<p>Consider the projective space <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> with underlying vector space <span class="SimpleMath">\(V(n+1,q)\)</span>. Consider a non-degenerate sesquilinear form <span class="SimpleMath">\(f\)</span>. Then <span class="SimpleMath">\(f\)</span> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<pSymmetricbilinear forms completely different geometric propertiesin in .On other, comparable orthogonal characteristic exist characteristic. The algebraic established quadratic forms a space forms a non-singular form<pan="">(\<span a spacespanclass"SimpleMath"\(V(n+1,))/> Theclassical withspan class""\(f)span theincidence geometrywhose elementsarethesubspaces ,\)<spanwhose vectorsubspace totallysingular with to< class""\q)<span connection with polar inoddcharacteristic clearsince quadratic and are. , with quadratic ineven <>orthogonalem>polarspaces/>
<div class="func"><table class="func" width="10&t;bilinear form >
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"lt space (
<p><var,x_3
< ="<>
<span="gap;<span class"> :=(poly,r);
[ [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), Z(5)^3, 0*Z(5) ],
*Z5,5,0Z) 0Z(5 ],[()2 0Z5,0*() 0Z5 ]java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
<span class=lt (3GF^):x_1+x_2x_30gt
< bilinear >
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PolarSpace [ 7^,*() 0Z(),0*7,*()]
<polar space in ProjectiveSpace(
3,GF(5^2)): x1*y4+Z(5)^3*x2*y3+Z(5)*x3*y2-x4*y1=0 > span ="GAPprompt">>> spanclass"> : PolynomialRing(GF(2,)
GF(20*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">poly := r.3*r.2+r.1*r.4;</span>
x_1+x_2x3
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByPolynomial(poly,r);</span>
<quadraticform>java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
<span class="GAPprompt">gap></spanlt; space ProjectiveSpace,GF7):x_1^+^2x_32+^2+520gt
<polar space in
<panclassGAPprompt">gap>">mat : IdentityMat(,GF(7));
[ [ Z(7)^0, 0*java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 0*Z(7), Z(7)^0, 0*Z(7), 0*Z(7), 0*Z(7) ],
[0Z7,*Z() Z(7))^, *Z7) 0() ],java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
[0*() 0*Z(7), 0*Z(77, Z7),0*() ],
[ 0*Z(7), 0*Z(7), 0*Z(7), 0*Z(7), Z(7)^0 ] ]
<span=GAPprompt>>/> <span="">phi:= PolarityOfProjectiveSpace
&;polarityof PG(,G() >
<spanspan classGAPprompt>>/> <span class""> :=PSL,)<span
<polar spaceinProjectiveSpace(4,GF():x_12x_22x_32+^2+x_5^2= &;
</pre</>
<p><strong class="pkg">FinInG</strong> relies on the package <strong class="pkg">Forms</strong> for its facility with sesquilinear and quadratic forms. One can specify a polar space with a user-defined form, and we refer to the documentation for <strong class="pkg">Forms</strong> for information on how one can create and use forms. Here we just display a worked example.</p>
<div class="example"><pre>
<span classspan="">&;/><spanclassGAPinputplanes= (ps );<spanjava.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103 span=GAPpromptgap;/pan <span=""form QuadraticFormByMatrix3)</>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PolarSpace( form );</span>
<polar space in
6,GF</pre</div>
<span class="GAPprompt">gap></span> <span class="GAPinput">psl32 :
Group[(,6(57,(,,)365 ]java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
<span class="GAPprompt">gap></span> <span class
<spanclass"">gap/> <span="GAPinput">ovoid :=Union List(reps,x-gtOrbit, x, Permuted ));;</span
<span class="GAPprompt">gap></span> <span class="GAPinput">ovoid := List(ovoid, x -> VectorSpaceToElement(ps, x));;</span>
< class"">gapgt</span><spanclass=GAPinput> =AsList Planes ps );/span>
#I Computing collineation group of canonical polar space...
