<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 class="SimpleMath">P</span>, not incident with a line <span class="SimpleMath">l</span>, <span class="SimpleMath">P</span> is collinear with exactly one point of <span class="SimpleMath">l</span> or with all points of <span class="SimpleMath">l</span>. The axiomatic treatment of polar spaces has its foundations in <a href="chapBib.html#biBVeldkamp">[Vel59]</a>, <a href="chapBib.html#biBTits74">[Tit74]</a>, and <a href="chapBib.html#biBBS1974">[BS74]</a>, the latter in which the one-or-all axiom is described. Polar spaces are axiomatically, point-line geometries, but may contain higher dimensional projective subspaces too. All maximal subspaces have the same projective dimension, and this determines the rank of the polar space.</p>
<p>Well known examples of <em>finite</em> polar spaces are the geometries attached to sesquilinear and quadratic forms of vector spaces over a finite field, these geometries are called the <em>finite classical polar spaces</em>. For a given sesquilinear, respectively quadratic, form <span class="SimpleMath">f</span>, the elements of the associated geometry are the totally isotropic, respectively totally singular, subspaces of the vectors space with relation to the form <span class="SimpleMath">f</span>. The treatment of the forms is done through the package <strong class="pkg">Forms</strong>.</p>
<p>From the axiomatic point of view, a polar space is a point-line geometry, and has rank at least 2. Considering a sesquilinear or quadratic form <span class="SimpleMath">f</span>, of Witt index 1, the associated geometry consists only of projective points, and is then in the axiomatic treatment, not a polar space. However, as is the case for projective spaces, we will consider the rank one geometries associated to forms of Witt index 1 as examples of classical polar spaces. Even the elliptic quadric on the projective line, a <em>geometry</em> associated to an elliptic quadratic form on a two dimensional vector space over a finite field, is considered as a classical polar space, though it has no singular subspaces. The reason for this treatment is that most, if not all, methods for operations applicable on these geometries, rely on the same algebraic methodology. So, in <strong class="pkg">FinInG</strong>, a classical polar space (sometimes abbreviated to polar space), is the geometry associated with a sesquilinear or quadratic form on a finite dimensional vector space over a finite field.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsClassicalPolarSpace</code></td><td class="tdright">( category )</td></tr></table></div>
<p>This category is a subcategory of <code class="code">IsLieGeometry</code>, and contains all the geometries associated to a non-degenerate sesquilinear or quadratic form.</p>
<p>The underlying vector space and matrix group are to our advantage in the treatment of classical polar spaces. We refer the reader to <a href="chapBib.html#biBHirschfeldThas">[HT91]</a> and <a href="chapBib.html#biBCameron">[Cam00b]</a> for the necessary background theory (if it is not otherwise provided), and we follow the approach of <a href="chapBib.html#biBCameron">[Cam00b]</a> to introduce all different flavours.</p>
<p>Consider the projective space PG(n,q) 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> is Hermitian, alternating or symmetric. When the characteristic of the field is odd, respectively even, a symmetric bilinear form is called orthogonal, respectively, pseudo. We do not consider the pseudo case, so we suppose that <span class="SimpleMath">f</span> is Hermitian, symplectic or orthogonal. The classical polar space associated with <span class="SimpleMath">f</span> is the incidence geometry whose elements are of the subspaces of PG(n,q) whose underlying vector subspace is totally isotropic with relation to <span class="SimpleMath">f</span>. We call a polar space <em>Hermitian</em>, respectively, <em>symplectic</em>, <em>orthogonal</em>, if the underlying sesquilinear form is Hermitian, respectively, symplectic, orthogonal.</p>
<p>Symmetric bilinear forms have completely different geometric properties in even characteristic than in odd characteristic. On the other hand, polar spaces geometrically comparable to orthogonal polar spaces in odd characteristic, do exist in even characteristic. The algebraic background is now established by quadratic forms on a vector space instead of bilinear forms. Consider a non-singular quadratic form <span class="SimpleMath">q</span> on a vector space <span class="SimpleMath">V(n+1,q)</span>. The classical polar space associated with <span class="SimpleMath">f</span> is the incidence geometry whose elements are the subspaces of PG(n,q) whose underlying vector subspace is totally singular with relation to <span class="SimpleMath">q</span>. The connection with orthogonal polar spaces in odd characteristic is clear, since in odd characteristic, quadratic forms 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><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. An error message will be displayed if <var class="Arg">pol</var> is a pseudo polarity. We refer to Chapter <a href="chap6.html#X87BA55CB86B110EC"><span class="RefLink">6</span></a> for more information on polarities of projective spaces, and more particularly to Section <a href="chap6.html#X83F8149B7D23301E"><span class="RefLink">6.3</span></a> for the connection between polarities and forms.</p>
