69,243,197,42,269,242,125,8,134,265,67,206,20,13,29,
182,205,36)(5,109,129,82,210,277,185,56,104,114,90,68,61,228,132,235,78,257,
10,238,145,184,241,170,153,263,45,179)(6,159,230,106,147,91,22,137,256,113,
117,180,7,133,279,100,55,156,168,86,122,131,12,35,273,264,254,152)(14,62,66,
268,51,233,253,218,172,130,144,25,169,83,234,127,171,221,34,190,21,46,272,224,
239,267,60,98)(15,40,278,128,160,215,87,178,203,166,247,119,209,84,255,271,
232,81,193,252,92,95,111,201,107,140,135,258)( [...] )
gap> q := 4;
4
gap> conic := ParabolicQuadric(2,q);
Q(2, 4)
gap> nucleus := NucleusOfParabolicQuadric(conic);
<a point in ProjectiveSpace(2, 4)>
gap> conic := ParabolicQuadric(2,q);
Q(2, 4)
gap> nucleus := NucleusOfParabolicQuadric(conic);
<a point in ProjectiveSpace(2, 4)>
gap> hyperoval := Union(List(Points(conic)),[nucleus]);
[ <a point in ProjectiveSpace(2, 4)>, <a point in Q(2, 4)>,
 <a point in Q(2, 4)>, <a point in Q(2, 4)>, <a point in Q(2, 4)>,
 <a point in Q(2, 4)> ]
gap> pg := PG(3,q);
ProjectiveSpace(3, 4)
gap> hyp := HyperplaneByDualCoordinates(pg,[1,0,0,0]*Z(q)^0);
<a plane in ProjectiveSpace(3, 4)>
gap> em := NaturalEmbeddingBySubspace(PG(2,q),pg,hyp);
<geometry morphism from <All elements of ProjectiveSpace(2,
4)> to <All elements of ProjectiveSpace(3, 4)>>
gap> O := List(hyperoval,x->x^em);
[ <a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)>,
 <a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)>,
 <a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)> ]
gap> points := Set(Filtered(Points(pg),x->not x in hyp));;
gap> lines := Union(List(O,x->Filtered(Lines(x),y->not y in hyp)));;
gap> inc := \*;
<Operation "*">
gap> gp := GeneralisedPolygonByElements(points,lines,inc);
<generalised quadrangle of order [ 3, 5 ]>
gap> coll := CollineationGroup(gp);
<permutation group of size 138240 with 8 generators>
gap> Order(coll);
138240
gap> Random(coll);
(1,29,60,40)(2,42,4,10,3,61,59,19,57,51,58,8)(5,21,17,25,52,13,64,48,44,36,9,
56)(6,34,41,55,50,45,63,27,20,14,11,24)(7,53,18,46,12,35,62,16,43,23,49,
26)(15,32,47,31,28,39,54,37,22,38,33,30)(65,74,83,111,66,117,149,104,70,151,
142,78)(67,135,139,136,68,109,98,125,69,95,120,137)(71,92,73,128,77,106,141,
105,145,150,88,155)(72,121,158,160,76,143,119,103,138,152,134,84)(75,153,133,
107,115,122,118,85,154,116,147,91)(79,110,101,159,126,90,157,81,112,100,89,
108)(80,99,97,86,156,129,144,94,127,114,148,82)(87,132,102,131,123,130,124,96,
93,113,146,140)

</pre></div>

In the third example, the use of an precomputed automorphism group is illustrated. It speeds up the construction of the underlying graph and the computation of the automorphism group of the underlying graph. However, as is also illustrated in the example, despite that the precomputed automorphism group of the generalised polygon is actually the full collineation group, still some time is needed to compute the automorphism group of the underlying graph. The timings after both <code class="file">CollineationGroup</code> commands are wrong. This is because GRAPE relies on an external binary to computed the automorphism group of a graph. The generalised quadrangle in this example is known as \(T_2^*(O)\).

<div class="example"><pre>
gap> q := 8;
8
gap> conic := ParabolicQuadric(2,q);
Q(2, 8)
gap> nucleus := NucleusOfParabolicQuadric(conic);
<a point in ProjectiveSpace(2, 8)>
gap> hyperoval := Union(List(Points(conic)),[nucleus]);
[ <a point in ProjectiveSpace(2, 8)>, <a point in Q(2, 8)>,
 <a point in Q(2, 8)>, <a point in Q(2, 8)>, <a point in Q(2, 8)>,
 <a point in Q(2, 8)>, <a point in Q(2, 8)>, <a point in Q(2, 8)>,
 <a point in Q(2, 8)>, <a point in Q(2, 8)> ]
gap> pg := PG(3,q);
ProjectiveSpace(3, 8)
gap> hyp := HyperplaneByDualCoordinates(pg,[1,0,0,0]*Z(q)^0);
<a plane in ProjectiveSpace(3, 8)>
gap> em := NaturalEmbeddingBySubspace(PG(2,q),pg,hyp);
<geometry morphism from <All elements of ProjectiveSpace(2,
8)> to <All elements of ProjectiveSpace(3, 8)>>
gap> O := List(hyperoval,x->x^em);
[ <a point in ProjectiveSpace(3, 8)>, <a point in ProjectiveSpace(3, 8)>,
 <a point in ProjectiveSpace(3, 8)>, <a point in ProjectiveSpace(3, 8)>,
 <a point in ProjectiveSpace(3, 8)>, <a point in ProjectiveSpace(3, 8)>,
 <a point in ProjectiveSpace(3, 8)>, <a point in ProjectiveSpace(3, 8)>,
 <a point in ProjectiveSpace(3, 8)>, <a point in ProjectiveSpace(3, 8)> ]
gap> points := Set(Filtered(Points(pg),x->not x in hyp));;
gap> lines := Union(List(O,x->Filtered(Lines(x),y->not y in hyp)));;
gap> inc := \*;
<Operation "*">
gap> gp := GeneralisedPolygonByElements(points,lines,inc);
<generalised quadrangle of order [ 7, 9 ]>
gap> time;
17466
gap> coll := CollineationGroup(gp);
<permutation group of size 5419008 with 9 generators>
gap> time;
69
gap> group := CollineationGroup(pg);
The FinInG collineation group PGammaL(4,8)
gap> stab := FiningSetwiseStabiliser(group,O);
#I Computing adjusted stabilizer chain...
<projective collineation group with 11 generators>
gap> time;
2045
gap> gp := GeneralisedPolygonByElements(points,lines,inc,stab,\^);
<generalised quadrangle of order [ 7, 9 ]>
gap> time;
394
gap> coll := CollineationGroup(gp);
<permutation group of size 5419008 with 9 generators>
gap> time;
62
gap> Order(coll);
5419008
gap> Order(stab);
5419008

</pre></div>

<a id="X8074EDB381D97984" name="X8074EDB381D97984"></a>

<h5>12.3-6 CollineationAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CollineationAction</code>( <var class="Arg">group</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a function

<var class="Arg">group</var> is a collineation group of a generalised polygon, computed using <code class="file">CollineationGroup</code>. The collineation group of classical generalised polygons will be a collineation group in the sense of FinInG. The natural action is <code class="file">OnProjectiveSubspaces</code>. The collineation group of any other generalised polygons will be a permutation group. The result of <code class="file">CollineationAction</code> for such a group is a function with input a pair <var class="Arg">(x,g)</var> where <var class="Arg">x</var> is an element of the generalised polygon, and <var class="Arg">g</var> is a collineation of the generalised polygon, so an element of <var class="Arg">group</var>. The example illustrates the use in the generalised quadrangle.

<div class="example"><pre>
gap> q := 4;
4
gap> conic := ParabolicQuadric(2,q);
Q(2, 4)
gap> nucleus := NucleusOfParabolicQuadric(conic);
<a point in ProjectiveSpace(2, 4)>
gap> hyperoval := Union(List(Points(conic)),[nucleus]);
[ <a point in ProjectiveSpace(2, 4)>, <a point in Q(2, 4)>,
 <a point in Q(2, 4)>, <a point in Q(2, 4)>, <a point in Q(2, 4)>,
 <a point in Q(2, 4)> ]
gap> pg := PG(3,q);
ProjectiveSpace(3, 4)
gap> hyp := HyperplaneByDualCoordinates(pg,[1,0,0,0]*Z(q)^0);
<a plane in ProjectiveSpace(3, 4)>
gap> em := NaturalEmbeddingBySubspace(PG(2,q),pg,hyp);
<geometry morphism from <All elements of ProjectiveSpace(2,
4)> to <All elements of ProjectiveSpace(3, 4)>>
gap> O := List(hyperoval,x->x^em);
[ <a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)>,
 <a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)>,
 <a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)> ]
gap> points := Set(Filtered(Points(pg),x->not x in hyp));;
gap> lines := Union(List(O,x->Filtered(Lines(x),y->not y in hyp)));;
gap> inc := \*;
<Operation "*">
gap> gp := GeneralisedPolygonByElements(points,lines,inc);
<generalised quadrangle of order [ 3, 5 ]>
gap> coll := CollineationGroup(gp);
<permutation group of size 138240 with 8 generators>
gap> act := CollineationAction(coll);
function( el, g ) ... end
gap> g := Random(coll);
(1,37,45,63,27,19)(2,53,13,64,11,51)(3,33,38,61,31,28)(4,49,6,62,15,60)(5,46,
47,59,20,17)(7,42,40,57,24,26)(8,58)(9,55)(10,39,41,56,25,23)(12,35,34,54,29,
32)(14,48,43,52,18,21)(16,44,36,50,22,30)(65,132,90,157,89,105)(66,68,131,143,
119,103)(67,135,76,123,130,106)(69,133,112,100,81,107)(70,134,150,88,155,
104)(71,99,79,144,93,149)(72,153,95,120,73,122)(74,125,115,128,140,87)(75,121,
136,117,113,91)(77,124,98,83,147,146)(78,145,84,118,85,142)(80,92,137,141,108,
97)(82,86,116,111,138,101)(94,127,126,102,109,96)(110,152,151,154,156,
129)(114,160,139,158,148,159)
gap> l := Random(Lines(gp));
<a line in <generalised quadrangle of order [ 3, 5 ]>>
gap> act(l,g);
<a line in <generalised quadrangle of order [ 3, 5 ]>>
gap> p := Random(Points(gp));
<a point in <generalised quadrangle of order [ 3, 5 ]>>
gap> act(p,g);
<a point in <generalised quadrangle of order [ 3, 5 ]>>
gap> stab := Stabilizer(coll,p,act);
<permutation group of size 2160 with 3 generators>
gap> List(Orbits(stab,List(Points(gp)),act),x->Length(x));
[ 45, 18, 1 ]
gap> List(Orbits(stab,List(Lines(gp)),act),x->Length(x));
[ 90, 6 ]

</pre></div>

<a id="X7D75521986B958DA" name="X7D75521986B958DA"></a>

<h5>12.3-7 BlockDesignOfGeneralisedPolygon</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlockDesignOfGeneralisedPolygon</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a block design

This method allows one to use the GAP package DESIGN to analyse a generalised polygon, so the user must first load this package. The argument <var class="Arg">gp</var> is a generalised polygon, and if it has a collineation group, the block design is computed with this extra information and thus the resulting design is easier to work with. Likewise, if <var class="Arg">gp</var> is an elation generalised quadrangle and it has an elation group, then we use the elation group's action to efficiently compute the block design. We should also point out that this method returns a mutable attribute of <var class="Arg">gp</var>, so that acquired information about the block design can be added. For example, the automorphism group of the block design may be computed after the design is stored as an attribute of <var class="Arg">gp</var>. Normally, attributes of GAP objects are immutable.

