<p>For most operations, <strong class="pkg">GAP</strong> only provides methods for finite matrix groups. Many calculations in finite matrix groups are done via so-called <q>nice monomorphisms</q> (see Section <a href="chap40_mj.html#X7FFD731684606BC6"><span class="RefLink">40.5</span></a>) that represent a faithful action on vectors.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DefaultFieldOfMatrixGroup</code>( <var class="Arg">mat-grp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Is a field containing all the matrix entries. It is not guaranteed to be the smallest field with this property.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FieldOfMatrixGroup</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The smallest field containing all the matrix entries of all elements of the matrix group <var class="Arg">matgrp</var>. As the calculation of this can be hard, this should only be used if one <em>really</em> needs the smallest field, use <code class="func">DefaultFieldOfMatrixGroup</code> (<a href="chap44_mj.html#X7D540083793CD496"><span class="RefLink">44.2-2</span></a>) to get (for example) the characteristic.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransposedMatrixGroup</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the transpose of the matrix group <var class="Arg">matgrp</var>. The transpose of the transpose of <var class="Arg">matgrp</var> is identical to <var class="Arg">matgrp</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFFEMatrixGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>tests whether all matrices in <var class="Arg">G</var> have finite field element entries.</p>
<h4>44.3 <span class="Heading">Actions of Matrix Groups</span></h4>
<p>The basic operations for groups are described in Chapter <a href="chap41_mj.html#X87115591851FB7F4"><span class="RefLink">41</span></a>, special actions for <em>matrix</em> groups mentioned there are <code class="func">OnLines</code> (<a href="chap41_mj.html#X86DC2DD5829CAD9A"><spanclass="RefLink">41.2-12</span></a>), <code class="func">OnRight</code> (<a href="chap41_mj.html#X7960924D84B5B18F"><span class="RefLink">41.2-2</span></a>), and <code class="func">OnSubspacesByCanonicalBasis</code> (<a href="chap41_mj.html#X85124D197F0F9C4D"><span class="RefLink">41.2-15</span></a>).</p>
<p>For subtleties concerning multiplication from the left or from the right, see <a href="chap44_mj.html#X868288377CFA8D1B"><span class="RefLink">44.7</span></a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveActionOnFullSpace</code>( <var class="Arg">G</var>, <var class="Arg">F</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a group of <var class="Arg">n</var> by <var class="Arg">n</var> matrices over a field contained in the finite field <var class="Arg">F</var>. <code class="func">ProjectiveActionOnFullSpace</code> returns the image of the projective action of <var class="Arg">G</var> on the full row space <span class="SimpleMath">\(\textit{F}^{\textit{n}}\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ProjectiveActionHomomorphismMatrixGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an action homomorphism for a faithful projective action of <var class="Arg">G</var> 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.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlowUpIsomorphism</code>( <var class="Arg">matgrp</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a matrix group <var class="Arg">matgrp</var> and a basis <var class="Arg">B</var> of a field extension <span class="SimpleMath">\(L / K\)</span> such that the entries of all matrices in <var class="Arg">matgrp</var> lie in <span class="SimpleMath">\(L\)</span>, <code class="func">BlowUpIsomorphism</code> returns the isomorphism with source <var class="Arg">matgrp</var> that is defined by mapping the matrix <span class="SimpleMath">\(A\)</span> to <code class="code">BlownUpMat</code><span class="SimpleMath">\(( A, \textit{B} )\)</span>, see <code class="func">BlownUpMat</code> (<a href="chap24_mj.html#X85923C107A4569D0"><span class="RefLink">24.13-4</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneralLinearGroup</code>( <var class="Arg">grp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGL</code>( <var class="Arg">grp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>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.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNaturalGL</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>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.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSpecialLinearGroup</code>( <var class="Arg">grp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSL</code>( <var class="Arg">grp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>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.