<!DOCTYPE":/www..org/TR/xhtml1/DTD/html1-strict.td"
:w3TRjava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 61
< xmlns=http/www.3o1999"xml:ang="en"
<head>
<script type="text/javascript"
src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (ref) - Chapter 69: The MeatAxe</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<="="java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 56
<script type="text/javascript">overwriteStyle();</script>
<body classjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
"<class="chlink1>oto <java.lang.StringIndexOutOfBoundsException: Range [75, 73) out of bounds for length 3028
<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a"java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 173
<p id="mathjaxlink" class=""><ahref=chap69.">[MathJax off]</a><p>
<>< id"7BF9D3CB81A8F8F9 name="X7BF9D3CB81A8F8F9"></a></p>
<<div ="
<div class="ContSect"><span class="tocline"><spanspan=Cjava.lang.StringIndexOutOfBoundsException: Range [21, 19) out of bounds for length 162
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> <
</span>
</div></div>
<div class="ContSect"
</span>
<div class= http:span=< /spann"&<>href=java.lang.StringIndexOutOfBoundsException: Range [90, 89) out of bounds for length 163
< "java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 163
</java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
< class=ContSSsrc=":/cdn.sdelivr./mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 141 spanclass=>br/><java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 147
<<anclass"ontSS><br ><class=""> n;span< <span class="ContSS"><br /><span class="nocss"> </span><a hrefContSS"<br/spanclass="nbsp;&;/> =htmlX7java.lang.StringIndexOutOfBoundsException: Range [108, 107) out of bounds for length 139
<<
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap69_mj.html#X7C77D22782C98D4E">69.3 <span class="Heading">Selecting a Different MeatAxe</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> />
<<java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<div class=C>spanclass="< title> (ef) -Chapter 69 MeatAxe<title>
</span>
<div class="ontSS"<r>span class=nocss>n&;span<href".tml"<eta -equiv"- "ext/ =UTF"/>
< /><<link rel="stylesheet" type="tex"java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 59
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 140
<span="<r /><pan </java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<divclass"hlinktop"<class</java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 7
<divC>">java.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 178
</span>
<div class="ContSSBlock">
<span class="ContSS"><span class="ContSS"><bre=java.lang.StringIndexOutOfBoundsException: Range [56, 52) out of bounds for length 62
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 154
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 151
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 12 divjava.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 182
</span>/
<divclass""
<span =Cjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 147 span =ContSS> /<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 146
<</a></span>
</div></div>
< "ContSect"><span class="tocline"><span class="nocss"> </span><=java.lang.StringIndexOutOfBoundsException: Range [96, 95) out of bounds for length 176
</span>
<div </div><><>
<span class="</span>
<span class="ContSS"><br /><span class="div =/>/>
<span =java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 146
<div"ContSSBlock>
<class"ontSS< ><span class="ContSS"><br /><span class=>
<class"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 143 spanclass=ContSS>br/<""nbsp;/java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 146
<span class="ContSS"><br /><span< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 152
>/>java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 153
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 150
<- java.lang.StringIndexOutOfBoundsException: Range [121, 120) out of bounds for length 148
<div<div
<div class="java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 12
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href=<span>
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 153
<span class="ContSS"><br /><span class="nocss"> </java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 12
java.lang.StringIndexOutOfBoundsException: Range [108, 107) out of bounds for length 153 spanclass"/>
/>/
<class"< "">java.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 178
<>
<div class="ContSSBlock"</div/div
<span">;;/java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 155
class=java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 155
<java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 149
<="br/<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 150
</>/>
<divspanclass""br>span"">nbsp&;/>ajava.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 151
</span>
<div class="ContSSBlock">
<"><br ><pan class""&bsp;n;<span>a href=".X858D2B0D7AE032D569.10- MTXIsEquivalent>/>
<="br >java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 148
<="nocss><java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 141
<.html#X7BC612D2860C582B6910- Homomorphisms>/java.lang.StringIndexOutOfBoundsException: Index 145 out of bounds for length 145
<;java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 143
</div></div>
< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 200
</span>
<div class="ContSSBlock">
<=ContSS<br>java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 153
