<p>The MeatAxe <a href="chapBib.html#biBPar84">[Par84]</a> is a tool for the examination of submodules of a group algebra. It is a basic tool for the examination of group actions on finite-dimensional modules.</p>
<p><strong class="pkg">GAP</strong> uses the improved MeatAxe of Derek Holt and Sarah Rees, and also incorporates further improvements of Ivanyos and Lux.</p>
<p>Please note that, consistently with the convention for group actions, the action of the <strong class="pkg">GAP</strong> MeatAxe is always that of matrices on row vectors by multiplication on the right. If you want to investigate left modules you will have to transpose the matrices.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GModuleByMats</code>( <var class="Arg">gens</var>, <var class="Arg">field<p<a id="X7D9B5B4E7F5A5FBD" name"7">/><java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
classfunc=""%>trtd class"dleft><code class="func">‣ GModuleByMats</code>( <var class="Arg">emptygens</var>, <var class="Arg">dim</var>, <var class="Arg">field</var> )</td><td class="tdright">( function )</td></tr></table></div>
> MeatAxe over <varclass"rg"><vara < class="rg">gens<var whichreflect groups the of java.lang.StringIndexOutOfBoundsException: Range [183, 182) out of bounds for length 244
<p>MeatAxe routines are on a level with Gaussian elimination. Therefore they do not deal with <strong class="pkg">GAP</strong> modules but essentially with lists of matrices. For the MeatAxe, a module is a record with components</p>
<dl>
<dt><strong class="Mark"><code class="code">generators</code></strong></dt>
<dd><p>A list of matrices which represent a group operation on a finite dimensional row vector space.</p>
</dd>
<dt><strong class="Mark"><code class="code">field</code></strong></dt>
<dd><p>The field over which the vector space is defined.</p>
</dd>
</dl>
<p>Once a module has been created its entries may not be changed. A MeatAxe may create a new component <var class="Arg">NameOfMeatAxe</var> in which it can store private information. By a MeatAxe <q>submodule</q> or <q>factor module</q> we denote actually the <em>induced action</em> on the submodule, respectively factor module. Therefore the submodules or factor modules are again MeatAxe modules. The arrangement of <code class="code">generators</code> is guaranteed to be the same for the induced modules, but<p>returns <arclass=Arg"modulevar>as sum java.lang.StringIndexOutOfBoundsException: Range [94, 93) out of bounds for length 413
<p><a id="X87B82250801A1BD0" <pcomputes the homogeneous components of< class=Arg>module<var given as sums of indecomposable components. The function returns a list, each entry of which is a record corresponding to one isomorphism type of indecomposable components. The record has the following components.</p>
<h5>69.2-1 <span class="Heading">java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 0
<div class="func"><table class="func" width="100%"><dd><p>the index numbers of the indecomposable compone as given by <code class""Indecomposition/>(a".html#781772FD865B9F9C">spanclass=RefLink">.-<span>a>), thehomogeneous ,</p>
<pp> theindecomposable <p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Called with a permutation group<dd><p>a list of the remaining indecomposable components, each given as a record with the components<code class"code">component</ode>thecomponent itself) and <code class="code">isomorphism</code> (an isomorphism from the defining component to this one).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; WedgeGModule</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<>code"func></ calculates the wedge product of a <var class="Arg">G</var>-module. That is the action on antisymmetric tensors.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>All MeatAxe routines are accessed via the global variablep><a id="X871D9AF87FABFB00" name="X871D9AF87FABFB00"></a></p>
<h>.- .<h5
<h4>69.4 <span class=
<p>Even though a MeatAxe module is a record, its components should never be accessed outside of MeatAxe functions. Instead the following operations should be used:</p>
<p =func>table=func"=100"<><="code ">#BasesMinimalSubmodulescode(var""module</var )/td>tdclass=tdright">( function )</td></tr></table></div>
<h5>69.4-1 MTX.java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.Generators</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of matrix generators of <var class="Arg">module</var>.</p>
<div class="func"><table class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns the dimension in which the matrices act.</p>
<div ="unc><table "func"width==100">trjava.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 221
<p>returns the field over which <var class="Arg">module</var> is defined.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.IsIrreducible</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>tests whether the module <var class="Arg">module</var> is irreducible (i.e. contains no proper submodules.)</p>
<>a"java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<h5>69.5-2 MTX.IsAbsolutelyIrreducible</h5>
<div class=func>table""width"%>>tdclass="tdleft"><code class="func">&