<div class="example"><pre>
gap> LoadPackage("design");
-----------------------------------------------------------------------------
Loading DESIGN 1.6 (The Design Package for GAP)
by Leonard H. Soicher (http://www.maths.qmul.ac.uk/~leonard/).
Homepage: http://www.designtheory.org/software/gap_design/
-----------------------------------------------------------------------------
true
gap> gh := SplitCayleyHexagon(2);
H(2)
gap> CollineationGroup(gh);
#I for Split Cayley Hexagon
#I Computing nice monomorphism...
#I Found permutation domain...
G_2(2)
gap> des := BlockDesignOfGeneralisedPolygon(gh);
rec( autSubgroup := <permutation group with 3 generators>,
 blocks := [ [ 1, 29, 52 ], [ 1, 34, 36 ], [ 1, 37, 48 ], [ 2, 13, 60 ],
 [ 2, 44, 53 ], [ 2, 45, 52 ], [ 3, 17, 35 ], [ 3, 22, 51 ],
 [ 3, 23, 48 ], [ 4, 16, 57 ], [ 4, 19, 36 ], [ 4, 54, 56 ],
 [ 5, 22, 63 ], [ 5, 31, 57 ], [ 5, 49, 52 ], [ 6, 7, 60 ],
 [ 6, 28, 57 ], [ 6, 35, 43 ], [ 7, 26, 27 ], [ 7, 33, 34 ],
 [ 8, 9, 53 ], [ 8, 22, 33 ], [ 8, 38, 56 ], [ 9, 25, 61 ],
 [ 9, 28, 37 ], [ 10, 18, 53 ], [ 10, 32, 35 ], [ 10, 36, 62 ],
 [ 11, 12, 63 ], [ 11, 26, 54 ], [ 11, 37, 42 ], [ 12, 41, 43 ],
 [ 12, 44, 50 ], [ 13, 15, 42 ], [ 13, 19, 51 ], [ 14, 15, 31 ],
 [ 14, 17, 61 ], [ 14, 34, 50 ], [ 15, 20, 38 ], [ 16, 23, 44 ],
 [ 16, 40, 59 ], [ 17, 45, 54 ], [ 18, 24, 26 ], [ 18, 30, 31 ],
 [ 19, 25, 41 ], [ 20, 21, 62 ], [ 20, 23, 27 ], [ 21, 28, 55 ],
 [ 21, 39, 45 ], [ 24, 29, 59 ], [ 24, 51, 55 ], [ 25, 27, 49 ],
 [ 29, 38, 43 ], [ 30, 39, 41 ], [ 30, 46, 48 ], [ 32, 40, 42 ],
 [ 32, 47, 49 ], [ 33, 39, 40 ], [ 46, 47, 56 ], [ 46, 58, 60 ],
 [ 47, 50, 55 ], [ 58, 59, 61 ], [ 58, 62, 63 ] ], isBlockDesign := true,
 v := 63 )
gap> f := GF(3);
GF(3)
gap> id := IdentityMat(2, f);;
gap> clan := List( f, t -> t*id );;
gap> clan := qClan(clan,f);
<q-clan over GF(3)>
gap> egq := EGQByqClan( clan );
#I Computed Kantor family. Now computing EGQ...
<EGQ of order [ 9, 3 ] and basepoint 0>
gap> HasElationGroup( egq );
true
gap> design := BlockDesignOfGeneralisedPolygon( egq );;
#I Computing orbits on lines of gen. polygon...
#I Computing block design of generalised polygon...
gap> aut := AutGroupBlockDesign( design );
<permutation group with 6 generators>
gap> NrBlockDesignPoints( design );
280
gap> NrBlockDesignBlocks( design );
112
gap> DisplayCompositionSeries(aut);
G (6 gens, size 26127360)
| Z(2)
S (5 gens, size 13063680)
| Z(2)
S (5 gens, size 6531840)
| Z(2)
S (4 gens, size 3265920)
| 2A(3,3) = U(4,3) ~ 2D(3,3) = O-(6,3)
1 (0 gens, size 1)

</pre></div>

<a id="X7A13D5EB82E01576" name="X7A13D5EB82E01576"></a>

<h4>12.4 Elements of generalised polygons</h4>

<a id="X7E7607CA7D59D086" name="X7E7607CA7D59D086"></a>

<h5>12.4-1 Collections of elements of generalised polygons</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementsOfIncidencStructure</code>( <var class="Arg">gp</var>, <var class="Arg">i</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Points</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Lines</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a collection of elements of a generalised polygon

<var class="Arg">gp</var> is any generalised polygon, <var class="Arg">i</var> is a natural number, necessarily \(1\) or \(2\). <code class="file">ElementsOfIncidenceStructure</code> returns the elements of type \(i\) of <var class="Arg">gp</var>, <code class="file">Points</code> and <code class="file">Lines</code> are the usual shortcuts.

<a id="X858ADA3B7A684421" name="X858ADA3B7A684421"></a>

<h5>12.4-2 Size</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Size</code>( <var class="Arg">els</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a number

<var class="Arg">els</var> is a collection of elements of a generalised polygon. This operation returns the number of element in <var class="Arg">els</var>.

<a id="X7E9B2A217DBF2849" name="X7E9B2A217DBF2849"></a>

<h5>12.4-3 Creating elements from objects and retrieving objects from elements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjectToElement</code>( <var class="Arg">gp</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjectToElement</code>( <var class="Arg">gp</var>, <var class="Arg">type</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingObject</code>( <var class="Arg">el</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a collection of elements of a generalised polygon

To create elements in <var class="Arg">gp</var> (of type <var class="Arg">type</var>), one of the versions of <code class="file">ObjectToElement</code> can be used. It is checked whether <var class="Arg">obj</var> represents an element (of type <var class="Arg">type</var>). To retrieve an underlying object of an element <var class="Arg">el</var>, <code class="file">UnderlyingObject</code> can be used.

<div class="example"><pre>
gap> mat := [ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ],
> [ 1, 0, 1, 0, 0, 0, 1 ], [ 0, 1, 1, 1, 0, 0, 0 ],
> [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],
> [ 0, 0, 1, 0, 1, 1, 0 ] ];
[ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ], [ 1, 0, 1, 0, 0, 0, 1 ],
 [ 0, 1, 1, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],
 [ 0, 0, 1, 0, 1, 1, 0 ] ]
gap> gp := GeneralisedPolygonByIncidenceMatrix(mat);
<projective plane order 2>
gap> p := Random(Points(gp));
<a point in <projective plane order 2>>
gap> UnderlyingObject(p);
7
gap> l := Random(Lines(gp));
<a line in <projective plane order 2>>
gap> UnderlyingObject(l);
[ 4, 6, 7 ]
gap> ObjectToElement(gp,1,4);
<a point in <projective plane order 2>>
gap> ObjectToElement(gp,2,5);
Error, <obj> does not represent a line of <gp> called from
<function "unknown">( <arguments> )
called from read-eval loop at line 18 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> ObjectToElement(gp,2,[1,2,3]);
Error, <obj> does not represent a line of <gp> called from
<function "unknown">( <arguments> )
called from read-eval loop at line 18 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> ObjectToElement(gp,[1,2,6]);
<a line in <projective plane order 2>>

</pre></div>

<a id="X83B0FA9E7AE3DF01" name="X83B0FA9E7AE3DF01"></a>

<h5>12.4-4 Incidence</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIncident</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \*</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true or false

Let <var class="Arg">v</var> and <var class="Arg">w</var> be two elements of a generalised polygon. It is checked if the ambient geometry of the two elements are identical, and true is returned if and only if the two elements are incident in their ambient geometry.

<a id="X875BE2957FAF6209" name="X875BE2957FAF6209"></a>

<h5>12.4-5 Span</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Span</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a line of a generalised polygon or fail

Let <var class="Arg">v</var> and <var class="Arg">w</var> be two elements of a generalised polygon. It is checked if the ambient geometries of the two elements are identical, and if the two elements are points. If <var class="Arg">v</var> and <var class="Arg">w</var> are incidence with a common line, this line is returned. Otherwise <code class="keyw">fail</code> is returned. For generalised polygons constructed with <code class="code">GeneralisedPolygonByBlocks</code>, <code class="code">GeneralisedPolygonByElements</code> an <code class="code">GeneralisedPolygonByInidenceMatrix</code>, the underlying graph is used. Note that the behaviour of <code class="file">Span</code> is different for elements of generalised polygons that belong to <code class="code">IsLieGeometry</code>, see <a href="chap4_mj.html#X875BE2957FAF6209">4.2-16</a>.

