<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</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; 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>
<p>creates a MeatAxe module over <var class="Arg">field</var> from a list of invertible matrices <var class="Arg">gens</var> which reflect a group's action. If the list of generators is empty, the dimension must be given as second argument.</p>
<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">dimension</code></strong></dt>
<dd><p>The dimension of the vector space (this is the common length of the row vectors (see <code class="func">DimensionOfVectors</code> (<a href="chap61.html#X8534A750878478D0"><span class="RefLink">61.9-6</span></a>))).</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 to obtain the complete relation to the original module, the bases used are needed as well.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NaturalGModule</code>( <var class="Arg">group</var>[, <var class="Arg">field</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates a MeatAxe module over <var class="Arg">field</var> from the generators of the matrix group <var class="Arg">group</var>. If <var class="Arg">field</var> is not provided then the value returned by <code class="func">DefaultFieldOfMatrixGroup</code> (<a href="chap44.html#X7D540083793CD496"><span class="RefLink">44.2-2</span></a>) is used instead.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PermutationGModule</code>( <var class="Arg">G</var>, <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Called with a permutation group <var class="Arg">G</var> and a field <var class="Arg">F</var> (<var class="Arg">F</var> may be infinite), <code class="func">PermutationGModule</code> returns the natural permutation module <span class="SimpleMath">M</span> over <var class="Arg">F</var> for the group of permutation matrices that acts on the canonical basis of <span class="SimpleMath">M</span> in the same way as <var class="Arg">G</var> acts on the points up to its largest moved point (see <code class="func">LargestMovedPoint</code> (<a href="chap42.html#X84AA603987C94AC0"><span class="RefLink">42.3-2</span></a>)).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TrivialGModule</code>( <var class="Arg">G</var>, <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Called with a group <var class="Arg">G</var> and a field <var class="Arg">F</var> (<var class="Arg">F</var> may be infinite), <code class="func">TrivialGModule</code> returns the trivial module over <var class="Arg">F</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TensorProductGModule</code>( <var class="Arg">m1</var>, <var class="Arg">m2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">TensorProductGModule</code> calculates the tensor product of the modules <var class="Arg">m1</var> and <var class="Arg">m2</var>. They are assumed to be modules over the same algebra so, in particular, they should have the same number of generators.</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>
<p><code class="func">WedgeGModule</code> 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 variable <code class="func">MTX</code>, which is a record whose components hold the various functions. It is possible to have several implementations of a MeatAxe available. Each MeatAxe represents its routines in an own global variable and assigning <code class="func">MTX</code> to this variable selects the corresponding MeatAxe.</p>
<>92 java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 29
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.Dimension</code>( <var class="Arg">java.lang.StringIndexOutOfBoundsException: Range [1, 146) out of bounds for length 144
< inwhichthe .p
<p>aid=X830C00887CE9323C X830C00887CE9323C"/<p>
<h5>69.4-3 MTX.Field</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.Field</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the field over which <var class="Arg">module</var> is defined.</p>
<><table"unc"width"%>tr>td class"><ode=">8227;MTX.IsIrreducible</code>( var ="Argmodule<var <td< class""> function)<td<tr<><div>
> java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 114
<div class="func</pan>
<>A module is absolutely irreducible it remains irreducible over the algebraic closure of the field. (Formally: If the tensor product <span class="SimpleMath">L ⊗_K M</span> is< "ontSS>br />span "">&; <span>< href=chap69.html#X84A93AC482A1946D">69.13-1 SMTX.Subbasis</a></span>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.DegreeSplittingFieldC>/<java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 142
<p>returns the degree of the splitting =ContSS>brspan ="nocss>nbsp;nbsp;/java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 147
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.IsIndecomposable</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></<div class="func"><table class"func"width=100"<>< =t"<java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 286
<<pcreates a module< =""field/> from list ofinvertiblematricesvar"">/>which a' action.If list of generators is empty, the dimension must be given as second argument.</p>
<p><a id="X781772FD865B9F9C" group operation on a finite dimensional row vector space.</p>
<h5>69.6-2 MTX.Indecomposition<
<iv class=func"<class" width"%"<tr<td class"">code=f>#8227 .<> var=module)>java.lang.StringIndexOutOfBoundsException: Range [175, 174) out of bounds for length 231 preturnsadecompositionof < class""></ adirect sum ofindecomposable 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>
</dd>
<dt><strong class="Mark"><code class="code">component</code></strong></dt>
