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< title>GAP (FinInG) - Chapter 12: Generalised Polygons</ title>
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<div class="ChapSects"><a href="chap12_mj.html#X7E1F10767D2A4D6A">12 <span class="Heading">Generalised Polygons</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 31
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X8141DFFF7B352E2D">12.1-1 IsGeneralisedPolygon</a></span>
class=C<br/>< =">&&java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 212
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X86C124B3804C7569">12.1-3 IsWeakGeneralisedPolygon</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X8540C04887CF8824">12.1-4 <span class="Heading">Subcategories in <code class="code">IsProjectivePlaneCategory 1, ,3,,5,6,7 , ,10,14,21 ,26,28,32, 34 40 ,44 ,
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7CF10DAE7847939D">12.1-5 <span class="Heading">Subcategories in <code class="code">IsGeneralisedQuadrangle</code></span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7A2ABB0B818E25C5">12.1-6 IsClassicalGeneralisedHexagon</a></span>
</div></div>
< class=">span class=>span class=n"&bsp;href"html#java.lang.StringIndexOutOfBoundsException: Range [119, 118) out of bounds for length 206
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X849A83597CC78B5F">12.2-1 GeneralisedPolygonByBlocks</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X8751B0FB8500D0B1">12.2-2 GeneralisedPolygonByIncidenceMatrix</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7A6169127CE8ABC0">12.2-3 GeneralisedPolygonByElements</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X864C966D8184A9C0">12.3 <span class="Heading">Attributes and operations for generalised polygons</span></a>
<
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X84F59A2687C62763">12.3-1 Order</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X836C15B57968A511">12.3-2 IncidenceGraphAttr</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X815BE6D57D623452">12.3-3 IncidenceGraph</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 17, 18, 19, 20, 26, 29, 34, 42, 43, 443 4 5,,78 9 , , 17,18,,2026 29 , ,43 44
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X83FF6FA0790D5747">12.3-5 CollineationGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 31
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7D75521986B958DA">12.3-7 BlockDesignOfGeneralisedPolygon</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7A13D5EB82E01576">12.4 <span class="Heading">Elements of generalised polygons</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 31
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X858ADA3B7A684421">12.4-2 Size</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7E9B2A217DBF2849">12.4-3 <span class="Heading">Creating elements from objects and retrieving objects from elements</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X83B0FA9E7AE3DF01">12.4-4 <span class="Heading">Incidence</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X875BE2957FAF6209">12.4-5 Span</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X8469B54180FE1E4C">12.4-6 Meet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X8154BB13844AA0FD">12.4-7 <span class="Heading">Shadow elements</span></a>
<span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7807682E86D3BE10">12.4-8 DistanceBetweenElements</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7934EB788049B533">12.5 <span class="Heading">The classical generalised hexagons</span></a>
</span>
<div class="ContSSBlock">
< ="ContSS"<br/<java.lang.StringIndexOutOfBoundsException: Range [40, 38) out of bounds for length 210
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7C35384980AA9B77">12.5-2 IsLieGeometry</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7A05FF0079F55291">12.5-3 SplitCayleyHexagon</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="[1 2 3 4 , , 7 9 10,15,17,18,23, 26, 31, 35, 38, 39, 40, 41,
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X807A60A28264BD93">12.5-5 vector spaceToElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7809B7C183FA7213">12.5-6 ObjectToElement</a></span>
<span "ontSS><br /><span class="nocss"> </span><a href="chap12_mj.html#X810D4D6D87069697">12.5-7 UnderlyingObject</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X87BDB89B7AAFE8AD"><code>12.5-8 \in</code></a></span>
<span class="[ 1, 2, 3 4,5 , 7,8 9 , , 17 18 , 29, 36,38, 39,40,41,44
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X83FF6FA0790D5747">12.5-10 CollineationGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12_mj.html#X7BA462527B2777BC">12.6 <span class="Heading">Elation generalised quadrangles</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X86BD86C77BAAF887">12.6-1 <span class="Heading">Elation generalised quadrangles and Kantor families</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7CC6903E78F24167">12.6-2 <span class="Heading">Categories</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X820A2D6A84A259FC">12.6-3 <span class="Heading">Kantor families</span></a>
<span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7B80E3AC7DEAF948">12.6-4 EGQByKantorFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X80C93974807A342B">12.6-5 <span class="Heading">Representation of elements and underlying objects</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7DCD7EAB839BD97F">12.6-6 <span class="Heading">Elation group and natural action on elements</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X8462F11584736E32">12.6-7 <span class="Heading">Kantor families, q-clans, and elation generalised quadrangles</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X806527387F1D5D42">12.6-8 qClan</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X858A1EA8843BEC13">12.6-9 <span class="Heading">Particular q-clans</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7C59CF8079EA33D3">12.6-10 KantorFamilyByqClan , 52 56 , 60,63 ]java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<span class="ContSS"><br /[ 1 2,3,4 , 6 , ,9,10 15 17,20,,26,29 ,39 ,42,,