irreducible remains irreducibleoverthealgebraic closure of the field. (Formally: If the tensor product <span class="SimpleMath">L ⊗_K M</span> is irreducible where <span class="SimpleMath">M</span> is the module defined over <span class="SimpleMath">K</span> and <span class="SimpleMath">L</span> is the algebraic closure of <span class="SimpleMath">K</span>.)</p>
<p><a id="X7E84E1927EBFD483
<h5>69.5-3 MTX.DegreeSplittingField</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.DegreeSplittingField</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.Indecomposition</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a decomposition of <var class="Arg">module</var> as a direct sum of indecomposable modules. It returns a list, each entry is a list of form [<var class="Arg">B</var>,<var class="Arg">ind</var>] where <var class="Arg">B</var> is a list of basis vectors for the indecomposable component and <var class="Arg">ind</var> the induced module action on this component. (Such a decomposition is not unique.)</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.HomogeneousComponents</code>( <var class="Arg">module</var> )</td><td class="tdright">(&<>returns a listgiving all java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 88
<p>computes the homogeneous components of <var class="Arg">module</var> given as sums of indecomposable components. The function returns a list, each entry of which is a record corresponding to one isomorphism type of indecomposable components. The record has the following components.</p>
<dl> dt<=Mark"<class=code"indices/>/>/>
<dd><p>the index numbers of the indecomposable components, as given by <code class="func">MTX.Indecomposition</code> (<a href="chap69.html#X781772FD865B9F9C"><span class="RefLink">69.6-2</
</dd>
<dt><strong class="Mark"><code class="code">componentjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<dd>pof thejava.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 48
</dd>
<><"">< class=code>java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 71
<dd><p>a list of the remaining indecomposable components, each given as a record with the components <code class="code">java.lang.StringIndexOutOfBoundsException: Range [0, 129) out of bounds for length 0
</dd>
<
<p><a id="X85A258567D96B9BE" name="X85A258567D96B9BEjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<> java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 62
<>697- MTX/>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.SubmoduleGModule</code>( <var class="Arg">module</var>, <var class="Arg">subspace</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.SubGModule</code>( <varjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><varclass"Arg">/>java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 371
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.ProperSubmoduleBasis</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<returnsthebasis of proper submodule <varclass=""module<var>and <code java.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 145
<div class="func"><table"func"width=%"<tr><td class="tdleft"><code class="func">‣ MTX.InducedActionSubMatrix</code>( <var class="Arg">mat</var>, <var class="Arg">sub</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of bases of all minimal submodules.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.BasesMaximalSubmodules</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of bases of all maximal submodules.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.BasisRadicaljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns a basis of the radical of <var class="Arg">module</var>.</p>
<div class="func"><table java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0 pjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 69
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.BasesMinimalSupermodules</code>( <var class="Arg">module</var>, <var class="Arg">sub</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of bases of all minimal supermodules of the submodule given by the basis <var class=
<div class="func"><table class="func" width="100%"><tr><td class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 88
<p><a id="p>returns a basisjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 300
<h5>69.7-10 MTX.CompositionFactors</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.CompositionFactors</code>( <var class="Arg">module</var> )</td
<p>returns a list of composition div class="func"><table class="func="00%><tr>< class="tdleft>class="func">‣ MTX.BasisModuleEndomorphisms</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><a id="X7E5038F384DBCAEC" name="X7E5038F384DBCAEC"></ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>69.7-11 MTX.CollectedFactors</h5>
<div class="func"><table class="func" width="100%"><tr><td
<p>returns a list giving all irreducible composition factors with their frequencies.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.NormedBasisAndBaseChange</code>( <var class="Arg">sub</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list <code class="code">[<var class="Arg">bas</var>, <var class="Arg">change</var> ]</code> where <var class="Arg">bas</var> is a normed basis (i.e. in echelon form with pivots normed to 1) for <var class="Arg">sub</var> and <var class="Arg">change</var> is the base change from <var class="Arg">bas</var> to <var class="Arg">sub</var> (the basis vectors of <var class="Arg">bas</var> expressed in coefficients for <var class="Arg">sub</var>).</p>