<dd><p>neof the indecomposable components,/pjava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
</dd>
<dt><strong class="Mark"><code class="code">images</code></strong></dt>
<=""/ode ( java.lang.StringIndexOutOfBoundsException: Range [152, 151) out of bounds for length 263
</dd>
</dl>
<p""85>/>/>
<h4>69.7 <span class="Heading">Finding java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [31, 4) out of bounds for length 265
< ">table class="func" width=100"<>tdclass""code class""&8227; MTX.</code>( <ar =">module<var>< class=Arg"subspace/>)/td< =java.lang.StringIndexOutOfBoundsException: Range [213, 211) out of bounds for length 259
<p><var class="Arg">subspace</var> should be a subspace of (or a vector in) the underlying vector space of <var class="Arg">module</var> i.e. the full row space of the same dimension and over the same field as <var class="Arg">module</var>. A normalized basis of the submodule of <var class="Arg">module</var> generated by <var class="Arg">subspace</var> is returned.</p>
<p><a id="java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<h5>69.7-2 MTX.ProperSubmoduleBasis</h5>
<div class="func"><table class=p><aid"""">/>p
<p>returns the basis of a proper java.lang.StringIndexOutOfBoundsException: Range [0, 42) 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.BasesSubmodules</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td><<>< class=""WedgeGModulecode>the java.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 157
<p>returns a list containing a basis for every submodule.</p>
<java.lang.StringIndexOutOfBoundsException: Range [3, 2) out of bounds for length 62
<56974MTX.asesMinimalSubmodules/>
<divclass"">table class"func"width"%>tr><td class=tdleft"<class=func"&8227; MTX.BasesMinimalSubmodules</>(< class=Arg>module</var> )<td>td ="java.lang.StringIndexOutOfBoundsException: Range [191, 190) out of bounds for length 238
<p>returns a list of bases of
<div class="func"><table class="func" width="100%"><tr>
<p>returns a basis of the radical of <vardivclass=""<tableclass= "%>tr><td class="tdleft"><code class="func">‣ MTX.Field</code>( <var class="Arg">module</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.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 <p>A module is absolutely ifit java.lang.StringIndexOutOfBoundsException: Range [83, 82) out of bounds for length 399
<div class="func"><table class="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 0
<p>returns a list of bases of submodules in a composition series in ascending order.</p>
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
<p>returns a list of composition factors of <var class="Arg">module</var> in ascending order.</p>
<div class="java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
< giving all irreducible composition factors with their frequencies.</p>
<h5>69.8-1 java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div class="func"><table class="func" width="100%"><trdd>p>one of the indecomposable components,</p>
<p>returns a list <code class="code"dt<strong class=Mark><code class"code">images</code></strong></dt>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.InducedActionFactorModule</code>( <var class=var =""subspace<<var>should be a subspace of (or a vector in) the underlying vector space of <var class="Arg">module</var> i.e. the full row space of the same dimension and over the same field as <var class="Arg">module</var>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>creates a new module corresponding to the action of <var class="Arg">module</var> on the factor of the 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 ofp> the a of var classArg>/ class="keyw">fail</code> if no proper submodule exists.</p>
<p>The basis <var class="Arg">sub</var> has to be given in normed form. (That is
<p><a id="X8753A03A7C7CBFF1" java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 0
<h5>69.8-4 MTX.InducedActionSubMatrix</h5>
<div class="func">< class="=100"java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 263
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.InducedActionSubMatrixNB<
<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
<p>work the same way as the above functions for modules, but take as input only a single matrix.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func"preturns a list of bases of submodules in a composition series in ascending order.</p>
<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 thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<" width="00%<td=""<code java.lang.StringIndexOutOfBoundsException: Range [240, 85) out of bounds for length 240
<p>returns a basis of all module homomorphisms from <var class="Arg">module</var> to <var class="Arg">module</var>.</p>
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 38
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.java.lang.StringIndexOutOfBoundsException: Index 117 out of bounds for length 0
<p>If <var class="Arg">module1</var> and <var class="Arg">module2</var> are isomorphic modules, this function returns an isomorphism from <var class="Arg">module1</var> to <var class="Arg">module2</var> in form of a matrix. It returns <code class="keyw">fail</code> if the modules are not isomorphic.</p>
<p><a id
<h5>69.9-4 MTXclass""< class="func =100"tr<class"tdleft"<codeclass=func"‣ MTX.InducedActionSubmoduleNB<code> var =Arg"module</var>, <var class="Arg">sub</var> )</td><td class="tdright">( function )</td></tr></table></div>
< classfunc>table"unc"width"100">tr>tdclass8227;MTX.oduleAutomorphisms/ode>( <varclass=Arg"module</var> </td>td =t" )/>/>/table>/div>