<div class="example"><pre>
gap> mat := [ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ],
> [ 1, 0, 1, 0, 0, 0, 1 ], [ 0, 1, 1, 1, 0, 0, 0 ],
> [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],
> [ 0, 0, 1, 0, 1, 1, 0 ] ];
[ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ], [ 1, 0, 1, 0, 0, 0, 1 ],
 [ 0, 1, 1, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],
 [ 0, 0, 1, 0, 1, 1, 0 ] ]
gap> gp := GeneralisedPolygonByIncidenceMatrix(mat);
<projective plane order 2>
gap> p := Random(Points(gp));
<a point in <projective plane order 2>>
gap> q := Random(Points(gp));
<a point in <projective plane order 2>>
gap> Span(p,q);
<a line in <projective plane order 2>>
gap> ps := ParabolicQuadric(4,3);
Q(4, 3)
gap> gp := GeneralisedPolygonByElements(Set(Points(ps)),Set(Lines(ps)),\*);
<generalised quadrangle of order [ 3, 3 ]>
gap> p := Random(Points(gp));
<a point in <generalised quadrangle of order [ 3, 3 ]>>
gap> q := Random(Points(gp));
<a point in <generalised quadrangle of order [ 3, 3 ]>>
gap> Span(p,q);
#I <x> and <y> do not span a line of gp
fail

</pre></div>

<a id="X8469B54180FE1E4C" name="X8469B54180FE1E4C"></a>

<h5>12.4-6 Meet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Meet</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a point of a generalised polygon or fail

Let <var class="Arg">v</var> and <var class="Arg">w</var> be two elements of a generalised polygon. It is checked if the ambient geometries of the two elements are identical, and if the two elements are lines. If <var class="Arg">v</var> and <var class="Arg">w</var> are incidence with a common point, this point is returned. Otherwise <code class="keyw">fail</code> is returned. For generalised polygons constructed with <code class="code">GeneralisedPolygonByBlocks</code>, <code class="code">GeneralisedPolygonByElements</code> an <code class="code">GeneralisedPolygonByInidenceMatrix</code>, the underlying graph is used. Note that the behavior of <code class="file">Meet</code> is different for elements of generalised polygons that belong to <code class="code">IsLieGeometry</code>, see <a href="chap4_mj.html#X8469B54180FE1E4C">4.2-17</a>

<div class="example"><pre>
gap> mat := [ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ],
> [ 1, 0, 1, 0, 0, 0, 1 ], [ 0, 1, 1, 1, 0, 0, 0 ],
> [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],
> [ 0, 0, 1, 0, 1, 1, 0 ] ];
[ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ], [ 1, 0, 1, 0, 0, 0, 1 ],
 [ 0, 1, 1, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],
 [ 0, 0, 1, 0, 1, 1, 0 ] ]
gap> gp := GeneralisedPolygonByIncidenceMatrix(mat);
<projective plane order 2>
gap> l := Random(Lines(gp));
<a line in <projective plane order 2>>
gap> m := Random(Lines(gp));
<a line in <projective plane order 2>>
gap> Meet(l,m);
<a point in <projective plane order 2>>
gap> ps := ParabolicQuadric(4,3);
Q(4, 3)
gap> gp := GeneralisedPolygonByElements(Set(Points(ps)),Set(Lines(ps)),\*);
<generalised quadrangle of order [ 3, 3 ]>
gap> l := Random(Lines(gp));
<a line in <generalised quadrangle of order [ 3, 3 ]>>
gap> m := Random(Lines(gp));
<a line in <generalised quadrangle of order [ 3, 3 ]>>
gap> Meet(l,m);
#I <x> and <y> do meet in a common point of gp
fail

</pre></div>

<a id="X8154BB13844AA0FD" name="X8154BB13844AA0FD"></a>

<h5>12.4-7 Shadow elements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ShadowOfElement</code>( <var class="Arg">geo</var>, <var class="Arg">v</var>, <var class="Arg">j</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Points</code>( <var class="Arg">el</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Lines</code>( <var class="Arg">el</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementsIncidentWithElementOfIncidenceStructure</code>( <var class="Arg">el</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A collection of elements

<var class="Arg">geo</var> is a generalised polygon, <var class="Arg">v</var> must be an element of <var class="Arg">geo</var>, <var class="Arg">j</var> is an integer equal to 1 or 2, since <var class="Arg">geo</var> is a rank two geometry. The operation <code class="file">ShadowOfElement</code> returns the collection of elements of <var class="Arg">geo</var> of type <var class="Arg">j</var>, incident with the element <var class="Arg">v</var>. The operations <code class="file">Points</code> and <code class="file">Lines</code> with argument are the usual shortcuts to <code class="file">ShadowOfElement</code> with <var class="Arg">j</var> respectively equal to 1, 2. The operation <code class="file">ElementsIncidentWithElementOfIncidenceStructure</code> is the usual shortcut to <code class="file">ShadowOfElement</code>.

<div class="example"><pre>
gap> blocks := [
> [ 1, 2, 3, 4, 5 ], [ 1, 6, 7, 8, 9 ], [ 1, 10, 11, 12, 13 ],
> [ 1, 14, 15, 16, 17 ], [ 1, 18, 19, 20, 21 ], [ 2, 6, 10, 14, 18 ],
> [ 2, 7, 11, 15, 19 ], [ 2, 8, 12, 16, 20 ], [ 2, 9, 13, 17, 21 ],
> [ 3, 6, 11, 16, 21 ], [ 3, 7, 10, 17, 20 ], [ 3, 8, 13, 14, 19 ],
> [ 3, 9, 12, 15, 18 ], [ 4, 6, 12, 17, 19 ], [ 4, 7, 13, 16, 18 ],
> [ 4, 8, 10, 15, 21 ], [ 4, 9, 11, 14, 20 ], [ 5, 6, 13, 15, 20 ],
> [ 5, 7, 12, 14, 21 ], [ 5, 8, 11, 17, 18 ], [ 5, 9, 10, 16, 19 ] ];; 
gap> gp := GeneralisedPolygonByBlocks( blocks ); 
<projective plane order 4>
gap> l := Random(Lines(gp)); 
<a line in <projective plane order 4>>
gap> pts := ShadowOfElement(gp,l,1);
<shadow points in <projective plane order 4>>
gap> List(pts);
[ <a point in <projective plane order 4>>,
 <a point in <projective plane order 4>>,
 <a point in <projective plane order 4>>,
 <a point in <projective plane order 4>>,
 <a point in <projective plane order 4>> ]
gap> p := Random(Points(gp));
<a point in <projective plane order 4>>
gap> lines := Lines(p);
<shadow lines in <projective plane order 4>>
gap> List(lines);
[ <a line in <projective plane order 4>>, <a line in <projective plane order
 4>>, <a line in <projective plane order 4>>,
 <a line in <projective plane order 4>>, <a line in <projective plane order
 4>> ]

</pre></div>

<a id="X7807682E86D3BE10" name="X7807682E86D3BE10"></a>

<h5>12.4-8 DistanceBetweenElements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DistanceBetweenElements</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a number

Let <var class="Arg">v</var> and <var class="Arg">w</var> be two elements of a generalised polygon. It is checked if the ambient geometry of the two elements are identical, and the distance between the two elements in the incidence graph of their ambient geometry is returned.

<div class="example"><pre>
gap> g := ElementaryAbelianGroup(27);
<pc group of size 27 with 3 generators>
gap> flist1 := [ Group(g.1), Group(g.2), Group(g.3), Group(g.1*g.2*g.3) ];;
gap> flist2 := [ Group([g.1, g.2^2*g.3]), Group([g.2, g.1^2*g.3 ]),
> Group([g.3, g.1^2*g.2]), Group([g.1^2*g.2, g.1^2*g.3 ]) ];;
gap> egq := EGQByKantorFamily(g, flist1, flist2);
<EGQ of order [ 3, 3 ] and basepoint 0>
gap> p := Random(Points(egq));
<a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>
gap> q := Random(Points(egq));
<a point of class 3 of <EGQ of order [ 3, 3 ] and basepoint 0>>
gap> DistanceBetweenElements(p,q);
2
gap> gh := SplitCayleyHexagon(3);
H(3)
gap> l := Random(Lines(gh));
#I for Split Cayley Hexagon
#I Computing nice monomorphism...
#I Found permutation domain...
<a line in H(3)>
gap> m := First(Lines(gh),x->DistanceBetweenElements(l,x)=6);
<a line in H(3)>

</pre></div>

<a id="X7934EB788049B533" name="X7934EB788049B533"></a>

<h4>12.5 The classical generalised hexagons</h4>

<a id="X7BF1D2E57B7630CB" name="X7BF1D2E57B7630CB"></a>

<h5>12.5-1 Trialities of the hyperbolic quadric and generalised hexagons</h5>

Consider the hyperbolic quadric \(\mathrm{Q}^+(7,q)\). This is a polar space of rank 4. It is well known that its generators fall into two systems. Each system contains exactly \((q^3+q^2+q+1)(q^2+q+1)\) generators, which is equal to the number of points of \(\mathrm{Q}^+(7,q)\). Generators from the same system meet each other in an empty subspace or in a line, Generators from a different system meet each other in a point or a plane. One defines the rank \(4\) geometry \(\Omega(7,q)\) as follows. The 0-points are the points of \(\mathrm{Q}^+(7,q)\), the 1-points are the generators of the first system, the 2-points are the generators of the second system, and the lines are the lines of \(\mathrm{Q}^+(7,q)\). The incidence is the natural incidence of the underlying projective space. Denote the set of \(i\)-points as \(P^{(i)}\) , \(i=0,1,2\).

A triality of \(\Omega(7,q)\) is a map \(\tau: P^{(i)} \to P^{(i+1)}\) (where \(i+1\) is computed modulo 3) preserving the incidence and for which \(\tau^3 = 1\). Note that the image of a line under \(\tau\) is determined by the image of the points incident with the line.

An \(i\)-point is absolute with respect to a fixed triality if it is incident with its image under the triality. Consequently, a line is absolute with if it is fixed by the triality.

A generalised hexagon can be constructed as geometry of absolute points of one kind and absolute lines with relation to a fixed triality. Note that not all trialities yield (thick) generalised quadrangles. There are different types of trialities, for some of them the absolute geometry is degenerate.