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7FAE48497B2F658A">12.6-12 <span class="Heading">BLT-sets, flocks, q-clans, and elation generalised quadrangles</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7A8438537D2F1374">12.6-13 IsEGQByBLTSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7ECC1871866AC286">12.6-14 BLTSetByqClan</a></span>
<span class"">br >span class="> nbsp;/>< =chap12_mj.html#X84700DDF80B39332">12.6-15 EGQByBLTSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X79186E29799355BD">12.6-16 DefiningPlanesOfEGQByBLTSet</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X80C93974807A342B">12.6-17 <span class="Heading">Representation of elements and underlying objects</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap12_mj.html#X7C57B0BD7A2E5877">12.6-18 CollineationSubgroup</a></span>
</div></div>
</div>
<h3>12 <span class="Heading">Generalised Polygons</span><[1 2 3, , , , 7 8 9 , 15 19,26, 31,35,36,39 40,41 46,47,java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
<p>A <em>generalised n-gon< , 55 56,57 58 , 62 ,
<p><strong class="pkg">FinInG</strong> provides some basic functionality to deal with generalised polygons as incidence geometries. A lot of non-trivial interaction with the package <strong class="pkg">GRAPE</strong> has been very useful and even necessary. Currently, generic functions to create generalised polygons, to create java.lang.StringIndexOutOfBoundsException: Index 337 out of bounds for length 77
<p><a id="X7CC6903E78F24167" name="X7CC6903E78F24167"></a></p>
<h4>12.1 <span class="Heading">Categories47,,54, ,57 62, ,
<p><a id="X8141DFFF7B352E2D" name="X8141DFFF7B352E2D"></a></p>
<h5>12.1-1 IsGeneralisedPolygon</h5>
<div class="func"><table 9,, ,22 , 25 2931 , 39,42,45,46,
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGeneralisedPolygonRep</code></td><td class="tdright">( representation )</td></tr></table></div>
<p>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>.</p>
<p><a id="X832E75AE7CCC5BB2" name="X832E75AE7CCC5BB2"></a></p>
<h5>12.1-2 <span class="Heading">Subcategories in <code class="code">IsGeneralisedPolygon</code></span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; IsGeneralisedOctagon49, ,52,, , 60 62],
<p>All generalised polygons in <strong class="pkg">FinInG</strong> 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 1,2,3, , 5, , 7, 8, 9, 10, 15, 26, 27,28,31 34 38,39 43 44,47
<p><a id="X86C124B3804C7569" name="X86C124B3804C7569"></a></[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 26, 31, 33, 36, 39, 43, 47 ,
<h5>12.1-3 IsWeakGeneralisedPolygon</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 31
<p><code class="code">IsWeakGeneralisedPolygon</code> is the category for weak generalised polygons.</p>
<p><a id="X8540C04887CF8824" name="X8540C04887CF8824"></a></p>
<h5>12.1-4 <span class="Heading">Subcategories in <code class="code">IsProjectivePlaneCategory</code></span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsDesarguesianPlane</code></td><td class="tdright">( category )</td></tr></table></div>
<p><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>.</p>
<p><a id="X7CF10DAE7847939D" name="X7CF10DAE7847939D"></a></p>
<eading""> in< class="ode"IsGeneralisedQuadrangle/</<h5
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsClassicalGQ</code></td><td class="tdright">( category )</td></tr></table></div>
< class=func"><tableclass=func" "100"><>td =""<class=func>‣ IsElationGQ<code>/< class="tdright"( category )</td></tr></table></div>
<p><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"><span class="RefLink">12.6</span></a></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := SymplecticSpace(3,2);</span>
W(3, 2)
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGeneralisedPolygon(gp);</span>
true
<span class=GAPprompt">gap&t;/span><spanclass"">IsGeneralisedQuadrangle(gp);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClassicalGQ(gp);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGeneralisedPolygonRep(gp);</span>
false
</pre></div>
<p><a id="X7A2ABB0B818E25C5" name="X7A2ABB0B818E25C5"></a></p>
<h5>12.1-6 IsClassicalGeneralisedHexagon</50,51,52 53 59, ,64 ,
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsClassicalGeneralisedHexagon</code></td><td class="tdright">( category )</td></1,2,3,4,5 , 7, 8, 99,10 16,18,, ,28,29,31 32,37,38 46
<p><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 <span class="SimpleMath">\(\mathrm{Q}^+(7,q)\)</span>. The Split Cayley hexagon is embedded in the parabolic quadric <span class="SimpleMath">\(\mathrm{Q}(6,q)\)</span>, or 1 2, 3 4,5,6,7 ,, 10,, 19 21,22 ,24 28 ,48 50,52
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := SplitCayleyHexagon(3);</span>
H(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGeneralisedHexagon(gp);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsClassicalGeneralisedHexagon(gp);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLieGeometry(gp);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGeneralisedPolygonRep(gp);</span>
true
</pre></div>
<p><a id="X8614D6A779F9B1AA" name="X8614D6A779F9B1AA"></a></p>
<h4>12.2 <span class="Heading">Generic functions to create generalised polygons</span></h4>
<p><a id="X849A83597CC78B5F" name="X849A83597CC78B5F"></a></p>
