tdclass=tdright"&;nbsptd<tr>/>
<p>Returns: a classical polar space</p>

<p><var class="Arg">form</var> must be a sesquilinear or quadratic form created by use of the GAP package <strong class="pkg">Forms</strong>. In the second variant, the argument <var class="Arg">pol</var> must be a polarity of a projective space


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[0,0,0,1],[0,0,-2,0],[0,2,0,0],[-1,0,0,0]]*Z(5)^0;</span>
[ [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), Z(5)^3, 0*Z(5) ], 
  [
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(matp> bilinear havecompletely  evencharacteristic than in oddcharacteristic.  the other handpolarspacesgeometrically  toorthogonal polar spaces inoddcharacteristic, do exist ineven backgroundisnow by   onavector instead of bilinear.Considernon-singular quadraticform < classSimpleMath>q)/> onvector < =">1q\>\f\</>is incidencegeometry whose   subspacesof <span class="SimpleMath">\(\mathrm{PG}(nq))/> whoseunderlying subspaceistotally singularwith relation span=SimpleMath>(\)/>.The orthogonal spaces   is,  in odd characteristic, quadratic forms symmetricbilinearformsare equivalentTherefore wecallpolarspaces anunderlying formin even characteristicalsoemorthogonal</em> polar spaces.</pjava.lang.StringIndexOutOfBoundsException: Index 1116 out of bounds for length 1116
&t  formgt
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PolarSpace(form);</span>
&;polar inProjectiveSpace
3,GF(5^2)): x1*y4+Z(5)^3*x2*y3+Z(5)*x3*y2-x4*y1=0 >
<span class="GAPprompt">gap></span> <span class="GAPinput">r
GF(2^5)[x_1,_2,,x_4]
<span class="GAPprompt">gap
x_1*x_4div class"example>pre>
<span class"GAPprompt">>/> <spanclass="GAPinput"form QuadraticFormByPolynomial
< quadratic form >[0Z() Z() *(5,*Z5)],  Z5^,*() 0Z5,*() ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PolarSpace(form);</span>
<polarspaceinProjectiveSpace,(25) *x_4x_2*= &;
<span class="GAPprompt">gap></span; form;
[[ Z()0 0Z7,*(7) 0*() 0Z7 ,java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
  [ 0*<spanclassGAPprompt>gapgt;</span<span ="GAPinput>r=PolynomialRing(3)4;/>
  [ 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7) ], 
  [ 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0, 0*Z(7) ], 
java.lang.StringIndexOutOfBoundsException: Range [4, 2) out of bounds for length 46
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := PolarityOfProjectiveSpace(mat*x_4x_2*_java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
<polarity;   &;
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PolarSpace(phi);</span>
<polar in(4,()) x_12x_2+^2x_4+_^= &;
 
</pre></div>

<s ="/pan>APinput=5GF


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">id := IdentityMat(7, GF(3));;</span>
<span class="GAPprompt">gap *() 0Z(7,Z70 00*,0Z7)], 
< quadratic form >
<span class="GAPprompt"> *7,7, () (^0 *Z7]
<polar < class""gap;<span< classGAPinput (mat,GF(7));</span>
6,GF(3)lt  PG4 F7)>
< =""gap;<span <span class=GAPinputpsl32 (32;/>
Group(;polar  ProjectiveSpace,GF7) x_1^+^+^x_4x_5=0gt
<span class><divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<span class="GAPprompt">gap></span> <span class="GAPinput">ovoid := Union( List(reps, x-> Orbit(psl32, x, Permuted)) );;</span>
<spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< classGAPprompt">apgt : AsList(Planes ps ));/>
#I  Computing collineation group of canonical polar space...
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll(planes, p -> Number(ovoid, x -> x in p) = 1)< class"">><s>  class"GAPinput> := QuadraticFormByMatrix(id, GF(3);/spanjava.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
true
 
<>/div

<p><a ([4),) 124(,,))

<h4>7.2 <span class="Heading">Canonical and standard Polar Spaces</span></h4>

<p>To introduce the classification of polar spaces, we span =GAPpromptgap></span> <span class="GAPinput">ovoid := Union( List(reps, x-> Orbit(psl32, x, Permuted)></span< class=GAPinputovoid (List(reps, &; (psl32 )) );/>

<p>Consider a vector space <span class="SimpleMath"<span =GAPprompt&;s> ""planes: ( ());<>

<"\f(vv,w) = lambda f'(\(),phi())\]

<p>for all vectors <span class="SimpleMath">\(v,w\)</span> and some non-zero <span class="SimpleMath">\(\lambda \in GF(q)\)</span> . Up to similarity, there is only one class of non-degenerate Hermitian forms, and one class of non-degenerate symplectic forms on a given vector space java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

<p>Consider now a vector space

<p class="center">\[q(v) = \lambda q'(\phi(v))),\]

an some(lambda\ GFq)</pan Forquadraticformsin characteristic numberof   depends thedimensionof<span="">\(V\<s>.  odd, there only similarityclass andnon-degenerate in class are called (bilinear . In even  dimension,there aretwosimilarityclasses andnon-degenerateformsareeither elliptic()formsorhyperbolic(bilinear) forms.</p>

<p>If <span class="SimpleMath">\(\phi\)</span> induces a similarity of a formed vector space such that <span class="SimpleMath">\(\lambda = 1\)</span>, then the similarity is called an <em>isometry</em> of the formed vector space. In almost all cases, each similarity class contains

