<!-- 
morphisms_app.xml    FinInG package documentation
                                                                   John Bamberg
                                                                   Anton Betten
                                                                  Philippe Cara
                                                                   Jan De Beule
                                                                 Michel Lavrauw
                                                                Max Neunhoeffer

  Copyright (C) 2018,	Colorado State University
                        Sabancı Üniversitesi
						Università degli Studi di Padova
						Universiteit Gent
						University of St. Andrews
						University of Western Australia
                        Vrije Universiteit Brussel


This appendix gives more (technical) information on the low level functions for morphisms.
 
-->
<Appendix Label="morphisms_app">
<Heading>Low level functions for morphisms</Heading>

<Section Label="field_red_forms">
<Heading>Field reduction and vector spaces</Heading>

<ManSection>
<Oper Name="ShrinkVec" Arg="f1, f2, v, basis"/>
<Oper Name="ShrinkVec" Arg="f1, f2, v"/>
<Returns>a vector</Returns>
<Description>
The argument <A>f2</A> is a subfield of <A>f1</A> and v is vector
in a vectorspace <M>V</M> over <A>f2</A>. The second flavour Returns
return the vector of length <M>d/t</M>, where <M>d=dim(V)</M>,
and <M>t=[f1:f2]</M>. The first flavour uses the natural basis
<F>Basis(AsVectorSpace(f2,f1))</F>. It is not checked whether
<A>f2</A> is a subfield of <A>f1</A>, but it is checked whether
the lengt of <A>v</A> is a multiple of the degree of the field extension.
</Description>
</ManSection>


<ManSection>
<Oper Name="ShrinkMat" Arg="basis, matrix"/>
<Oper Name="ShrinkMat" Arg="f1, f2, matrix"/>
<Returns>a matrix</Returns>
<Description>
Let <M>K</M> be the field &gfq; and let <M>L</M> be the field &gfqt;.
Asumme that <M>B</M> is a basis for <M>L</M> as <M>K</M> vector space. Let
<Alt Only="HTML MathJax"><M>A=(a_{ij})</M></Alt><Alt Only="HTML noMathJax">A=(a&lt;sub>ij &lt;/sub>)</Alt>
<Alt Not="HTML">A=(a&lt;sub>ij &lt;/sub>)</Alt> be a matrix over <M>L</M>.
The result of <F>BlownUpMat(B,A)</F> is the matrix
<Alt Only="HTML MathJax"><M>M=(m_{ij})</M></Alt><Alt Not="HTML"><M>M=(m_{ij})</M></Alt>
<Alt Only="HTML noMathJax">M=(m&lt;sub>ij &lt;/sub>)</Alt>, where each entry
<Alt Only="HTML MathJax"><M>a=a_{ij}</M></Alt><Alt Not="HTML"><M>a=a_{ij}</M></Alt>
<Alt Only="HTML noMathJax">a=a&lt;sub>ij &lt;/sub></Alt> is replaced by
the <M>t \times t</M> matrix <Alt Only="HTML MathJax"><M>M_a</M></Alt><Alt Not="HTML"><M>M_a</M></Alt>
<Alt Only="HTML noMathJax">M&lt;sub>a&lt;/sub></Alt>, representing the linear map
<M>x \mapsto ax</M> with respect to the basis <M>B</M>. This means that if <Alt Only="HTML MathJax"><M>B=\{b_1,b_2,\ldots,b_t\}</M></Alt>
<Alt Not="HTML"><M>B=\{b_1,b_2,\ldots,b_t\}</M></Alt>
<Alt Only="HTML noMathJax">B={b&lt;sub>1&lt;/sub>,b&lt;sub>2&lt;/sub>,...,b&lt;sub>t&lt;/sub>}</Alt>, then row <M>j</M>
is the row of the <M>t</M> coefficients of <Alt Only="HTML MathJax"><M>ab_j</M></Alt><Alt Not="HTML"><M>ab_j</M></Alt>
<Alt Only="HTML noMathJax">ab&lt;sub>j&lt;/sub></Alt> with resepect to the basis
<M>B</M>. The operation <F>ShrinkMat</F> implements the converse of <F>BlownUpMat</F>.
It is checked if the input is a blown up matrix as follows. Let <M>A</M> be a
<M>tm \times tn</M> matrix. For each <M>t \times t</M> block, say <M>M</M>, we need to check that the
set <Alt Only="HTML MathJax"><M>\{b_i^{-1} \sum_{j=1}^{t} m_{ij} b_j: i \in \{1,..,t\}\}</M></Alt>
<Alt Not="HTML"><M>\{b_i^{-1} \sum_{j=1}^{t} m_{ij} b_j: i \in \{1,..,t\}\}</M></Alt>
<Alt Only="HTML noMathJax">{b&lt;sub>i&lt;/sub>&lt;sup>-1&lt;/sup><M>\sum</M>&lt;sub>1&lt;/sub>&lt;sup>t&lt;/sup>m&lt;sub>ij&lt;/sub>b&lt;sub>j&lt;/sub>: <M>i \in \{1,\ldots,t\}</M>}</Alt>.
has size one, since the unique element in that case is the element <M>a \in L</M> represented as a linear map by M with respect to the basis <M>B</M>.<P/>
The first flavour of this operation requires a given basis as first argument. The second flavour requires a field <A>f1</A> and a subfield <A>f2</A> as first two arguments
and calls the first flavour with <F>Basis(AsVectorSpace(f2,f1))</F> as basis. It is not checked whether <A>f2</A> is a subfield of <A>f1</A>.
</Description>
</ManSection>