<span class="GAPpromptcenter>[f(,w\lambda(\phiv\w),</pjava.lang.StringIndexOutOfBoundsException: Range [63, 64) out of bounds for length 63
true
<p>for all vectors <span class="SimpleMath">\(v,w\)</> andsome non-zero <span class="SimpleMath">\\ inGF(\)</> . forms even, thenumber ofsimilarity classes on of <span classSimpleMath(\)/panIn dimension is one , forms this are parabolic)forms even there , either bilinear
<p>Consider now a vector space <span class="SimpleMath">\(V\)</span> and a quadratic form <span class="SimpleMath">\(q\)</span> on <span class="SimpleMath"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<t=tdleft <td
< ="odd /java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
<p>If <span classtdleftortd
<divtr caption="b>able />finite lassicalpolarspaces/aptionjava.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82
<tr>
<td class="tdleft">polar space</td>
<td class="tdleft">standard form</td>
< class"characteristic< ="SimpleMath\p)/span
<td class="tdleft">projective dimension</td>
</tr>
<tr
<td class="tdleft">hermitian polar space</java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 42
<td class="tdleft"><span class="SimpleMath">\(X_0^{q+1} + X_1^{q+1} + \ldots + X_
<td class="tdleft">odd or even</td>
<td class="tdleft">odd or eventd =tdleft<td>
</tr<td
<tr>
< class="tdleft">symplectic space</td>
<td class="tdleft"><span class="SimpleMath">\( X_0 Y_1td="tdleft"< class"SimpleMath>(pequiv,3\{8\)) orSimpleMath">\(p=2\)
<td class="tdleft">odd or even</td>
<tdtdclasstdleft></>
</tr>
<tr>
<td class="tdleft">hyperbolic quadric< ="">parabolicquadric/>
<td class="tdleft"><span class="SimpleMath">\( X_0 X_1td=""><span=""\(X_0 X_1X_2\+{}\/><panSimpleMath>(t\)/ primitiveof=SimpleMath>({}))span>/
<td class="td ="tdleft"SimpleMath>\(p \equiv 5,7 \pmod{8}\)</span></td>
<class"">odd</tdjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
</tr>
<tr>
< class""hyperbolic<td>
<td class="tdleft"><span class="SimpleMath">\(2(X_0 X_1 + \ldots + X_{n-1}X_n)\)</span></tdtd="tdleft">/tdjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27 td< =""\( equiv{4\<span>/>
<td class="tdleft">odd</td>
</tr>
<tr>
<tdclasstdleft </>
<td class="tdleft">X<sub>0</sub><sup>tclasstdleftodd/>
<td class="tdleft"><span ="">ellipticquadric
< classtdlefteven</>
</tr>
<>
<td class="tdleft">parabolic quadric</td>
<td class="tdleft"><span=""><>
<td class="tdleft"><span>
<td class="java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 6
<>
<tr>
<td class="tdleft">elliptic quadric</td>
<td class
<td=tdleftspan"SimpleMath">(\ 3 \{})s>/td>
<td class
<trjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<tr>
<td class="tdleft">elliptic class"< class=""width="100%" class="< ="func>#227;SymplecticSpace("d var =java.lang.StringIndexOutOfBoundsException: Index 249 out of bounds for length 249
<td class="tdleft"><span class="SimpleMath">\(X_0div=func< class"" ="0%"">tr>td class"tdleft< ="> <ddvar q/> </>td class"">&;operation;)/><tr<table/>
=tdleft ="\((\equiv\{4)>
<td class="tdleft">odd</td>
</trclassexample>pre
<trspa =GAPpromptgap;</pan < class"GAPinput">ps= (3,4;/>
<td class="tdleft">elliptic quadric</td>
<tdclass""><spanclass="SimpleMath">\X_02 X_0X_1=)<span>
<td class=W(,)
<td class="tdleft">odd</td>
</tr>
</table><br />
</ .. .