<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b>finite classical polar spaces</caption>
<tr>
<td class="tdleft">polar space</td>
<td class="tdleft">standard form</td>
<td class="tdleft">characteristic <span class="SimpleMath">\(p\)</span></td>
< class"tdleft">projectivedimension/>
</tr>
<tr>
<td class="tdleft">hermitian polar space</td>
<td class="tdleft"><span class="SimpleMath">\(X_0^{q+1} + X_1^{q+1} + \ldots + X_{n}^{q+1}\)</span></td>
<td class="tdleft>odd or ven
<td class="tdleft">odd</td>
</tr>
<tr>
<td class="tdleft">hyperbolic quadric</td>
<td class="tdleft"><span class="SimpleMath">\(2(X_0 X_1 + \ldots + X_{n-1}X_n)\)</span></td>
java.lang.StringIndexOutOfBoundsException: Range [81, 82) out of bounds for length 79
<tdclass"">odd/tdjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
</tr>
<tr>
<td class="tdleft">parabolic quadric/td>
<td class="tdleft"td
< class>span=""\ \ 1 pmod}\)span>  span java.lang.StringIndexOutOfBoundsException: Range [92, 91) out of bounds for length 124
< =""eventd
</tr>
<tr>
<d classtdleftparabolic <tdjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
< classtdleft> class"impleMath>(tt(^2+ + ldots X_n-1X_n)) aprimitive element of "\\mathrmGF(p\
<class> classSimpleMath">\(java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 81
<tdtd =tdleft/>
</tr>
<tr>
<td class="tdleft">elliptic quadric</td>
<td class
<td="tdlefttdleft>hyperbolic quadric
< class="odd
</tr>
<tr>
<td class="tdleft">elliptic quadric</td>
<td class="tdleft"><span class="SimpleMath">\(X_0^2 + tX_1^2 + X_2X_3 + \ldots + X_{n-1}X_n\)</span>, <span class=<td class="tdleft">spanclass="impleMath">(\equiv 1 \pmod})/span>td
< ="">parabolicquadrictd
<d =""><td
</tr>
<tr>
<tdclasstdleft />
<td td="">even<td
<td class="tdleft">eventrjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<td classtdleft>odd/tdjava.lang.StringIndexOutOfBoundsException: Range [27, 28) out of bounds for length 27
</tr>
</table><br />
</div>

<p>We refer to Appendix <a href="chapB_mj.html#X866C644987E43DF8"><span class="RefLink">B</span></a> for information on the operations that construct gram matrices that/tr

<p>The class""><span class=SimpleMath\p equiv3 \pmod4\</pan<td

<p>>

<p><a id="X850980EF8607BDA7" name="X850980EF8607BDA7"></a></p>

<h5>7.2-1 SymplecticSpace</h5>

<div="func>table=func width"><tr<td"tdleft">>codeclass"">82 SymplecticSpacecode> <classArg>/,<class"Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table></div>
< class""><table=funcwidth=10%<<=">code class"func>‣ SymplecticSpace</code>  ""></>,class""<var )td<td =tdright(nbspoperation <td</>/><div>
<p>Returns: a symplectic polar space</p>

<p>This function returns the symplectic<tdclass""><spanclass"SimpleMath>\p 1 pmod}\td


<div ="example"<>
<spanclass"">><s>span=GAPinput :SymplecticSpace)<span
W(3, 4)
<span =tdleftspan class class"SimpleMath>\^2+X_0X_1 + dX_1^2 + X_2X_3 + \ldots + X_{n-1}X_n\), \(\mathrm{Tr}(d)=1\)
W(, java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
Symplectic form
Gram Matrix:
 . 1 . .
 1  . .
 . . . 1
 . . 1 .
Witt Index: 2
 
</pre>div>

<id=>/a/java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<>.2-2<>

<div class:asymplectic spacepjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
< classclassfunc=0%" =tdleft"><codeclass=func&82;HermitianPolarSpacecode(< class""><varvar =Argq/> )</td<d ="tdright">&;operationnbsp)</>/></>/>
<p>Returns: a Hermitian polar space</p>

<p>This function returns the Hermitian polar space of dimension <var class="Arg">d</var> over <var class="Arg">F</var> for a field <var class="Arg">F</var> or over GF(<var class="Arg">q</var>) for a prime power <var class="Arg">q</var>.</p>


<div class="example"><pre>
< =GAPprompt></>< classGAPinput> :22)<span
H(2, 5^2)
<span class="GAPprompt">gap></span><panclass="GAPprompt">></pan class"APinput"Display);/span
H(2 25
Hermitian form
Gram Matrix Matrixjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 1 . .
 . 1 .
 . . 1
Polynomialx_1+6+6 ]
Witt Index: 1
 
</pre></div>

<p><a id="X7CEDD5CC7A6B0E64" name="X7CEDD5CC7A6B0E64"></a></p>

<h5>7.2-3 ParabolicQuadric

<div class="func"< class=func width"00%>tr>>/table>
Q+(5, 4)
<span class="GAPprompt">gap></span> <span class
java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 1
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Range [0, 64) out of bounds for length 0
W( 8)
<span class="GAPprompt">gap></span> <span class="GAPinput">Rank(ps);</span>
4
 
</pre></div>

<p>a id"" name"X7BCBA564829D9E89"></></>

<h5>7.3-5 BaseField</h5>

< =""><table class="func"" width="100%><r>tdclass=tdleft>< class"func>&827 BaseField( s( operation& )/div>
<p>Returns: theW(, 1java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8


