<!-- groups_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 construction of finite classical groups and related topics.
-->
<Appendix Label="groups_app">
<Heading>The finite classical groups in <Package>FinInG</Package> </Heading>
<Section Label="groups_app_forms">
<Heading>Standard forms used to produce the finite classical groups.</Heading>
An overview of operations is given that produce gram matrices to construct standard forms. The notion
<E>standard form</E> is explained in Section <Ref Sect="can_standard"/>, in the context of
canonical and standard polar spaces.
<ManSection>
<Oper Name="CanonicalQuadraticForm" Arg="type, d, f"/>
<Returns>a Gram matrix usable as input to construct a quadratic form</Returns>
<Description>
The arguments <A>d</A> and <A>f</A> are the vector dimension and the finite field
respectively. The argument <A>type</A> is either "hyperbolic", "elliptic" or "parabolic".
The matrix returned can be used to construct a quadratic form.
<P/>
If <A>type</A> equals "hyperbolic", the Gram matrix returned will result in the quadratic form
<Alt Only="HTML">x<sub>1</sub>x<sub>2</sub>+...+ x<sub>d-1</sub>x<sub>d</sub></Alt>
<Alt Not="HTML"><M>x_1x_2+ x_3x_4+\ldots+ x_{d-1}x_d</M></Alt><P/>
If <A>type</A> equals "elliptic", the Gram matrix returned will result in the quadratic form
<Alt Only="HTML">x<sub>1</sub><sup>2</sup>+x<sub>1</sub>x<sub>2</sub>+νx<sub>2</sub><sup>2</sup>
...+ x<sub>d-1</sub>x<sub>d</sub></Alt><Alt Not="HTML"><M>x_1^2+x_1x_2+\nu x_2^2 + x_3x_4+\ldots+ x_{d-1}x_d</M></Alt>with
<Alt Only="HTML">ν=α<sup>i</sup></Alt><Alt Not="HTML"><M>\nu=\alpha^i</M></Alt>, with <Alt Only="HTML">α</Alt>
<Alt Not="HTML">\alpha</Alt> the primitive element of the multiplicative group of <M>GF(q)</M>, which is in GAP <F>Z(q)</F>, and <M>i</M>
the first number in <M>[0,1,...,q-2]</M> for which <Alt Only="HTML">x<sup>2</sup>+x+ν</Alt><Alt Not="HTML"><M>x^2+x+\nu</M>
</Alt> is irreducible over <M>GF(q)</M>.
<P/>
If <A>type</A> equals "parabolic", the Gram matrix returned will result in the quadratic form
<Alt Only="HTML">x<sub>1</sub><sup>2</sup>+x<sub>2</sub>x<sub>3</sub>+...
+ x<sub>d-1</sub>x<sub>d</sub></Alt><Alt Not="HTML"><M>x_1^2+x_2x_3 + \ldots x_{d-1}x_d</M></Alt><P/>
This function is intended to be used only when the characteristic of <A>f</A> is two, but there is no error message is this is not the case.
For this reason, <F>CanonicalQuadraticForm</F> is not an operation designed for the user.
</Description>
</ManSection>
</Section>
<Section Label="groups_app_forms">
<Heading>Direct commands to construct the projective classical groups in <Package>FinInG</Package></Heading>
As explained in Chapter <Ref Chap="classicalpolarspaces"/>, Section <Ref Sect="polar:classicalgroups"/>, we have assumed that
the user asks for the projective classical groups in an indirect way, i.e. as a (subgroup) of the collineation group of
a classical polar space. However, shortcuts to these groups exist. More information on the notations can be found in
Section <Ref Sect="polar:classicalgroups"/>.
