<h3>6 <span class="Heading">Polarities of Projective Spaces</span></h3>
<p>A <em>polarity</em> of a incidence structure is an incidence reversing, bijective, and involutory map on the elements of the incidence structure. It is well known that every polarity of a projective space is just an involutory correlation of the projective space. The construction of correlations of a projective space is described in Chapter <a href="chap5_mj.html#X816FCFB683915E8A"><span class="RefLink">5</span></a>. In this chapter we describe methods and operations dealing with the construction and use of polarities of projective spaces in <strong class="pkg">FinInG</strong>.</p>
<h4>6.1 <span class="Heading">Creating polarities of projective spaces</span></h4>
<p>Since polarities of a projective space necessarily have an involutory field automorphism as companion automorphism and the standard duality of the projective space as the companion projective space isomorphism, a polarity of a projective space is determined completely by a suitable matrix <span class="SimpleMath">\(A\)</span>. Every polarity of a projective space <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> is listed in the following table, including the conditions on the matrix <span class="SimpleMath">\(A\)</span>.</p>
<p>A hermitian polarity of the projective space <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> exists if and only if the field <span class="SimpleMath">\(\mathrm{GF}(q)\)</span> admits an involutory field automorphism.</p>
<p>It is well known that there is a correspondence between polarities of projective spaces and non-degenerate sesquilinear forms on the underlying vector space. Consider a sesquilinear form <span class="SimpleMath">\(f\)</span> on the vector space <span class="SimpleMath">\(V(n+1,q)\)</span>. Then <span class="SimpleMath">\(f\)</span> induces a map on the elements of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> as follows: every element with underlying subspace <span class="SimpleMath">\(\alpha\)</span> is mapped to the element with underlying subspace <spanclass="SimpleMath">\(\alpha^\perp\)</span>, i.e. the subspace of <span class="SimpleMath">\(V(n+1,q)\)</span> orthogonal to <span class="SimpleMath">\(\alpha\)</span> with respect to the form <span class="SimpleMath">\(f\)</span>. It is clear that this induced map is a polarity of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span>. Also the converse is true, with any polarity of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> corresponds a sesquilinear form on <span class="SimpleMath">\(V(n+1,q)\)</span>. The above classification of polarities of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> follows from the classification of sesquilinear forms on <span class="SimpleMath">\(V(n+1,q)\)</span>. For more information, we refer to <a href="chapBib_mj.html#biBHirschfeldThas">[HT91]</a> and <a href="chapBib_mj.html#biBKleidmanLiebeck">[KL90]</a>. We mention that the implementation of the action of correlations on projective points (see <a href="chap5_mj.html#X7EBA895D7A501CE0"><span class="RefLink">5.8</span></a>) guarantees that a sesquilinear form with matrix <span class="SimpleMath">\(M\)</span> and field automorphism <span class="SimpleMath">\(\theta\)</span> corresponds to a polarity with matrix <span class="SimpleMath">\(M\)</span> and field automorphism <span class="SimpleMath">\(\theta\)</span> and vice versa.</p>
<p>In <strong class="pkg">FinInG</strong>, polarities of projective spaces are always objects in the category <code class="code">IsPolarityOfProjectiveSpace</code>, which is a subcategory of the category <code class="code">IsProjGrpElWithFrobWithPSIsom</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolarityOfProjectiveSpace</code>( <var class="Arg">mat</var>, <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a polarity of a projective space</p>
<p>The underlying correlation of the projective space is constructed using matrix <var class="Arg">mat</var>, field <var class="Arg">f</var>, the identity mapping as field automorphism and the standard duality of the projective space. It is checked whether the matrix <var class="Arg">mat</var> satisfies the necessary conditions to induce a polarity.</p>
<p>The underlying correlation of the projective space is constructed using matrix <var class="Arg">mat</var>, field automorphism <var class="Arg">frob</var>, <var class="Arg">f</var> and the standard duality of the projective space. It is checked whether the <var class="Arg">mat</var> satisfies the necessary conditions to induce a polarity, and whether <var class="Arg">frob</var> is a non-trivial involutory field automorphism. The second operation only needs the arguments <var class="Arg">mat</var> and <var class="Arg">f</var> to construct a hermitian polarity of a projective space, provided the field <var class="Arg">f</var> allows an involutory field automorphism and <var class="Arg">mat</var> satisfies the necessary conditions. The latter is checked by constructing the underlying hermitian form.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolarityOfProjectiveSpace</code>( <var class="Arg">form</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a polarity of a projective space</p>