<div 
<span=">gap>/span GAPinput : 57;/>
Q+(5, 7)
<span class="GAPprompt">gap></span> <span class="GAPinput">BaseField(ps);</span>
GF(7)
< class="">gapgt</> span class"GAPinput>ps := HermitianPolarSpace((2,26)
H(2, 16^2)
<span class="GAPprompt">gap></span> <span class="GAPinput">BaseField(ps);<<>7.3-4 </>
GF28)
 
</pre></div>

<p><a id="X8412E6307ACC82C2" name="X8412E6307ACC82C2"></<p: the ofvarclassArg>ps/><p>

<h5>7.3-6 IsHyperbolicQuadric</h5>

<div class""><table="" width=10"< class=tdleft"<codeclass"">RIsHyperbolicQuadriccode>( < class"">ps</>)/><td class="tdright>(nbsp;propertynbsp;;/>
<p>Returns: true or false</p>

<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


d classexample<>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := IdentityMat(6,GF(5));<="GAPprompt"gap></> <spanclass=GAPinput>(ps</pan>
&t;mutable  matrix6x6over () gtjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
<span="GAPprompt"><span=GAPinputform: BilinearFormByMatrix)<spanjava.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
<bilinear form>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PolarSpace(form);</span>
<polar spacein ProjectiveSpace
5,GF(5)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6java.lang.StringIndexOutOfBoundsException: Range [1, 2) out of bounds for length 1
<span class="GAPprompt">gap></span> <span class
true
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := IdentityMat(6,GF(7));</span>
< mutable compressed matrix 6x6 over GF(7) >
<span class="GAPprompt">gap></spanh5>73-- BaseField<//h5>
< bilinear form > class"func">tableclass""  width"10%><>td =tdleft>< class"func&##27 </>(<varclass=Arg>s/ar )/><td class"">( operation&;)</></tr></table></div>
< class""gapgt;/>< ="">ps=PolarSpace(form;/>
<polar space in ProjectiveSpace(
5,GF(7)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0 >
<spanclass"GAPprompt">gapgt<span s class=""GAPinput>IsHyperbolicQuadric();<</>
false
GF()
</pre></div>

<p><a id="X85386DAB8056E316" name="X85386DAB8056E316"></a><<spanclass"">gapgt;span spanc="GAPinput"ps (2,2556));/spanjava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102

<h5>7.3-7 IsEllipticQuadric</h5>

<div class="func"><table class="func" width="100%
<p>Returns: true or false</p>

<p>returnstrueifandonlyif<var class=""ArgArg></var isanellipticquadric.<p>


<div class="example"><pre>
< classGAPpromptgap&;</span>< ="GAPinput"> := (6GF(5));/>
< mutable compressed matrix 6x6 over GF(5) >
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat,GF(5));</span>
&
<span="GAPprompt">gapgt;</span << class="GAPinput""ps:= PolarSpace(form);
<polar space in ProjectiveSpace(
5,GF(5)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0 >
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEllipticQuadric(ps);</span>
false
<span class=GAPprompt>>/> <span class"">mat :=IdentityMat(6,(7))/span
< mutable compressed matrix 6x6 over GF(7) >
< ="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat,GF(7));</span>
<bilinearformgt
<spanclass="APprompt>gapgt
<polar space in ProjectiveSpace(
5,GF(7)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0 >
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEllipticQuadric(ps);</span>
true
 
</pre></div>

<>aidX87ADB8B5790776C7=>/a></java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

<h5>7.3-8 IsParabolicQuadric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsParabolicQuadric</code>( <var class="Arg">ps</var> )</td>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<>Returns:true or false</pjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

<p>returns true if and only if <var class="Arg">ps</var> is a parabolic quadric.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := IdentityMat(5,GF(9));<,GF(7):x_12x_2+x_3^2+x_4+x_5^2+x_6^=0&;
[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3) ], 
  [ 0*Z(3), Z(3)java.lang.StringIndexOutOfBoundsException: Range [5, 6) out of bounds for length 5
  [ 0*Z(3
  <p>a ="X85386DAB8056E316" name="X85386DAB8056E316"></a></p>
  [ 0*Z(3), 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := BilinearFormByMatrix(mat,GF(9));</span>
< bilinear form >h5>7.33- IsEllipticQuadric/h5>
<span class="GAPprompt">gap></span> <span class<div ="func">tableclass="" width="100%"<tr><td class=tdleft"< 82;IsEllipticQuadric><var)/>< classnbsp&;)/><tr</></div>
<polar space in ProjectiveSpace(4java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">IsParabolicQuadric(psjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
true
<span< class=">> span lass=GAPinput> :=IdentityMat(,GF5);
[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
  [nclass"GAPprompt">gap><</span><span class"">form: BilinearFormByMatrix(matGF())<span>
  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], 
  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], 
  [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ]ltbilinearform>
<<span="GAPprompt"gap;</> <span="GAPinput">ps: PolarSpace(form;</>
< quadratic ltpolar space inProjectiveSpace(
<span class,GF5):x_1^2+x_2^+x_3^2+x_4+x_52x_6^= >
<polar space in ProjectiveSpace(4,GF(2^3)): x_1^2+x_2*x_3+x_4*x_5=0 >
<pan"gap>> >IsParabolicQuadric);</panjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
true
 
/>/iv

<p><a id="X787E0AEA8284B34B" name="X787E0AEA8284B34B"></a></p>

<h4>7.4 <span class="Heading">Subspaces of finite classical polar spaces</span></h4>