The triality used in FinInG to construct the classical generalised hexagons is fixed. It is described explicitly in <a href="chapBib_mj.html#biBHVM">[VM98]</a>. To describe the triality, a trilinear form expressing the incidence between \(i\)-points of \(\Omega(7,q)\) is used. Given the fact that, because of the existence of a triality, the role of the 0, 1 and 2 points are the same, each of the 1 and 2 points can be labelled the same way as the 0-points, which are effectively labelled by 8-tuples \((x_0,\ldots,x_7) \in V(8,q)=V\) where each 8-tuples represents a projective point of \(\mathrm{Q}^+(7,q)\).

Consider the hyperbolic quadric determined by the quadratic form \(X_0X_4+X_1X_5+X_2X_6+X_3X_7\) . Consider the trilinear map \(T\),

\(T(x,y,z) = \left|\begin{array}{ccc} x_0 & x_1 & x_2 \\ y_0 & y_1 & y_2 \\ z_0 & z_1 & z_2 \\ \end{array}\right| + \left|\begin{array}{ccc} x_4 & x_5 & x_6 \\ y_4 & y_5 & y_6 \\ z_4 & z_5 & z_6 \\ \end{array}\right| + x_3(z_0y_4+z_1y_5+z_2y_6) + x_7(y_0z_4+y_1z_5+y_2z_6) + y_3(x_0z_4+x_1z_5+x_2z_6) + \\ y_7(z_0x_4+z_1x_5+z_2x_6) + z_3(y_0x_4+y_1x_5+y_2x_6) + z_7(x_0y_4+x_1y_5+x_2y_6) - x_3y_3z_3 - x_7y_7z_7.\)

Now a pair \((x,y) \in V \times V\) represents an incident 0-1 pair of points if and only if \(T(x,y,z)\) vanishes in the variable \(z\), and similarly for any cyclic permutation of the letters \(x,y,z\). So given a \(1\)-point \(y\), \(T(x,y,z) = 0\), where \(z\) is a variable, and \(x\) is an unknown, gives a set of equations representing a generator of \(\mathrm{Q}^+(7,q)\) this is the generator of \(\mathrm{Q}^+(7,q)\) represented by \(y\) as label of a \(1\)-point.

Let \(\sigma\) be an automorphism of \(\mathrm{GF}(q)\) of order \(3\), or the identity. Consider the map

\(\tau_{\sigma}: P^{(i)} \to P^{(i+1)}\)

\((x_j) \mapsto (x_j^{\sigma})\), \(j=0\ldots 7\) .

This map clearly preserves \(T(x,y,z)\), so preserves the incidence, and has order three, so it is a triality of \(\Omega(7,q)\). We call an element \(p\) absolute with respect to a triality \(\tau\) if and only if \(p \mathrm{I} p^\tau\) . Consequently, a line is absolute if and only if it is fixed by the triality. Denote the set of \(i\)-points that are absolute with respect to the triality as \(P_{\mathrm{abs}}^{(i)}\) , and the set of absolute lines with respect to the triality as \(L_{\mathrm{abs}}\). Then a famous theorem of Tits (<a href="chapBib_mj.html#biBTits1959">[Tit59]</a>) says that for the triality \(\tau_{\sigma}\) , the point-line geometry \(\Gamma^{(i)} = (P^{(i)}_{\mathrm{abs}},L_{\mathrm{abs}},\mathrm{I})\) is a generalised hexagon of order \((|K|,|L|)\), \(K\) the field \(\mathrm{GF}(q)\) and \(L\) the subfield of invariant elements of \(K\) under the field automorphism \(\sigma\). Note that a finite field has a field automorphism of order three if and only if its order equals \(q^3\). So, for \(K\) equal to \(\mathrm{GF}(q)\) and \(\sigma=1\), \(\Gamma^{(i)}\) is a generalised hexagon of order \(q\), which is called the split Cayley hexagon of order \(q\), denoted \(H(q)\). For \(K\) equal to \(\mathrm{GF}(q^3)\), and \(\sigma\) a non-trivial field automorphism of order \(3\), \(\Gamma^{(i)}\) is a generalised hexagon of order \((q^3,q)\) , which is called the twisted triality hexagon of order \((q^3,q)\) , denoted \(T(q^3,q)\) . Note that for a given triality, the hexagons \(\Gamma^{(i)}\), \(i=0,1,2\) are isomorphic. Consequently, \(\Gamma^{(0)}\) is a point-line geometry of which the point set, line set respectively, is a subset of the points, lines respectively of \(\mathrm{Q}^+(7,q)\). Finally, we mention the following important theorem, which was shown by Tits (<a href="chapBib_mj.html#biBTits1959">[Tit59]</a>): the split Cayley hexagon, obtained by the triality with \(\sigma=1\), is contained in the hyperplane with equation \(X_3+X_7=0\) , which intersects the hyperbolic quadric in the parabolic quadric \(\mathrm{Q}(6,q)\). The points of the split Cayley hexagon are the points of \(\mathrm{Q}(6,q)\).

This above description of the triality and the associated generalised hexagons, contains sufficient analytical information to implement the split Cayley hexagon and the twisted triality hexagon in an efficient way. The user is allowed to choose a representation for the ambient polar space. For \(q=2^h\) the polar spaces \(\mathrm{Q}(6,q)\) and \(\mathrm{W}(5,q)\) are isomorphic. Consequently, the user may choose \(\mathrm{W}(5,q)\) as ambient polar space for the split Cayley hexagon of even order. This embedding in \(\mathrm{W}(5,q)\) is called the perfect symplectic embedding of the split Cayley hexagon. Finally, <a href="chapBib_mj.html#biBHVM">[VM98]</a> contains an explicit description of the generators of the collineation groups of both generalised hexagons.

<a id="X7C35384980AA9B77" name="X7C35384980AA9B77"></a>

<h5>12.5-2 IsLieGeometry</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLieGeometry</code></td><td class="tdright">( category )</td></tr></table></div>
Recall that the classical generalised hexagons are constructed as an object in <code class="file">IsLieGeometry</code>. This makes most operations described in the appropriate chapters on Lie geometries, projective spaces and polar spaces applicable.

<a id="X7A05FF0079F55291" name="X7A05FF0079F55291"></a>

<h5>12.5-3 SplitCayleyHexagon</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SplitCayleyHexagon</code>( <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SplitCayleyHexagon</code>( <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SplitCayleyHexagon</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a generalised hexagon

<div class="example"><pre>
gap> hexagon := SplitCayleyHexagon( 3 );
H(3)
gap> AmbientPolarSpace(hexagon);
Q(6, 3): -x_1*x_5-x_2*x_6-x_3*x_7+x_4^2=0
gap> ps := ParabolicQuadric(6,3);
Q(6, 3)
gap> hexagon := SplitCayleyHexagon( ps );
H(3) in Q(6, 3)
gap> AmbientPolarSpace(hexagon);
Q(6, 3)
gap> hexagon := SplitCayleyHexagon( 4 );
H(4)
gap> AmbientPolarSpace(hexagon);
W(5, 4): x1*y4+x2*y5+x3*y6+x4*y1+x5*y2+x6*y3=0
gap> ps := ParabolicQuadric(6,4);
Q(6, 4)
gap> hexagon := SplitCayleyHexagon( ps );
H(4) in Q(6, 4)
gap> AmbientPolarSpace(hexagon);
Q(6, 4)

</pre></div>

<a id="X7E17977384011587" name="X7E17977384011587"></a>

<h5>12.5-4 TwistedTrialityHexagon</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwistedTrialityHexagon</code>( <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwistedTrialityHexagon</code>( <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwistedTrialityHexagon</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a generalised hexagon

<div class="example"><pre>
gap> hexagon := TwistedTrialityHexagon(2^3);
T(8, 2)
gap> AmbientPolarSpace(hexagon);
<polar space in ProjectiveSpace(
7,GF(2^3)): x_1*x_5+x_2*x_6+x_3*x_7+x_4*x_8=0 >
gap> ps := HyperbolicQuadric(7,2^3);
Q+(7, 8)
gap> hexagon := TwistedTrialityHexagon(ps);
T(8, 2) in Q+(7, 8)
gap> AmbientPolarSpace(hexagon);
Q+(7, 8)

</pre></div>

<a id="X807A60A28264BD93" name="X807A60A28264BD93"></a>

<h5>12.5-5 vector spaceToElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ vector spaceToElement</code>( <var class="Arg">gh</var>, <var class="Arg">vec</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: an element of a classical generalized hexagon

The argument <var class="Arg">vec</var> is one vector or a list of vectors from the underlying vector space of <var class="Arg">gh</var>. This operation checks whether <var class="Arg">vec</var> represents a point or a line of <var class="Arg">gh</var>. Note that vectors and matrices in different representations are allowed as argument.

<div class="example"><pre>
gap> ps := ParabolicQuadric(6,9);
Q(6, 9)
gap> gh := SplitCayleyHexagon(ps);
H(9) in Q(6, 9)
gap> vec := [ Z(3)^0, Z(3^2), 0*Z(3), Z(3^2), Z(3^2)^3, Z(3^2)^5, 0*Z(3) ];
[ Z(3)^0, Z(3^2), 0*Z(3), Z(3^2), Z(3^2)^3, Z(3^2)^5, 0*Z(3) ]
gap> p := VectorSpaceToElement(gh,vec);
<a point in H(9) in Q(6, 9)>
gap> vec := [ [ Z(3)^0, 0*Z(3), Z(3^2)^7, 0*Z(3), Z(3)^0, Z(3^2)^2, Z(3^2)^2 ], 
> [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), Z(3^2)^3, 0*Z(3) ] ];
[ [ Z(3)^0, 0*Z(3), Z(3^2)^7, 0*Z(3), Z(3)^0, Z(3^2)^2, Z(3^2)^2 ],
 [ 0*Z(3), Z(3)^0, 0*Z(3), Z(3)^0, 0*Z(3), Z(3^2)^3, 0*Z(3) ] ]
gap> line := VectorSpaceToElement(gh,vec);
Error, <x> does not generate an element of <geom> called from
<function "unknown">( <arguments> )
called from read-eval loop at line 14 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;

</pre></div>

<a id="X7809B7C183FA7213" name="X7809B7C183FA7213"></a>

<h5>12.5-6 ObjectToElement</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjectToElement</code>( <var class="Arg">gh</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: an element of a classical generalized hexagon

The argument <var class="Arg">obj</var> is one vector or a list of vectors from the underlying vector space of <var class="Arg">gh</var>. This operation checks whether <var class="Arg">obj</var> represents a point or a line of <var class="Arg">gh</var>. Note that vectors and matrices in different representations are allowed as argument.