<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 class="GAPprompt">gap></span> <span class="GAPinput">id := IdentityMat(7, GF(3));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">form := QuadraticFormByMatrix(id, GF(3));</span>
< quadratic form >
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := PolarSpace( form );</span>
<polar space in ProjectiveSpace(
6,GF(3)): x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2=0 >
<span class="GAPprompt">gap></span> <span class="GAPinput">psl32 := PSL(3,2);</span>
Group([ (4,6)(5,7), (1,2,4)(3,6,5) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">reps:=[[1,1,1,0,0,0,0], [-1,1,1,0,0,0,0], [1,-1,1,0,0,0,0], [1,1,-1,0,0,0,0]]*Z(3)^0;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ovoid := Union( List(reps, x-> Orbit(psl32, x, Permuted)) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ovoid := List(ovoid, x -> VectorSpaceToElement(ps, x));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">planes := AsList( Planes( ps ) );;</span>
#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);</span>
true
<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 of the underlying forms in similarity classes. We follow mostly the approach and terminology of <a href="chapBib.html#biBKleidmanLiebeck">[KL90]</a>, as we did in the manual of the package <strong class="pkg">Forms</strong>.</p>
<p>Consider a vector space <span class="SimpleMath">V=V(n+1,q)</span> and a sesquilinear form <span class="SimpleMath">f</span> on <span class="SimpleMath">V</span>. The pair <span class="SimpleMath">(V,f)</span> is called a formed space. Consider now two formed spaces <span class="SimpleMath">(V,f)</span> and <span class="SimpleMath">(V,f'), where f and f'</span> are two sesquilinear forms on <span class="SimpleMath">V</span>. A non-singular linear map φ from <span class="SimpleMath">V</span> to itself induces a <em>similarity</em> of the formed space <span class="SimpleMath">(V,f)</span> to the formed space <span class="SimpleMath">(V,f') if and only if
<p class="pcenter">f(v,w) = λ (f'(φ(v),φ(w))),
<p>for all vectors <span class="SimpleMath">v,w</span> and some non-zero λ in <span class="SimpleMath">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 <span class="SimpleMath">V</span>. For symmetric bilinear forms in odd characteristic, the number of similarity classes depends on the dimension of <span class="SimpleMath">V</span>. In odd dimension, there is only one similarity class, and non-degenerate forms in this class are called parabolic (bilinear) forms. In even dimension, there are two similarity classes, and non-degenerate forms are either elliptic (bilinear) forms or hyperbolic (bilinear) forms.</p>
<p>Consider now a vector space <span class="SimpleMath">V</span> and a quadratic form <span class="SimpleMath">q</span> on <span class="SimpleMath">V</span>. The pair <span class="SimpleMath">(V,q)</span> is called a formed space. Consider now two formed spaces <span class="SimpleMath">(V,q)</span> and <span class="SimpleMath">(V,q'), where q and q'</span> are two quadratic forms on <span class="SimpleMath">V</span>. A non-degenerate linear map φ from <span class="SimpleMath">V</span> to itself induces a <em>similarity</em> of the formed space <span class="SimpleMath">(V,q)</span> to the formed space <span class="SimpleMath">(V,q') if and only if
<p class="pcenter">q(v) = λ (q'(φ(v))),
<p>for all vectors <span class="SimpleMath">v</span> and some non-zero λ in <span class="SimpleMath">GF(q)</span>. For quadratic forms in even characteristic, the number of similarity classes depends on the dimension of <span class="SimpleMath">V</span>. In odd dimension, there is only one similarity class, and non-degenerate forms in this class are called parabolic (bilinear) forms. In even dimension, there are two similarity classes, and non-degenerate forms are either elliptic (bilinear) forms or hyperbolic (bilinear) forms.</p>
<p>If φ induces a similarity of a formed vector space such that <span class="SimpleMath">λ = 1</span>, then the similarity is called an <em>isometry</em> of the formed vector space. In almost all cases, each similarity class contains exactly one isometry class. Only the orthogonal sesquilinear forms (in odd characteristic) have two isometry classes. Consequently, if an isometry exists between formed vector spaces, they are called <em>isometric</em>. Projectively, a formed vector space becomes a classical polar space embedded in a projective space. Obviously, forms in the same similarity class determine exactly the same classical polar space. Conversely, it is well known that a classical polar space determines a form up to a constant factor, i.e. it determines a similarity class of forms. In <strong class="pkg">FinInG</strong>, the word <em>canonical</em> is used in the mathematical sense, i.e. a classical polar space is <em>canonical</em> if its determining form belongs to a fixed similarity class. A classical polar space is called <em>standard</em> if its determining form is the fixed representative of the canonical similarity class. Hence a <em>standard</em> classical polar space is always a <em>canonical</em> classical polar space, a canonical polar space is determined by a standard form up to a constant factor. In the following table, we summaries the above information on polar spaces, together with the standard forms that are chosen in <strong class="pkg">FinInG</strong>. Note that Tr refers to the absolute trace map from GF(q) to GF(p).</p>
