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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Frank Celler.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the operations for matrix groups.
##
#############################################################################
##
#C IsMatrixGroup(<grp>)
##
## <#GAPDoc Label="IsMatrixGroup">
## <ManSection>
## <Filt Name="IsMatrixGroup" Arg='grp' Type='Category'/>
##
## <Description>
## The category of matrix groups.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareSynonym( "IsMatrixGroup", IsRingElementCollCollColl and IsGroup );
#############################################################################
##
#M IsHandledByNiceMonomorphism( <mat-grp> )
##
## For finite matrix groups, there is a default method for
## `NiceMonomorphism' based on the action on vectors from the right.
##
InstallTrueMethod( IsHandledByNiceMonomorphism, IsMatrixGroup and IsFinite );
#############################################################################
##
#M CanComputeSize( <mat-grp> )
##
InstallTrueMethod(CanComputeSize,IsMatrixGroup and IsFinite);
#############################################################################
##
## Operations of Matrix Groups
## <#GAPDoc Label="[1]{grpmat}">
## The basic operations for groups are described
## in Chapter <Ref Chap="Group Actions"/>,
## special actions for <E>matrix</E> groups mentioned there are
## <Ref Func="OnLines"/>, <Ref Func="OnRight"/>,
## and <Ref Func="OnSubspacesByCanonicalBasis"/>.
## <!-- what about acting directly on subspace objects via <C>OnRight</C>? -->
## <P/>
## For subtleties concerning multiplication from the left or from the
## right,
## see <Ref Sect="Acting OnRight and OnLeft"/>.
## <#/GAPDoc>
##
#############################################################################
##
#F ProjectiveActionHomomorphismMatrixGroup(<G>)
##
## <#GAPDoc Label="ProjectiveActionHomomorphismMatrixGroup">
## <ManSection>
## <Func Name="ProjectiveActionHomomorphismMatrixGroup" Arg='G'/>
##
## <Description>
## returns an action homomorphism for a faithful projective action of
## <A>G</A> on the underlying vector space.
## (Note: The action is not necessarily on the full space,
## if a smaller subset can be found on which the action is faithful.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("ProjectiveActionHomomorphismMatrixGroup");
#############################################################################
##
#A DefaultFieldOfMatrixGroup( <mat-grp> )
##
## <#GAPDoc Label="DefaultFieldOfMatrixGroup">
## <ManSection>
## <Attr Name="DefaultFieldOfMatrixGroup" Arg='mat-grp'/>
##
## <Description>
## Is a field containing all the matrix entries. It is not guaranteed to be
## the smallest field with this property.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute(
"DefaultFieldOfMatrixGroup",
IsMatrixGroup );
InstallSubsetMaintenance( DefaultFieldOfMatrixGroup,
IsMatrixGroup and HasDefaultFieldOfMatrixGroup, IsMatrixGroup );
#############################################################################
##
#A DimensionOfMatrixGroup( <mat-grp> )
##
## <#GAPDoc Label="DimensionOfMatrixGroup">
## <ManSection>
## <Attr Name="DimensionOfMatrixGroup" Arg='mat-grp'/>
##
## <Description>
## The dimension of the matrix group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute(
"DimensionOfMatrixGroup",
IsMatrixGroup );
InstallSubsetMaintenance( DimensionOfMatrixGroup,
IsMatrixGroup and HasDimensionOfMatrixGroup, IsMatrixGroup );
#############################################################################
##
#A FieldOfMatrixGroup( <matgrp> )
##
## <#GAPDoc Label="FieldOfMatrixGroup">
## <ManSection>
## <Attr Name="FieldOfMatrixGroup" Arg='matgrp'/>
##
## <Description>
## The smallest field containing all the matrix entries of all elements
## of the matrix group <A>matgrp</A>.
## As the calculation of this can be hard, this should only be used if one
## <E>really</E> needs the smallest field,
## use <Ref Attr="DefaultFieldOfMatrixGroup"/> to get (for example)
## the characteristic.
## <Example><![CDATA[
## gap> DimensionOfMatrixGroup(m);
## 3
## gap> DefaultFieldOfMatrixGroup(m);
## GF(3)
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute(
"FieldOfMatrixGroup",
IsMatrixGroup );