<div class="example"><pre>
gap> mat := IdentityMat(8,GF(5^3));
< mutable compressed matrix 8x8 over GF(125) >
gap> form := BilinearFormByMatrix(mat,GF(5^3));
< bilinear form >
gap> ps := PolarSpace(form);
<polar space in ProjectiveSpace(
7,GF(5^3)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2=0 >
gap> gh := TwistedTrialityHexagon(ps);
T(125, 5) in Q+(7, 125): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+x_8^2
gap> vec := [ Z(5)^0, Z(5^3)^55, Z(5^3)^99, Z(5^3)^107, Z(5^3)^8, Z(5^3)^35, Z(5^3)^73, 
> Z(5^3)^115 ];
[ Z(5)^0, Z(5^3)^55, Z(5^3)^99, Z(5^3)^107, Z(5^3)^8, Z(5^3)^35, Z(5^3)^73,
 Z(5^3)^115 ]
gap> p := ObjectToElement(gh,vec);
<a point in T(125, 5) in Q+(7, 125): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2
+x_8^2>
gap> vec := [ [ Z(5)^0, 0*Z(5), Z(5^3)^76, Z(5^3)^117, Z(5^3)^80, Z(5^3)^19, Z(5^3)^48, 
> Z(5^3)^100 ], 
> [ 0*Z(5), Z(5)^0, Z(5^3)^115, Z(5^3)^14, Z(5^3)^40, Z(5^3)^67, Z(5^3)^123, 
> Z(5^3)^3 ] ];
[ [ Z(5)^0, 0*Z(5), Z(5^3)^76, Z(5^3)^117, Z(5^3)^80, Z(5^3)^19, Z(5^3)^48,
 Z(5^3)^100 ],
 [ 0*Z(5), Z(5)^0, Z(5^3)^115, Z(5^3)^14, Z(5^3)^40, Z(5^3)^67, Z(5^3)^123,
 Z(5^3)^3 ] ]
gap> line := ObjectToElement(gh,vec);
<a line in T(125, 5) in Q+(7, 125): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2+
x_8^2>

</pre></div>

<a id="X810D4D6D87069697" name="X810D4D6D87069697"></a>

<h5>12.5-7 UnderlyingObject</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingObject</code>( <var class="Arg">gh</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a vector or a matrix

<a id="X87BDB89B7AAFE8AD" name="X87BDB89B7AAFE8AD"></a>

<h5><code>12.5-8 \in</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \in</code>( <var class="Arg">x</var>, <var class="Arg">gh</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true or false

<div class="example"><pre>
gap> ps := HyperbolicQuadric(7,5^3);
Q+(7, 125)
gap> gh := TwistedTrialityHexagon(ps);
T(125, 5) in Q+(7, 125)
gap> repeat
> p := Random(Points(ps));
> until p in gh;
gap> time;
18399
gap> p in gh;
true
gap> q := ElementToElement(gh,p);
<a point in T(125, 5) in Q+(7, 125)>
gap> l := Random(Lines(p));
<a line in Q+(7, 125)>
gap> l in gh;
false
gap> List(Lines(q),x->x in gh);
[ true, true, true, true, true, true ]

</pre></div>

<a id="X7B1380878358938C" name="X7B1380878358938C"></a>

<h5>12.5-9 Span and meet of elements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Span</code>( <var class="Arg">x</var>, <var class="Arg">y</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Meet</code>( <var class="Arg">x</var>, <var class="Arg">y</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a subspace of a projective space

<var class="Arg">x</var> and <var class="Arg">y</var> are two elements of a classical generalised hexagon. The operation <code class="file">Span</code> returns the projective line spanned by <var class="Arg">x</var> and <var class="Arg">y</var>. The operation <code class="file">Meet</code> returns the intersection of the elements <var class="Arg">x</var> and <var class="Arg">y</var>. Note that the classical generalised hexagons are Lie geometries, so their elements belong to a subcategory of <code class="code">IsSubspaceOfProjectiveSpace</code>. Therefore, the operations <code class="file">Span</code> and <code class="file">Meet</code> behave as described in <a href="chap7_mj.html#X875BE2957FAF6209">7.5-2</a> and <a href="chap7_mj.html#X8469B54180FE1E4C">7.5-3</a>.

<div class="example"><pre>
gap> ps := SymplecticSpace(5,8);
W(5, 8)
gap> gh := SplitCayleyHexagon(ps);
H(8) in W(5, 8)
gap> vec := [ Z(2)^0, Z(2^3)^6, Z(2^3)^5, Z(2^3)^6, Z(2)^0, Z(2^3) ];
[ Z(2)^0, Z(2^3)^6, Z(2^3)^5, Z(2^3)^6, Z(2)^0, Z(2^3) ]
gap> p := VectorSpaceToElement(gh,vec);
<a point in H(8) in W(5, 8)>
gap> vec := [ Z(2)^0, Z(2^3)^2, Z(2^3), Z(2^3)^3, Z(2^3)^5, Z(2^3)^5 ];
[ Z(2)^0, Z(2^3)^2, Z(2^3), Z(2^3)^3, Z(2^3)^5, Z(2^3)^5 ]
gap> q := VectorSpaceToElement(gh,vec);
<a point in H(8) in W(5, 8)>
gap> span := Span(p,q);
<a line in ProjectiveSpace(5, 8)>
gap> ElementToElement(gh,span);
<a line in H(8) in W(5, 8)>
gap> vec := [ [ Z(2)^0, 0*Z(2), Z(2^3)^6, Z(2)^0, 0*Z(2), Z(2^3) ], 
> [ 0*Z(2), Z(2)^0, Z(2^3)^6, Z(2^3)^4, Z(2^3)^4, 0*Z(2) ] ];
[ [ Z(2)^0, 0*Z(2), Z(2^3)^6, Z(2)^0, 0*Z(2), Z(2^3) ],
 [ 0*Z(2), Z(2)^0, Z(2^3)^6, Z(2^3)^4, Z(2^3)^4, 0*Z(2) ] ]
gap> l := VectorSpaceToElement(gh,vec);
<a line in H(8) in W(5, 8)>
gap> vec := [ [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2^3), 0*Z(2), Z(2^3) ], 
> [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2^3)^2, Z(2^3)^4, Z(2^3)^4 ] ];
[ [ Z(2)^0, 0*Z(2), Z(2)^0, Z(2^3), 0*Z(2), Z(2^3) ],
 [ 0*Z(2), Z(2)^0, Z(2)^0, Z(2^3)^2, Z(2^3)^4, Z(2^3)^4 ] ]
gap> m := VectorSpaceToElement(gh,vec);
<a line in H(8) in W(5, 8)>
gap> Meet(l,m);
< empty subspace >
gap> DistanceBetweenElements(l,m);
6

</pre></div>

<a id="X83FF6FA0790D5747" name="X83FF6FA0790D5747"></a>

<h5>12.5-10 CollineationGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CollineationGroup</code>( <var class="Arg">gh</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: a group of collineations

<var class="Arg">gh</var> is a classical generalised hexagon. This attribute returns the full collineation group, equipped with a nice monomorphism.

<div class="example"><pre>
gap> mat := IdentityMat(7,GF(9));
< mutable compressed matrix 7x7 over GF(9) >
gap> form := BilinearFormByMatrix(mat,GF(9));
< bilinear form >
gap> ps := PolarSpace(form);
<polar space in ProjectiveSpace(
6,GF(3^2)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2=0 >
gap> gh := SplitCayleyHexagon(ps);
H(9) in Q(6, 9): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2
gap> group := CollineationGroup(gh);
#I for Split Cayley Hexagon
#I Computing nice monomorphism...
#I Found permutation domain...
<projective collineation group with 18 generators>
gap> time;
19602
gap> HasNiceMonomorphism(group);
true
gap> gh := TwistedTrialityHexagon(2^3);
T(8, 2)
gap> group := CollineationGroup(gh);
#I Computing nice monomorphism...
#I Found permutation domain...
3D_4(8)

</pre></div>

<a id="X7BA462527B2777BC" name="X7BA462527B2777BC"></a>

<h4>12.6 Elation generalised quadrangles</h4>

<a id="X86BD86C77BAAF887" name="X86BD86C77BAAF887"></a>

<h5>12.6-1 Elation generalised quadrangles and Kantor families</h5>

Suppose \(S=(P,B,I)\) is a generalised quadrangle of order \((s,t)\) for which there exists a point \(p\) and a group of collineations \(G\) fixing \(p\) and each line through \(p\), with the extra property that \(G\) acts regularly on the points not collinear with \(p\). Then \(S\) is called an elation generalised quadrangle with base-point \(p\) and elation group \(G\), and \(G\) has order \(s^2t\). Let \(y\) be a fixed point of \(S\), not collinear with \(p\). Denote the \(t+1\) lines incident with \(p\) as \(L_i, i=0 \ldots t\) . Define for each line \(L_i\) the unique point-line pair \((z_i,M_i)\) such that \(L_i I z_i I M_i I y\) . Define the groups \(S_i\) as the subgroups of \(G\) fixing the lines \(M_i\), and define the groups \(S_i^*\) as the subgroups of \(G\) fixing the point \(z_i\) . Define the set \(J = \{S_i: i=0 \ldots t\}\) , and the set \(J^* = \{S_i: i=0 \ldots t\}\) . Since \(S\) is an elation generalised quadrangle, \(J\) is a collection of \(t+1\) subgroups of \(G\) of order \(s\), and each \(S_i^*\) contains has order \(st\) and contains \(S_i\) as a subgroup. Furthermore, the following two conditions are satisfied.

(K1) \(S_iS_j \cap S_k = \{1\}\) , for distinct \(i,j,k\).

(K2) \(S_i \cap S_j^* = \{1\}\) , for distinct \(i,j\).

The pair \((J,J^*)\) is called a 4-gonal family or Kantor family in \(G\).

Remarkably, each Kantor family in a group of order \(s^2t\) gives rise to an elation generalised quadrangle. Kantor families and elation generalised quadrangles are equivalent objects, and one of the motivations to study Kantor families in groups was to find examples of non-classical elation generalised quadrangles.