<p>We refer to Appendix <a href="chapB.html#X866C644987E43DF8"><span class="RefLink">B</span></a> for information on the operations that construct gram matrices that are used to obtain the above standard forms.</p>
<p>The <strong class="pkg">FinInG</strong> provides a wealth of flexibility in constructing polar spaces. The user may choose a particular quadratic or sesquilinear form, but may also chose to construct polars spaces that have one of the above mentioned forms as underlying form. Furthermore, <strong class="pkg">FinInG</strong> will detect when necessary if the user-constructed polar space is canonical. This mechanism gives the user complete flexibility while avoiding unnecessary computations when, for example, constructing the collineation group of a user-defined polar space.</p>
<p>The following five operations always return polar spaces induced by one of the above standard forms.</p>
<p>This function returns the symplectic polar space of dimension <var class="Arg">d</var> over <varclass="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>
<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>
<p><var class="Arg">d</var> must be an even positive integer. This function returns the parabolic 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>
<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>
<p><var class="Arg">d</var> must be an odd positive integer. This function returns the elliptic 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>
<p>This attribute returns true when a polar space with a particular underlying form is canonical. The execution of this attribute on a general user constructed polar space needs to check the type of <var class="Arg">ps</var>. The obtained extra information is stored automatically as attribute for <var class="Arg">ps</var>, as can be noted by the different printing of <var class="Arg">ps</var> before and after execution.</p>
<p>the canonical polar space isometric to the given polar space <var class="Arg">P</var> or the classical polar space with underlying form <var class="Arg">form</var>.</p>
<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>
<p>The polar space <var class="Arg">ps</var> is the geometry associated with a sesquilinear or quadratic form <span class="SimpleMath">f</span>. The vector space on which <span class="SimpleMath">f</span> is acting is returned.</p>
<p>When <var class="Arg">ps</var> is a polar space, this operation returns the ambient projective space, i.e. the underlying projective space of the sesquilinear or quadratic form that defines <var class="Arg">ps</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveDimension</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Dimension</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the dimension of the ambient projective space of <var class="Arg">ps</var></p>
<p>When <var class="Arg">ps</var> is a polar space, an ambient projective space <span class="SimpleMath">P</span> is uniquely defined and can be asked using <code class="file">AmbientSpace</code>. This operation and its synonym <code class="file">Dimension</code> returns the dimension of <span class="SimpleMath">P</span>.</p>
<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">ps</var></p>
<p>When <var class="Arg">ps</var> is a polar space, its rank, i.e. the number of different types, equals the Witt index of the defining sesquilinear or quadratic form.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BaseField</code>( <var class="Arg">ps</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the base field of the polar space <var class="Arg">ps</var></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">P</span> are the subspaces of the ambient projective space that are totally isotropic with relation to the sesquilinear or quadratic form that defines <span class="SimpleMath">P</span>. Constructing subspaces of finite classical polar spaces is done as in the projective space case, except that additional checks are implemented in the methods to check that the subspace of the vector space is totally isotropic. The empty subspace, also called the trivial subspace, which has dimension -1, corresponds with the zero dimensional vector space of the underlying vector space of the ambient projective space of <span class="SimpleMath">P</span>, and is of course totally isotropic. As such, is is considered as a subspace of a finite classical polar space in the mathematical sense, but not as an element of the incidence geometry, and hence do in <strong class="pkg">FinInG</strong> not belong to the category <code class="code">IsSubspaceOfClassicalPolarSpace</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VectorSpaceToElement</code>( <var class="Arg">ps</var>, <var class="Arg">v</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: an element of the polar space <var class="Arg">geo</var></p>