<p>We
<p>The <strong class/pre></div
<p>The following five operations always<p<a id="X7A8CF840833D3D8B" name"X7A8CF840833D3D8B"></a><p>
<p>This function returns the symplectic polar space of dimension <var class="Arg">d</var> over <varclass="Arg">Fdiv="func"><table class="" width"10%><>tdclass"tdleft class="">#827; HermitianPolarSpace</>( varclass=Argd/>, <varclass"">q<var)td>tclasstdright(nbspoperation&;)td<tr<table<divjava.lang.StringIndexOutOfBoundsException: Index 253 out of bounds for length 253
<p><a id=": [ [ x_1^6x_2^x_3^] java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
<h5>7.2-2 HermitianPolarSpace</h5>
<divclass""table="" =1"<< =func#27 ParabolicQuadriccode(< "">/,varArgF< <> =tdrightnbspoperation;)/d<tr/>divjava.lang.StringIndexOutOfBoundsException: Index 250 out of bounds for length 250
< class""<class"" width"00%"<>< class""><codeclass"func>#827; (( q/var>>)/td< class"tdright>; </d>/tr</table>/div>
<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>
<divjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< ="">>> span=GAPinput = HermitianPolarSpace(22)<span
H(2, 5^2)
<pan=GAPprompt>></spanspan=GAPinput>(ps<span
H(2, 25)
Hermitian java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
Gram Matrix java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
1 .
. 1 .
. . 1
Polynomial: [span=gapspan< class()>
(1
<p>/iv
<p><a id="X7CEDD5CC7A6B0E64" java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 10
<h5
<. .
<div class="func"><table class="func" width="100%"><tr
<p>Returns: a parabolic quadric</p>
<p><var
<div class="example"><pre>
<classgap/ < ""ps= ParabolicQuadric<spanjava.lang.StringIndexOutOfBoundsException: Range [97, 98) out of bounds for length 97
Q(,9
<span class="GAPprompt">gap>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
Q(2, 9)
Parabolic bilinear form class="">gapgt;<spanspan="">ps=HyperbolicQuadric(5)<spanjava.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 98
Gram Matrix:
1 . .
. . 2
. 2 .
Polynomial: [ [ x_1 :
Witt . .
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ParabolicQuadric(4,16);</span>
Q(4, 16)
<spanclass"GAPprompt"">gap>Display(ps);<
(,16)
Parabolic quadratic form
Gram Matrix:
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
.=GAPprompt></pan spanclass"GAPinput>(ps);/>
. . . . .
.. . 1
. . . . .
Polynomial: [ [ x_1^2+x_2*x_3+x_4*x_5 ] ]
Witt Index: 2
Bilinear Matrix:
Gram Matrix:
. .java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
. . 1 .
. 1 Polynomial: [ x_1*_x_3*x_4 ]]]
. . . . 1
. . . 1 .
<p><var class="Arg">d</var> must be an odd positive integer. This function returns the hyperbolic quadric 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>
<span class=>gapgtspanspanclass =(,)<span
Q+(5, 3)
<span class="GAPprompt"<>Returns an quadric</p>
Q+(5, 3)
Hyperbolic bilinear form
Gram Matrix:
. 2 . . . .
2 . . . . .
.. 2..
. . 2 . . .
. . . . . 2
. .. .2 .
Polynomial: [ [ x_1*x_2+x_3*x_4+x_5*x_6 ] ]
Witt Index: 3
<span ="GAPprompt"&;/span< =GAPinput> : HyperbolicQuadric3,,4;/>
Q+(3, 4)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(psQ-(3, 27java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
Q+(3, 4)
yperbolic form
Gram Matrix:
. 1 . .
. . . .
. . . 1
. . . .
Polynomial: [ [ x_1*x_2+x_3*x_4 .java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
Witt Index Index java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
Bilinear form
Gram Matrix:
. 1 . .
1 . . .
. 1
. . 1 .