Given a group \(G\), together with a Kantor family \((J,J^*)\) , a generalised quadrangle is defined as follows.

The points are of three types:

(i) points of type 1 are the elements of \(G\);

(ii) points of of type 2 are the right cosets \(S^*g, s^* \in J^*\)

(iii) the unique point of type (iii) is the symbol \((\infty)\) .

The lines are of two types:

(a) lines of type (a) are the right cosets \(Sg\), \(S \in J\);

(b) Lines of type (b) are the symbols \([S], S \in J\).

Incidence is defined as follows. A point \(g\) of type (i) is incident with each line \(Sg, S \in J\) of type (a). A point of type (ii) \(S^*g\) is incident the line \([S]\) of type (b) and the \(t\) lines of type (a) for which \(Sh \subset S^*g\) . Finally, the unique point of type (iii) is incident with the lines of type (b), and there are no further incidences.

It is shown, see e.g. the standard work in this field of Payne and Thas <a href="chapBib_mj.html#biBFGQ">[PT84]</a>, that this point-line geometry is a generalised quadrangle of order \((s,t)\).

FinInG provides functions to construct elation generalised quadrangles directly from a Kantor family. The constructed generalised quadrangles are generalised polygons in the sense of FinInG, i.e. all generic operations described in Sections <a href="chap12_mj.html#X864C966D8184A9C0">12.3</a> and <a href="chap12_mj.html#X7A13D5EB82E01576">12.4</a>.

<a id="X7CC6903E78F24167" name="X7CC6903E78F24167"></a>

<h5>12.6-2 Categories</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEGQByKantorFamily</code></td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsElementOfKantorFamily</code></td><td class="tdright">( category )</td></tr></table></div>
<code class="code">IsEGQByKantorFamily</code> is a subcategory of <code class="code">IsElationGQ</code>. It contains all elations generalised quadrangles that are constructed from a Kantor family. <code class="code">IsElementOfKantorFamily</code> is a subcategory of <code class="code">IsElementOfGeneralisedPolygon</code>. It contains the elements from generalised quadrangles in the category <code class="code">IsEGQByKantorFamily</code>.

<a id="X820A2D6A84A259FC" name="X820A2D6A84A259FC"></a>

<h5>12.6-3 Kantor families</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsKantorFamily</code>( <var class="Arg">g</var>, <var class="Arg">f</var>, <var class="Arg">fstar</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: true or false

There is no specific way to construct a Kantor family in FinInG. However, given a group \(G\) and two collections of subgroups, <code class="file">IsKantorFamily</code> will check whether the input satisfies the conditions of a Kantor family. If so, the input can be used directly for the operation <code class="file">EGQByKantorFamily</code>.

<a id="X7B80E3AC7DEAF948" name="X7B80E3AC7DEAF948"></a>

<h5>12.6-4 EGQByKantorFamily</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EGQByKantorFamily</code>( <var class="Arg">g</var>, <var class="Arg">f</var>, <var class="Arg">fstar</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a generalised quadrangle

Let <var class="Arg">g</var> be a group and <var class="Arg">f</var> and <var class="Arg">fstar</var> two collections of subgroups, satisfying the conditions of a Kantor family. This operation returns the corresponding elation generalised quadrangle. Note that this operation does not check if the input satisfies the conditions to be a Kantor family, it only checks whether the group <var class="Arg">f[i]</var> is a subgroup of the group <var class="Arg">fstar[i]</var>. In the example below, the use of <code class="file">IsKantorFamily</code> is also demonstrated, and some categories are displayed.

<div class="example"><pre>
gap> g := ElementaryAbelianGroup(27);
<pc group of size 27 with 3 generators>
gap> flist1 := [ Group(g.1), Group(g.2), Group(g.3), Group(g.1*g.2*g.3) ];;
gap> flist2 := [ Group([g.1, g.2^2*g.3]), Group([g.2, g.1^2*g.3 ]), 
> Group([g.3, g.1^2*g.2]), Group([g.1^2*g.2, g.1^2*g.3 ]) ];; 
gap> IsKantorFamily( g, flist1, flist2 );
#I Checking tangency condition...
#I Checking triple condition...
true
gap> egq := EGQByKantorFamily(g, flist1, flist2);
<EGQ of order [ 3, 3 ] and basepoint 0>
gap> CategoriesOfObject(egq);
[ "IsIncidenceStructure", "IsIncidenceGeometry", "IsGeneralisedPolygon",
 "IsGeneralisedQuadrangle", "IsElationGQ", "IsElationGQByKantorFamily" ]
gap> p := Random(Points(egq));
<a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>
gap> CategoriesOfObject(p);
[ "IsElementOfIncidenceStructure", "IsElementOfIncidenceGeometry",
 "IsElementOfGeneralisedPolygon", "IsElementOfKantorFamily" ]

</pre></div>

<a id="X80C93974807A342B" name="X80C93974807A342B"></a>

<h5>12.6-5 Representation of elements and underlying objects</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjectToElement</code>( <var class="Arg">egq</var>, <var class="Arg">t</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjectToElement</code>( <var class="Arg">egq</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasePointOfEGQ</code>( <var class="Arg">egq</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingObject</code>( <var class="Arg">el</var> )</td><td class="tdright">( operation )</td></tr></table></div>
For technical reasons, the underlying objects of the elements of an elation generalised quadrangle constructed from a Kantor family, are not exactly the mathematical objects from the definition. However, these technicalities are almost completely hidden for the user, except for the representation of lines of type (b), which are represented in FinInG by the elements of the collection \(J^*\) (instead of the elements of the collection \(J\)). This change from the original definition has no mathematical implications, since there is a bijective correspondence between the elements of \(J^*\) and \(J\). Notice also that it is only possible to obtain the base-point of an elation generalised quadrangle constructed from a Kantor family through calling the attribute <code class="file">BasePointOfEGQ</code>.

<div class="example"><pre>
gap> g := ElementaryAbelianGroup(27);
<pc group of size 27 with 3 generators>
gap> flist1 := [ Group(g.1), Group(g.2), Group(g.3), Group(g.1*g.2*g.3) ];;
gap> flist2 := [ Group([g.1, g.2^2*g.3]), Group([g.2, g.1^2*g.3 ]),
> Group([g.3, g.1^2*g.2]), Group([g.1^2*g.2, g.1^2*g.3 ]) ];;
gap> egq := EGQByKantorFamily(g, flist1, flist2);
<EGQ of order [ 3, 3 ] and basepoint 0>
gap> h := Random(g);
f1*f2^2
gap> p := ObjectToElement(egq,h);
<a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>
gap> coset := RightCoset(flist1[1],h);
RightCoset(Group( [ f1 ] ),f1*f2^2)
gap> l := ObjectToElement(egq,coset);
<a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>
gap> p * l;
true
gap> S := flist2[2];
<pc group of size 9 with 2 generators>
gap> m := ObjectToElement(egq,S);
<a line of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>
gap> q := BasePointOfEGQ(egq);
<a point of class 3 of <EGQ of order [ 3, 3 ] and basepoint 0>>
gap> m * q;
true
gap> lines := List(Lines(p));
[ <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>> ]
gap> pts1 := List(Points(m));
[ <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 3 of <EGQ of order [ 3, 3 ] and basepoint 0>> ]
gap> pts2 := List(Points(l));
[ <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ]
gap> List(pts2,x->UnderlyingObject(x));
[ f2^2, f1*f2^2, f1^2*f2^2, RightCoset(Group( [ f1, f2^2*f3 ] ),f3^2) ]
gap> UnderlyingObject(q);
0

</pre></div>

<a id="X7DCD7EAB839BD97F" name="X7DCD7EAB839BD97F"></a>

<h5>12.6-6 Elation group and natural action on elements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElationGroup</code>( <var class="Arg">egq</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OnKantorFamily</code>( <var class="Arg">g</var>, <var class="Arg">el</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CollineationAction</code>( <var class="Arg">g</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
The attribute <code class="file">ElationGroup</code> is naturally set upon creation of an elation generalised quadrangle from a Kantor family. The elements of the elation group act "naturally" on the elements of the elation generalised quadrangle. This natural action is implemented in the action function <code class="file">OnKantorFamily</code>. When <var class="Arg">g</var> is the elation group of an elation generalised quadrangle constructed from a Kantor family, the attribute <code class="file">CollineationAction</code> will return the action function <code class="file">OnKantorFamily</code>. The action function makes use of generic GAP operations possible, as is demonstrated in the example.

<div class="example"><pre>
gap> g := ElementaryAbelianGroup(27);
<pc group of size 27 with 3 generators>
gap> flist1 := [ Group(g.1), Group(g.2), Group(g.3), Group(g.1*g.2*g.3) ];;
gap> flist2 := [ Group([g.1, g.2^2*g.3]), Group([g.2, g.1^2*g.3 ]),
> Group([g.3, g.1^2*g.2]), Group([g.1^2*g.2, g.1^2*g.3 ]) ];;
gap> egq := EGQByKantorFamily(g, flist1, flist2);
<EGQ of order [ 3, 3 ] and basepoint 0>
gap> group := ElationGroup(egq);
<pc group of size 27 with 3 generators>
gap> CollineationAction(group) = OnKantorFamily;
true
gap> l := ObjectToElement(egq,RightCoset(flist1[1],One(g)));
<a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>
gap> stab := Stabilizer(group,l,OnKantorFamily);
Group([ f1 ])
gap> pts := List(Points(egq));
[ <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 3 of <EGQ of order [ 3, 3 ] and basepoint 0>> ]
gap> Orbits(group,pts,OnKantorFamily);
[ [ <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a point of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a point of class 3 of <EGQ of order [ 3, 3 ] and basepoint 0>> ] ]
gap> lines := List(Lines(egq));
[ <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ]
gap> Orbits(group,lines,OnKantorFamily);
[ [ <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>>,
 <a line of class 1 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a line of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a line of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a line of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ],
 [ <a line of class 2 of <EGQ of order [ 3, 3 ] and basepoint 0>> ] ]

</pre></div>

<a id="X8462F11584736E32" name="X8462F11584736E32"></a>

<h5>12.6-7 Kantor families, q-clans, and elation generalised quadrangles</h5>

Let \(C=\{A_i: i=1 \ldots q\}\) be a set of \(q\) distinct upper triangle \(2 \times 2\) matrices over the finite field \(\mathrm{GF}(q)\). Then \(C\) is called a q-clan if \(A_r - A_t\) is anisotropic for \(r \neq t\).