<p>The elements of a finite classical polar space <span class="SimpleMath">\(<spanclass"">gap;<> span=GAPinput : (form)<s>

<p><aspan=GAPpromptgap;<span> span=GAPinputIsEllipticQuadric()</>

<h5>7.4-1 VectorSpaceToElement</h5>

<div
<p>Returns: an element of thep><a id="X87ADB8B5790776C7" name=X87ADB8B5790776C7></</>

<p>Let <var class="Arg">ps</var> be a polar space, and <var class="Arg">v</var> is either a row vector (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span <p>Returns:true false/>
W(3, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">v := [1,0,1,0]*Z(4)^0;</span>
[ Z(2)^0, 0*Z(2), Z(p> true and  ifvarclass"">ps</var is a  quadric</>
<span class="GAPprompt">gap></span> <span class="GAPinput">
<a point in W(3, 4)&<divclass""><prejava.lang.StringIndexOutOfBoundsException: Range [26, 27) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">mat[ [ Z(3^,0*(3, 0*Z3) *Z3) *Z() ], 
[ [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ] ]
< classGAPprompt>>/pan< class"">line= VectorSpaceToElement(ps,mat);/span>
Error, <x> does not generate an element of <geom> called from
<function "unknown">(  ltarguments> )
 called from read-eval loop at line 12 of *stdin*
you can 'quit;' to quit to outer loop, or
you 'return'tocontinue
<span class *(3,*() *(,Z3^ ] ]
< =GAPpromptgapgt/> << class="">mat=[[,1,][,,,0]*4^0</>
[ [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z<bilinearform >
<span ="">gapgt;/span> <span="GAPinput">line :=  VectorSpaceToElement(psmat;span
&; line(, )gt
< =GAPpromptgt/>< =GAPinput()<span
< empty subspace
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
</pre  0Z2,0*(,Z()0 0Z(2)) 0*ZZ(() ] 

<p><a id="X8461BCEF862B9A7B" name="X8461BCEF862B9A7B"></a></p>

<h5>7.4-2 EmptySubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EmptySubspace</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation )</td></tr[0Z(2) *() 0Z2,0Z2,0Z2 ] ]
<p>Returnslt quadraticformgt

<>he returnedby this is in every subspace theprojectivespace<var class=Arg></> but not < classArgps> Hence, testing, testing   error/


<div class="example">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
< classGAPpromptgap;</span<span class=""> : HermitianPolarSpace0,9;/span
H(10, 7^2)
<span class="GAPprompt">gap>h474<span class"">Subspaces finite polar</span<h>
< empty subspace >
 
</pre></div>

<p><a id="X84FDF25D797B874B" name="X84FDF25D797B874B"></a></p>

<h5>7.4-3 ProjectiveDimension</h5>

<div ="">tableclass=funcwidth"100%>tdleft<code class=func&87 </code(var="Arg"sub/>)</td<tdclass"tdright">&;operation )/td></tr<table>/>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Dimension</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>>:  projective of a subspace  apolar space  operation<ode="">ProjectiveDimension<code>is alsoapplicable onon theEmptySubspace/p>


<divclass="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := <p>Let< ="">ps/var beapolar <varclass"Arg> iseither a rowvector(orpoints or an spanclass"SimpleMath>\m)/>x<class"\n)/>matrix (for an < =""))-subspace a polar space with an ((n-1)\)-dimensionalambient projective space In thecase that < class>v isa matrix,therows represent basis vectors for the subspace. Anexceptionalcasei class="""v</>  a zero-vector,wherebythe subspace is.  ischeckedthat thesubspacedefined by<span ="SimpleMath"">(v\)/span>is totally isotropic with relationto theform defining s/>

Q-(7, 8)
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0]]*Z(8)^0;</span>
[ [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(
  [ 0<divclass=""example">pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">line := VectorSpaceToElement(ps,mat);</span>
<aline in Q-(7,8)>
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveDimension(line);</span>
1
<span class="W(3 4)
1
<span class class=GAPprompt>>><classGAPinput> : 1010*()0<span
lt emptysubspace&;
<span class="GAPprompt">gap></span class"gap&;"p=VectorSpaceToElementpsv;/panjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
-1
 
</pre></div>

<p><a id, lt; doesgenerate  element <geom

<h5>7.4 called  loop  1  stdin

<divyoucanreturntojava.lang.StringIndexOutOfBoundsException: Range [29, 30) out of bounds for length 29
<:the of  elements   projective < classArg></var>of type< classArg>j</><p>

<p>For the projective space <var class="Arg">ps</var> of dimension <span class="SimpleMath">  ()0()0 *() 0()] [0Z() *()Z(2)0 0()] ]


<lt linein W(3 4)>
<span classspan class=GAPprompt>&;/span spanclass=GAPinputp :=VectorSpaceToElementps00,,]Z4^;</>
H(8lt; emptysubspacegt;
<span class= 
<planes of H8, 13^)>
<span class="GAPprompt">gap></span> <span class="GAPinput">solids := ElementsOfIncidenceStructure(ps,4);</span>
<solids H(8, 3^2&;
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementsOfIncidenceStructure(ps,5);</span>
Error, <geo> has no elements of type <j> called from
<function "unknown">( <arguments> )
 called from read-eval loop at line 11 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>
 