<ManSection>
<Oper Name="BlownUpProjectiveSpace" Arg="basis, pg1"/>
<Returns>a projective space</Returns>
<Description>
Let <A>basis</A> be a basis of the field <Alt Not="HTML"><M>GF(q^t)</M></Alt><Alt Only="HTML">GF(q&lt;sup>t&lt;/sup>)</Alt>
that is an extension of the basefield of the <M>r-1</M> dimensional projective space <A>pg1</A>.
This operation returns the <M>rt-1</M> dimensional projective space over <M>GF(q)</M>. The basis itself is only used to determine
the field <Alt Not="HTML"><M>GF(q^t)</M></Alt><Alt Only="HTML">GF(q&lt;sup>t&lt;/sup>)</Alt>.
</Description>
</ManSection>

<ManSection>
<Oper Name="BlownUpProjectiveSpaceBySubfield" Arg="subfield, pg"/>
<Returns>a projective space</Returns>
<Description>
Blows up a projective space <A>pg</A> with respect to the standard basis of the basefield of <A>pg</A> over the <A>subfield</A>.
</Description>
</ManSection>

<ManSection>
<Oper Name="BlownUpSubspaceOfProjectiveSpace" Arg="basis, subspace"/>
<Oper Name="BlownUpSubspaceOfProjectiveSpace" Arg="basis, space"/>
<Returns>a subspace of a projective space</Returns>
<Description>
The first flavour blows up a <A>subspace</A> of a projective space with respect to the <A>basis</A> using
field reduction and returns a subspace of the projective space obtained from blowing
up the ambient projective space of <A>subspace</A> with respect to <A>basis</A> using field reduction.
This operation relies on <F>BlownUpMat</F>.
</Description>
</ManSection>

<ManSection>
<Oper Name="BlownUpSubspaceOfProjectiveSpaceBySubfield" Arg="subfield,subspace"/>
<Returns>a subspace of a projective space</Returns>
<Description>
Blows up a <A>subspace</A> of a projective space with respect to the
standard basis of the basefield of <A>subspace</A> over the <A>subfield</A>, using field reduction and
returns it a subspace of the projective space obtained from blowing up the ambient
projective space of <A>subspace</A> over the subfield.
</Description>
</ManSection>


<ManSection>
<Oper Name="IsDesarguesianSpreadElement" Arg="basis,subspace"/>
<Returns>true or false</Returns>
<Description>
Checks wether the <A>subspace</A> is a subspace which is obtained from a blowing up a projective point using field reduction with respect to <A>basis</A>.
</Description>
</ManSection>

</Section>


<Section Label="field_red_forms">
<Heading>Field reduction and forms</Heading>

The embedding of polar spaces by field reduction is explained in detail in Section <Ref Sect="polar_red"/>,
and relies on the following three operations.