< ">br >class"java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 157
<span""< <class=n"n;<span class="ContSS"><br /><span class="nocss"> "/< java.lang.StringIndexOutOfBoundsException: Range [40, 38) out of bounds for length 146
<span span classjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 152
<</span>
</div><div
<div class="ContSect"><span class="tocline"><spanjava.lang.StringIndexOutOfBoundsException: Range [22, 19) out of bounds for length 148
</span>
<div class="java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 12
<span class="ContSS"><br /><java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 12 span=>/<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 151
<"><
<span class="ContSS"><br /><spandiv =>
<<span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap69_mj.html<iv ="ContSSBlock>
<class"ContSSspan =CntSS"< >""java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 153
< >classnjava.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 159
</iv>/>
<<div class""><span class="tocline"><span class="nocss"> </span><a href="chap69_mj.java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 7
< class"ontSS>br /span class=n"n;;/>ahref".#java.lang.StringIndexOutOfBoundsException: Range [109, 107) out of bounds for length 147
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><<span class="ContSS"><br /><span class="nocss"> </span><a href="chapX78EE1274825D9E036992 B</a<span>
</span>
<div class="/>/>
<">br >span =nocss>n;n;/span>a ="html#84"6913- SMTX.ubbasisa>/>
<span class="ContSS"><br /><span class="nocss"> </span><a hrefspan class="">br /><spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<class="ContSS"><br /><span class=nocss"> &bsp</pan>ahref"html#">.13- .</a></span>
<span class="span class="ContSS/<java.lang.StringIndexOutOfBoundsException: Range [33, 5) out of bounds for length 143
<< C" >span>n;<<href="#>.- java.lang.StringIndexOutOfBoundsException: Range [144, 142) out of bounds for length 153
<chap69_mjhtmljava.lang.StringIndexOutOfBoundsException: Range [108, 107) out of bounds for length 150
<span class="ContSS"><br /><span class="nocss">&<span class="ContSS"><br /><span class="nocss"> <span><a href="chap69_mj.html#X7BC612D2860C582B">69.10-4 MTX.Homomorphisms</a></span>
<span class="ontSS"< /< class="nocss"&;nbsp;/panahref=.html#"69.138 SMTX.CentMat/>/>
<span class="ContSS"<div""java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 173
<div>div
</div>
<h3>s class" /span ="nocss>;&bsp<span>href>
<" >"&;<java.lang.StringIndexOutOfBoundsException: Range [67, 65) out of bounds for length 144
<p><strong class="ContSS< ><an="java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 146
<p>Please note that, consistently with the span class="ContSS"><br /><span class="nocss">>/>
<p><a id="X85B05BBA78ED7BE2" name=>
<h4>69.1 <span class="Heading<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 151
<p><a id="<span class="ContSS"><br java.lang.StringIndexOutOfBoundsException: Range [45, 38) out of bounds for length 147
<h5>69.1-1 <span class="span java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 151
<div class="func"><table class=< ""> < java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 155
<div class=func>table class=func"width=100pan class""<r/<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 149
</var> which reflect group's action. If the list of generators is empty, the dimension must be given as second argument.</p> "><br/<=">&;java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 139
<dl>
<dt><strong class="divclass="ContSSBlockjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 147
<<pofjava.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 105
</dd>
<sp"ontSS></>java.lang.StringIndexOutOfBoundsException: Range [40, 32) out of bounds for length 144
<ddCjava.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 178
</dd>
<dt> class"ark">code<java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<d><p>The field over which the which the vector<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 138
</dd>
</dl>
<p>Once a module has been created its entries <span class="ContSS"><br /><span class="nocss">&n/<ajava.lang.StringIndexOutOfBoundsException: Range [90, 84) out of bounds for length 141
<div class"java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
<p>creates a MeatAxe module over <var <span class="ContSS"><br /><span class="noc" java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p8233134java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
<h5>69.2-
<<spa"" classunc =00<>tdclass">class"func>‣</(< "rg">mptygens/,<varjava.lang.StringIndexOutOfBoundsException: Range [168, 167) out of bounds for length 286
< (< class""><var> be ) < class=<codethepermutation< =impleMath>(\/>over<"">F/>forthejava.lang.StringIndexOutOfBoundsException: Range [307, 306) out of bounds for length 637
<h5>69.2-p>MeatAxe routines are alevelwith Gaussian elimination.Therefore they do not deal with <strong class="pkg">GAP</strong> modules but essentially with lists of matrices. For java.lang.StringIndexOutOfBoundsException: Range [0, 182) out of bounds for length 0
<div class="func<d<p>A list of matrices which represent a group operation on a finite dimensional row vector space.</p>
<p>Called<dd>
<div class=
<p><code</code> calculates thetensor 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, theydd><>The over which the vector space is defined.</p>
<p<spanclass"ContSS><br /<span class""&; bsp;</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div div class"ontSSBlock">