#############################################################################
##
#A TransposedMatrixGroup( <matgrp> ) . . . . . . transpose of a matrix group
##
## <#GAPDoc Label="TransposedMatrixGroup">
## <ManSection>
## <Attr Name="TransposedMatrixGroup" Arg='matgrp'/>
##
## <Description>
## returns the transpose of the matrix group <A>matgrp</A>. The transpose of
## the transpose of <A>matgrp</A> is identical to <A>matgrp</A>.
## <Example><![CDATA[
## gap> G := Group( [[0,-1],[1,0]] );
## Group([ [ [ 0, -1 ], [ 1, 0 ] ] ])
## gap> T := TransposedMatrixGroup( G );
## Group([ [ [ 0, 1 ], [ -1, 0 ] ] ])
## gap> IsIdenticalObj( G, TransposedMatrixGroup( T ) );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "TransposedMatrixGroup", IsMatrixGroup );
#############################################################################
##
#F NaturalActedSpace( [<G>, ]<acts>, <veclist> )
##
## <ManSection>
## <Func Name="NaturalActedSpace" Arg='[G, ]acts, veclist'/>
##
## <Description>
## returns the space in which the action of <A>G</A> via the matrix list
## <A>acts</A>,
## acting on the orbits of the vectors in <A>veclist</A> takes place. This
## function is used for example by orbit calculations to obtain a suitable
## domain for hashing.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("NaturalActedSpace");
#############################################################################
##
#P IsGeneralLinearGroup( <grp> )
#P IsGL(<grp>)
##
## <#GAPDoc Label="IsGeneralLinearGroup">
## <ManSection>
## <Prop Name="IsGeneralLinearGroup" Arg='grp'/>
## <Prop Name="IsGL" Arg='grp'/>
##
## <Description>
## The General Linear group is the group of all invertible matrices over a
## ring. This property tests, whether a group is isomorphic to a General
## Linear group. (Note that currently only a few trivial methods are
## available for this operation. We hope to improve this in the future.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsGeneralLinearGroup", IsGroup );
DeclareSynonymAttr( "IsGL", IsGeneralLinearGroup );
InstallTrueMethod( IsGroup, IsGeneralLinearGroup );
#############################################################################
##
#P IsNaturalGL( <matgrp> )
##
## <#GAPDoc Label="IsNaturalGL">
## <ManSection>
## <Prop Name="IsNaturalGL" Arg='matgrp'/>
##
## <Description>
## This property tests, whether a matrix group is the General Linear group
## in the right dimension over the (smallest) ring which contains all
## entries of its elements. (Currently, only a trivial test that computes
## the order of the group is available.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsNaturalGL", IsMatrixGroup );
InstallTrueMethod(IsGeneralLinearGroup,IsNaturalGL);
#############################################################################
##
#P IsSpecialLinearGroup( <grp> )
#P IsSL(<grp>)
##
## <#GAPDoc Label="IsSpecialLinearGroup">
## <ManSection>
## <Prop Name="IsSpecialLinearGroup" Arg='grp'/>
## <Prop Name="IsSL" Arg='grp'/>
##
## <Description>
## The Special Linear group is the group of all invertible matrices over a
## ring, whose determinant is equal to 1. This property tests, whether a
## group is isomorphic to a Special Linear group. (Note that currently
## only a few trivial methods are available for this operation. We hope
## to improve this in the future.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSpecialLinearGroup", IsGroup );
DeclareSynonymAttr( "IsSL", IsSpecialLinearGroup );
InstallTrueMethod( IsGroup, IsSpecialLinearGroup );
#############################################################################
##
#P IsNaturalSL( <matgrp> )
##
## <#GAPDoc Label="IsNaturalSL">
## <ManSection>
## <Prop Name="IsNaturalSL" Arg='matgrp'/>
##
## <Description>
## This property tests, whether a matrix group is the Special Linear group
## in the right dimension over the (smallest) ring which contains all
## entries of its elements. (Currently, only a trivial test that computes
## the order of the group is available.)
## <Example><![CDATA[
## gap> IsNaturalGL(m);
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsNaturalSL", IsMatrixGroup );
InstallTrueMethod(IsSpecialLinearGroup,IsNaturalSL);
#############################################################################
##
#P IsSubgroupSL( <matgrp> )
##
## <#GAPDoc Label="IsSubgroupSL">
## <ManSection>
## <Prop Name="IsSubgroupSL" Arg='matgrp'/>
##
## <Description>
## This property tests, whether a matrix group is a subgroup of the Special
## Linear group in the right dimension over the (smallest) ring which
## contains all entries of its elements.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsSubgroupSL", IsMatrixGroup );
InstallTrueMethod(IsSubgroupSL,IsNaturalSL);
#############################################################################
##
#A InvariantBilinearForm( <matgrp> )