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNaturalSL</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>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.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubgroupSL</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>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.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantBilinearForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute describes a bilinear form that is invariant under the matrix group <var class="Arg">matgrp</var>. The form is given by a record with the component <code class="code">matrix</code> which is a matrix <span class="SimpleMath">\(F\)</span> such that for every generator <span class="SimpleMath">\(g\)</span> of <var class="Arg">matgrp</var> the equation <span class="SimpleMath">\(g \cdot F \cdot g^{tr} = F\)</span> holds.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullSubgroupGLorSLRespectingBilinearForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property tests, whether a matrix group <var class="Arg">matgrp</var> is the full subgroup of GL or SL (the property <code class="func">IsSubgroupSL</code> (<a href="chap44_mj.html#X7ED43D4F7E993A31"><span class="RefLink">44.4-5</span></a>) determines which it is) respecting the form stored as the value of <code class="func">InvariantBilinearForm</code> (<a href="chap44_mj.html#X7C08385A81AB05E1"><span class="RefLink">44.5-1</span></a>) for <var class="Arg">matgrp</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantSesquilinearForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This attribute describes a sesquilinear form that is invariant under the matrix group <var class="Arg">matgrp</var> over the field <span class="SimpleMath">\(F\)</span> with <span class="SimpleMath">\(q^2\)</span> elements. The form is given by a record with the component <code class="code">matrix</code> which is a matrix <span class="SimpleMath">\(M\)</span> such that for every generator <span class="SimpleMath">\(g\)</span> of <var class="Arg">matgrp</var> the equation <span class="SimpleMath">\(g \cdot M \cdot (g^{tr})^f = M\)</span> holds, where <span class="SimpleMath">\(f\)</span> is the automorphism of <span class="SimpleMath">\(F\)</span> that raises each element to its <span class="SimpleMath">\(q\)</span>-th power. (<span class="SimpleMath">\(f\)</span> can be obtained as a power of the <code class="func">FrobeniusAutomorphism</code> (<a href="chap59_mj.html#X8758E4AB7D0A1955"><span class="RefLink">59.4-1</span></a>) value of <span class="SimpleMath">\(F\)</span>.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullSubgroupGLorSLRespectingSesquilinearForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property tests, whether a matrix group <var class="Arg">matgrp</var> is the full subgroup of GL or SL (the property <code class="func">IsSubgroupSL</code> (<a href="chap44_mj.html#X7ED43D4F7E993A31"><span class="RefLink">44.4-5</span></a>) determines which it is) respecting the form stored as the value of <code class="func">InvariantSesquilinearForm</code> (<a href="chap44_mj.html#X82F22079852130C9"><span class="RefLink">44.5-3</span></a>) for <var class="Arg">matgrp</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantQuadraticForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a matrix group <var class="Arg">matgrp</var>, <code class="func">InvariantQuadraticForm</code> returns a record containing at least the component <code class="code">matrix</code> whose value is a matrix <span class="SimpleMath">\(Q\)</span>. The quadratic form <span class="SimpleMath">\(q\)</span> on the natural vector space <span class="SimpleMath">\(V\)</span> on which <var class="Arg">matgrp</var> acts is given by <span class="SimpleMath">\(q(v) = v Q v^{tr}\)</span>, and the invariance under <var class="Arg">matgrp</var> is given by the equation <span class="SimpleMath">\(q(v) = q(v M)\)</span> for all <span class="SimpleMath">\(v \in V\)</span> and <span class="SimpleMath">\(M\)</span> in <var class="Arg">matgrp</var>. (Note that the invariance of <span class="SimpleMath">\(q\)</span> does <em>not</em> imply that the matrix <span class="SimpleMath">\(Q\)</span> is invariant under <var class="Arg">matgrp</var>.)</p>