<p>The following operations are applicable on any classical ; (,():*+x_3java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
pa=name>>pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
>. </h5
<div class="func"><table classlt gt
<>: or<pjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
java.lang.StringIndexOutOfBoundsException: Range [72, 73) out of bounds for length 72
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<class"gap>s lass"> [01,0,0000,0001,[0,,00]Z(5^0;/>
[ [ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), 0*Z(5), 0*Z(5) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat,GF(5));</span>
&; gtaclassicalspace
<span tothe polar < class=""P or the classical underlying formform><p
<polar space ina ="" =X7EE25EDF7ADC30D7><a</java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
< classGAPpromptgapgt<spanspan"GAPinput>(ps);/>
true
<span class="GAPprompt">gap></span> <span> class>&82;StandardPolarSpace<code(< classArgP/ <td ="">&;operation;<td>/>/>
Q+(3, 5): x_1*x_2+x_3*x_4=0
<span class="GAPprompt">gap></span> <span class="GAPinput">matp>Returns: a polar</>
[[Z()0 0*Z(,0Z(3 ] [0Z() 0Z(, Z()0,
[ 0*Z(3), Z(3)^0, 0*Z(3) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat,GF(3));</span>
<quadraticform gtjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
< classGAPpromptgap;/> < class"GAPinput">ps=PolarSpace(orm<s>
<polar space in ProjectiveSpace
<span class>7.- UnderlyingVectorSpaceh5
false
<span class="GAPprompt"<iv class=">
<p>the canonical polar space isometric to theGF(22^
<p><a id="3 )
<h5>7.2-8 StandardPolarSpace</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardPolarSpacespan="GAPprompt"gap;</span < class""> : UnderlyingVectorSpace);/>
<div class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>Returns space
<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 class="Arg">form</var>.</p>
<div class= =G"gapgt;<>java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 86
<p>Returns: a vector space</p>
<(,8
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
-5, )
<span class="GAPprompt">gap></span> <span class="GAPinput">vs := UnderlyingVectorSpace(ps);</span>
( GF(2^2)^6 )
<class""gap>/>< classGAPinput> := SymplecticSpace(3,81;/panjava.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
W3 1java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
<class=GAPpromptgap&;/><span =GAPinput"v: (ps);
( GF(3^4)^4 )
<h5>7.3-3 ProjectiveDimension classGAPprompt>/spanspan =GAPinputps : EllipticQuadric(,)<spanjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
<div class tr"code =func"&827Dimension></var<td=""(;peration/tdtr<table>
<p>Returns: the dimension of the ambient projective space of <var class="Arg">ps</var></p>
<p>When <var class="Arg">ps</var> is
<div class="example"><pre>
<span class="GAPprompt">gap></W7,1java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
Q-(5, 4)
<span<id=X7BCBA564829D9E89=X7BCBA564829D9E89a>pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
5
<span class="GAPprompt"divclassfunctableclassfunc""codeclass="#2;BaseField var>/>/>tdtdright;operation)/td/tr/<div
W(8)
<span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveDimension(ps);</span>
3
<div class="func"><table(^java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<Returns rank < =""><var</java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
<p><a id="X7BCBA564829D9E89" name="X7BCBA564829D9E89"></a>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
<>7.35BaseField<h5
<div="< =func"=0%"
">&82;BaseField/ode> class""><v><tdtd=tdright&; </tdtr/table>
<p>Returns: the base field of the polar space <var class="Arg">ps</var></p>
<div class="example"><pre>
<span class="GAPprompt"span=GAPprompt>&<spanspanclassGAPinput : olarSpaceform)<span
Q+(5, 7)
<span class="GAPprompt">gap></span> <span =GAPpromptgap&;/><span="GAPinput"IsHyperbolicQuadricps;/span
GF7java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
< =GAPprompt></>< lass"GAPinput> :=HermitianPolarSpace(2,26;<>
H(2, 16^2)
<span class="GAPprompt">gap></span> <span class="GAPinput">BaseField(ps);</span>
GF
</pre></div>
<p><a id var=""ps> /
<h5>7.3-6 IsHyperbolicQuadric</h5>
<div class="func"><table class="func"span="">gapgtspanspanclassGAPinputmat IdentityMat,GF;<spanjava.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
<p>Returns: true or false</p>