<ManSection>
<Oper Name="SOdesargues" Arg="e, d, f"/>
<Returns>the special isometry group of a canonical orthogonal polar space</Returns>
<Description>
The argument <A>e</A> determines the type of the orthogonal polar space, i.e. -1,0,1 for
an elliptic, hyperbolic, parabolic orthogonal space, respectively. The argument <A>d</A> is
the dimension of the underlying vector space, <A>f</A> is the finite field. The method relies
on <F>SO</F>, a GAP command returning the appropriate matrix group. Internally, the invariant form
is asked, and the base change to our canonical form is obtained using the package <Package>form</Package>
<Example>
<#Include SYSTEM "../examples/include/groups_app_sodesargues.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GOdesargues" Arg="e, d, f"/>
<Returns>the isometry group of a canonical orthogonal polar space</Returns>
<Description>
The argument <A>e</A> determines the type of the orthogonal polar space, i.e. -1,0,1 for
an elliptic, hyperbolic, parabolic orthogonal space, respectively. The argument <A>d</A> is
the dimension of the underlying vector space, <A>f</A> is the finite field. The method relies
on <F>GO</F>, a GAP command returning the appropriate matrix group. Internally, the invariant form
is asked, and the base change to our canonical form is obtained using the package <Package>form</Package>
<Example>
<#Include SYSTEM "../examples/include/groups_app_godesargues.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="SUdesargues" Arg="d, f"/>
<Returns>the special isometry group of a canonical hermitian polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>SU</F>, a GAP command returning the appropriate matrix group. Internally, the invariant form
is asked, and the base change to our canonical form is obtained using the package <Package>form</Package>
<Example>
<#Include SYSTEM "../examples/include/groups_app_sudesargues.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GUdesargues" Arg="d, f"/>
<Returns>the isometry/similarity group of a canonical hermitian polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>GU</F>, a GAP command returning the appropriate matrix group. Internally, the invariant form
is asked, and the base change to our canonical form is obtained using the package <Package>form</Package>
<Example>
<#Include SYSTEM "../examples/include/groups_app_gudesargues.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="Spdesargues" Arg="d, f"/>
<Returns>the (special) isometry group of a canonical symplectic polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>Sp</F>, a GAP command returning the appropriate matrix group. Internally, the invariant form
is asked, and the base change to our canonical form is obtained using the package <Package>form</Package>
<Example>
<#Include SYSTEM "../examples/include/groups_app_spdesargues.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GeneralSymplecticGroup" Arg="d, f"/>
<Returns>the isometry group of a canonical symplectic form</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field. Internally, the invariant form
is asked, and the base change to our canonical form is obtained using the package <Package>form</Package>
<Example>
<#Include SYSTEM "../examples/include/groups_app_generalsymplectic.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GSpdesargues" Arg="d, f"/>
<Returns>the similarity group of a canonical symplectic polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>Sp</F>, a GAP command returning the appropriate matrix group. Internally, the invariant form
is asked, and the base change to our canonical form is obtained using the package <Package>form</Package>
<Example>
<#Include SYSTEM "../examples/include/groups_app_gspdesargues.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GammaSp" Arg="d, f"/>
<Returns>the collineation group of a canonical symplectic polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>GeneralSymplecticGroup</F>, and adds the frobenius automorphism.
<Example>
<#Include SYSTEM "../examples/include/groups_app_gammasp.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="DeltaOminus" Arg="d, f"/>
<Returns>the similarity group of a canonical elliptic orthogonal polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>GOdesargues</F>, and computes the generators to be added.
<Example>
<#Include SYSTEM "../examples/include/groups_app_deltaominus.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="DeltaOplus" Arg="d, f"/>
<Returns>the similarity group of a canonical hyperbolic orthogonal polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>GOdesargues</F>, and computes the generators to be added.
<Example>
<#Include SYSTEM "../examples/include/groups_app_deltaoplus.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GammaOminus" Arg="d, f"/>
<Returns>the collineation group of a canonical elliptic orthogonal polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>DeltaOminus</F>, and computes the generators to be added.
<Example>
<#Include SYSTEM "../examples/include/groups_app_gammaominus.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GammaO" Arg="d, f"/>
<Returns>the collineation group of a canonical parabolic orthogonal polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>GO</F>, a GAP command returning the appropriate matrix group. Internally, the invariant form
is asked, and the base change to our canonical form is obtained using the package <Package>form</Package>. Furthermore,
the generators to be added are computed.