</pre></div>

<p><a id="X8606750A8586DF8D" name="X8606750A8586DF8D"></a></p>

<h5>7.4-5 AmbientSpace</h5>

<div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
<>: returns ambient spaceof an element <var="">el/var>ofa  polarspace/p

<p>This operation is also applicable on the trivial subspace. For a Lie geometry, the ambient space of an element is defined as the ambient space of the Lie geometry, i.e. a projective space.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := HermitianPolarSpace(3,7^2);</span>
H(3, 7^2)
<span class="GAPprompt">gapp>The objectreturned by thisoperationiscontainedineveryprojective subspaceof the projective space <var="Arg">ps</var>,  is  an element of<var class"Arg">ps/var>. , testingincidence results in error</pjava.lang.StringIndexOutOfBoundsException: Index 243 out of bounds for length 243
<span class="GAPprompt">></span> <span class=">gapgt<span> <span class=""GAPinput">psps:=HermitianPolarSpace(10,49);
<a line in H(3, 7^2)>
<span class="GAPprompt">gap></span> <span class="GAPinput">AmbientSpace(line);</span>
ProjectiveSpace3, 9)
 
<></div

<p><a id="X84E3985A8700B302" name="X84E3985A8700B302"></a></p>

<h5>7.4-6 Coordinates</h5>

<ivclass"