<p>Let <var class="Arg">ps</var> be a polar space, and <var class="Arg">v</var> is either a row vector (for points) or an <span class="SimpleMath">m</span>x<span class="SimpleMath">n</span> matrix (for an <span class="SimpleMath">(m-1)</span>-subspace of a polar space with an <span class="SimpleMath">(n-1)</span>-dimensional ambient projective space. In the case that <var class="Arg">v</var> is a matrix, the rows represent basis vectors for the subspace. An exceptional case is when <var class="Arg">v</var> is a zero-vector, whereby the trivial subspace is returned. It is checked that the subspace defined by <span class="SimpleMath">v</span> is totally isotropic with relation to the form defining <var class="Arg">ps</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := SymplecticSpace(3,4);</span>
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(2)^0, 0*Z(2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">p := VectorSpaceToElement(ps,v);</span>
<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), Z(2)^0 ], [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">line := VectorSpaceToElement(ps,mat);</span>
Error, <x> does not generate an element of <geom> called from
<function "unknown">( <arguments> )
called from read-eval loop at line 12 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">quit;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">mat := [[1,1,0,0],[0,0,1,0]]*Z(4)^0;</span>
[ [ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">line := VectorSpaceToElement(ps,mat);</span>
<a line in W(3, 4)>
<span class="GAPprompt">gap></span> <span class="GAPinput">p := VectorSpaceToElement(ps,[0,0,0,0]*Z(4)^0);</span>
< empty subspace >
<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></table></div>
<p>Returns: the trivial subspace in the projective <var class="Arg">ps</var></p>
<p>The object returned by this operation is contained in every projective subspace of the projective space <var class="Arg">ps</var>, but is not an element of <var class="Arg">ps</var>. Hence, testing incidence results in an error message.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveDimension</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Dimension</code>( <var class="Arg">sub</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the projective dimension of a subspace of a polar space. The operation <code class="file">ProjectiveDimension</code> is also applicable on the EmptySubspace</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementsOfIncidenceStructure</code>( <var class="Arg">ps</var>, <var class="Arg">j</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the collection of elements of the projective space <var class="Arg">ps</var> of type <var class="Arg">j</var></p>
<p>For the projective space <var class="Arg">ps</var> of dimension <span class="SimpleMath">d</span> and the type <var class="Arg">j</var>, <span class="SimpleMath">1 ≤ j ≤ d</span> this operation returns the collection of <span class="SimpleMath">j-1</span> dimensional subspaces.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := HermitianPolarSpace(8,13^2);</span>
H(8, 13^2)
<span class="GAPprompt">gap></span> <span class="GAPinput">planes := ElementsOfIncidenceStructure(ps,3);</span>
<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>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ps := HyperbolicQuadric(7,7);</span>
Q+(7, 7)
<span class="GAPprompt">gap></span> <span class="GAPinput">p := VectorSpaceToElement(ps,[1,0,1,0,0,0,0,0]*Z(7)^0);</span>
<a point in Q+(7, 7)>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := VectorSpaceToElement(ps,[[1,0,1,0,0,0,0,0],[0,-1,0,1,0,0,0,0]]*Z(7)^0);</span>
<a line in Q+(7, 7)>
<span class="GAPprompt">gap></span> <span class="GAPinput">p * l;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">l * p;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsIncident(p,l);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">p in l;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">l in p;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">e := EmptySubspace(ps);</span>
< empty subspace >
<span class="GAPprompt">gap></span> <span class="GAPinput">e * l;</span>
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `*' on 2 arguments called from
<function "HANDLE_METHOD_NOT_FOUND">( <arguments> )
called from read-eval loop at line 17 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
<span class="GAPbrkprompt">brk></span> <span class="GAPinput">quit;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">e in l;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">l in ps;</span>
true
<p><var class="Arg">u</var> and <var class="Arg">v</var> are elements of a projective or polar space. This function returns the join of the two elements, that is, the span of the two subspaces.</p>
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