<p><span class="SimpleMath">\(q\)</span> is defined relative to an invariant symmetric bilinear form <span class="SimpleMath">\(f\)</span> (see <code class="func">InvariantBilinearForm</code> (<a href="chap44_mj.html#X7C08385A81AB05E1"><span class="RefLink">44.5-1</span></a>)), via the equation <span class="SimpleMath">\(q(\lambda x + \mu y) = \lambda^2 q(x) + \lambda \mu f(x,y) + \mu^2 q(y)\)</span>, see <a href="chapBib_mj.html#biBCCN85">[CCN+85, Chapter 3.4]</a>. If <spanclass="SimpleMath">\(f\)</span> is represented by the matrix <span class="SimpleMath">\(F\)</span> then this implies <span class="SimpleMath">\(F = Q + Q^{tr}\)</span>. In characteristic different from <span class="SimpleMath">\(2\)</span>, we have <span class="SimpleMath">\(q(x) = f(x,x)/2\)</span>, so <span class="SimpleMath">\(Q\)</span> can be chosen as the strictly upper triangular part of <span class="SimpleMath">\(F\)</span> plus half of the diagonal part of <span class="SimpleMath">\(F\)</span>. In characteristic <span class="SimpleMath">\(2\)</span>, <span class="SimpleMath">\(F\)</span> does not determine <span class="SimpleMath">\(Q\)</span> but still <span class="SimpleMath">\(Q\)</span> can be chosen as an upper (or lower) triangular matrix.</p>
<p>Whenever the <code class="func">InvariantQuadraticForm</code> value is set in a matrix group then also the <code class="func">InvariantBilinearForm</code> (<a href="chap44_mj.html#X7C08385A81AB05E1"><span class="RefLink">44.5-1</span></a>) value can be accessed, and the two values are compatible in the above sense.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFullSubgroupGLorSLRespectingQuadraticForm</code>( <var class="Arg">matgrp</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>This property tests, whether the matrix group <var class="Arg">matgrp</var> is the full subgroup of GL or SL (the property <code class="func">IsSubgroupSL</code> (<a href="chap44_mj.html#X7ED43D4F7E993A31"><span class="RefLink">44.4-5</span></a>) determines which it is) respecting the <code class="func">InvariantQuadraticForm</code> (<a href="chap44_mj.html#X7BCACC007EB9B613"><span class="RefLink">44.5-5</span></a>) value of <var class="Arg">matgrp</var>.</p>
<h4>44.6 <span class="Heading">Matrix Groups in Characteristic 0</span></h4>
<p>Most of the functions described in this and the following section have implementations which use functions from the <strong class="pkg">GAP</strong> package <strong class="pkg">CaratInterface</strong>. If <strong class="pkg">CaratInterface</strong> is not installed or not compiled, no suitable methods are available.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNaturalGLnZ</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>tests whether <var class="Arg">G</var> is <span class="SimpleMath">\(GL_n(ℤ)\)</span> in its natural representation by <span class="SimpleMath">\(n \times n\)</span> integer matrices. (The dimension <span class="SimpleMath">\(n\)</span> will be read off the generating matrices.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNaturalSLnZ</code>( <var class="Arg">G</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>tests whether <var class="Arg">G</var> is <span class="SimpleMath">\(SL_n(ℤ)\)</span> in its natural representation by <span class="SimpleMath">\(n \times n\)</span> integer matrices. (The dimension <span class="SimpleMath">\(n\)</span> will be read off the generating matrices.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ InvariantLattice</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a matrix <span class="SimpleMath">\(B\)</span>, whose rows form a basis of a <span class="SimpleMath">\(ℤ\)</span>-lattice that is invariant under the rational matrix group <var class="Arg">G</var> acting from the right. It returns <code class="keyw">fail</code> if the group is not unimodular. The columns of the inverse of <span class="SimpleMath">\(B\)</span> span a <span class="SimpleMath">\(ℤ\)</span>-lattice invariant under <var class="Arg">G</var> acting from the left.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizerInGLnZ</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is an attribute used to store the normalizer of <var class="Arg">G</var> in <span class="SimpleMath">\(GL_n(ℤ)\)</span>, where <var class="Arg">G</var> is an integer matrix group of dimension <var class="Arg">n</var>. This attribute is used by <code class="code">Normalizer( GL( n, Integers ), G )</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CentralizerInGLnZ</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is an attribute used to store the centralizer of <var class="Arg">G</var> in <span class="SimpleMath">\(GL_n(ℤ)\)</span>, where <var class="Arg">G</var> is an integer matrix group of dimension <var class="Arg">n</var>. This attribute is used by <code class="code">Centralizer( GL( n, Integers ), G )</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ZClassRepsQClass</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The conjugacy class in <span class="SimpleMath">\(GL_n(ℚ)\)</span> of the finite integer matrix group <var class="Arg">G</var> splits into finitely many conjugacy classes in <span class="SimpleMath">\(GL_n(ℤ)\)</span>. <code class="code">ZClassRepsQClass( <var class="Arg">G</var> )</code> returns representative groups for these.