<p>returns the module automorphisms of <var class="Arg">module</var> (the set of all isomorphisms from <var class="Arg">module</var> to itself) as a matrix group.</p>
<="X850324FF7912A541"<a<p
<java.lang.StringIndexOutOfBoundsException: Range [21, 3) out of bounds for length 88
<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>
<p><a id="<p>creates a new the <ar ="rg><>on java.lang.StringIndexOutOfBoundsException: Range [113, 112) out of bounds for length 344
<h5>69.10-1 MTX.IsEquivalent</h5>
<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> )</td><td class=java.lang.StringIndexOutOfBoundsException: Index 0 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>
<>returnsan from<var ="Arg"<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>, java.lang.StringIndexOutOfBoundsException: Range [0, 394) out of bounds for length 265
<div class="func"><table class="func" width="100%"><trjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>var class=Arg>> <class="×span><var class=Arg">im2/var matrixdefininga homomorphism from < class""module1<var to <var class"Arg"module2<var.Thisfunction verifies that<ar=A"mat> a java.lang.StringIndexOutOfBoundsException: Range [348, 347) out of bounds for length 495
<< =X8519B3C486AC8C7E"name=="85193C486AC8C7E>/a><p
<p>returns a basis of the space of all homomorphisms69.-3MTX.IsomorphismModules/h5
<>a =X81A6ECB078D4441Cname""<a>/>
<h5>69.10-5 MTX.Distinguish</h5>
class""< =func width=100%"<tr><">code=func"#8227 .istinguish</code> var class="Argcf/> <var class"Arg"nr/>)</><tdclass=tdright"(nbsp;unction&)/d>tr>/>/iv
<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
<java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.java.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 26
<p>returns either the matrix of an invariant quadratic form of the absolutely irreducible module <var class="Arg">module</var>, or <code class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<> is odd then < 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 -
<p>An error is signalled if <var class="Arg">module</var> is not absolutely irreducible.</p>
<div class="func"><table class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns a basis of the underlying vector space of <
<p><a id="X8168EB348474046B" name="X8168EB348474046B"></a></java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<h5>69
<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</<>the action .<>
<p>An error is signalled if <var class="Arg">module</var> is not absolutely irreducible.</p>
<pThe code =code">SMTX</code> implementation uses an algorithm due to Jon Thackray.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= GeneratorsOfGroup( GO(1,4,2) );;</span>
<span class="GAPprompt">pfinds element with minimal dimension < class""module/>is known tobe </java.lang.StringIndexOutOfBoundsException: Index 126 out of bounds for length 126 1
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= GeneratorsOfGroup( GO(-1,4,2) );;</span>
< class=GAPprompt">gap><span <pan =GAPinput">TX.rthogonalSignGModuleByMats( GF2 ))<spanjava.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
-1
<span class=">69.12-9 SMTX.MinimalSubGModule
<span class
0divclass=class" ><tdclass=tdleft "">8227</(<java.lang.StringIndexOutOfBoundsException: Range [144, 136) out of bounds for length 288
<span class="GAPprompt">gap></span> <span class="GAPinput">mats:= GeneratorsOfGroup( SP(4,2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MTX.OrthogonalSign( GModuleByMats( mats, GF(2) ) );</span>
fail
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.RandomIrreducibleSubGModule</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the module action on a random irreducible submodule.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.GoodElementGModule</code>( <var class="<h4>69.3 spanclass="Heading">Smash MeatAxe Flags</span></h4>
<p>finds an element with minimal possible nullspace dimension if <var class="Arg">module</var> is known to be irreducible.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 102
<p>returns a setter function for the component <code class="<<a "" =X7D6C947A7C8C14B2">/a<p>
<> =X7E60EBC57FFDF7BD =X7E60EBC57FFDF7BD<a<
<5>.-6SMTXGetter/>
<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</divf>""java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 229
<>Tests for irreducibility setsasubbasis ifreducibleIt neither irreducibilityflag, tests it the routinealso simply be toobtain random .<p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.java.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 0
<p>java.lang.StringIndexOutOfBoundsException: Index 5 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.MatrixSum</code>( <var class
<p>creates the direct sum of two matrix lists.</p>
<<di class"> =func ="00"<class=<java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 224
<p>The following getter routines access internalp>centralising generators which computed a of<odeclass""AbsoluteIrreducibilityTestcode(a"hap69.html#7"<pan"128/span><></a>./>
<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.AlgElNullspaceVec</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>nullspace of the matrix evaluated under this factor.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SMTX.CentMat</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>matrix centralising all generators which is computed as a byproduct of <code class="func">SMTX.AbsoluteIrreducibilityTest</code> (<a href="chap69.html#X7E692DC97AFB661E"><span class="RefLink">69.12-8</span></a>).</p>
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