<h5>12.2-1 GeneralisedPolygonByBlocks</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GeneralisedPolygonByBlocks</code>( <var class="Arg">l</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a generalised polygon</p>
<p>The argument <var class="Arg">l</var> is a finite homogeneous list consisting of ordered sets of a common size <span class="SimpleMath">\(n+1\)</span>. This operation will assume that each element of <var class="Arg">l</var> represents a line of the generalised polygon. Its,53 ,59, ,63],
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">blocks := [ </span>
<span class="GAPprompt">></ 52, 53, 59, 61 62, 6364]
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 14, 15, 16, 17 ], [ 1, 18, 19, 20, 21 ], [ 2, 6, 10, 14, 18 ], </span>
<span class="n class="GAPprompt">></span> <span class="GAPinput"> [ 2, 7, 11, 15, 19 ], [ 2, 8, 12, 16, 20 ], [ 2, 9, 13, 17, 21 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 3, 6, 11, 16, 21 ], [ 3, 7, 10, 17, 20 ], [ 3, 8, 13, 14, 19 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 3, 9, 12, 15, 18 ], [ 4, 6, 12, 17, 19 ], [ 4, 7, 13, 16, 18 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 4, 8, 10, 15, 21 ], [ 4, 9, 11, 14, 20 ], [ 5, 6, 13, 15, 20 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 5, 7, 12, 14, 21 ], [ 5, 8, 11, 17, 18 ], [ 5, 9, 10, 16, 19 ] ];;</span>
<span 48, 49, 52,59 63 64],
<projective plane order 4>
</pre></div>
<p><a id="X8751B0FB8500D0B1" name="X8751B0FB8500D0B1"></a 1 2,3 4 5,6789 ,16 19 21,22 , , 28 3246 47, ,
<h5>12.2-2 GeneralisedPolygonByIncidenceMatrix</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GeneralisedPolygonByIncidenceMatrix</code>( <var class="Arg">incmat</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a generalised polygon</p>
<p>The argument <span class="SimpleMath">\(incmat\)</span> 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>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">incmat := [ </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [1, 2, ,45 ,7 ,9 10 ,19 21 22, 25,,28 ,
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 1, 0, 0, 0, 0, 1, 0, 48 ,52, 53, 58,,60, 63 ],
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, , 51,,53 62 ,64 ,
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0,[ , 29,30,,34, 3537, ,
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 0, 1, 0, 1, 0, [ 1 2 3,4,5 67 8 , 10,16,,,29, ,32 34,, 37,,46java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
<,2, 3 ,5,,7 , , ,16, ,24 , ,32,37, 44, 46, 53, 54,java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1 ], </span>
<pan class"APprompt">>/span> <span class="GAPinput"> [ 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0 ] ];;</span>
<span class="[ ,2 3 4 , 6, ,8 9 ,17,18,20 ,25 29 30 32 33,, ,java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
<projective plane order 4>
</pre></div>
<p><a id="X7A6169127CE8ABC0" name="X7A6169127CE8ABC0"></a></p>
<h5>12.2-3 GeneralisedPolygonByElements</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GeneralisedPolygonByElements</code>( <var class="Arg">pts</var>, <var class="Arg">lns</var>, <var class="Arg">inc<[ 1 2,3 ,,6 , 89,,17,, , 22 23 25 , 32 33,35 ,
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GeneralisedPolygonByElements</code ,,3,4,5 ,7 ,9,10, 17 , , ,31, ,39 ,44,47
<p>Returns: a generalised java.lang.StringIndexOutOfBoundsException: Range [2, 33) out of bounds for length 31
<p>The argument <var class="Arg">pts</var>, <var class="Arg">lns</var> and <var class="Arg">inc</var> are respectively a set[1,2 3 4,5 6, 7 8, 9 , 17 19, 20 25 26, 28,3133 34,36 39,
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(2,25);</span>
ProjectiveSpace(2,[,2 3 ,5,7,8,9,10,17, 19,20, 25, 31,33 35,38 41,4243,
<span class="GAPprompt">gap></span> <span class="GAPinput">pts := Set(Points(pg));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">lns := Set(Lines46 52,,55 ,61 ,
<span class="GAPprompt">gap></span> <span class="GAPinput">inc := \*;</span>
<Operation "*">
< class="GAPprompt>/><spanclass=GAPinput> : java.lang.StringIndexOutOfBoundsException: Range [97, 96) out of bounds for length 117
<projective plane order 25>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
26427
<span class="GAPprompt">gap></span> <span class="GAPinput">grp := CollineationGroup(pg);</span>
The FinInG collineation group PGammaL(3,25)
<span class="GAPprompt">gap></span> <span class="GAPinput">act := OnProjSubspaces;</span>
java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 27
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByElements(pts,lns,inc,grp,act);</span>
<48, 50, 52,, ,63 ,
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
127
<span class="GAPprompt">gap></span> <span class="GAPinput">q := 4;</span>
4
<span class="GAPprompt">gap></[ ,2 , 4 5,6, 7, 8,9, 10, 18,20 23, 27 29, 30, 36,42 44,46,
[ <a point in Q(2, 4)>, <a point in Q(2, 4)>, <a point in Q( 50 52 54,, 58,59, , 63],
<a point in Q(2, 4)>, <a point in Q(2, 4)> ]
(3q;<>
ProjectiveSpace(3, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">hyp := HyperplaneByDualCoordinates(pg,[1,0,0,0]*Z(q)^0);</span>
<a plane in ProjectiveSpace(3, 4)>
<span class="44 49,52,53 ,59 63]java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<geometry morphism from <All elements of ProjectiveSpace(2,
4)> to <All elements of ProjectiveSpace(3, 4)>>
<spanclass=GAPprompt"gap&t;/span><span ="O=java.lang.StringIndexOutOfBoundsException: Range [72, 71) out of bounds for length 98
<span class="GAPprompt">gap></span> <span class="GAPinput">group := CollineationGroup(pg);</span>
The FinInG collineation group PGammaL(4,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">stab := FiningSetwiseStabiliser(group,O);</span>
#I Computing adjusted stabilizer chain...