Quelle  chap7_mj.html   Sprache: HTML
products/Sources/formale Sprachen/GAP/pkg/fining/doc/chap7_mj.html java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <!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 7: Finite Classical Polar Spaces</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=" <script=textjavascript>(;/> </head> <body class < ="&;a hrefref=chap0_mj.html"[op ]</a&;  ahref"chap0_mjhtml##contents>Contents>Previous<a>nbsp   a =chap8_mjhtml>NextChapter/>nbsp</> <div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap6_mj.html">[Previous Chapter]</a>    <a href="chap8_mj.html">[Next Chapter]</a>   </div> <p id="mathjaxlink" class="pcenter"><a href="chap7.html">[MathJax off]</a></p> <>< =X7F96B1327C022A28 name"">><a>/p> <div class="ChapSects"><a href="chap7_mj.html#X7F96B1327C022A28">7 <span class="Heading">Finite Classical Polar Spaces <div class class""><br/< class""> &;</span<a="chap7_mj..html#X83A92A177BE4E0BB">7.1- IsClassicalPolarSpace/></span /span <div class="ContSSBlock"> <span class="ContSS"><br /><<div>/> < class""><br>spanclass=nocss>nbsp; </span>a =".html#X863BC8E57C98A471">712PolarSpace/>/span> </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a /> <span <div class=" class= />< class="nocss">nbsp;nbsp;.htmlX7A8CF840833D3D8B>722HermitianPolarSpacea<span <span class""><br/<span class"">nbspnbsp</><a="chap7_mj.html#X7CEDD5CC7A6B0E64">7.2-</>/span =chap7_mj2HermitianPolarSpace/>/> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7CEDD5CC7A6B0E64">7.2-3 ParabolicQuadric</a></span>  class="ontSS">br<spanclass"" ;</>a href="chap7_mj.html#X7CF248A2864BCD8F">7.2-4 HyperbolicQuadric</a></span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X85EEF70480D7149E">7.2-5 EllipticQuadric</a></span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7F6D25D57BC697B5">7.2-6 IsCanonicalPolarSpace</a></span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X8240380A7F8B6EC3"><panclass""><br /><span class""> &;<spana hrefchap7_mjhtmlX7F6D25D57BC697B5>.- IsCanonicalPolarSpace/a>/> < class"ContSS"><br />span="nocss"&; /><a href"chap7_mj..html#X7EE25EDF7ADC30D7">7.2-8 StandardPolarSpace/a>/span> </div></div> <divclass"ContSect"><span ="">spanclass""&;<s><ahref".html#X7A04340A7EC9215B">7.3 s class="">Basic for  classicalpolar spaces</pan></a>> </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7D544D7985A4572D">7.3-1 UnderlyingVectorSpace</a></span> s="ContSS"<br /span ="nocss>nbsp;nbsp;/spanspan>ahref=chap7_mjhtml#X8606750A8586DF8D">7.3-2 AmbientSpace</a></span> <span class="ContSS"><<br /<span class="nocss">  </><ahref".html#X84FDF25D797B874B"">7.3-3 ProjectiveDimension <span class="ContSS">/span <span="ontSS">br/< class"nocss>nbsp;nbspchap7_mj.htmlX7BCBA564829D9E89>.- BaseField/a></span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X8412E6307ACC82C2">7.3-6 IsHyperbolicQuadric</a></span> <span class="ContSS"><br /><spanclass" &; < class"< />span class="nocss>nbsp</a ref"hap7_mj.html#">.- <a>/> </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X787E0AEA8284B34B">7.4 <span class="Heading">Subspaces of finite classicalspanjava.lang.StringIndexOutOfBoundsException: Range [11, 6) out of bounds for length 144 <span <div class="ContSSBlock"> < =" > class">&;&;</span>< href".html#">7.41</>/> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X8461BCEF862B9A7B">7.4-2 EmptySubspace</a></span> <span class="ContSS"><br /><span class="nocss"> &spanclass" < class=">;;/<hrefchap7_mj.#X87ADB8B5790776C77- IsParabolicQuadric</a<span span=ContSS =>;&;><ajava.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 153 <span class ="">br >spanclassnocss&;nbspspan<hrefchap7_mjX8461BCEF862B9A7B.- mptySubspace</<s> span="ContSS>> &;<span< href=chap7_mjhtmlX84E3985A8700B30274 Coordinates/>/> </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><span="">< / class"&;<.-4<a</java.lang.StringIndexOutOfBoundsException: Index 153 out of bounds for length 153 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7904128479BDFCC9">7.5-1 <span class="Heading">ncidenceand containment</></a> </span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X875BE2957FAF6209">7.5-2 Span</a></span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X8469B54180FE1E4C">7.5-3 Meet</a></span> <span="ContSS"><r /><span class="nocss>  chap7_mj.htmlX8176044E7D2242D9>.- </a>/> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7EBD8C07802562B7">7.5-5 PolarityOfProjectiveSpace</a></span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X863F1924805F6C63">7.5-6 TypeOfSubspace</a></span> <span class""><br/>span=nocss&;nbsp<spana hrefchap7_mjhtmlX7904128479BDFCC9>.- < class""> and<span>/a> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html < class"ontSS>< />span ="nocss>nbsp;nbsp</span><ah="chap7_mj.htmlX0AB7B9840ABC27">7.5-9 EvaluateForm</a></span> </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7AA14EDF7B0B1569">7.6 <span class="Heading">Shadow of elements</span><span classContSS< >spanclass"nocss""> nbsp7.- Meet </span> <div class="ContSSBlock"> < class""><spannocss;span=chap7_mj76-ShadowOfElementa>/> <span class="ContSS"><br /><span="">br >< class>nbspnbsp/><a href="chap7_mj.html#X82C38639870C0DD6">7.5-7 TangentSpace/a><//span </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X7988AF9978E75E37">7.7 <span class="Heading">Projective Orthogonal/Unitary/Symplectic groups in <strong class="pkg">FinInG</strong></span><spanclass""><br />spanclass=nbspnbsp</span< =chap7_mj#X808AB7B9840ABC277-9 EvaluateForma<s spanContSSBlock <div class="ContSSBlock"> <span class="ContSS"><br /><span classspan="< >span=nocss&;&;/><="java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 140 <span class="ContSS"><br /><span class=</div></div> ass"nocss" nbsp<>a="chap7_mj..#X812836E48364AE6D">7.7-3 SimilarityGroup</a></span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X83FF6FA0790D5747">7.7-4 CollineationGroup</a></span> </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X855D48A07E0BBCDB">7.8 <span class="Heading">Enumerating subspaces of polar spaces</span></a> </spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss">  </span<ahref.html#X7AB1BA95825BDE71>.