<div class="func"><table class="func" width="
<div ="func><ablefunc" width=100%%"><tr><d class=">code =>8227;MTX/(<class"rg">java.lang.StringIndexOutOfBoundsException: Range [211, 161) out of bounds for length 268
<p>creates a new module correspondingdiv ="func"< class=" width="100%"><>< ="tdleft"><code class="func">‣MTXM</ class="modulevar)td<class"dright"( unctionnbsp;<td>/r<<div
<h5>69.8-3 MTX.h4>69.10 <span class="Heading">Module Homomorphisms for irreducible modules</span></h4>
<div java.lang.StringIndexOutOfBoundsException: Index 248 out of bounds for length 247
modulecorrespondingto actionof < class="rg"module/var on the factorofthe proper submodule <var class="Arg">sub</var>. If <var class="Arg">compl</var> is given, it has to be a basis of a (vector space-)complement of <var class="Arg">
<p>The basis <var class="Arg">sub</var> has to be given in normed form. (That is it must be in echelon form with pivots normed to 1, see <code class="func">MTX.NormedBasisAndBaseChange</code> (<a href="chap69.html#X79EA05D4822C2668"><span class="RefLink">69.8-1</span></a>))</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><codep> isomorphism <class""module1/java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 709
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.InducedActionSubMatrixNB</code>( <var class="Arg">mat</var>, <var class="Arg">sub</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.InducedActionFactorMatrix</code>( <var class="Arg">mat</var>, <var class="Arg">sub</var>[, <var class="Arg">compl</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>work the same way as the above functions for modules, but take as input only a single matrix.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.InducedAction</code>( <var class="Arg">module</var>, <var class="Arg">sub</var>[, <var class="Arg">type</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Computes induced actions on submodules or factor modules and also returns the corresponding bases. The action taken is binary encoded in <var class="Arg">type</var>: <code class="code">1</code> stands for subspace action, <code class="code">2</code> for factor action, and <code class="code">4</code> for action of the full module on a subspace adapted basis. The routine returns the computed results in a list in sequence (<var class="Arg">sub</var>,<var class="Arg">quot</var>,<var class="Arg">both</var>,<var class="Arg">basis</var>) where <var class="Arg">basis</var> is a basis for the whole space, extending <var class="Arg">sub</var>. (Actions which are not computed are omitted, so the returned list may be shorter.) If no <var class="Arg">type</var> is given, it is assumed to be <code class="code">7</code>. The basis given in <var class="Arg">sub</var> must be normed!</p>
<p>All these routines return <code class="keyw">fail</code> if <var class="Arg">sub</var> is not a proper subspace.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.BasisModuleHomomorphisms</code>( <var class="Arg">module1</var>, <var class="Arg">module2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a basis of all module homomorphisms from <var class="Arg">module1</var> to <var class="Arg">module2</var>. Homomorphisms are by matrices, whose rows give the images of the standard basis vectors of <var class="Arg">module1</var> in the standard basis of <var class="Arg">module2</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.BasisModuleEndomorphismsp< ="mat</var> should be a <var class="Arg">dim1</var> <pan =SimpleMath></ var "rg><> ahomomorphismfrom<ar=Arg><>to =Arg>/> functionverifies < class"rg></var>really does define amodule homomorphism,and then returns the corresponding homomorphism between the underlying row spaces of the modules. This can be used for computing kernels, images and pre-images.</p>
<h5>104MTX.Homomorphismsh5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><java.lang.StringIndexOutOfBoundsException: Range [0, 79) out of bounds for length 0
<p>returns the module automorphisms of <var class="Arg">module</var> (java.lang.StringIndexOutOfBoundsException: Range [0, 73) out of bounds for length 0
<h4>69.10 <span class="Heading">Module Homomorphisms for irreducible modules</span></h4>