##
## <#GAPDoc Label="InvariantBilinearForm">
## <ManSection>
## <Attr Name="InvariantBilinearForm" Arg='matgrp'/>
##
## <Description>
## This attribute describes a bilinear form that is invariant under the
## matrix group <A>matgrp</A>.
## The form is given by a record with the component <C>matrix</C>
## which is a matrix <M>F</M> such that for every generator <M>g</M> of
## <A>matgrp</A> the equation <M>g \cdot F \cdot g^{tr} = F</M> holds.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "InvariantBilinearForm", IsMatrixGroup );
#############################################################################
##
#P IsFullSubgroupGLorSLRespectingBilinearForm(<matgrp>)
##
## <#GAPDoc Label="IsFullSubgroupGLorSLRespectingBilinearForm">
## <ManSection>
## <Prop Name="IsFullSubgroupGLorSLRespectingBilinearForm" Arg='matgrp'/>
##
## <Description>
## This property tests, whether a matrix group <A>matgrp</A> is the full
## subgroup of GL or SL (the property <Ref Prop="IsSubgroupSL"/> determines
## which it is) respecting the form stored as the value of
## <Ref Attr="InvariantBilinearForm"/> for <A>matgrp</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsFullSubgroupGLorSLRespectingBilinearForm", IsMatrixGroup );
InstallTrueMethod( IsGroup, IsFullSubgroupGLorSLRespectingBilinearForm );
#############################################################################
##
#A InvariantSesquilinearForm( <matgrp> )
##
## <#GAPDoc Label="InvariantSesquilinearForm">
## <ManSection>
## <Attr Name="InvariantSesquilinearForm" Arg='matgrp'/>
##
## <Description>
## This attribute describes a sesquilinear form that is invariant under the
## matrix group <A>matgrp</A> over the field <M>F</M> with <M>q^2</M>
## elements.
## The form is given by a record with the component <C>matrix</C>
## which is a matrix <M>M</M> such that for every generator <M>g</M> of
## <A>matgrp</A>
## the equation <M>g \cdot M \cdot (g^{tr})^f = M</M> holds,
## where <M>f</M> is the automorphism of <M>F</M> that raises each element
## to its <M>q</M>-th power.
## (<M>f</M> can be obtained as a power of the
## <Ref Attr="FrobeniusAutomorphism"/> value of <M>F</M>.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "InvariantSesquilinearForm", IsMatrixGroup );
#############################################################################
##
#P IsFullSubgroupGLorSLRespectingSesquilinearForm(<matgrp>)
##
## <#GAPDoc Label="IsFullSubgroupGLorSLRespectingSesquilinearForm">
## <ManSection>
## <Prop Name="IsFullSubgroupGLorSLRespectingSesquilinearForm" Arg='matgrp'/>
##
## <Description>
## This property tests, whether a matrix group <A>matgrp</A> is the full
## subgroup of GL or SL (the property <Ref Prop="IsSubgroupSL"/> determines
## which it is) respecting the form stored as the value of
## <Ref Attr="InvariantSesquilinearForm"/> for <A>matgrp</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsFullSubgroupGLorSLRespectingSesquilinearForm",
IsMatrixGroup );
InstallTrueMethod( IsGroup, IsFullSubgroupGLorSLRespectingSesquilinearForm );
#############################################################################
##
#A InvariantQuadraticForm( <matgrp> )
