<p>The argument <var class="Arg">f2</var> is a subfield of <var class="Arg">f1</var> and v is vector in a vector space <span class="SimpleMath">\(V\)</span> over <var class="Arg">f2</var>. The second flavour Returns return the vector of length <span class="SimpleMath">\(d/t\)</span>, where <span class="SimpleMath">\(d=dim(V)\)</span>, and <span class="SimpleMath">\(t=[f1:f2]\)</span>. The first flavour uses the natural basis <code class="file">Basis(AsVectorSpace(f2,f1))</code>. It is not checked whether <var class="Arg">f2</var> is a subfield of <var class="Arg">f1</var>, but it is checked whether the length of <var class="Arg">v</var> is a multiple of the degree of the field extension.</p>
<p>Let <span class="SimpleMath">\(K\)</span> be the field <span class="SimpleMath">\(\mathrm{GF}(q)\)</span> and let <span class="SimpleMath">\(L\)</span> be the field <span class="SimpleMath">\(\mathrm{GF}(q^t)\)</span>. Assume that <span class="SimpleMath">\(B\)</span> is a basis for <span class="SimpleMath">\(L\)</span> as <span class="SimpleMath">\(K\)</span> vector space. Let <span class="SimpleMath">\(A=(a_{ij})\)</span> be a matrix over <span class="SimpleMath">\(L\)</span>. The result of <code class="file">BlownUpMat(B,A)</code> is the matrix <span class="SimpleMath">\(M=(m_{ij})\)</span> , where each entry <span class="SimpleMath">\(a=a_{ij}\)</span> is replaced by the <span class="SimpleMath">\(t \times t\)</span> matrix <span class="SimpleMath">\(M_a\)</span> , representing the linear map <span class="SimpleMath">\(x \mapsto ax\)</span> with respect to the basis <span class="SimpleMath">\(B\)</span>. This means that if <span class="SimpleMath">\(B=\{b_1,b_2,\ldots,b_t\}\)</span> , then row <span class="SimpleMath">\(j\)</span> is the row of the <span class="SimpleMath">\(t\)</span> coefficients of <span class="SimpleMath">\(ab_j\)</span> with respect to the basis <span class="SimpleMath">\(B\)</span>. The operation <code class="file">ShrinkMat</code> implements the converse of <code class="file">BlownUpMat</code>. It is checked if the input is a blown up matrix as follows. Let <span class="SimpleMath">\(A\)</span> be a <span class="SimpleMath">\(tm \times tn\)</span> matrix. For each <span class="SimpleMath">\(t \times t\)</span> block, say <span class="SimpleMath">\(M\)</span>, we need to check that the set <span class="SimpleMath">\(\{b_i^{-1} \sum_{j=1}^{t} m_{ij} b_j: i \in \{1,..,t\}\}\)</span> . has size one, since the unique element in that case is the element <span class="SimpleMath">\(a \in L\)</span> represented as a linear map by M with respect to the basis <span class="SimpleMath">\(B\)</span>.</p>
<p>The first flavour of this operation requires a given basis as first argument. The second flavour requires a field <var class="Arg">f1</var> and a subfield <var class="Arg">f2</var> as first two arguments and calls the first flavour with <code class="file">Basis(AsVectorSpace(f2,f1))</code> as basis. It is not checked whether <var class="Arg">f2</var> is a subfield of <var class="Arg">f1</var>.</p>
<p>Let <var class="Arg">basis</var> be a basis of the field GF(q<sup>t</sup>) that is an extension of the base field of the <span class="SimpleMath">\(r-1\)</span> dimensional projective space <var class="Arg">pg1</var>. This operation returns the <span class="SimpleMath">\(rt-1\)</span> dimensional projective space over <span class="SimpleMath">\(GF(q)\)</span>. The basis itself is only used to determine the field GF(q<sup>t</sup>).</p>
<p>Blows up a projective space <var class="Arg">pg</var> with respect to the standard basis of the base field of <var class="Arg">pg</var> over the <var class="Arg">subfield</var>.</p>
<p>The first flavour blows up a <var class="Arg">subspace</var> of a projective space with respect to the <var class="Arg">basis</var> using field reduction and returns a subspace of the projective space obtained from blowing up the ambient projective space of <var class="Arg">subspace</var> with respect to <var class="Arg">basis</var> using field reduction. This operation relies on <code class="file">BlownUpMat</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlownUpSubspaceOfProjectiveSpaceBySubfield</code>( <var class="Arg">subfield</var>, <var class="Arg">subspace</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a subspace of a projective space</p>