<p>The following are lower-level functions that provide homomorphism functionality for irreducible modules. Generic code should use the functions in Section <a href="chap69.html#X8040270F791514C8"><span class="RefLink">69.9</span></a> instead.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.IsEquivalent</code>( <var class="Arg">module1</var>, <var class="Arg">module2</var> )</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>tests two irreducible modules for equivalence.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.IsomorphismIrred</code>( <var class="Arg">module1</var>, <var class="Arg">module2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an isomorphism from <var class="Arg">module1</var> to <var class="Arg">module2</var> (if one exists), and <code class="keyw">fail</code> otherwise. It requires that one of the modules is known to be irreducible. It implicitly assumes that the same group is acting, otherwise the results are unpredictable. The isomorphism is given by a matrix <span class="SimpleMath">M</span>, whose rows give the images of the standard basis vectors of <var class="Arg">module1</var> in the standard basis of <var class="Arg">module2</var>. That is, conjugation of the generators of <var class="Arg">module2</var> with <span class="SimpleMath">M</span> yields the generators of <var class="Arg">module1</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func<p><a id="X7E1F430278A334E1" name="X7E1F430278A334E1"></a></p>
<p><var class="Arg">mat</var> should be a <var class="Arg">dim1</var> <span class="SimpleMath">×</span> <var class="Arg">dim2</var> matrix defining a homomorphism from <var class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.Homomorphisms</code>( <var class="Arg">module1</var>, <varclass="Arg">module2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a basis of the space of all homomorphisms from the irreducible module <var class="Arg">module1</var> to <var class="Arg">module2</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.Distinguish</code>( <var class="Arg">cf</var>, <var class="Arg">nr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">cf</var> be the output of <code class="func">MTX.CollectedFactors</code> (<ahref="chap69.html#X7E5038F384DBCAEC"><span class="RefLink">69.7-11</span></a>). This routine tries to find a group algebra element that has nullity zero on all composition factors except number <var class="Arg">nr</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.InvariantBilinearForm</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an invariant bilinear form, which may be symmetric or anti-symmetric, of <var class="Arg">module</var>, or <code class="keyw">fail</code> if no such form exists.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.InvariantSesquilinearForm</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an invariant hermitian (= self-adjoint) sesquilinear form of <var class="Arg">module</var>, which must be defined over a finite field whose order is a square, or <code class="keyw">fail</code> if no such form exists.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.InvariantQuadraticForm</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns either the matrix of an invariant quadratic form of the absolutely irreducible module <var class="Arg">module</var>, or <code class="keyw">fail</code>.</p>
<p>If the characteristic of <var class="Arg">module</var> is odd then <code class="keyw">fail</code> is returned if there is no nonzero invariant bilinear form, otherwise a matrix of the bilinear form divided by <span class="SimpleMath">2</span> is returned; note that this matrix may be antisymmetric and thus describe the zero quadratic form. If the characteristic of <var class="Arg">module</var> is <span class="SimpleMath">2</span> then <code class="keyw">fail</code> is returned if <var class="Arg">module</var> does not admit a nonzero quadratic form, otherwise a lower triangular matrix describing the form is returned.</p>
<p>An error is signalled if <var class="Arg">module</var> is not absolutely irreducible.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.BasisInOrbit</code>( <var class="Arg">module</var> )</td><tdclass="tdright">( function )</td></tr></table></div>
<p>returns a basis of the underlying vector space of <var class="Arg">module</var> which is contained in an orbit of the action of the generators of module on that space. This is used by <code class="func">MTX.InvariantQuadraticForm</code> (<a href="chap69.html#X7ADE65997F16EE63"><span class="RefLink">69.11-3</span></a>) in characteristic 2.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.OrthogonalSign</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">module</var> be an absolutely irreducible <span class="SimpleMath">G</span>-module. If <var class="Arg">module</var> does not fix a nondegenerate quadratic form see <code class="func">MTX.InvariantQuadraticForm</code> (<a href="chap69.html#X7ADE65997F16EE63"><span class="RefLink">69.11-3</span></a>) then <code class="keyw">fail</code> is returned. Otherwise the sign <span class="SimpleMath">ϵ ∈ { -1, 0, 1 }</span> is returned such that <span class="SimpleMath">G</span> embeds into the general orthogonal group <span class="SimpleMath">GO^ϵ(d, q)</span> w.r.t. the invariant quadratic form, see <code class="func">GeneralOrthogonalGroup</code> (<a href="chap50.html#X7C2051CB7B94CEB1"><span class="RefLink">50.2-6</span></a>). That is, <code class="code">0</code> is returned if <var class="Arg">module</var> has odd dimension, and <code class="code">1</code> or <code class="code">-1</code> is returned if the orthogonal group has plus or minus type, respectively.</p>
<p>An error is signalled if <var class="Arg">module</var> is not absolutely irreducible.</p>
<p>The <code class="code">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= GeneratorsOfGroup( GO(1,4,2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MTX.OrthogonalSign( GModuleByMats( mats, GF(2) ) );</span> 1