<>MeatAxeroutinesareaccessedtheglobaljava.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 388
<div class="func"><table class="func<p>Called witha group varclass=Arg"G/>java.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for length 637
<returnslistofmatrix < classclass=>java.lang.StringIndexOutOfBoundsException: Range [26, 24) out of bounds for length 147
<p><a id="X7DF2D6C07D7B09CD" name="java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 0
<569.4-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<divjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
<returns dimension java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 55
<p><a java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
<h5>69.4-3 MTX.Field</h5>
<iv java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 221
<p> class><java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 255
<p><a id="X84D04C7E8423EB5D" name
<h4>69.5 <span class="java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
">=<div class="func"><table class="func
<p>returns degree of the field as extension of p> MeatAxe routines areaccessed via global <=">TX<code,hich whose componentshold the .It ispossible have implementations of a MeatAxe .Each its routinesinan own global variable and assigning <code class="func">MTX</code> to this variable
<p><a id="X791BA495829669C4" name="X791BA495829669C4"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h4>69.6 <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>A module
<p><a id/java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
<h5>69.6-1 MTX.IsIndecomposable</h5>
<div=func<><pThe fieldover thejava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
<preturns varjava.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 71
<p><a id="X781772FD865B9F9C
<h5>69.6-2 MTX
<div class> the java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<">module<<
<div java.lang.StringIndexOutOfBoundsException: Range [0, 10) out of bounds for length 0
<>the components <arclass"rg"module/>given as sums indecomposable components function returns a list
<dl>
<dt><strong class="Mark"><<>< id=X84D04C7E8423EB5D" name="X84D04C7E8423EB5D">
<><p>the index numbers of thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</dd>
<<p>a id="83BEDF86784A6491" name="X83BEDF86784A6491"></a></p>
<java.lang.StringIndexOutOfBoundsException: Range [37, 3) out of bounds for length 48
</dd>
<dt><strong <5>69.-1MTXIsIrreducible</h5>
<dd><p>a"X8233134A81D58DA3"""java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h4>69.7 <span class="java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 0
<p><a id="X80FFB229852B24E9" name="java.lang.StringIndexOutOfBoundsException: Index 42 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;java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
( <var class"Arg"module</var>, <var class="Arg">subspace</var> )</td><td class="tdright">( function )</td></tr></java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p><var class"Arg">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X81326D84845C206F" <p>returns degree of splitting field as <p>Called with a group <var class="Arg">G</var> and a<var class=""Fvar
<div class="func"><table class="func" width="100%"><tr><td class="
<p>eturns a list containing a basis for every submodule.</p>
<><>a of<=rg"java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 413
<h5>69.7-4 MTX.BasesMinimalSubmodules</h5
<class"unc"< class""width="00%"<>td tdleft<
<returns list ofjava.lang.StringIndexOutOfBoundsException: Range [1, 33) out of bounds for length 0
<p>< ="86452777A359195"name="/></java.lang.StringIndexOutOfBoundsException: Range [62, 63) out of bounds for length 62
<<> components of var class="<var>given java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 289
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.p>All MeatAxe routines are accessed via the global variable <code class="func">MTX</code>, which is a record whose components
<p>returns a list of components java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>69.7-6 MTX.BasisRadical</h5java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" java.lang.StringIndexOutOfBoundsException: Range [0, 43) out of bounds for length 0
<>returns basis of the var class"rg>module</ar>./p>
<p><a id=div<div class
<h5>69.7-7 java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 0
<h5>69.43 java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 25
<>java.lang.StringIndexOutOfBoundsException: Range [10, 10) out of bounds for length 0
<p><a id="X7F7FB6687ADE3FD8" name
<h5>h>695spanp<class="subspace<var> java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 371
<div class="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
<p>returns a list of bases ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>69.7-9 MTX.java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
<func> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 p
<p><a id="X7E77F9A97EA855E2" name="X7E77F9A97EA855E2"p> p<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 62
<java.lang.StringIndexOutOfBoundsException: Range [35, 34) out of bounds for length 39
<div .4java.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 42
<div"">able=func 100"<><td class=t>code class=func"&8227;MTXBasesMinimalSubmodules>( var=Arg>< </tdclasstdright>&bsp )/td<tr/><div
<p><a p>returns a list of bases of all minimal submodules.</p>
<h5>69.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0 preturns degreeof splitting fieldas extension theprime .