Define \(G = \{(\alpha,c,\beta): \alpha,\beta \in GF(q)^2, c \in GF(q)\}\) , and define the binary operator \(\times\) as \((\alpha,c,\beta) \times (\alpha',c',\beta') = (\alpha + \alpha',c+c'+\beta \cdot \alpha'^{T},\beta + \beta')\) . Then \(G,\times\) is a group with center \(Z(G)=\{(0,c,0): c \in GF(q)\}\). Consider a q-clan \(C=\{A_i: i=1 \ldots q\}\), define \(K_i = A_i+A_i^T\). Now define the following subgroups of \(G\)

\(A(i) = \{(\alpha,\alpha A_t,\alpha K_t): \alpha \in GF(q)^2, i=1\ldots q\}\) and \(A(\infty) = \{(0,0,\gamma): \gamma \in GF(q)^2\}\) , and

\(A^*(i) = \{(\alpha,b,\alpha K_t): \alpha \in GF(q)^2, b \in GF(q), i=1\ldots q\}\) and \(A^*(\infty) = \{(0,b,\gamma): \gamma \in GF(q)^2, b \in GF(q)\}\)

Define \(J=\{A(i):i=1\ldots q\} \cup \{A(\infty)\}\) and \(J^*=\{A^*(i):i=1\ldots q\} \cup \{A^*(\infty)\}\)

A combination of results of Payne and Kantor yield the famous theorem that (J,J^*) is a Kantor family in \(G\) if and only if \(C\) is a q-clan. FinInG provides functionality to construct q-clans and to construct the corresponding Kantor family. As such, elation generalised quadrangles can directly constructed from q-clans.

<a id="X806527387F1D5D42" name="X806527387F1D5D42"></a>

<h5>12.6-8 qClan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ qClan</code>( <var class="Arg">list</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a q-clan.

Given a list <var class="Arg">list</var> of \(2 \times\) matrices over the field <var class="Arg">f</var>, it is checked if the matrices in the list satisfy the condition to constitute a q-clan over \(f\). If so, the q-clan is returned.

<a id="X858A1EA8843BEC13" name="X858A1EA8843BEC13"></a>

<h5>12.6-9 Particular q-clans</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinearqClan</code>( <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FisherThasWalkerKantorBettenqClan</code>( <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KantorMonomialqClan</code>( <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KantorKnuthqClan</code>( <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FisherqClan</code>( <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: a q-clan

Some famous q-clans are built in. We refer to ... for more information on these.

<a id="X7C59CF8079EA33D3" name="X7C59CF8079EA33D3"></a>

<h5>12.6-10 KantorFamilyByqClan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ KantorFamilyByqClan</code>( <var class="Arg">clan</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A Kantor family.

The Kantor family constructed from a q-clan will be a matrix group together with the corresponding collections \(J\) and J^* .

<a id="X830222B084BF866D" name="X830222B084BF866D"></a>

<h5>12.6-11 EGQByqClan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EGQByqClan</code>( <var class="Arg">clan</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: An elation generalised quadrangle constructed from a q-clan.

Given a q-clan <var class="Arg">clan</var>, the operation <code class="file">KantorFamilyByqClan</code> will be used to construct the Kantor family from <var class="Arg">clan</var>, followed by the construction of the elation generalised quadrangle using the operation <code class="file">EGQByKantorFamily</code>. The first example shows also the use of <code class="file">qClan</code>, and shows that a linear q-clan yields a classical generalised quadrangle.

<div class="example"><pre>
gap> f := GF(3);
GF(3)
gap> id := IdentityMat(2, f);;
gap> list := List( f, t -> t * id );;
gap> clan := qClan(list,f);
<q-clan over GF(3)>
gap> egq := EGQByqClan(clan);
#I Computed Kantor family. Now computing EGQ...
<EGQ of order [ 9, 3 ] and basepoint 0>
gap> incgraph := IncidenceGraph(egq);;
#I Computing incidence graph of generalised polygon...
#I Using elation of the collineation group...
gap> group := AutomorphismGroup(incgraph);
<permutation group with 6 generators>
gap> Order(group);
26127360
gap> Order(CollineationGroup(HermitianPolarSpace(3,9)));
26127360
gap> clan := KantorKnuthqClan(9);
<q-clan over GF(3^2)>
gap> egq := EGQByqClan(clan);
#I Computed Kantor family. Now computing EGQ...
<EGQ of order [ 81, 9 ] and basepoint 0>
gap> clan := FisherThasWalkerKantorBettenqClan(11);
<q-clan over GF(11)>

</pre></div>

<a id="X7FAE48497B2F658A" name="X7FAE48497B2F658A"></a>

<h5>12.6-12 BLT-sets, flocks, q-clans, and elation generalised quadrangles</h5>

A flock is a partition of the points of a quadratic cone in \(\mathrm{PG}(3,q)\) minus its vertex into conics. Each conic is determined by a plane, and each plane is determined uniquely by a triple of elements of the field \(\mathrm{GF}(q)\). So a flock is determined by \(q\) such triples. Remarkably, as was shown by J.A. Thas, the conditions for these triples to constitute a flock, are exactly the same conditions for these triples to constitute \(q\) upper triangle matrices making a q-clan over \(\mathrm{GF}(q)\). Hence, q-clans and flocks, and thus flocks and elation generalised quadrangles, are equivalent objects.

The quadratic cone can be embedded as a hyperplane section into the parabolic quadric \(\mathrm{Q}(4,q)\). L. Bader, G. Lunardon and J.A. Thas observed that a set of \(q\) points of \(\mathrm{Q}(4,q)\) can be constructed from the \(q\) planes determining the flock, and this set of points satisfies certain geometric conditions. Such a set is called a BLT-set. Dually, a BLT-set corresponds to a set of lines of \(\mathrm{W}(3,q)\). Furthermore, from this BLT-set of lines, it is possible to construct directly an elation generalised quadrangle from carefully selecting points and lines from the symplectic space \(\mathrm{W}(5,q)\). This construction is called the Knarr construction.

Consider the symplectic polar space \(\mathrm{W}(5,q)\) and choose a point \(P \in W(5,q)\). Its tangent space is a \(4\)-dimensional space \(F\). Embed \(\mathrm{W}(3,q)\) in a solid of \(F\) not on the point \(p\), and let \(L\) be the set of BLT lines. Each line spans with \(p\) a plane of \(\mathrm{W}(5,q)\), call these \(q\) planes the BLT-planes. Now we can define the points an lines of the elation generalised quadrangle as follows.

Points of type (i) are the points of \(W(5,q) \setminus F\), points of type (ii) are the lines of each BLT-plane not on \(p\) and the unique point of type (iii) is the point \(p\). Lines of type (a) are the planes of \(\mathrm{W}(5,q)\) meeting a BLT-plane in a line. Note that no plane of \(\mathrm{W}(5,q)\) meeting two BLT planes in a line can exist. Lines of type (b) are the BLT-planes. Incidence is the natural incidence (so the incidence inherited) from the polar space \(\mathrm{W}(5,q)\), and this geometry is a elation generalised quadrangle with base-point \(p\) and of order \((q^2,q)\) .

FinInG provides functions to construct elation generalised quadrangles using this model from BLT-sets, and provides a function to compute a BLT set from a q-clan directly. The advantage of constructing a elation generalised from elements of Lie geometries is the availability of the underlying projective groups and their action on elements of Lie geometries.

<a id="X7A8438537D2F1374" name="X7A8438537D2F1374"></a>

<h5>12.6-13 IsEGQByBLTSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEGQByBLTSet</code></td><td class="tdright">( category )</td></tr></table></div>
<code class="code">IsEGQByBLTSet</code> is a subcategory of <code class="code">IsElationGQ</code>. It contains all elations generalised quadrangles that are constructed from a BLT set.

<a id="X7ECC1871866AC286" name="X7ECC1871866AC286"></a>

<h5>12.6-14 BLTSetByqClan</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BLTSetByqClan</code>( <var class="Arg">clan</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: A BLT-set.

The BLT-set is a set of points of the parabolic quadric in \(PG(4,q)\) with particular equation \(2X_1x_5+2X_2X_4+w^{(q+1)/2}=0\) , where \(w\) is a primitive element of the underlying field of <var class="Arg">clan</var>.

<div class="example"><pre>
gap> clan := KantorKnuthqClan(9);
<q-clan over GF(3^2)>
gap> blt := BLTSetByqClan(clan);
[ <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0> ]
gap> clan := FisherThasWalkerKantorBettenqClan(11);
<q-clan over GF(11)>
gap> blt := BLTSetByqClan(clan);
[ <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0> ]

</pre></div>

<a id="X84700DDF80B39332" name="X84700DDF80B39332"></a>

<h5>12.6-15 EGQByBLTSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EGQByBLTSet</code>( <var class="Arg">blt</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Returns: An elation generalised quadrangle.

<var class="Arg">blt</var> is a BLT-set, this operation returns an elation generalised quadrangle constructed as described above consisting of elements of \(\mathrm{W}(5,q)\). Notice in the example that computing the full collineation group of the GQ constructed directly from the q-clan (hence a group coset geometry) is substantially slower than computing the full collineation group of the GQ constructed from the BLT-set.