##
## <#GAPDoc Label="InvariantQuadraticForm">
## <ManSection>
## <Attr Name="InvariantQuadraticForm" Arg='matgrp'/>
##
## <Description>
## For a matrix group <A>matgrp</A>, <Ref Attr="InvariantQuadraticForm"/>
## returns a record containing at least the component <C>matrix</C>
## whose value is a matrix <M>Q</M>.
## The quadratic form <M>q</M> on the natural vector space <M>V</M> on which
## <A>matgrp</A> acts is given by <M>q(v) = v Q v^{tr}</M>,
## and the invariance under <A>matgrp</A> is given by the equation
## <M>q(v) = q(v M)</M> for all <M>v \in V</M> and <M>M</M> in
## <A>matgrp</A>.
## (Note that the invariance of <M>q</M> does <E>not</E> imply that the
## matrix <M>Q</M> is invariant under <A>matgrp</A>.)
## <P/>
## <M>q</M> is defined relative to an invariant symmetric bilinear form
## <M>f</M> (see <Ref Attr="InvariantBilinearForm"/>),
## via the equation
## <M>q(\lambda x + \mu y) =
## \lambda^2 q(x) + \lambda \mu f(x,y) + \mu^2 q(y)</M>,
## see <Cite Key="CCN85" Where="Chapter 3.4"/>.
## If <M>f</M> is represented by the matrix <M>F</M> then this implies
## <M>F = Q + Q^{tr}</M>.
## In characteristic different from <M>2</M>, we have <M>q(x) = f(x,x)/2</M>,
## so <M>Q</M> can be chosen as the strictly upper triangular part of
## <M>F</M> plus half of the diagonal part of <M>F</M>.
## In characteristic <M>2</M>, <M>F</M> does not determine <M>Q</M>
## but still <M>Q</M> can be chosen as an upper (or lower) triangular matrix.
## <P/>
## Whenever the <Ref Attr="InvariantQuadraticForm"/> value is set in a
## matrix group then also the <Ref Attr="InvariantBilinearForm"/> value
## can be accessed,
## and the two values are compatible in the above sense.
## <!-- So wouldn't it be natural to store the inv. bilinear form in the-->
## <!-- record of the invariant quadratic form?-->
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "InvariantQuadraticForm", IsMatrixGroup );
#############################################################################
##
#P IsFullSubgroupGLorSLRespectingQuadraticForm( <matgrp> )
##
## <#GAPDoc Label="IsFullSubgroupGLorSLRespectingQuadraticForm">
## <ManSection>
## <Prop Name="IsFullSubgroupGLorSLRespectingQuadraticForm" Arg='matgrp'/>
##
## <Description>
## This property tests, whether the matrix group <A>matgrp</A> is the full
## subgroup of GL or SL (the property <Ref Prop="IsSubgroupSL"/> determines
## which it is) respecting the <Ref Attr="InvariantQuadraticForm"/> value
## of <A>matgrp</A>.
## <Example><![CDATA[
## gap> g:= Sp( 2, 3 );;
## gap> m:= InvariantBilinearForm( g ).matrix;
## [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ]
## gap> [ 0, 1 ] * m * [ 1, -1 ]; # evaluate the bilinear form
## Z(3)
## gap> IsFullSubgroupGLorSLRespectingBilinearForm( g );
## true
## gap> g:= SU( 2, 4 );;
## gap> m:= InvariantSesquilinearForm( g ).matrix;
## [ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, 0*Z(2) ] ]
## gap> [ 0, 1 ] * m * [ 1, 1 ]; # evaluate the bilinear form
## Z(2)^0
## gap> IsFullSubgroupGLorSLRespectingSesquilinearForm( g );
## true
## gap> g:= GO( 1, 2, 3 );;
## gap> m:= InvariantBilinearForm( g ).matrix;
## [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ] ]
## gap> [ 0, 1 ] * m * [ 1, 1 ]; # evaluate the bilinear form
## Z(3)^0
## gap> q:= InvariantQuadraticForm( g ).matrix;
## [ [ 0*Z(3), Z(3)^0 ], [ 0*Z(3), 0*Z(3) ] ]
## gap> [ 0, 1 ] * q * [ 0, 1 ]; # evaluate the quadratic form
## 0*Z(3)
## gap> IsFullSubgroupGLorSLRespectingQuadraticForm( g );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsFullSubgroupGLorSLRespectingQuadraticForm",
IsMatrixGroup );
InstallTrueMethod( IsGroup, IsFullSubgroupGLorSLRespectingQuadraticForm );
#############################################################################
##
#F AffineActionByMatrixGroup( <M> )
##
## <ManSection>
## <Func Name="AffineActionByMatrixGroup" Arg='M'/>
##
## <Description>
## takes a group <A>M</A> of <M>n \times n</M> matrices over the finite
## field <M>F</M> and returns an affine permutation group
## <M>F^n :</M> <A>M</A>
## for the natural action of <A>M</A> on the vector space <M>F^n</M>.
## The labelling of the points of the resulting group is not guaranteed.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "AffineActionByMatrixGroup" );
#############################################################################
##
#F BlowUpIsomorphism( <matgrp>, <B> )
##
## <#GAPDoc Label="BlowUpIsomorphism">
## <ManSection>
## <Func Name="BlowUpIsomorphism" Arg='matgrp, B'/>
##
## <Description>
## For a matrix group <A>matgrp</A> and a basis <A>B</A> of a field
## extension <M>L / K</M> such that the entries of all matrices in
## <A>matgrp</A> lie in <M>L</M>,
## <Ref Func="BlowUpIsomorphism"/> returns the isomorphism with source
## <A>matgrp</A> that is defined by mapping the matrix <M>A</M> to
## <C>BlownUpMat</C><M>( A, <A>B</A> )</M>,
## see <Ref Func="BlownUpMat"/>.
## <Example><![CDATA[
## gap> g:= GL(2,4);;
## gap> B:= CanonicalBasis( GF(4) );; BasisVectors( B );
## [ Z(2)^0, Z(2^2) ]
## gap> iso:= BlowUpIsomorphism( g, B );;
## gap> Display( Image( iso, [ [ Z(4), Z(2) ], [ 0*Z(2), Z(4)^2 ] ] ) );
## . 1 1 .
## 1 1 . 1
## . . 1 1
## . . 1 .
## gap> img:= Image( iso, g );
## <matrix group with 2 generators>
## gap> Index( GL(4,2), img );
## 112
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "BlowUpIsomorphism" );
DeclareGlobalFunction( "BasisVectorsForMatrixAction" );
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