- span> forajava.lang.StringIndexOutOfBoundsException: Index 175 out of bounds for length 175 <span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7EF8910F82B45EC7">7.8-2 Enumerator</a></span> <span class="ContSS"><br /><span class="nocss">  <panclass""><br/< class="nocss">> nbsp&;</span<a href".html#X83FF6FA0790D5747"">7.7-4 CollineationGroup<><> <span> <span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X83ADF8287ED0668E">7.8-4 Iterator</a></span> spanclassContSS /span=nocss;&</>a  =chap7_mjhtml5AsLista></span> </div></div> </div> <h3>7 <span class="Heading">Finite Classical Polar Spaces</span></h3> <p>In this chapter we <div=ContSSBlock"> <p><a id="X7F96B1327C022A28" name="X7F96B1327C022A28"</> <h4>7.1 <span class="Heading">Finite Classical Polar Spaces</spanspan class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X861463147B738DF1">7.8-3 <span class="Heading">Iterators for polar spaces</span></a> <p>A <em>polar space</em> is a point-line incidence geometry, satisfying the famous one-or-all axiom, i.e. for any point <span <pan ="ContSS>br/nocss>nbspnbsp</span><ahrefchap7_mjX8289FCCC8274C89D>85AsList</a></> <p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <p>From< =X7F96B1327C022A28nameX7F96B1327C022A28>a<pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62 < <h5>7.1-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <div class="func"><table class="func" width="100%"><tr><td class <pThis isa  subcategoryof class"">IsLieGeometry/code,andcontains the associatedtoa non-degeneratesesquilinear orquadratic.</> <p>The  <p>Consider the projective>7.- </h5> icOn the ,  comparable orthogonal spaces  ,do exist     background  quadratic  a  instead   forms a non-singularquadratic  <pan=""\(q)/> avector>Vn+,q)\)/>.Theclassical space withspan=SimpleMath>f\)<>thewhose    spanclass"\{PG}(nq\v subspaceis with relation to \  withpolar odd isclear since odd quadraticforms and symmetric bilinear forms are equivalent. Therefore, we call polar spaces with an underlying quadratic form in even characteristic also <em>orthogonal</em> polar spaces.</p> <p><a id="X863BC8E57C98A471" name="X863BC8E57C98A471"></a></p> <h5>7.1-2 PolarSpace</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolarSpace</code><>This category  asubcategory <code ="">IsLieGeometry<code>,and all geometries associatedtoanon-degenerate  orquadratic.</p < =functable ="" width1%>trtdclass"dleft"><codeclass=func&827 </code class=">pol/var )	<td class="tdleft">odd or even</td> </tr> <tr> <td class="tdleft">symplectic space</td> <td class="tdleft"><span class="SimpleMath">\( X_0 Y_1 - Y_0X_1+ \ldots + X_{n-1}Y_n - Y_{n-1}X_n\)</span></td <td class="tdleft">odd or even</td> <td class="tdleft">odd</td> </tr> <tr> <td class="tdleft">hyperbolic quadric< class"GAPDocTable"<b>:<bc  <c> <td class="tdleft"><span class="SimpleMath">\( X_0 X_1 + \ldots + X_{java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 4 <td class="tdleft"><span class<td class="tdleft> span class"">(\)/span>	tdleft>codeclass"func">R ParabolicQuadric</>( varclass=Arg"d F )/td"tdright">(  </d>/></table</> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ParabolicQuadric</code>( <var class="Arg">d</var>, <var class="Arg">q</var> )</td><td class="tdright">( &lt;div=func>table =funcwidth=1%>tr<d=tdleft>< =""&2 HermitianPolarSpace/> var=Arg>/> <c"Arg"><var <>td="tdright">( operation;)td</></></> <p>Returns: a parabolic quadric</p> <p><var class= <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ps := ParabolicQuadric(<panclassGAPpromptgap;</span< class"">ps:HermitianPolarSpace2,5;/> Q(2s class"GAPprompt"gapgt;</span> <span class""Display);/> <span class="GAPprompt">gap& form Q(2, 9) Parabolic bilinear form Gram Matrix:  1. .  . . 2  . 2 . Polynomial: [ [  x_1^2+x_2*x_3 ] ] Witt Index: 1 <span 1 .java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6 Q(4, 16) < class="APprompt">gap></span> <spanspan class="GAPinput">Display(ps;</span Q(4, 16 Parabolic quadratic form Gram Matrix/re<div>  1 . . . .  . . 1 . .  . . . . .  . . . . 1  . . . . . Polynomial: [ [  x_1^2+x_2*x_3+x_4*x_5 ] ] Witt Index: 2 Bilinear form Gram Matrix:  . . . . .  . . 1 . .  . 1 . . .  . . . . 1  .  . 11 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10   </pre></div> <p><a id="X7CF248A2864BCD8F" name="java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 35 <h5>7.2-4 HyperbolicQuadric</h5> <div class="func"><table class="func" width="100%"><tr><java.lang.StringIndexOutOfBoundsException: Range [0, 58) out of bounds for length 0 <div class="func"><table class="func" width="1GAPprompt">>ps : ParabolicQuadric(2,9);/> <p>Returns: a hyperbolic quadric</p> <p><var class="Arg">d</var> must be an2, 9 <div class="example"><pre> <span=GAPprompt&;>< classGAPinput : HyperbolicQuadric53;/> Q+(5, 3) <span class="GAPprompt">gap></span> <span class="GAPinput">Display(ps);</span> Q+(5, 3) Hyperbolic bilinear form   Gram Matrix  . 2 . . . .  . ... java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12  . . . 2 . .   =GAPprompt<spanGAPinput)</span  4 6  . . . . 2 . Polynomial Witt Index: 3 <span class="GAPprompt">gap></span> <span class="GAPinput">ps := HyperbolicQuadric(3,4);</span> Q+(3, 4) <span class"GAPprompt">gap;</> < class="GAPinput"Displayps<span Q+(3, 4) Hyperbolic quadratic  . .java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10 Gram  . 1.. .  . . . .  . . . 1  .. .. Polynomial[x_1*_+*  java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36 Witt Index rform Gram Matrix:  . 1 . .  1 . . .  . . . 1  . . 1 .   </pre><div="func">table=""width10%<trtd class=tdleft"#22;HyperbolicQuadric varclass"">d,  name="X85EEF70480D7149E"></a></p> <h5>7.2-5 EllipticQuadric</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EllipticQuadric</code>( <var class="Arg">d</var>, <var class="Arg">F</var> )</td><td class="tdright">( operation )</td></tr></table><<>Returns   pjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36 <div class="func"><<span class="GAPprompt">gap"GAPprompt">gap&;</> <span class="GAPinput">ps: HyperbolicQuadric53;/span> <:  ellipticjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35  .   <div ..  java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 <span classclass"gapgt spanclass""ps=(34) Q-) <span class="GAPprompt">gap></span> <span class="GAPinput">Display(ps);</span> Q-(3, 27) quadraticform Elliptic bilinear form Gram Matrix:  1 . . .  . 1 . .  . .  2  . . 2 . Polynomial: [ [  x_1^2+x_2^2+x_3*x_4 ] ] Witt: 1 <span class="GAPprompt">gap></span> <span class="GAPinput">ps := EllipticQuadric(5,8);</span> Q-(5, 8) <span class . .java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 Q-(5, 8) Elliptic form Gram Matrix:  1 1 . . . .  . 1 . . . .  . . . 1. java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12  .. . ..  .  . . . . . 