<Example>
<#Include SYSTEM "../examples/include/groups_app_gammao.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GammaOplus" Arg="d, f"/>
<Returns>the collineation group of a canonical hyperbolic orthogonal polar space</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>DeltaOplus</F>, and computes the generators to be added.
<Example>
<#Include SYSTEM "../examples/include/groups_app_gammaoplus.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="GammaU" Arg="d, f"/>
<Returns>the collineation group of a canonical hermitian variety</Returns>
<Description>
The argument <A>d</A> is the dimension of the underlying vector space, <A>f</A> is the finite field.
The method relies on <F>GU</F>, a GAP command returning the appropriate matrix group. Internally, the invariant form
is asked, and the base change to our canonical form is obtained using the package <Package>form</Package>. Furthermore,
the generators to be added are computed.
<Example>
<#Include SYSTEM "../examples/include/groups_app_gammau.include">
</Example>
</Description>
</ManSection>
<ManSection>
<Oper Name="G2fining" Arg="d, f"/>
<Returns>the Chevalley group G_2(q)</Returns>
<Description>
This group is the group of projectivities stabilising the split Cayley hexagon embedded in the parabolic
quadric <M>Q(6,q):</M><Alt Not="HTML"><M>X_0X_4+X_1X_5+X_2X_6=X_3^2</M></Alt>
<Alt Only="HTML">X<sub>0</sub>X<sub>4</sub>+X<sub>1</sub>X<sub>5</sub>+X<sub>2</sub>X<sub>6</sub>=X<sub>3</sub><sup>2</sup></Alt>.
<A>f</A> must be a finite field and <A>d</A> must be 5 or 6. When <A>d</A> is 5, <A>F</A> must be a field of even order, and then
the returned group consists of projectivities of <M>W(5,q)</M>. The generators of this group are described explicitly in
<Cite Key="HVM"/>, Appendix D. A correction can be found in <Cite Key="PhDOffer"/>. However, also this source contains
a mistake.
</Description>
</ManSection>
<ManSection>
<Oper Name="3D4fining" Arg="f"/>
<Returns>the Chevalley group 3D4(q)</Returns>
<Description>
The argument <A>f</A> must be a field of order <Alt Not="HTML"><M>q^3</M></Alt><Alt Only="HTML">q<sup>3</sup></Alt>
This group is the group of collineations stabilising the twisted triality hexagon embedded in the
hyperbolic quadric <Alt Not="HTML"><M>Q+(7,q)</M></Alt><Alt Only="HTML">Q<sup>+</sup>(7,q)</Alt>:
<Alt Not="HTML"><M>X_0X_4+X_1X_5+X_2X_6+X_3X_7</M></Alt>
<Alt Only="HTML">X<sub>0</sub>X<sub>4</sub>+X<sub>1</sub>X<sub>5</sub>+X<sub>2</sub>X<sub>6</sub>+X<sub>3</sub>X<sub>7</sub></Alt>
The generators of this group are described explicitly in <Cite Key="HVM"/>, Appendix D.
</Description>
</ManSection>
</Section>
<Section Label="groups_app_forms">
<Heading>Basis of the collineation groups</Heading>
The <Package>GenSS</Package> uses a function <F>FindBasePointCandidates</F> taking a group as one of the arguments.
From a geometrical point of view, it is straightforward to construct a basis for a collineation group for the action
on projective points.
<ManSection>
<Oper Name="FindBasePointCandidates" Arg="g, opt, i, parentS"/>
<Returns>a record</Returns>
<Description>
The returned record contains the base points for the action, and some other fields.
The information in the other fields is determined from the arguments <A>opt</A> and <A>i</A>.
More information on these details can be found in the manual of <Package>GenSS</Package>.
</Description>
</ManSection>
Variations on this version of <F>BasePointCandidates</F> are found in <Package>FinInG</Package> used in previous
versions of <Package>GenSS</Package>. These variations are already or will become obsolete in the (near) future.
</Section>
</Appendix>
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