<projective collineation group with 6 generators>
<span class="GAPprompt">gap></ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 19, 2022, 25, 31,3233, 35, 38,40,41
<span class="GAPprompt">gap></span> <span class="GAPinput">tangents := List(conic,x->TangentSpace(x)^em);</span>
[ <a line in ProjectiveSpace(3, 4 , ,49, 54 58 61]
<a line in ProjectiveSpace(3, 4)>, <a line in ProjectiveSpace(3, 4)>,
<[ 1,2 3, , 5, 6, 7 , 9,10,19 20, ,26 31 ,44,
<span class="GAPprompt">gap></span> <span class="GAPinput">planes := List(tangents,x->Filtered(Planes(x),y->not y in hyp));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">points2 := Union(planes);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">points3 := [hyp];</span>
[ <a plane in ProjectiveSpace(3, 4)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">linesa := Union(List(O,x->java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 31
<span class="GAPprompt">gap></span> <span class="GAPinput">linesb := Set(O);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pts :=[ ,2 , , 5, , 7, , ,1019, 26,28,31,34,39,40,42,44, 45
<span class="GAPprompt">gap></span> <span class="GAPinput">lns := Union(linesa,linesb);;</span>47,48, ,54, 55, 56,63],
<span class="GAPprompt">gap></span> <span class="GAPinput">inc := \*;</span>
<Operation "*">
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByElements(pts,lns,inc,stab,\^);</span>
<generalised quadrangle of order [ 4, 4 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
50
</pre></div>
<p><a id="X864C966D8184A9C0" name="X864C966D8184A9C0"></a 48,51 52 54,55,56 57],
<h4>12.3 <span class="Heading">Attributes and operations for generalised polygons</span></h4>
ion areonthe category <code class=code">IsGeneralisedPolygon</code>.</p>
<p><a id="X84F59A2687C62763" name="X84F59A2687C62763"></a></p>
<h5>12.3-1 Order</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Order</code>( <var class="Arg">gp</var> )</td><td class="tdright">(&48,50,51 , 55 ,60]java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<p>Returns: a pair of positive integers</p>
<p>This method returns the parameters <span class="SimpleMath">\((s,t)\)</span> of the generalised polygon <var class="Arg">gp</var>. That is, <span class="SimpleMath">\(s+1\)</span> is the number of points on any line of <var class="Arg">gp</var>, and <span class="SimpleMath">\(t+1\)</span> is the number of lines incident with any point of <var class="Arg">gp</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := TwistedTrialityHexagon(2^3);</span>
T(8, 2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(gp);</span>
[ 8, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := HermitianPolarSpace(4,25);</span>
H(4, 5^2)
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(gp);</span>
[ 25, 125 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := EGQByqClan(LinearqClan(3));</span>
#I Computed Kantor 51, 52, 53, 55, 58, 61 ,
<EGQ of order [ 9, 3 ] and basepoint 0>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(gp);</span>
[9 3 ]
</pre></div>
<p><a id="X836C15B57968A511" name="X836C15B57968A511"></a></p>
<h5>12.3-2 IncidenceGraphAttr</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IncidenceGraphAttr</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute&java.lang.StringIndexOutOfBoundsException: Index 199 out of bounds for length 77
<p>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">IncidenceGraph49,52 53 55 ,61,62 ,
<p><a id="X815BE6D57D623452" name="X815BE6D57D623452"></a></p>
<h5>12.3-3 IncidenceGraph</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IncidenceGraph</code>( <var class="Arg">gp</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a graph</p>
<p>The argument <var class=[1 ,3 , 5 , 7 8 ,1021 23 ,26,27,28 32,,40 ,45,
<p>We should also point out that this method returns a <em>mutable</em> attribute of <var class="Arg">gp</var>, so that acquired information about the incidence graph can be added. For 48,49 ,535558 ,
<p>Note that the factor 2 as difference in the order of the collineation group of <span class="SimpleMath">\(Q(4,4)\)</span> and the order of the automorphism group of its incidence graph is easily explained by the fact that the <span class="SimpleMath">\(Q(4,4)\)</span> is self dual.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">blocks := [ </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 2, 3, 4, 5 ], [ 1, 6, 7, 8, 9 ], [ 1, 10, 11, 12, 13 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 14, 15, 16, 17 ], [ 1, 18, 19, 20, 21 ], [ 2, 6, 10, 14, 18 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> 1 52,53,5558,62,64],
<span class="GAPprompt">></span> <span class="GAPinput"> [ 3, 6, 11, 16, 21 ], [ 3, 7, 10, 17, 20 ], [ 3, 8, 13, 14, 19 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 3, 9, 12, 15, 18 ], [ 4, 6, 12, 17, 19 ], [ 4, 7, 13, 16, 18 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 4, 8, 10, 15, 21 ], [ 4, 9, 11, 14, 20 ], [ 5, 6, 13, 15, 20 ], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 5, 7, 12, 14, 21 ], [ 5, 8, 11, 17, 18 ], [ 5, 9, 10, 16, 19 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByBlocks( blocks );</span>
<projective plane order 4>
<span class="GAPprompt">gap></span> <span class="GAPinput">incgraph := IncidenceGraph( gp );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Diameter( incgraph );</span>
3
<span class="GAPprompt">gap></span> <span class="GAPinput">Girth( incgraph );</span>
6
<span class="GAPprompt">gap></span> <span class=" 4646 51 ,60,61, 62 63]java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
[ 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">aut := AutGroupGraph( incgraph );</span>
<permutation group with 9 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayCompositionSeries(aut);</span>
G (9 gens, size 241920)
| Z(2)
S (5 gens, size 120960)
| Z(2)
5 ,size 60480
| Z(3)
S (4 gens, size 20160)
| A(2,4) = L(3,4)
1(0 ,size1)
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := ParabolicQuadric(4,4);</span>
Q(4, 51,52, 59, 60, , 63 ]java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<span class="GAPprompt">gap></span> <span class="GAPinput">incgraph := IncidenceGraph( gp );;</span>
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
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">quit;</span>
class class="GAPinput">CollineationGroup(gp);</span>
PGammaO(5,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(last);</span>
1958400
<span class="GAPprompt">gap></span> <span class="GAPinput">incgraph := IncidenceGraph( gp );;</span>
# Computing incidence graphof generalised .java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
<span class="GAPprompt">gap></span> <span class="GAPinput">aut := AutGroupGraph( incgraph );</span>
<permutation group with 10 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(aut);</span>
3916800
</pre></div>
<p><a id="X81F531BB7DEA127D"53 55, 58 59,6162 64 ],
<h5>12.3-4 IncidenceMatrixOfGeneralisedPolygon</h5>
<div class="func"><table class="func" width="100%"><tr><td< 1, 2,3,4 5 6,7, 8, 9,, 10,21,26,, ,32 , 37,40,42 43,
<p>Returns: a matrix</p>
<p>This method returns the incidence matrix of the generalised polygon via the operation <code class="code">CollapsedAdjacencyMat</code> in the <strong class="pkg">GRAPE</strong> 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 1,2 3, ,5 6 7,8,,, ,25,26,27,28 , 37 ,42,43,
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span [ 1,2,3,4 ,,7 ,9,21 26, 27 28,32 34,37,40, 42 43,46
W(3, 2)
<span class="GAPprompt">gap></span> <span class="GAPinput">CollineationGroup(gp);</span>
PGammaSp(4,2)
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := IncidenceMatrixOfGeneralisedPolygon(gp);</span>
#I Computing incidence graph of generalised polygon...