<p>returns classGAPprompt>/> span"GAPinput> : java.lang.StringIndexOutOfBoundsException: Range [79, 78) out of bounds for length 92
<div class="example"c""gap;<span=GAPinput=IdentityMatGF);<>
<span class="GAPprompt">gap></span> spanclassGAPprompt
< &;
<span""&;/span"GAPinput>: PolarSpaceform
< bilinear form >
<span class="GAPprompt">gap></span
<polar space in ProjectiveSpace(
5,GF(5)): x_1^2+x_2p< ="X87ADB8B5790776C7" name"X87ADB8B5790776C7"<a/>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsHyperbolicQuadric(ps);</spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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></span> <span class="GAPinput">form := BilinearFormByMatrix(mat,GF(7))pReturns: orp>
< bilinear form >
<span class="GAPprompt"
<polar space in ProjectiveSpace(
5GF)) ^2+^2x_3^2x_5+x_62 gt
<span class="GAPprompt">gap></span> <span class="GAPinput">IsHyperbolicQuadric(ps);</span>
false
</pre></div>
<<idjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<p>returns true if and only if <var class="Arg">ps</var> is an elliptic quadric.</p>
<div class="example"><pre> span=GAPpromptgap;span<c""mat 6()/>
< mutable compressed matrix 6x6 over GF(5) >
< =GAPpromptgap;> s =GAPinput =BilinearFormByMatrix,GF5;/java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
&; >
< classGAPprompt>>span class =PolarSpace);<span
&;polar ProjectiveSpace
5(): x_1^2x_3^2^+^20>
<span class="GAPprompt">gap></span> <span class="span class=GAPprompt">apspan=GAPinput(pss>
false
<span class="GAPprompt">gap></span> <span class=true
< mutable compressed matrix 6x6 over </re</>
<span class="GAPprompt">gap></span> <span class="GAPinput">form := java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 62
< java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
<span =GAPpromptgap><span < class"">ps=PolarSpace);/pan
<polar space in ProjectiveSpace(
5,GF(7)): x_1
< class"">><span>< class"">IsEllipticQuadric(ps;/span
true
</pre></div>
<id"name"">pjava.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">&
> or<pjava.lang.StringIndexOutOfBoundsException: Range [29, 30) out of bounds for length 29
<preturns if only < =Arg<> isparabolic./pjava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
d =example>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := IdentityMat(5,GF(9));</span>
[[()0 Z)(,0Z(),0Z3 ,java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
[ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ],span=""gap;<s> span=GAPinput :(psmat;<span>
[ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3),;function;(&;&;java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
[ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) can; to
[ 0*Z(3),0*Z() 0Z3,0Z3) ()0] ]
<span class="GAPprompt">gap>spanclass"">&;<spanspanclassGAPinput : 1100,0010]*()0;span
& &;
<span class="GAPprompt">gap></spanclassGAPprompt&;<span>< classGAPinputline =VectorSpaceToElement,)</span>
<polar space in ProjectiveSpace(4,GF< line in W(3 4>
<span class"GAPprompt">gap></spanspanclass"">IsParabolicQuadricps;/>
true
<
[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ],
*() Z() 2^,0*(2,0*Z2,java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
[ 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(java.lang.StringIndexOutOfBoundsException: Range [38, 39) out of bounds for length 0
*2,0Z2,*() *() *() java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(mat,GF(8));</span>
< &;
<spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<polar space in ProjectiveSpace(4,GF(2^3)): x_1^2+x_2*x_3+x_4*x_5=0 >
<span class="GAPprompt">gap>p> object by operation contained every projective of < class=""ps/ar,but is an element of<var ="Arg">ps</var>.Hencetesting incidenceresultsinan error message.<p>
true
</pre></div>
<p><a idspan="">>> <spanclass"GAPinputps:= (1,)
<>7 span=Heading of classical spaces>/4java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
<p>The elements of a finite classical polar space <span class=
<div class="func"><table class="func" width="100%"><trpReturnsthe dimension subspace of polar.Theoperation < classfile/>is on <p>
<p>Returns: class>pre
<> <varclassArg</> space, and< =""v/var either ( )oran< ="">(\<spanxspan classlass="SimpleMath">(n\<span>matrixanspanclassSimpleMath>\((m-1)\<span ofclass>(n-1)/pan ambient.In case<var="Arg"><var owsbasisforthesubspace swhen<var=Arg>/ar is zero-vector trivial returnedIt defined spanclassSimpleMath\\< iswith orm classArg>s<var./java.lang.StringIndexOutOfBoundsException: Index 755 out of bounds for length 755
