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<p>Consider the projective space <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> with un <pSymmetricbilinear forms completely different geometric propertiesin in .On other, compar <p><a id="X863BC8E57C98A471" name="X863BC8E57C98A471"></a></p> <h5>7.1-2 PolarSpace</h5> <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 <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 <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 [ 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 [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> fo <div class="example"><pre> <span classspan="">&;/><spanclassGAPinputplanes= (ps );<spanjava.lang.StringIndexOutOfBoun 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 <span class="GAPprompt">gap></span> <span class <spanclass"">gap/> <span="GAPinput">ovoid :=Union List(reps,x-gtOrbit, x, Permuted ));;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">ovoid := List(ovoid, x -> VectorSpace < class"">gapgt</span><spanclass=GAPinput> =AsList Planes ps );/span> #I Computing collineation group of canonical polar space... <span class="GAPprompt true </pre></div> <p><a id="X850CD32686B0656B" name="X850CD32686B0656B"></a></p> <h4>7.2 <span class="Heading">Canonical and standard Polar Spaces</span></h4> <p>To introduce the classification of polar spaces, we use the classification <p>Consider a vectorjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds <p class="center">\[f(v,w) = \lambda f'(\phi(v),\phi(w)),\] <p>for all vectors <span class="SimpleMath">\(v,w\)</> andsome non-zero <span class="SimpleMath <p>Consider now a vector space <span class="SimpleMath">\(V\)</span> and a quadratic form <span cla <<t=tdleft <td < ="odd /java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for le <p>If <span classtdleftortd <divtr caption="b>able />finite lassicalpolarspaces/aptionjava.lang.StringIndexOutOfBou <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: I <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>(pequi | < 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 l <> <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(" <td class="tdleft"><span class="SimpleMath">\(X_0div=func< class"" ="0%"">tr>td class"tdl =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">< <p><a id="X850980EF8607BDA7" name="X850980EF8607BDA7"></a></p> <h5 <divh57.- HermitianPolarSpace<h5> <div class="func"><table <p>Returns polarspace</> <p>This function returns the symplectic polar space of dimension <var class="Arg">d</var> over <var <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ps := SymplecticSpace(3,4);</span> W(3, spanclass"GAPprompt">gapgt;<span span=""ps= HermitianPolarSpace(,5;/> < classgap;s> <span=">(ps W(3, 4) Symplectic form,2) Gram: . 1 . . 1 . . . . . . 1 . . 1 . Witt Index: 2 </pre></div> <p><a id=": [ [ x_1^6x_2^x_3^] java.lang.StringIndexOutOfBoundsException: Index 38 <h5>7.2-2 HermitianPolarSpace</h5> <divclass""table="" =1"< < class""<class"" width"00%"<>< class""><codeclass"func>#827; (( q/var>>)/td< class"tdright>; </<p>Returns: a Hermitian polar space</p> <p>This function returns the Hermitian polar space of dimension <var class="Arg">d</var> over <var c <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 l Gram Matrix java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for l 1 . . 1 . . . 1 Polynomial: [span=gapspan< class()> (1 <p>/iv <p><a id="X7CEDD5CC7A6B0E64" java.lang.StringIndexOutOfBoundsException: Index 32 out <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: Ran 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;<span span="">ps=HyperbolicQuadric(5)<spanjava.la Gram Matrix: 1 . . . . 2 . 2 . Polynomial: [ [ x_1 : Witt . . <span class="GAPprompt">gap></span> <span class="GAPinput">ps := ParabolicQuadric(4,16);</spa 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 1 . . 1 . . 1 Polynomial: [ x_1*_x_3*x_4 ]]] . . . . 1 . . . 1 . </pre></div> <p><a id="X7CF248A2864BCD8F" name="X7CF248A2864BCD8F"></a></p> <h5>7.2 java.lang.StringIndexOutOfBoundsException: Range [13, 14) out of bounds for < classfunc< classfunc ="00">><td"">code""&87 HyperbolicQuadric(< =Arg,var"F/var <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Hy <Returns:ahyperbolicquadric</> <p><var class="Arg">d</var> must be an odd positive integer. This function returns the hyperbolic q <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.Stri Q+(3, 4) yperbolic form Gram Matrix: . 1 . . . . . . . . . 1 . . . . Polynomial: [ [ x_1*x_2+x_3*x_4 .java.lang.StringIndexOutOfBoundsException: Index 8 Witt Index Index java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds Bilinear form Gram Matrix: . 1 . . 1 . . . . 1 . . 1 . </pre></div> <p><a quadratic <h5>7.2-5 EllipticQuadric</h5> <div class="func"><table . . <div class="func"><table class="func" width="1 . . . . <p>Returns: p>Returns: an <p><var class="Arg">d</var> must be an odd positive integer. This function returns the elliptic qua <div class="example"><pre> . . .. <span class="GAPprompt">gap> .. Q-(3, 27) <span class= .. . 