<p><a id="X7AE730FB81ED86FE><id="" name""><a>/>
<h4>69.8 <java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
<>a<> abasisof radicalof < class"rg>module</><p>
<h5>69<5>69.6- .
<div class="func"><table classdiv =""><tableclass="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MTX.BasisSocle</code
<p<<a id
<ph5>ajava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 413
<h5>69.<returns listof basesof all minimal supermodules of the
<div class<>.-9.java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<divclass="< class=f ="%>tr<tdleft>code=>#227;.nducedActionSubmoduleNB</code>( <var class="Arg">module</varcomputes homogeneous components of < class="Arg">module</var> given as sums of indecomposable components. The function returns a list, each
<p to the of varclass=rg"module</var> on the non-trivial submodule <var class="Arg">sub</var>. In the <code class="code">NB</code> version the basis <var class="Arg">subjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a<<java.lang.StringIndexOutOfBoundsException: Range [5, 8) out of bounds for length 0
<h5>69
< "unc"< class"unc ="00%<r<class
<p>createsa new
<p>The basis <var class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div classdiv =f>=func"=100%>tr>td =tdleft"< class""&8227;MTXNormedBasisAndBaseChange>( var=""></ar <td>td =""(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< class=""< =func"=100%>tr>< =tdleft>code =func"&8227 .nducedActionSubMatrixNB(<var =Arg>/,var class""<)/< java.lang.StringIndexOutOfBoundsException: Range [209, 208) out of bounds for length 265 divf">table class= =100%>trtd =tdleft"<class""&8227;java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>work the same way as the above<5>>69.-2 MTX.InducedActionSubmodule</h5>
<="" name="7B137BE5877A7FA1"<a<p
<h5>69.8-5 MTX.InducedAction</h5>
<div classjava.lang.StringIndexOutOfBoundsException: Range [44, 40) out of bounds for length 45
<p>Computes induced>reates newjava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 344
<p>All these routines return <code <5>69.-2 MTX.
<p><a id="X8040270F791514C8" namejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h4>69.9 <span class="<p>returns the basis of a proper of <var class="Arg">module</var> and <code class="keyw">fail</code> if no proper submodule exists.</p>
<< the way asthe abovefunctions formodules,but take as inputonlya matrix<p
< class=func"idth%><tr>d ="tdleft> classfunc"‣MTXBasisModuleEndomorphisms</ode>( <var =""module</var> )</>td
<p>returns a basis of all module homomorphisms
<>"dleft =func"&227 MTX.ducedAction/ varclass=Arg"<var, <varclass="rg>sub</>,varclass"Arg"><var>] </pjava.lang.StringIndexOutOfBoundsException: Range [221, 11) out of bounds for length 886
<h5>69.9-3 MTX.IsomorphismModules</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MTX.IsomorphismModules</code>( <var class="Arg">module1</var>, <var class="Arg">module2</var> )</td><td class="tdright">( function )</td></tr></table></div
<p>If <var class="Arg">module1</var> java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 0
<p><a id="X8442D91F7C4D724F" <p><a id="X8292535D8533671C" namejava.lang.StringIndexOutOfBoundsException: Range [56, 52) out of bounds for length 62
<h5>=">table div ="< f java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 273
<unc" =100"<<classt"< =func"&8227;MTXModuleAutomorphisms(< =Arg></> <td>tdjava.lang.StringIndexOutOfBoundsException: Range [180, 178) out of bounds for length 235
<p>returns the module automorphisms of <var class="Arg">module</var> (the set of all isomorphisms from <var class="Arg">module</var> <>returnsa bases of <p>
<pdiv =f"<<ble java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 238
<4>910< =Heading> forirreduciblemodules<span>h4
<p>The following are p><a id="X8519B3C486AC8C7=XBCA"<a</>
<>a id"858D2B0D7AE032D5" name""java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>69.10-1 MTX.h5>69.7-6 MTX.BasisRadical>
<"< class=func" width"%"tr>class=dleft< class""&8227; .Equivalent/> class"Arg><varclass"module2</><java.lang.StringIndexOutOfBoundsException: Range [195, 194) out of bounds for length 261
<p>tests two
<><a="java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
<h5>69.10-2 MTX.IsomorphismIrred<java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 0
< class=func"<table class="func" java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>returns an isomorphismp<a id="850324<p><a id="X850324FF7912A541 of <var class=Arg"module/>.pjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