[ [ , 0, 1 0,0, 0, 0,0, 0,1 , 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, 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 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>
<p><a id="X83FF6FA0790D5747" name="X83FF6FA0790D5747"></a></p>
<h5>12.3-5 CollineationGroup</h5>
<div ="func>< class=""100"><tdclass="tdleft"><code class="func">‣ CollineationGroup</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a group</p>
<p>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 49 51 57 59 ,62 ,
<p>The collineation groups computed for classical generalised quadrangles and classical generalised hexagons are collineation groups in the sense of <strong class="
<p>Note that the computation of the automorphism group of the underlying graph can be time consuming, also if the complete 1, , ,4,,6 , ,20 31 ,,4244
<p>The first example illustrates that <code class="file">CollineationGroup</code> is naturally applicable to all classical generalised Polygons.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := PG(2,2);</span>
ProjectiveSpace(2, 2[ ,2 3,4 5678 , ,12 13 15, 17,19 , , , ,3942,java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
<span class="GAPprompt">gap></span> <span class="GAPinput">CollineationGroup(gp);</span>
he FinInG groupPGL(,)
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := EllipticQuadric(5,4);</span>
Q-(5, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">[1 2 3 4,5 6,7 8 , ,12, 15 17,19 ,313233 35 36,
PGammaO-(6,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := TwistedTrialityHexagon(3^3);38,39 45 52 53,58,62 ,
T(27, 3)
<span class="GAPprompt">gap></span> <span class="GAPinput">CollineationGroup(gp);</span>
#I Computing nice monomorphism...
#I Found permutation domain...
3D_4(27)
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
40691
</pre></div>
<p>The second example illustrates the computation of collineation groups of generalised polygons constructed using different objects.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 0, 1, 0, 0, 0, 1 ], [ 0, 1, 1, 1, 0, 0, 0 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],</span>
<span class="GAPprompt">></java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 31
[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByIncidenceMatrix(mat);</span>
<projective plane order 2>
<span class="GAPprompt">gap></span> <span class="GAPinput">group := CollineationGroup(gp);</span>
roup[ (3,4)(5,7)(9,10)13,),(3,7)(4,5(1112(13,4), (2,3)(6,7)(8,9)
(12,13), (2,6)(4,5)(11,13)(12,14), (1,2)(4,7)(9,11)(10,12) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := EGQByqClan(FisherqClan(3));</span>
#I Computed Kantor family. Now computing EGQ...
<EGQ of order [ 9, 3 ] and basepoint 0>
<span class="GAPprompt">gap></span> <span class="GAPinput">group := CollineationGroup(gp);</span>
#I Computing incidence graph of generalised polygon...
#I Using elation of the collineation group...
<permutation group of size 26127360 with 8 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(group);</span>
26127360
<span class="GAPprompt">gap></span> <span class="GAPinput">Random(group);</span>
(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,,208,9,43,197,42,269,42,25,,134,,,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,
268,51,233,253,218,172,30,44,25169,,234,27,71,,,,21,6java.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
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)( [...] )
<span class"GAPprompt"gap></span> <span class="GAPinput">q := 4;</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">conic := ParabolicQuadric(2,q);</span>
Q(2, 4)
<span class="GAPprompt">gap></span> , 15 ,23 , , , , ,39
<a point in ProjectiveSpace(2, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">conic := ParabolicQuadric(2,q);</span>
Q(2, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">nucleus := NucleusOfParabolicQuadric(conic);</span>
<a point in ProjectiveSpace(2, 4)>
<panclass="GAPprompt">gap></span> <span class="GAPinput">hyperoval := Union(List(Points(conic)),[nucleus]);</span>
[ <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)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(3,q);</span>
ProjectiveSpace(3, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">hyp := HyperplaneByDualCoordinates(pg,[1,0,0,0]*Z(q)^0);</span>
<a plane in ProjectiveSpace(3, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">em := NaturalEmbeddingBySubspace(PG(2,q),pg,hyp);</span>
<geometry morphism from <All elements of ProjectiveSpace(2,
4)> to <All elements of ProjectiveSpace(3, 4)>>
<span class="GAPprompt">gap></span> <span class="GAPinput">O := List(hyperoval,x->x^em);</span>
[ <a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)>,
&t;a in (3 4)gt;,<a point in ProjectiveSpace(3, 4)>,
<a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">points := Set(Filtered(Points(pg),x->not x in hyp));;</span>
span class="">ap>/span> class="GAPinput">lines := Union(List(O,x->Filtered(Lines(x),y->not y in hyp)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inc := \*;</span>
<Operation "*">
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByElements(points,lines,inc);</span>
<generalised quadrangle of order [ 3, 5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">coll := CollineationGroup(gp);</span>
<permutation group[1 2 ,4, ,67 8,9,11,12,13, 15 18, 20, 23 27 29, 30,42 47,java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(coll);</span>
138240
<span class="GAPprompt">gap></span> <span class="GAPinput">Random(coll);</span>
(,296040(,42,4,0,,611944,369
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>
<p>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 java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 31
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">q := 8;</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">conic := ParabolicQuadric(2,q);</span>
Q(2, 8)
<span class="GAPprompt">gap></span> <span class="GAPinput">nucleus := NucleusOfParabolicQuadric(conic);</span>
<a point in ProjectiveSpace(2, 8)>
<span class="">/<class"java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 119
[ <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)>,
lt;a point in Q(2,8)gt;, lt;a Q(,8&t java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(3,q);</span>
ProjectiveSpace(3, 8)
<span class="GAPprompt">gap></span> <span class="GAPinput">hyp := HyperplaneByDualCoordinates(pg,[1,0,0,0]*Z(q)^0);</span>
l; in 3,8)gt
<span class="GAPprompt">gap></span> <span class="GAPinput">em := NaturalEmbeddingBySubspace(PG(2,q),pg,hyp);</span>
<geometry morphism from <All elements44,45 46, , , , ]java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
8)> to <All elements of ProjectiveSpace(3, 8)>>
<span class="GAPprompt">gap></span> <span class="GAPinput">O := List(hyperoval,x->x^em);</span>
veSpace(, )gt;,java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
<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)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">points := Set(Filtered(Points(pg),x->not x in hyp));;[ 1 , ,4,,6 ,8 , ,12 , 15, 20 ,27 31 ,42,44,45,
<span class="GAPprompt">gap></span> <span class="GAPinput">lines := Union(List(O,x->Filtered(Lines(x),y->not y in hyp)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">inc := \*;</span>
<Operation "*">
<span class="GAPprompt">gap></span> <span[ 1 2 3 4 5 , ,,9 ,,, 15 23 25,27,29,31 32,36 38,
<generalised quadrangle of order [ 7, 9 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
17466
<span class="GAPprompt">gap></span> <span class="GAPinput">coll := CollineationGroup(gp);</span>
<permutation group of size 5419008 with 9 generators>
<span class="GAPprompt">gap></span> <span class=" 1 ,3 ,5 ,7, , , , 12 ,15, 30,31, 34, 36, 39, 43,
69
<span class="GAPprompt">gap></span> <span class="GAPinput">group := CollineationGroup(pg);</span>
The FinInG 44, 48, 50, 51, 52, ]java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<span class="GAPprompt">gap></span> <span class="GAPinput">stab := FiningSetwiseStabiliser(group,O);</span>
#I Computing adjusted stabilizer chain...