<div class="example"><pre>
gap> clan := LinearqClan(3);
<q-clan over GF(3)>
gap> bltset := BLTSetByqClan( clan);
[ <a point in Q(4, 3): -x_1*x_5-x_2*x_4+x_3^2=0>,
 <a point in Q(4, 3): -x_1*x_5-x_2*x_4+x_3^2=0>,
 <a point in Q(4, 3): -x_1*x_5-x_2*x_4+x_3^2=0>,
 <a point in Q(4, 3): -x_1*x_5-x_2*x_4+x_3^2=0> ]
gap> egq := EGQByBLTSet( bltset );
#I Now embedding dual BLT-set into W(5,q)...
#I Computing elation group...
<EGQ of order [ 9, 3 ] and basepoint in W(5, 3 ) >
gap> p := BasePointOfEGQ(egq);
<a point in <EGQ of order [ 9, 3 ] and basepoint in W(5, 3 ) >>
gap> UnderlyingObject(p);
<a point in W(5, 3)>
gap> l := Random(Lines(egq));
<a line in <EGQ of order [ 9, 3 ] and basepoint in W(5, 3 ) >>
gap> UnderlyingObject(l);
<a plane in W(5, 3)>
gap> group := ElationGroup(egq);
<projective collineation group with 5 generators>
gap> Order(group);
243
gap> CollineationGroup(egq);
#I Using elation group to enumerate elements
#I Using elation group to enumerate elements
#I Computing incidence graph of generalised polygon...
#I Using elation of the collineation group...
#I Using elation group to enumerate elements
<permutation group of size 26127360 with 7 generators>
gap> time;
147
gap> egq := EGQByqClan(clan);
#I Computed Kantor family. Now computing EGQ...
<EGQ of order [ 9, 3 ] and basepoint 0>
gap> CollineationGroup(egq);
#I Computing incidence graph of generalised polygon...
#I Using elation of the collineation group...
<permutation group of size 26127360 with 6 generators>
gap> time;
1139

</pre></div>

<a id="X79186E29799355BD" name="X79186E29799355BD"></a>

<h5>12.6-16 DefiningPlanesOfEGQByBLTSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DefiningPlanesOfEGQByBLTSet</code>( <var class="Arg">egq</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
For an elation generalised quadrangle in the category <code class="code">IsEGQByBLTSet</code> (constructed from a BLT-set), the planes of the polar space \(\mathrm{W}(5,q)\), as described in the introduction, determine the generalised quadrangle completely. This attribute returns these \(q\) planes of \(\mathrm{W}(5,q)\).

<div class="example"><pre>
gap> clan := KantorKnuthqClan(9);
<q-clan over GF(3^2)>
gap> blt := BLTSetByqClan(clan);
[ <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0>,
 <a point in Q(4, 9): -x_1*x_5-x_2*x_4+Z(3^2)^5*x_3^2=0> ]
gap> egq := EGQByBLTSet(blt);
#I Now embedding dual BLT-set into W(5,q)...
#I Computing elation group...
<EGQ of order [ 81, 9 ] and basepoint in W(5, 9 ) >
gap> DefiningPlanesOfEGQByBLTSet(egq);
[ <a plane in W(5, 9)>, <a plane in W(5, 9)>, <a plane in W(5, 9)>,
 <a plane in W(5, 9)>, <a plane in W(5, 9)>, <a plane in W(5, 9)>,
 <a plane in W(5, 9)>, <a plane in W(5, 9)>, <a plane in W(5, 9)>,
 <a plane in W(5, 9)> ]

</pre></div>

<a id="X80C93974807A342B" name="X80C93974807A342B"></a>

<h5>12.6-17 Representation of elements and underlying objects</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjectToElement</code>( <var class="Arg">egq</var>, <var class="Arg">t</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ObjectToElement</code>( <var class="Arg">egq</var>, <var class="Arg">obj</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingObject</code>( <var class="Arg">el</var> )</td><td class="tdright">( operation )</td></tr></table></div>
The underlying objects of the elements of an elation generalised quadrangle in the category <code class="code">IsEGQByBLTSet</code> are elements of the polar space \(\mathrm{W}(5,q)\) in its standard representation. These elements can be used as underlying object to construct elements of <var class="Arg">egq</var>. The example also demonstrates the use of <code class="file">DistanceBetweenElements</code>.

<div class="example"><pre>
gap> clan := FisherThasWalkerKantorBettenqClan(11);
<q-clan over GF(11)>
gap> blt := BLTSetByqClan(clan);
[ <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0>,
 <a point in Q(4, 11): Z(11)*x_1*x_5+Z(11)*x_2*x_4+Z(11)^6*x_3^2=0> ]
gap> egq := EGQByBLTSet(blt);
#I Now embedding dual BLT-set into W(5,q)...
#I Computing elation group...
<EGQ of order [ 121, 11 ] and basepoint in W(5, 11 ) >
gap> planes := DefiningPlanesOfEGQByBLTSet(egq);
[ <a plane in W(5, 11)>, <a plane in W(5, 11)>, <a plane in W(5, 11)>,
 <a plane in W(5, 11)>, <a plane in W(5, 11)>, <a plane in W(5, 11)>,
 <a plane in W(5, 11)>, <a plane in W(5, 11)>, <a plane in W(5, 11)>,
 <a plane in W(5, 11)>, <a plane in W(5, 11)>, <a plane in W(5, 11)> ]
gap> p := BasePointOfEGQ(egq);
<a point in <EGQ of order [ 121, 11 ] and basepoint in W(5, 11 ) >>
gap> up := UnderlyingObject(p);
<a point in W(5, 11)>
gap> ps := SymplecticSpace(5,11);
W(5, 11)
gap> uq := VectorSpaceToElement(ps,[1,1,0,0,0,0]*Z(11)^0);
<a point in W(5, 11)>
gap> q := ObjectToElement(egq,1,uq);
<a point in <EGQ of order [ 121, 11 ] and basepoint in W(5, 11 ) >>
gap> DistanceBetweenElements(p,q);
4
gap> l := ObjectToElement(egq,2,planes[1]);
<a line in <EGQ of order [ 121, 11 ] and basepoint in W(5, 11 ) >>
gap> DistanceBetweenElements(p,l);
1
gap> DistanceBetweenElements(q,l);
3
gap> um := VectorSpaceToElement(ps,[[1,0,0,0,1,1],[0,1,0,9,1,0],[0,0,1,9,9,9]]*Z(11)^0);
<a plane in W(5, 11)>
gap> m := ObjectToElement(egq,2,um);
<a line in <EGQ of order [ 121, 11 ] and basepoint in W(5, 11 ) >>
gap> DistanceBetweenElements(p,m);
3
gap> DistanceBetweenElements(q,m);
3
gap> DistanceBetweenElements(l,m);
2

</pre></div>

<a id="X7C57B0BD7A2E5877" name="X7C57B0BD7A2E5877"></a>

<h5>12.6-18 CollineationSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CollineationSubgroup</code>( <var class="Arg">egq</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
For an elation generalised quadrangle in the category <code class="code">IsEGQByBLTSet</code> (constructed from a BLT set), the planes of the polar space \(\mathrm{W}(5,q)\), as described in the introduction, determine the generalised quadrangle completely. The setwise stabiliser of these planes in the collineation group of \(\mathrm{W}(5,q)\) is a subgroup of the completely collineation group of the elation generalised quadrangle, and can be computed much faster than the complete collineation group. This attribute returns this setwise stabiliser. The returned group is equipped with the <code class="file">CollineationACtion</code> attribute. If <code class="file">CollineationSubgroup</code> is computed, this group will be used instead of the elation group to compute the incidence graph.

<div class="example"><pre>
gap> clan := FisherThasWalkerKantorBettenqClan(5);
<q-clan over GF(5)>
gap> blt := BLTSetByqClan(clan);
[ <a point in Q(4, 5): Z(5)*x_1*x_5+Z(5)*x_2*x_4+Z(5)^3*x_3^2=0>,
 <a point in Q(4, 5): Z(5)*x_1*x_5+Z(5)*x_2*x_4+Z(5)^3*x_3^2=0>,
 <a point in Q(4, 5): Z(5)*x_1*x_5+Z(5)*x_2*x_4+Z(5)^3*x_3^2=0>,
 <a point in Q(4, 5): Z(5)*x_1*x_5+Z(5)*x_2*x_4+Z(5)^3*x_3^2=0>,
 <a point in Q(4, 5): Z(5)*x_1*x_5+Z(5)*x_2*x_4+Z(5)^3*x_3^2=0>,
 <a point in Q(4, 5): Z(5)*x_1*x_5+Z(5)*x_2*x_4+Z(5)^3*x_3^2=0> ]
gap> egq := EGQByBLTSet(blt);
#I Now embedding dual BLT-set into W(5,q)...
#I Computing elation group...
<EGQ of order [ 25, 5 ] and basepoint in W(5, 5 ) >
gap> coll := CollineationSubgroup(egq);
#I Computing adjusted stabilizer chain...
<projective collineation group with 13 generators>
gap> Order(coll);
9000000
gap> act := CollineationAction(coll);
function( el, x ) ... end
gap> orbs := Orbits(coll,Points(egq),act);;
#I Using elation group to enumerate elements
gap> List(orbs,x->Length(x));
[ 1, 3125, 150 ]
gap> el := ElationGroup(egq);
<projective collineation group with 5 generators>
gap> orbs := Orbits(el,Points(egq),act);;
#I Using elation group to enumerate elements
gap> List(orbs,x->Length(x));
[ 1, 3125, 25, 25, 25, 25, 25, 25 ]

</pre></div>

<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap11_mj.html">[Previous Chapter]</a> <a href="chap13_mj.html">[Next Chapter]</a> </div>

<div class="chlinkbot">Goto Chapter: <a href="chap0_mj.html">Top</a> <a href="chap1_mj.html">1</a> <a href="chap2_mj.html">2</a> <a href="chap3_mj.html">3</a> <a href="chap4_mj.html">4</a> <a href="chap5_mj.html">5</a> <a href="chap6_mj.html">6</a> <a href="chap7_mj.html">7</a> <a href="chap8_mj.html">8</a> <a href="chap9_mj.html">9</a> <a href="chap10_mj.html">10</a> <a href="chap11_mj.html">11</a> <a href="chap12_mj.html">12</a> <a href="chap13_mj.html">13</a> <a href="chap14_mj.html">14</a> <a href="chapA_mj.html">A</a> <a href="chapB_mj.html">B</a> <a href="chapC_mj.html">C</a> <a href="chapBib_mj.html">Bib</a> <a href="chapInd_mj.html">Ind</a> </div>

<hr />
generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a>
</body>
</html>

Messung V0.5 in Prozent

¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.55Angebot (Wie Sie bei der Firma Beratungs- und Dienstleistungen beauftragen können 2026-05-01) ¤

*Eine klare Vorstellung vom Zielzustand

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.