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= GeneratorsOfGroup( GO(-1,4,2) );;</span>
<span class="GAPprompt">gap></span> <span class="p>If the characteristic of <var class="Arg">module</var code ">java.lang.StringIndexOutOfBoundsException: Range [94, 93) out of bounds for length 631
-
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<div class="func"><table classp><codeclass"java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<>findsanelement possiblenullspaceifvarclass=Arg"<var> known to irreducible./>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.MinimalSubGModules</code>( <var class="Arg">module1</var>, <var class="Arg">module2</var>[, <var class="Arg">max</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns (at most <var class="Arg">max</var>) bases of submodules of <var class="Arg">module2</var> which are isomorphic to the irreducible module <var class="Arg">module1</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.Setter</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a setter function for the component <code class="code">smashMeataxe.(string)</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.Getter</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a getter function for the component <code class="code">smashMeataxe.(string)</code>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.IrreducibilityTest</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Tests for irreducibility and sets a subbasis if reducible. It neither sets an irreducibility flag, nor tests it. Thus the routine also can simply be called to obtain a random submodule.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AbsoluteIrreducibilityTest</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Tests for absolute irreducibility and sets splitting field degree. It neither sets an absolute irreducibility flag, nor tests it.</p>
<div"func"><table class=func"width="100%"><tr>td class""><codeclass="func">#; SMTX.MinimalSubGModulecode> var class="Arg">module</var>, <var class="Arg">cf</var>, <var class="Arg">nr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the basis of a minimal submodule of <var class="Arg">module</var> containing the indicated composition factor. It assumes <code class="code">Distinguish</code> has been called already.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.MatrixSum</code>( <var class="Arg">matrices1</var>, <var class="Arg">matrices2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates the direct sum of two matrix lists.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.CompleteBasis</code>( <var class="Arg">module<<pThe standard MeatAxe provided in the <strong class="pkg">GAP</strong> library is based on the MeatAxe in the <strong class="pkg">GAP</strong> 3 package <strong class="pkg">Smash</strong>, originally written by Derek Holt and Sarah Rees <a href=
<p>extends the partial basis <var class="Arg">pbasis</var> to a basis of the full space by action of <var class="Arg">module</var>. It returns whether it succeeded.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.Subbasis</code>( <var class="Arg">module</var> )</td><td class=<>unction tosort the of< class="code">Homomorphisms</code>.</p>
<p>Basis of a submodule.</p
<p>
<5>69.3- SMTX.lgEl</5
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AlgEl</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>list <code class="code">[newgens,coefflist]</code> giving an algebra element used for chopping.</p>
<>a id=X7D6C947A7C8C14B2name"7">/a>/p>
<h5>69.13-3 SMTX.AlgElMat</h5p<aid""name="">/>/p>
<div class="func"><table class="func"<5>912-6 SMTX.etter<h5java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
<p>matrix of <code class="func">SMTX.AlgEl</code> (<a href="chap69.html#X7ABCD69880772B2D"><span class="RefLink">69.13-2</span></a>).</p>
< class="unc"<table class=func"width="100%"><tr><td class="tdleft"><code class="func">‣ SMTX.AlgElCharPol</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>minimal polynomial of <code class="func">SMTX.AlgEl</code> (<a href="chap69.html#X7ABCD69880772B2D"><span class="RefLink">69.13-2</span></a>).</pTestsfor irreducibilityand a subbasis if . It setsan flag, nor .Thus routine cansimply be called obtaina submodule<p>
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<p< id=X8367java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<lspaceVec/
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code classp<aid"" name=X79EF16677C2EE095<a<p
<p>nullspace of the java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AlgElNullspaceDimension</code>( <var java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>dimension of the nullspace.</p>
<vclass="unc"<table class"func"width="00%"><tr>td class=tdleft">code class="func">‣ SMTX.CentMat</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<matrix all is computedasabyproduct <ode =func"SMTX.AbsoluteIrreducibilityTest</code> < href=chap69.htmlX7E692DC97AFB661E>< class=RefLink">69.-8<span)<pjava.lang.StringIndexOutOfBoundsException: Index 219 out of bounds for length 219
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