<projective collineation group with 11 generators>
<pan class=GAPprompt"gap&t</span>< =>time;</span
2045
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByElements(points,lines,inc,stab,\^);</span>
<generalised quadrangle of order [ 7, 9 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">time;</span>
394
<span class="GAPprompt">gap></span> <span class="GAPinput">coll := CollineationGroup(gp);</span[ 1 , , , , 6, 7 8,9, 11, 12,13 15,24,27 29, 31, 44 ,
<permutation group of size 5419008 with 9 generators>
<span 49, 51,,59, ,62, 63]
62
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(coll);</span>
5419008
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(stab);</span>
5419008
</pre></div>
<p><a id="X8074EDB381D97984" name="X8074EDB381D97984"></a></p>
<h5> ,2 ,4, , ,7, ,9,11, 12 ,15 ,29 ,31,36,43, 44 ,
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CollineationAction</code>( <var class="Arg">group</var> )</td><td class="tdright">( attribute )</td></tr></table 48 51 52,, , 57 59],
<p>Returns: a function</p>
<p><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 49,,55, 56 ,64]
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">q := 4;</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">conic := ParabolicQuadric(2,q);</span>
Q(2, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">nucleus := NucleusOfParabolicQuadric(conic);</span>
<a point in ProjectiveSpace(2, 4)>
<spanclass=GAPprompt">gap></span> <span class="GAPinput">hyperoval := Union(List(Points(conic)),[nucleus]);</span>
[ <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)> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">pg := PG(3,q);</span>
3 )
<span class="GAPprompt">gap></span> <span class="GAPinput">hyp := HyperplaneByDualCoordinates(pg,[1,0,0,0]*Z(q)^0);</span>
<a plane in ProjectiveSpace(3, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">em := NaturalEmbeddingBySubspace(PG(2,q),pg,hyp);</span>
<geometry morphism from <All elements of ProjectiveSpace(2,
4)> to <All elements of ProjectiveSpace(3, 4)>>
<span class="GAPprompt">gap></span> <span class="GAPinput">O := List(hyperoval,x->x^em);</span>
[ <a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)>,
<a point in ProjectiveSpace(3, 4)>, <a point in ProjectiveSpace(3, 4)>,
&ta in ProjectiveSpace(3 )gt;,&ta point java.lang.StringIndexOutOfBoundsException: Range [76, 74) out of bounds for length 86
<span class="GAPprompt">gap></span> <span class="GAPinput">points := Set(Filtered(Points(pg),x->not x in hyp));;</span>
<pan class="GAPprompt">gap></span> <span class="GAPinput">lines := Union(List(O,x->Filtered(ines)y-¬ in hyp);/
<span class="GAPprompt">gap></span> <span class="GAPinput">inc := \*;</span>
<Operation "*">
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByElements(points,lines,inc);</span>
<generalised quadrangle of order [ 3, 5 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">coll := java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 31
<permutation group of size 138240 with 8 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">act := CollineationAction(coll);</span>
function( el, g ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">g := Random(coll);</span>
(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,
119,03(67,135,6,123130,106)(,,112100,1,)70134,,,55,
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)(80,48 ,52 ,57 ,64 ,
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)
<span class="GAPprompt">gap></span> <span class="GAPinput">l := Random(Lines(gp));</span>
<a line in <generalised quadrangle of order [ 3, 5 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">act(l,g);</span>
<a line &t;generalised quadrangle oforder[3, 5 ]gt&;
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(gp));</span>
<a point in <generalised quadrangle of order [ 3, 5 ]>>
<span class= 50,52,55 57,61,6364],
<a point in <generalised quadrangle of order [ 3, 5 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">stab := Stabilizer(coll,p,act);</span>
<permutation group of size 2160 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(Orbits(stab,List(Points(gp)),act),x->Length(x));</span>
[ 45, 18, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(Orbits(stab,List(Lines(gp)),act),x->Length(x));</span>
[ 90, 6 ]
</pre></div>
<p><a id="X7D75521986B958DA" name="X7D75521986B958DA"></a></p>
<h5>12.3-7 BlockDesignOfGeneralisedPolygon</h5>
<div class="func"><table 1 , ,4 , 6 , ,9 11 1213 ,25,31, 34, 35 36 37,39 43,java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
<p>Returns: a block design</p>
<p>This method allows one to use the GAP package <strong class="pkg">DESIGN</strong> to analyse a generalised polygon[1, 2, 3 4,5, 6 7,8, 9 11 12, 13, , 27,28 29 37 , , 48,49
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("design");</span>
-----------------------------------------------------------------------------
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
<span class="GAPprompt">gap></span> <span class="GAPinput">gh := SplitCayleyHexagon(2);</span>
H(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">CollineationGroup(gh);</span>
java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 28
#I Computing nice monomorphism...