1. Q-(3, 27) Elliptic form Gram Matrix: 1. .. . 1 . . . . . 2 . . 2 . Polynomial: [ [ x_1 Witt Indexjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for le < class"GAPprompt">gapgt/span <span class=GAPinputps := EllipticQuadric8;/span Q-(5, 8) <span class="GAPprompt">gap></span> <span class="GAPinput">Display(ps);</span> Q-(5This true withparticular is execution on needs type var"/ obtainedextraisstoredautom Elliptic quadratic form Gram Matrix: 1 1 . . . . . 1 . . . . . . . 1 . . . . . . . . . . . . . 1 . . . .. Polynomial: [ [ x_1^2+x_10*();/pan Witt Index: 2 Bilinear form Gram Matrix: . 1 . 1 . . . . . .. 1. . . 1 . . . . . . . . 1 . . . . 1 . <; &; <p>The following operations are applicable on any classical ; (,():*+x_3java.lang.StringInd pa=name>>pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for le >. </h5 <div class="func"><table classlt gt <>: or<pjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length java.lang.StringIndexOutOfBoundsException: Range [72, 73) out of bounds for leng 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 &; gtaclassicalspace <span tothe polar < class=""P or the classical underlying form <polar space ina ="" =X7EE25EDF7ADC30D7><a</java.lang.StringIndexOutOfBoundsException: I < classGAPpromptgapgt<spanspan"GAPinput>(ps);/> true <span class="GAPprompt">gap></span> <span> class>&82;StandardPolarSpace<code(< classArgP/ <td ="">&;oper 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 <quadraticform gtjava.lang.StringIndexOutOfBoundsException: Index 24 < classGAPpromptgap;/> < class"GAPinput">ps=PolarSpace(orm<s> <polar space in ProjectiveSpace <span class>7.- UnderlyingVectorSpaceh5 false <span class="GAPprompt"<iv class=">
Q(2, 3): x_1^2-x_2*x_3=0 </pre></div> <p><a id"" name=" <h5>7.2-7 CanonicalPolarSpace</h5> < <div class="func(, ) <>Returnsclassical polar polar space</> <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">‣ St <div class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for l <>Returns space <p>the polar space induced by a standard form and similar to the given polar space <var class="Arg"> <p><a id="X7A04340A7EC9215B" name="X7A04340A7EC9215B"></a></p> <h4>7 <p><a <h5(,) <div class= =G"gapgt;<>java.lang.StringIndexOutOfBoundsException: Range [45, 44) out <p>Returns: a vector space</p> <(,8 <div class="example"><pre> <span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException -5, ) <span class="GAPprompt">gap></span> <span class="GAPinput">vs := UnderlyingVectorSpace(ps);</ ( GF(2^2)^6 ) <class""gap>/>< classGAPinput> := SymplecticSpace(3,81;/panjava.lang.StringIndexOutOfBou W3 1java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for len <class=GAPpromptgap&;/><span =GAPinput"v: (ps); ( GF(3^4)^4 ) </pre></div> <p><a id="X8606750A8586DF8D" name="X8606750A8586DF8D"></a></p> <h5>7.3-2 spanclass="GAPprompt">gap></span> <span class="GAPinput">ps := EllipticQuadric(5,4) <div class="func"><table class="func" width="100%"><tr><td class(5, 4 <>eturns the projective</ <p>When <var class="Arg">ps</var> is a polar space, java.lang.StringIndexOutOfBoundsException <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ps := EllipticQuadric(5,4);</span> Q- ) <span class="GAPprompt">gap></ ProjectiveSpace(5, 4) <span class="GAPprompt"java.lang.StringIndexOutOfBoundsException: Index 0 out of b W(3, 81) <span class="GAPprompt">gap></span> <span class="GAPinput">AmbientSpace(ps);</span> ProjectiveSpace(3, 81) <>/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12 <p><a id="X84FDF25D797B874B" name="X84FDF25D797B874B"></a> <h5>7.3-3 ProjectiveDimension classGAPprompt>/span span =GAPinput <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 Q-(5, 4) <span<id=X7BCBA564829D9E89=X7BCBA564829D9E89a>pjava.lang.StringIndexOutOfBoundsExce 5 <span class="GAPprompt"divclassfunctableclassfunc" W(8) <span class="GAPprompt">gap></span> <span class="GAPinput">ProjectiveDimension(ps);</span> 3 </pre></div class"GAPprompt"gap<>< =">ps=HyperbolicQuadric(,) <p><a id="X827146F37E2AA841" name="X827146F37E2AA841"></a></spanclassGAPpromptgap&;span<span h5 Rankh5 <div class="func"><table(^java.lang.StringIndexOutOfBoundsException: Index 7 out of <Returns rank < =""><var</java.lang.StringIndexOutOfBoundsException: Index 53 out of bou <p>When class=func< classfuncwidth"0%>trtd">code =func#87 </code(<var=Argps<var <td"&property&bsp)/td> <div class="example"><pre> <span class="GAPprompt">gap> Q-<iv="example">pre <span class"">gap>/pan "">ank);/pan> 2 <span class="l; mutablecompressed x6 GF5&; Q+(5, 4) <span span class">gapgt; class"">form =BilinearFormByMatrix(mat,GF(5);/> 3 <span class="GAPprompt">gap<bilinear ; W(7, 81) <span class="GAPprompt">gap></spanltpolar in( 4 </pre></div> <p><a id="X7BCBA564829D9E89" name="X7BCBA564829D9E89"></a>java.lang.StringIndexOutOfBoun <>7.35BaseField<h5 <div="< =func"=0%" >&82;BaseField/ode> class""><v><tdtd=tdright&; </tdtr/table> | ||||||||||||||||||
2026-04-02