<h5>69.10-3 MTX.h5>69.7-8 MTX.BasesMinimalSupermodules</java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
< "
<>><varshouldbe <var class"Arg>dim1</ar> <<pan class="impleMath>(\imes)</span><var class="Arg">dim2</var> matrix defining a homomorphism from <var class="Arg">module1</var> to <var class="Arg">module2</var>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
="BC612D2860C582B/>/
<h5>>
<div
<>java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 150
<p><a id="X81A6ECB078D4441C
<h5>69.div"
<div class="func"><table class="<p>returns
<p>et<arclass"rg></ar>be =java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p><a id="java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 0
<h4>h>10 java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 34
<> java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 101
<==java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
<h4>98class"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<class""java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 237
<p>returns an invariant bilinear
<p><a id="X7E1F430278A334E1" name
<pjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
<div class="func"><p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns an invariant hermitian (= self-adjoint) java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 42
<div class=""<table class"func"width="100%tr><td classp>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>
<p>returns either the matrix of an invariant quadratic form h5>698-3 MTX.InducedActionFactorModule/
<<>returns invariant hermitian (=selfadjoint sesquilinear form of < class""panewtothe of< =Arg><varonthefactor theproper varclass"rg"/>Ifjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>An error is signalled if <var class="Arg">module</var> is not absolutely irreducible.</p>
<div class="example"><<div class="func"><table class
<span class="GAPprompt">gap&ofan invariant quadratic form of the absolutely java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 0
<span class=div =""table class="" width=100"<tr><dclass="tdleft"><code class="func">&<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣
=>&;spanspan class=GAPinput"Display( MTX.InvariantQuadraticForm(m );<spanjava.lang.StringIndexOutOfBoundsException: Index 112 out of bounds for length 112
.. .. 1 . . .
..1.
. . 11
<span class="GAPprompt">gap></span> <span class="GAPinput">g: .. ..
< =>&; <class">:=NaturalGModule g );/>
<span class="GAPprompt">gap></span> <span class="GAPinput">MTX.p>Computes induced actions on submodules or factor modules and also returns bases action java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 8
ail
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Sp(4, 3);;</span>
<span class=GAPprompt>gtpan class=""gap&t</span><span class=GAPinput"m: NaturalGModule( g );;</span>
<span class="GAPprompt">gap><pan ="">ap&t;<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [30, 5) out of bounds for length 130
truespan class=GAPprompt">gap></span> <span class="GAPinput">m:= NaturalGModule( g );;</span>
</pre></div>
<div class=<><a id"" ="78E60EFE802AEBC1"></a></p>
<p> a basis of the
<p><a <p><a id="X78EE1274825D9E03" <iv ="unc" class="func" width="100%"><tr><td class="tdleft"><codeclass="func">‣ MTX.BasisInOrbit</code>( <var class="Arg">module</var> )</td><td class="tdright">( functionp>eturnsa of underlyingunderlyingvector space of<var class="rg"module/var>which is contained inan of theaction of the generators of module
<h5>69.12-1 SMTX.java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
<div class="<p><a id="X807AEAC7E9B7CFF name="<p>returns submodule.<p
<p>returns the module java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<p><a id="X7EA698517A19D35B" h569.- be a <="Arg"dim1/ar>spanclass"SimpleMath"\\\</>< =Arg"dim2/> matrix defining a <var class"></ar <var =Arg
<h5>69.12-2 SMTX.GoodElementGModule</h5>
<div class="func"><table class="func" <p><a id="X7BC612D2860C582B" name="X7BC612DC582B">/>finds an with minimal possible nullspace dimension if <var class="Arg">module</var> is known to be irreducible.</p>
<p>finds an element
< class"func"<table class="func"width=100">tr>td class="tdleft"><code class="func">‣ SMTX.SortHomGModule</code>( <var class="Arg">module1</var>, <var class="Arg">module2</var>, <var class="Arg">homs</var> )</td><td class=<>Function to sort the output of <code class="code">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>p><a id name"7B426E4679C1AF25">a>/p>