#I Found permutation domain...
G_2(2)
<span class="GAPprompt">gap></span> <span class="GAPinput">des := BlockDesignOfGeneralisedPolygon(gh);</span>
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 ],
1,12 ,[ ,26 ], ,37 42] [12 ,43 ,
[ 12, 44, 50 ], [ 13, 15, 42 ], [ 13, 19, 51 ], [ 14, 15, 31 ],
,17 61 , 14 ,50],[ , ,38 ,[ 16 23, 44 ],
[ 16, 40, 59 ], [ 17, 45, 54 ], [ 18, 24, 26 ], [ 18, 30, 31 ],
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 )
<span class="GAPprompt">gap></span> <span class="GAPinput">f := GF(3);</span>
GF(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">id := IdentityMat(2, f);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">clan := List( f, t -> t*id );;</span>
<span class"GAPprompt">gap></span> <span class="GAPinput">clan := qClan(clan,f);</span>
<q-clan over GF(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">egq := EGQByqClan( clan );</span>
#I Computed Kantor family. Now computing EGQ...
<EGQ of order [ 9, 3 ] and basepoint 0>
<span class="GAPprompt">gap></span> <span class="GAPinput">HasElationGroup( java.lang.StringIndexOutOfBoundsException: Range [0, 82) out of bounds for length 77
true
<span class="GAPprompt">gap></span> <span class="GAPinput">design := BlockDesignOfGeneralisedPolygon( egq );;</span>
#I Computing orbits on lines of gen. polygon...
#I Computing block design of generalised polygon...
<span class="GAPprompt">gap></span> <span class="[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 17, 20, 25, 28, 314 ,6,, ,9,11, 12 13,17 20,25 28, 31, 33 35 4041,
<permutation group with 6 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrBlockDesignPoints( design );</span>
280
<span class="GAPprompt">gap></span> <span class="GAPinput">NrBlockDesignBlocks( design );</span>
112
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayCompositionSeries(aut);</span>
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)
|2(3,) =U(4,3) ~ 2D(33=O-(63
1 (0 gens, size 1)
</pre></div>
<p><a id="X7A13D5EB82E01576" name="X7A13D5EB82E01576"></a></p>
<h4>12.4 <span class="Heading">Elements of generalised polygons</span></h4>
<p><a id="X7E7607CA7D59D086" name="X7E7607CA7D59D086"></a></p>
<h5>12.4-1 <span class="Heading">Collections of elements of generalised polygons</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; Points</code>( <var class="Arg">gp</var> )</td>[ 1, 2 , 4, 5, 6, 7, 8,9 11 12132223 ,, ,39 42,4344
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Lines</code>( <var class="Arg">gp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a collection of elements of a generalised polygon</p>
<p><var class="Arg">gp</var> is any generalised polygon, <var class="Arg">i</var> is a natural number, necessarily <span class="SimpleMath">\(1\)</span> or <49 53 54, ,56,61, ,
<p><a id="X858ADA3B7A684421" name="[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 1528,29 34,35,37 46,
<h5>12.4-2 Size</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Size</code>( <var class="Arg">els</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a number</p>
<p><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>.</p>
<p><a id="X7E9B2A217DBF2849" name="X7E9B2A217DBF2849"></a></p>
<h5>12.4-3 <span class="Heading">Creating elements from objects and retrieving objects from elements</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ObjectToElement</code>( <var class="Arg">gp</var>, <var class="Arg">obj</var> )</td><td class="java.lang.StringIndexOutOfBoundsException: Index 200 out of bounds for length 31
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; UnderlyingObject</code>( <var class="Arg">el</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a collection of elements of a generalised polygon</p>
<p>To create elements in <var class="Arg">gp</var> (of type <var class="Arg">type</var>), one of the versions of
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 0, 1, 0, 0, 0, 1 ], [ 0, 1, 1, 1, 0, 0, 0 ],</span>
<span class, 49,51, 58, 59,61 63 ]
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 1, 0, 1, 1, 0 ] ];</span>
[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByIncidenceMatrix(mat);</span>
<projective plane order 2>
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(gp));</span>
<a point in <projective plane order 2>>
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingObject(p);</span>
7
< class="APprompt">gap></span> <span class="GAPinput">l := Random(Lines(gp));</span>
<a line in <projective plane order 2>>
<span class="GAPprompt">gap></span> <span class="GAPinput">UnderlyingObject(l);</span>
[ 4, 6, 7 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ObjectToElement(gp,1,4);</span>
<a point in <projective plane order 2>>
<45 46 ,5859, 61, 62 ,
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
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">quit;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ObjectToElement(gp,2,[1,2,3]);</span>
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
<pan ">&java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 77
< , ,56 58 ,60 ,
<a line in <projective plane order 2>>
</pre></div>
<><a id=X83B0FA9E7AE3DF01" name="X83B0FA9E7AE3DF01"></a></p>
<h5>12.4-4 <span class="Heading">Incidence</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsIncident</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( operation )</td></tr></table>[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, , 15 19 , 32 ,35 36,39 ,46
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; \*</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: true or false</p>
<p>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.</p>
<p><a id="X875BE2957FAF6209" name="X875BE2957FAF6209"></a></p>
<5>12.- </h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Span</code>( <var class="Arg">v</var>, 50 52 ,, 58 , ],
<p> ,2, , 4, 5 6 , 8, 9,11,12, 15,25 ,28 34 35 36 ,46
<p>Let <var class="Arg">v</var> and <var 47 49, 50 ,565759],
<iv class=example"<pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 0, 1, 0, 0, 0, 1 ], [ 0, java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 77
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],<51,52 ,57,,61,62]
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 0, 1, 0, 1, 1, 0 ] ];</span>
[ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ], [ 1, 053, 54, 56 , , , 62]
[ 0, 1, 1, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 1, 030,31 , 36 39 44 , ,
[ 0, 0, 1, 0, 1, 1, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByIncidenceMatrix(mat);</span>