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BravaisGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the Bravais group of a finite integer matrix group <var class="Arg">G</var>. If <span class="SimpleMath">\(C\)</span> is the cone of positive definite quadratic forms <span class="SimpleMath">\(Q\)</span> invariant under <span class="SimpleMath">\(g \mapsto g Q g^{tr}\)</span> for all <span class="SimpleMath">\(g \in \textit{G}\)</span>, then the Bravais group of <var class="Arg">G</var> is the maximal subgroup of <span class="SimpleMath">\(GL_n(ℤ)\)</span> leaving the forms in that same cone invariant. Alternatively, the Bravais group of <var class="Arg">G</var> can also be defined with respect to the action <span class="SimpleMath">\(g \mapsto g^{tr} Q g\)</span> on positive definite quadratic forms <span class="SimpleMath">\(Q\)</span>. This latter definition is appropriate for groups <var class="Arg">G</var> acting from the right on row vectors, whereas the former definition is appropriate for groups acting from the left on column vectors. Both definitions yield the same Bravais group.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BravaisSubgroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the subgroups of the Bravais group of <var class="Arg">G</var>, which are themselves Bravais groups.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BravaisSupergroups</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the subgroups of <span class="SimpleMath">\(GL_n(ℤ)\)</span> that contain the Bravais group of <var class="Arg">G</var> and are Bravais groups themselves.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalizerInGLnZBravaisGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the normalizer of the Bravais group of <var class="Arg">G</var> in the appropriate <span class="SimpleMath">\(GL_n(ℤ)\)</span>.</p>
<h4>44.7 <span class="Heading">Acting OnRight and OnLeft</span></h4>
<p>In <strong class="pkg">GAP</strong>, matrices by convention act on row vectors from the right, whereas in crystallography the convention is to act on column vectors from the left. The definition of certain algebraic objects important in crystallography implicitly depends on which action is assumed. This holds true in particular for quadratic forms invariant under a matrix group. In a similar way, the representation of affine crystallographic groups, as they are provided by the <strong class="pkg">GAP</strong> package <strong class="pkg">CrystGap</strong>, depends on which action is assumed. Crystallographers are used to the action from the left, whereas the action from the right is the natural one for <strong class="pkg">GAP</strong>. For this reason, a number of functions which are important in crystallography, and whose result depends on which action is assumed, are provided in two versions, one for the usual action from the right, and one for the crystallographic action from the left.</p>
<p>For every such function, this fact is explicitly mentioned. The naming scheme is as follows: If <code class="code">SomeThing</code> is such a function, there will be functions <code class="code">SomeThingOnRight</code> and <code class="code">SomeThingOnLeft</code>, assuming action from the right and from the left, respectively. In addition, there is a generic function <code class="code">SomeThing</code>, which returns either the result of <code class="code">SomeThingOnRight</code> or <code class="code">SomeThingOnLeft</code>, depending on the global variable <code class="func">CrystGroupDefaultAction</code> (<a href="chap44_mj.html#X7D1318A6780CD88B"><span class="RefLink">44.7-1</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrystGroupDefaultAction</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>can have either of the two values <code class="code">RightAction</code> and <code class="code">LeftAction</code>. The initial value is <code class="code">RightAction</code>. For functions which have variants OnRight and OnLeft, this variable determines which variant is returned by the generic form. The value of <code class="func">CrystGroupDefaultAction</code> can be changed with the function <code class="func">SetCrystGroupDefaultAction</code> (<a href="chap44_mj.html#X792D237385977BE6"><span class="RefLink">44.7-2</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SetCrystGroupDefaultAction</code>( <var class="Arg">action</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>allows one to set the value of the global variable <code class="func">CrystGroupDefaultAction</code> (<a href="chap44_mj.html#X7D1318A6780CD88B"><span class="RefLink">44.7-1</span></a>). Only the arguments <code class="code">RightAction</code> and <code class="code">LeftAction</code> are allowed. Initially, the value of <code class="func">CrystGroupDefaultAction</code> (<a href="chap44_mj.html#X7D1318A6780CD88B"><span class="RefLink">44.7-1</span></a>) is <code class="code">RightAction</code>.</p>
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