ame"B6092681221D7A<>/p>
<</h5
<preturns(t <var java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
867983"X87E49FCD867983B5"/java.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 62
<h5>69.12-5 SMTX.Setter</h5>
<<pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>returns a setter function<h5>69..11- MTX.nvariantSesquilinearForm</h5>
<p><a id="X808345D784E0AC85" name="X808345D784E0AC85"><<><>returnseither thematrix an invariant quadraticformof theabsolutely irreducible var =Arg"module<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>69.12-7 SMTX.IrreducibilityTest</h5>
<div class="unc>table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SMTX.IrreducibilityTest</code>( <var class="Arg">module</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p irreducibility and sets a subbasis if reducible. It neither sets an irreducibility flag, nor tests it. Thus theroutine alsocan be to a submodule.<p>
<p><a id"X7E692DC97AFB661E<var class="Arg">module</var> is not absolutely irreducible.</p>
<h5>69.12-8 SMTX.AbsoluteIrreducibilityTest</h5>
<div class="func"><table class="func" width="100%<h5>69.12-8 SMTX.AbsoluteIrreducibilityTest</h5>
<p>Tests for absolute irreducibility and sets splitting fieldjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code
<p>returns the basis of
<p><a<returns the basis of a ..11
<h5>69.12-10 SMTX.java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
<div class="func"><table class="func"width="100%"<tr<td class=tdleft"<code class="func">‣ SMTX.MatrixSum</code>( <var class="Arg">matrices1</var>, <var class="java.lang.StringIndexOutOfBoundsException: Range [0, 173) out of bounds for length 101
<reatesthe direct sum of two matrix listsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
<h5>69.12-11 SMTX.CompleteBasis</h5<iv class=func"<<pan =GAPprompt">gap></span> <span class="GAPinput">m:= NaturalGModule( g );;</span>
<div class="func"><table class="func" width= pspan java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 130
<p><a id="java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 4
<followinggetter routines access internal flags.For each , theappropriate setter' nameis prefixed with <code class="code">Set</code>.</p>
<p>The following asisInOrbit
<p><a
<<p> ofa .</>
<div classp>a id=X7ABCD69880772B2D" name="X7ABCD69880772B2D"></a></p>
<p>Basis of
<72B2D"name"7ABCD69880772B2D"></a></p>
<h5
< class=func"><table class="func" width="100%"><tr><td class="tdleft"><code impleMath">\(\)span>module. class"Arg">java.lang.StringIndexOutOfBoundsException: Range [139, 138) out of bounds for length 1039
< algebra elementusedfor java.lang.StringIndexOutOfBoundsException: Range [98, 97) out of bounds for length 102
<< = =X7</>/>
<h5>69.13-3 SMTX.java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0
<div
<>code=func.lgEl</code> (<spanG"java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 120
<div class="func"><table <span class="GAPprompt">gap>
<p>minimal polynomial of <code class=span ="GAPprompt">gap&;/span <spanjava.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 106
<>a java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>69
<p ="java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
<> factor of<codeclass"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><a id="X78A6B95686671067" name="X78A6B95686671067"></a></java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<h5>69.13-8 SMTX.CentMat</h5>
<divclass=func>table class="func" width="100%"><tr><<divclass"func">table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SMTX.SortHomGModule</code>( <var classpminimalpolynomial <code class="unc>SMTX.CentMat</code> (<a href="chap69_mj.html#X78A6B95686671067"><span class="RefLink">69.13-8</span></a<p>Function to sort the output of <code classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<ducibilityTest< ( href".tml#java.lang.StringIndexOutOfBoundsException: Range [175, 174) out of bounds for length 222
<p>
<h5>69.13-9 SMTX.CentMatMinPoly</h5>
00>>tdleft<java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 231
<p>minimal polynomial of <code class="func">java.lang.StringIndexOutOfBoundsException: Range [0, 48) out of bounds for length 7
<div class="java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
<class=">span "java.lang.StringIndexOutOfBoundsException: Range [69, 43) out of bounds for length 3028
¤ Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.0.16Bemerkung:
¤
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.