<p>The polarity of the projective space is constructed using a non-degenerate sesquilinear form <var class="Arg">form</var>. It is checked whether the given form is non-degenerate.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PolarityOfProjectiveSpace</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a polarity of a projective space</p>
<p>The polarity of the projective space is constructed using the non-degenerate sesquilinear form that defines the polar space <var class="Arg">ps</var>. When <var class="Arg">ps</var> is a parabolic quadric in even characteristic, no polarity of the ambient projective space can be associated to <var class="Arg">ps</var>, and an error message is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := HermitianPolarSpace(4,64);</span>
H(4, 8^2)
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := PolarityOfProjectiveSpace(ps);</span>
<polarity of PG(4, GF(2^6)) >
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := ParabolicQuadric(6,8);</span>
Q(6, 8)
<span class="GAPprompt">gap></span> <span class="GAPinput">PolarityOfProjectiveSpace(ps);</span>
Error, no polarity of the ambient projective space can be associated to <ps> 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>
<p>The polarity <var class="Arg">phi</var> is an orthogonal polarity of a projective space if and only if the underlying matrix is symmetric and the characteristic of the field is odd.</p>
<p>The polarity <var class="Arg">phi</var> is a pseudo-polarity of a projective space if and only if the underlying matrix is symmetric, not all elements on the main diagonal are zero and the characteristic of the field is even.</p>
<h4>6.3 <span class="Heading">Polarities, absolute points, totally isotropic elements and finite classical polar
spaces</span></h4>
<p>We already mentioned the equivalence between polarities of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> and sesquilinear forms on <span class="SimpleMath">\(V(n+1,q)\)</span>, hence there is a relation between polarities of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> and polar spaces induced by sesquilinear forms. The following concepts express these relations geometrically.</p>
<p>Suppose that <span class="SimpleMath">\(\phi\)</span> is a polarity of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> and that <span class="SimpleMath">\(\alpha\)</span> is an element of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span>. We call <span class="SimpleMath">\(\alpha\)</span> a <em>totally isotropic element</em> or an <em>absolute element</em> if and only if <spanclass="SimpleMath">\(\alpha\)</span> is incident with <span class="SimpleMath">\(\alpha^\phi\)</span>. An absolute element that is a point is also called an <em>absolute point</em> or an <em>isotropic point</em>. It is clear that an element of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span> is absolute if and only if the underlying vector space is totally isotropic with respect to the sesquilinear form equivalent to <span class="SimpleMath">\(\phi\)</span>. Hence the absolute elements induce a <em>finite classical polar space</em>, the same that is induced by the equivalent sesquilinear form. When <span class="SimpleMath">\(\phi\)</span> is a pseudo-polarity, the set of absolute elements are the elements of a hyperplane of <span class="SimpleMath">\(\mathrm{PG}(n,q)\)</span>.</p>
<p>We restrict our introduction to finite classical polar spaces in this section to the following examples. Many aspects of these geometries are extensively described in Chapter <a href="chap7_mj.html#X7F96B1327C022A28"><span class="RefLink">7</span></a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeometryOfAbsolutePoints</code>( <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a polar space or a hyperplane</p>
<p>When <var class="Arg">f</var> is not a pseudo-polarity, this operation returns the polar space induced by <var class="Arg">f</var>. When <var class="Arg">f</var> is a pseudo-polarity, this operation returns the hyperplane containing all absolute elements.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbsolutePoints</code>( <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a set of points</p>
<p>This operation returns all points that are absolute with respect to <var class="Arg">f</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := IdentityMat(4,GF(3));</span>
[ [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ],
[ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">phi := PolarityOfProjectiveSpace(mat,GF(3));</span>
<polarity of PG(3, GF(3)) >
<span class="GAPprompt">gap></span> <span class="GAPinput">points := AbsolutePoints(phi);</span>
<points of Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(points);</span>
[ <a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0>,
<a point in Q+(3, 3): x_1^2+x_2^2+x_3^2+x_4^2=0> ]
<p><strong class="pkg">FinInG</strong> constructs polarities of projective spaces as correlations. This allows polarities to be multiplied easily, resulting in a collineation. The resulting collineation is constructed in the correlation group but can be mapped onto its unique representative in the collineation group. We provide an example with two commuting polarities.</p>
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