<projective plane order 2>
< =Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 93
<a point in <projective plane order 2>>
ompt"gap;<span> span =GAPinput> = (Points();<span
<a point in <projective plane order 2>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Span(p,q);</span>
<alinein <rojective order >gtjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ParabolicQuadric(4,3);</span>
4,3java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByElements(Set(Points(ps)),Set(Lines(ps)),\*);</span>
<generalised quadrangle of order [ 3, 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(gp));</span>
<a point in <generalised quadrangle of order [ 3, 3 ]>>
pt">gap></span> <span class="GAPinput">q := Random(Points(gp));</span>
<a point in <generalised quadrangle of order [ 3, 3 ]>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Span(p,q);</span>
#I <x> and <y[ 1 2 3 4 , 6,7 ,,11,12, 16,17,18, 21, 22,25 35,39, 44 ,
fail
</pre></div>
<p><a id="X8469B54180FE1E4C" name=[1 ,3 4 , ,7,8,, , 12,16,17,18 21 22 25, 35 39 47 48,java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
<h5>12.4-6 Meet</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Meet</code>( <var class="Arg">v</var>, <var class="Arg">w</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a point of a generalised polygon or fail</p>
<p>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 java.lang.StringIndexOutOfBoundsException: Index 347 out of bounds for length 77
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [ [ 1, 1, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 1, 1, 0, 0 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 0, 1, 0, 0, 0, 1 ], [ 0, 1, 1, 1, 0, 0, 0 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 1, 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0, 1, 1 ],</java.lang.StringIndexOutOfBoundsException: Index 122 out of bounds for length 31
<span class[1,2 3 4 ,6,7, 8 9,11,12,,17, ,22, 25, , 35,41 , ,
[ [ 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 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByIncidenceMatrix(mat);</span>
<projective plane order 2>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := Random(Lines(gp));</span>
<a line in <projective plane order 2>>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := Random(Lines(gp));</span>
<a line in <projective plane order 2>>
<span class="GAPprompt">gap></span> <span class="GAPinput">Meet(l,m);</span>
<a point in <projective plane order 2>>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ParabolicQuadric(4,3);</span>
Q(4, 3)
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByElements(Set(Points(ps)),Set(Lines(ps)),\*);</span>
<generalised quadrangle of order [ 3, 3 ]>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := Random(Lines(gp));</span>
<a line in <generalised quadrangle of order [ 3, 3 ]>,32,34,37,
<span class="GAPprompt">gap></span> <span class="GAPinput">m := Random(Lines(gp));</span>
<a line in <generalised ,, 58 59,60,6263],
<span class="GAPprompt">gap></span> <span class="GAPinput">Meet(l,m);</span>
[1, 2,3 4,5 ,7 8,9 , 12 17, 20 21 , , 29 32 ,44 46,
fail
</pre></div>
<p><a id="X8154BB13844AA0FD" name="X8154BB13844AA0FD"></a></p>
<h5>12.4-7 <span class="Heading">Shadow elements</span></h5>
<iv class="func"><tableclass="func" width="100%"><tr><td class="tdleft"><code class="func">&#n style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; ElementsIncidentWithElementOfIncidenceStructure</code>( <var class="Arg">el</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td> 45 49 59 61 62 , ,
<p>Returns: A collection of elements</p>
<p><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 <java.lang.StringIndexOutOfBoundsException: Range [0, 581) out of bounds for length 77
< class="example><pre>java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">blocks := [</span>
< class="span> <span class="APinput" [1,2 3,4,5], [1 6 7 8,9,[ 1, 10,11 12,13],</pan>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 14, 15, 16, 17 ], [ 1, 18, 19, 20, 21 ], [ 2, 6, 10, 14, 18 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 2, 7, 11, 15, 19 ], [ 2, 8, 12, 16, 20 ], [ 2, 9, 13, 17, 21 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 3, 6, 11, 16, 21 ], [ 3, 7, 10, 17, 20 ], [ 3, 8, 13, 14, 19 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 3, 9, 12, 15, 18 ], [ 4, 6, 12, 17, 19 ], [ 4, 7, 13, 16, 53,54,58,60 61 62, ],
<span class="GAPprompt">></span> <span class="GAPinput"> [ 4, 8, 10, [1,2 ,4,5,,7 8, 9, 11, 12, 21, 23, 24,25, 29 32 34, 37 40,42
<span class=" 43, 44, 46, 49, 58, 62, 63]
<span class="GAPprompt">gap></span> <span class="GAPinput">gp := GeneralisedPolygonByBlocks( blocks ); </span>
lt; &;
<span class="GAPprompt">gap></span> <span class="GAPinput">l := Random(Lines(gp)); </span>
&t;a line < plane order4gt&tjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
<span class="GAPprompt">gap></span> <span class="GAPinput">pts := ShadowOfElement(gp,l,1);</span>
java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 57
<span class="GAPprompt">gap></span> <span class="GAPinput">List(pts);</span>
[ <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>>,
l point in <projective plane order 4>> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">p := Random(Points(gp));</span>
<a point in <projective plane order 4>>
<span class="GAPprompt">gap></span> <span class="GAPinput">lines := Lines(p);</span>
<shadow lines in <projective plane order 4>>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(lines);</span>
[ <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>
<>< id="X7807682E86D3BE10"name="7807682E86D3BE10"></a></p>
<h5>12.4-8 DistanceBetweenElements</h5>
<div class=" 48,49, 60, ,62,63 64]java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<p>Returns: a number</p>
<pLet var class="Arg"></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.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := ElementaryAbelianGroup(27);</span>
<pc group of size 27 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">flist1 := [ Group(g.1), Group(g.2), Group(g.3), Group(g.1*g.2*g.3) ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">flist2 := [ Group([g.1, g.2^2*g.3]), Group([g.2, g.1^2*g.3 ]),</span>
<span class="GAPprompt">></span> <span java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 31
<span class="GAPprompt">gap></span> <span class="GAPinput">egq := EGQByKantorFamily(g, flist1, flist2);</ | |