1   Polynomial: [ [  x_1^2+x_1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 Witt: java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13 Bilinear form Gram Matrix:  . 1 . . . .  1 . . . java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12  . .. . 1  .  . . 1 . . .  . . . . . 1  .  . .   </ bilinear form <p>The following operations are  .  . java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 <p><a id="X7F6D25D57BC697B5" name="java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 8 <h5>7.2-6 IsCanonicalPolarSpace</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCanonicalPolarSpace</code>( span=GAPprompt&;<><="">ps(5,)</> <p>Returns: true or false</p> <p>This attributereturns when a polarspace a particular underlyingform canonical. The execution of thisattributeon a generaluserconstructedpolarspace needs to check the of<var class="Arg>ps.The obtained extra information is stored automatically asattribute or Arg>ps</var>, ascan noted by different printingof<var classlass"">ps</var> before after execution<pjava.lang.StringIndexOutOfBoundsException: Index 427 out of bounds for length 427 <div class="example"><pre  . ...  . <span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,,0]]Z(55^0</pan> [ [ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ],   . .. .   [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z.  ..1  java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 <span class="GAPprompt">gap lt quadratic form gt <span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Range [0, 60) out of bounds for length 0 <polar spacein ProjectiveSpace3,GF5) x_1x_2+x_3*x_4=0 > <span class="GAPprompt">gap></span> <span class="GAPinput">IsCanonicalPolarSpace(ps);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">ps;</span> Q+(3, 5): x_1*x_2+x_3*x_4=0 <span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[1,0,0],[0,0,1<p>< id=""X7F6D25D57BC697B5" name="X7F6D25D57BC697B5">/a [ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), 0*Z(3), Z(3)^0 ],    [ 0*Z(3), Z(<h5>7.26 IsCanonicalPolarSpace> <span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 &; quadratic form &; <span class="GAPprompt">gap></span> <span class="GAPinput">ps := PolarSpacep>Returnstrue  false/> <polar space in ProjectiveSpace(2,GF(3)): x_1^2-x_2*x_3=0 > <span class="GAPprompt">gap></span> <span class="GAPinput">IsCanonicalPolarSpace(ps);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">ps;</span> Q(2, 3): x_1^2-x_2*x_3=0   </pre></div> <p><a id="X8240380A7F8B6EC3" name="X8240380A7F8B6EC3"></a></p> <h5>7.2-7 CanonicalPolarSpace</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanonicalPolarSpace</code>( <var class="Arg">form</var> )</td><td classspan class="GAPprompt>gap;mat:=[,,0,][,,,][,,,],[0,,,]*Z()^0<spanjava.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 125 <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CanonicalPolarSpace</lt quadraticform> <p>Returns: a classical polar space</p> <p>the canonicalpolarspaceisometricto the given spacevarclassArg>P/varorthepolarspacewith <class"> <p><idX7EE25EDF7ADC30D7name"">/><p> <h5>7.2-8 StandardPolarSpace</h5> spanclass="GAPprompt">gap>/> <span class=""IsCanonicalPolarSpace<spanjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95 <div class="func"><table class="func" width="100%"><tr><td class="tdleft">code="func" /code> var=""><var>)/><tdclasstdright(nbsp )</td></tr<table<div <Returnsa classical spacep> <p>the polar space induced by a standard form and similar to the given polar space <var class="Arg">P</var> or the classical polar space with underlying form <var [ 3^,0*() *(), *(3,*() 3^ ]  <p><a id;   &; <h4>7.3 <span class="Heading">Basic operations for finite classical polar spaces</span></h4> <p><<pan="">><spanspan=GAPinput : f);/pan <h531 </> <classfunc" <p>Returns:  <p>The=X8240380A7F8B6EC3name=X8240380A7F8B6EC3>/>/java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62 <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ps := EllipticQuadric(5,4);</span> Q-54) <span class="GAPprompt">gap></span> <span class="GAPinput">vs :<p>: a polar<p ( (^)6) <span W(, 1java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 < class>>> span=GAPinputvs=(ps<span ( GF(3^4)^4 )   </pre></div> <p><a id="X8606750A8586DF8D" name="X8606750A8586DF8D"></a></p> <h5>7.3-2 AmbientSpace</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AmbientSpace</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation )</<p: a classical polar</p> <p>Returns: the ambient projective space</p> <p>When <var class="Arg">ps</var> is a polar spacejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ps := EllipticQuadric(5,4);</span> Q-(5 4java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 <spanclass"APprompt>>/span> AmbientSpace(ps); ProjectiveSpace(5, 4) <span class="GAPprompt">gap></span> <span class="GAPinput">ps := SymplecticSpace(3,81);</span> W(3, 81) <span class="GAPprompt">gap><< rojectiveSpace(33 8)   </pre></div> <p><a id="X84FDF25D797B874Bjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <h5>7.3-3 ProjectiveDimension</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveDimension</code>( <var class="Arg"(54 <div class="func"><table class="func" width="span =GAPprompt">><span span=""ps,1)<s> <p>Returns(, 8) <p>When <var class="Arg">ps</var> isspan class"">><span <spanclass">s =UnderlyingVectorSpace(java.lang.StringIndexOutOfBoundsException: Range [93, 92) out of bounds for length 101 <div class="example"><pre> <span java.lang.StringIndexOutOfBoundsException: Range [96, 11) out of bounds for length 96 Q-) <span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveDimension(ps);p>:  ambient space/> 5 <span class="GAPprompt">gap></span> <span class="GAPinput">ps := SymplecticSpace(3,81);</span> W(3, 81) <span class="GAPprompt">gap></span> <span class="GAPinput"java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0 3  (5, 4 </pre></div> <p><a id="X827146F37E2AA841" name="X827146F37E2AA841"></a></p> <h5>7.3-4 Rank</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Rank</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation )</td></tr></table></div> <p>Returns: the rank of <var class="Arg/pre> <p>When <var class="Arg">ps</var> is a polar space,  <div class="example"><pre> <span="GAPprompt">gap;</><spanclass"">ps:== EllipticQuadric54;</> Q-(5, 4) <span class="GAPprompt">gap></span> <span class="GAPinput">Rank(ps);</span> 2 <span class="GAPprompt">gap></span> <span class=">	#82; Dimension( )>	operation )

‣ Coordinates( p )

( operation )