<p>Blows up a <var class="Arg">subspace</var> of a projective space with respect to the standard basis of the base field of <var class="Arg">subspace</var> over the <var class="Arg">subfield</var>, using field reduction and returns it a subspace of the projective space obtained from blowing up the ambient projective space of <var class="Arg">subspace</var> over the subfield.</p>
<p>Checks wether the <var class="Arg">subspace</var> is a subspace which is obtained from a blowing up a projective point using field reduction with respect to <var class="Arg">basis</var>.</p>
<h4>C.2 <span class="Heading">Field reduction and forms</span></h4>
<p>The embedding of polar spaces by field reduction is explained in detail in Section <a href="chap10_mj.html#X7823BA95797898CE"><span class="RefLink">10.5-3</span></a>, and relies on the following three operations.</p>
<p>Let <span class="SimpleMath">\(f\)</span> be quadratic form determining a polar space over the field <span class="SimpleMath">\(L\)</span> This operation returns the quadratic form <span class="SimpleMath">\(T_{\alpha} \circ f \circ \Phi^{-1}\)</span> over a subfield <span class="SimpleMath">\(K\)</span> of <span class="SimpleMath">\(L\)</span>, as explained in Section <a href="chap10_mj.html#X7823BA95797898CE"><span class="RefLink">10.5-3</span></a>.</p>
<p>Let <span class="SimpleMath">\(f\)</span> be bilinear form determining a polar space over the field <span class="SimpleMath">\(L\)</span> This operation returns the bilinear form <span class="SimpleMath">\(T_{\alpha} \circ f \circ \Phi^{-1}\)</span> over a subfield <span class="SimpleMath">\(K\)</span> of <span class="SimpleMath">\(L\)</span>, as explained in Section <a href="chap10_mj.html#X7823BA95797898CE"><span class="RefLink">10.5-3</span></a>.</p>
<p>Let <span class="SimpleMath">\(f\)</span> be bilinear form determining a polar space over the field <span class="SimpleMath">\(L\)</span> This operation returns the hermitian form <span class="SimpleMath">\(T_{\alpha} \circ f \circ \Phi^{-1}\)</span> over a subfield <span class="SimpleMath">\(K\)</span> of <span class="SimpleMath">\(L\)</span>, as explained in Section <a href="chap10_mj.html#X7823BA95797898CE"><span class="RefLink">10.5-3</span></a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PluckerCoordinates</code>( <var class="Arg">matrix</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">‣ InversePluckerCoordinates</code>( <var class="Arg">vector</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The first operation can also take a matrix representing a line of <span class="SimpleMath">\(\mathrm{PG}(3,q)\)</span> as argument. No checks are performed in this case. It returns the plucker coordinates of the argument as list of finite field elements. The second operation is the inverse of the first. No check is performed whether the argument represents a point of the correct hyperbolic quadric. Both operations are to be used internally only.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPolarSpacesProjectionFromNucleus</code>( <var class="Arg">quadric</var>, <var class="Arg">w</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>This operation returns the isomorphism between a parabolic quadric and a symplectic polar space. Although it is checked whether the base field and rank of both polar spaces are equal, this operation is meant for internal use only. This operation is called by the operation <code class="file">IsomorphismPolarSpaces</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismPolarSpacesNC</code>( <var class="Arg">ps1</var>, <var class="Arg">ps2</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">‣ IsomorphismPolarSpacesNC</code>( <var class="Arg">ps1</var>, <var class="Arg">ps2</var>, <var class="Arg">boolean</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="file">IsomorphismPolarSpacesNC</code> is the version of <code class="file">IsomorphismPolarSpaces</code> where no checks are built in, and which is only applicable when the two polar spaces are equivalent. As no checks are built in, this operation is to be used internally only.</p>
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