=example<java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := SymplecticSpace(3,4);lt lineQ-7 >java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
W,java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<span="gapgt;"v=[,,,]Z4^;/span>
[ Z(2)^0, 0*Z(2), Z(2)^0, 0*<empty subspace &tjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
<span="GAPprompt>>span>span =GAPinput> : (,)
<a point in W(3, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[1,1,0,1],[0,0,1,0]]*Z(4)^0;</span>
[ [ Z(2)^0, Z(2)^0, 0*Z(2), java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<span class="GAPprompt"
Error &;x> not generatean oflt;> called from
<function "unknown">
fromread-evalloop atline2of**
you can 'quit;' to quit
you can ';' to continue
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">quit;</span>
<span class="GAPprompt">gap></span><>Returns the collectionof ofthe spacevar="">ps/varvar=""j<var/java.lang.StringIndexOutOfBoundsException: Index 125 out of bounds for length 125
[[Z2^0 Z2^,0Z2,0Z2 , *(,0Z2, (2^,0Z2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">line := VectorSpaceToElement(ps,mat);</span>
<a in(3,);
<span class=""gapgt<span>< class""> : (,[,,00*()0);/span
< &;
</pre;planesof(,32gtjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
<p><a id="X8461BCEF862B9A7B" name of12>
<h5>7.4-2 EmptySubspacejava.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 64
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EmptySubspace</code>( <var class="Arg">ps</var> )</tdp>eturns the of anelement classArg<var a <>
<p>Returns: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<The by projective classpsvar but not element < =Arg<var.Hence incidence an message./>
<div class="example"><pre>
=GAPprompt&;/pan<span="GAPinput> java.lang.StringIndexOutOfBoundsException: Range [88, 87) out of bounds for length 102
H(10(3, 49))
<span class="GAPprompt">gap></span> <span<pre<div>
< empty subspace >
</pre></div>
<p><a id="X84FDF25D797B874B" name="X84FDF25D797B874B"></a> ="func>java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 219
<h5>7.4-3 ProjectiveDimension</h5>
<div class="func">div classexamplepre
s="func">‣ Dimensioncode( < =Arg/var<td =tdright&)<td
<p>Returns: the projective dimension of a subspace of a polar space. The operation <code class="file">ProjectiveDimension</code> is also applicable6 java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<div class="example"><pre>
< =GAPprompt&;> < =GAPinput = EllipticQuadric8;/spanjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
Q-(7, 8)
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[0
[ [ 0*Z(2), 0*Z(2), Z(2)^0java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
[ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(
<assgapgt/><span=""line(,)<span
<a line in Q-
<span class="GAPprompt">gap&h>.-1span=Heading containment
1
< ""&;/ spanGAPinput)<span
1
<spanclass">&;> spanc="> :=(ps/>
< empty subspace >
<span=GAPprompt>/>< =">ProjectiveDimensione)/java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
-java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<p>For the projective space <var class="Arg">ps</var> of dimension
=">
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
H8 32
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<planes of H(8, 13^2)>
<span class="GAPprompt">gap></span> <span class="GAPinput">solids := ElementsOfIncidenceStructure(ps,4);</span>
<solids of H(8, 13^2)>
<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AmbientSpace</code>( <var class="Arg">el</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: returns the ambient space of an element <var class="Arg">el</var> of a polar space</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>
<p>Recall that for projective spaces, incidence is symmetrized containment, where the empty subspace and the whole projective space are excluded as arguments for this operation, since they are not considered as elements of the geometry, but both the empty subspace and the whole projective space are allowed as arguments for <code class="file">\in</code>.</p>
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