<ManSection>
<Oper Name="QuadraticFormFieldReduction" Arg="qf1, f2, alpha, basis"/>
<Oper Name="QuadraticFormFieldReduction" Arg="qf1, f2, alpha"/>
<Returns>a quadratic form</Returns>
<Description>
Let <M>f</M> be quadratic form determining a polar space over the field <M>L</M> This operation returns
the quadratic form <Alt Only="HTML noMathJax">T&lt;sub>&#945;&lt;/sub> &#8226; f &#8226; &#934; &lt;sup>-1&lt;/sup></Alt>
<Alt Only="HTML MathJax"><M>T_{\alpha} \circ f \circ \Phi^{-1}</M></Alt>
<Alt Not="HTML"><M>T_{\alpha} \circ f \circ \Phi^{-1}</M></Alt> over a subfield <M>K</M> of <M>L</M>, as explained
in Section <Ref Sect="polar_red"/>.
</Description>
</ManSection>

<ManSection>
<Oper Name="BilinearFormFieldReduction" Arg="bil11, f2, alpha, basis"/>
<Oper Name="BilinearFormFieldReduction" Arg="bil11, f2, alpha"/>
<Returns>a bilinear form</Returns>
<Description>
    Let <M>f</M> be bilinear form determining a polar space over the field <M>L</M> This operation returns
    the bilinear form <Alt Only="HTML noMathJax">T&lt;sub>&#945;&lt;/sub> &#8226; f &#8226; &#934; &lt;sup>-1&lt;/sup></Alt>
    <Alt Only="HTML MathJax"><M>T_{\alpha} \circ f \circ \Phi^{-1}</M></Alt>
    <Alt Not="HTML"><M>T_{\alpha} \circ f \circ \Phi^{-1}</M></Alt> over a subfield <M>K</M> of <M>L</M>, as explained
    in Section <Ref Sect="polar_red"/>.
</Description>
</ManSection>

<ManSection>
<Oper Name="HermitianFormFieldReduction" Arg="hf1, f2, alpha, basis"/>
<Oper Name="HermitianFormFieldReduction" Arg="hf1, f2, alpha"/>
<Returns>a hermitian form</Returns>
<Description>
    Let <M>f</M> be bilinear form determining a polar space over the field <M>L</M> This operation returns
    the hermitian form <Alt Only="HTML noMathJax">T&lt;sub>&#945;&lt;/sub> &#8226; f &#8226; &#934; &lt;sup>-1&lt;/sup></Alt>
    <Alt Only="HTML MathJax"><M>T_{\alpha} \circ f \circ \Phi^{-1}</M></Alt>
    <Alt Not="HTML"><M>T_{\alpha} \circ f \circ \Phi^{-1}</M></Alt> over a subfield <M>K</M> of <M>L</M>, as explained
    in Section <Ref Sect="polar_red"/>.
</Description>
</ManSection>

</Section>


<Section>
    <Heading>Low level functions</Heading>
    
    <ManSection>
        <Oper Name="PluckerCoordinates" Arg="matrix"/>
        <Oper Name="InversePluckerCoordinates" Arg="vector"/>
        <Description>
            The first operation can also take a matrix representing a line of &pg3q; 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.
        </Description>
    </ManSection>
    
    <ManSection>
        <Oper Name="IsomorphismPolarSpacesProjectionFromNucleus" Arg="quadric, w, boolean"/>
        <Description>
            This operation returns the isomorphism between a parabolic quadric and a sympletic polar space. Although it
            is checked whether the base field and rank of both polar spaces are equal, this operation is menat for internal use
            only. This operation is called by the operation <F>IsomorphismPolarSpaces</F>.
        </Description>
    </ManSection>
    
    <ManSection>
        <Oper Name="IsomorphismPolarSpacesNC" Arg="ps1, ps2"/>
        <Oper Name="IsomorphismPolarSpacesNC" Arg="ps1, ps2, boolean"/>
        <Description>
            <F>IsomorphismPolarSpacesNC</F> is the version of <F>IsomorphismPolarSpaces</F> 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.
        </Description>
    </ManSection>
    
    <ManSection>
        <Oper Name="NaturalEmbeddingBySubspaceNC" Arg="geom1, geom2, v"/>
        <Description>
            The operation <F>NaturalEmbeddingBySubspaceNC</F> is the ``no check'' version of <F>NaturalEmbeddingBySubspace</F>.
        </Description>
    </ManSection>

<ManSection>
<Oper Name="NaturalProjectionBySubspaceNC" Arg="ps, v"/>
<Description>
The operation <F>NaturalEmbeddingBySubspaceNC</F> is the ``no check'' version of <F>NaturalEmbeddingBySubspace</F>.
</Description>
</ManSection>

</Section>
</Appendix>
