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<div class="ContSect"><span class="tocline"><span class="nocss">#  NaturalDualitySymplectic
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<h3>1 <span class="Heading">Introduction</span></h3>

<>This   introduction#most of thecode isrecycledfrom  operations nddocumentedthere

<p><a id="X8389AD927B74BA4A" name="X8389AD927B74BA4A"></a></p>

<h4>1.1 <span class="Heading">Overview</span></h4>

<p>The package <strong class="pkg">WPE</strong> (<b><u>W</u></b>reath <b><u>P</b></u>roduct <b><u>E</b></u>lements) provides methods to work with elements 

<java.lang.StringIndexOutOfBoundsException: Range [8, 2) out of bounds for length 11

<p>Additionally, this representation may have computational benefits over other representations. Note, that just by loading the package <strong class="pkg">WPE</strong# operationi ajava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 2

<p>In particular, this package java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 15

<, the ofall <class"SimpleMath"\886\640)/span conjugacy 


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">LoadPackage("WPE");;</span>
<pan="APprompt"gap&;s <class">:=22);/pan>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := AlternatingGroup(9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := WreathProduct(K,  ines: [;
< classclass="GAPprompt">gap&;/><anclass="C : ConjugacyClassesG;</java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(C);</span>
886640
</pre></div1 (,)- (,)by inverse projection (twinerprefunof Isomorphism..)

<p><a id="X7950ED88879EE36F" name="X7950ED88879EE36F"></a></p>

<h4>1.2 <        lines =[[]id],id2PGO(,)-PsP,  inversePlucker (twinerprefunof NaturalDualitySymplectic

<p>One of the two main goals of the package is to provide the User with tools to conduct <b>intuitive research</b> in wreath products of groups on the computer.</p>

<p>In this section we present an example session which demonstrates how we can access the generic representation of a wreath product. As noted in the introduction, no additional setup is required if one wants to benefit java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

< we  wreath<span class""\G \{}5 w textrm}7\span>( ahref="chap2_mj.html#7DF2AEBC8518FFA4> class""2java.lang.StringIndexOutOfBoundsException: Index 188 out of bounds for length 71

<p class=""for a symplecticGQ and  boolean"java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
\vert G \vert =List(x-*)
\]</p>


<div class="[ IsClassicalGQIsBool ,
<span class="GAPprompt">gap></span> <span class="GAPinput">K := AlternatingGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := SymmetricGroup(7);;</span>
<span class=GAPprompt>&;/span><class="> : (K H);/pan>
<permutation  4generators&t
</pre></div>

<p>Now we construct an isomorphism<p>Now we construct an isomorphism  data , cw, ,q ,


<div class="example"><pre>
<spanipts[]j[]=-ptsi[];       func,formw,delta ,coll1 hom,ones,can_formjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
<span class="GAPprompt" = java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 50
<group of size 14108774400000000 with 4 generators>
/>/divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<p>Let us compare how <strong class="pkg">GAP</strong> displays elements of <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(W\)</span> respectively. Elements


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g := (1,13,vec[]6]: -vec[[1;
<span class="GAPprompt">></span> <span class="GAPinput">        (6,31,21,7,35,25,9,33,23,8,34,24,10,32,22)</span>
<span class="GAPprompt">></span> <span class="GAPinput">        (18,19,20);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g in G;</span>
true
</pre></div>

<p>Elements of <span class="SimpleMath">\(W\)</span>end;


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">w := g ^ iso;</span>
< wreath
<span class="GAPprompt">gap></span> <span class="GAPinput">Print(w);</span>
[ (1,3,4,2,5), (2,5)(3,4), (), (3,4,5), (1,2)(4,5), (), (), (1,3)(2,7,5) ]
<class=GAPprompt>gapgt;/pan> <span class="GAPinput">Display()<span
       1           2       3      4         5       6   7       top     
( (1,3,4,2,5), (2,5)(3, SetIsBijective(map, true )
</pre></div>

<p>Furthermore, we can display and access each component easily with the provided commands.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">BaseComponentOfWreathProductElement(w, 2);</span>
(2,5)(3,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">TopComponentOfWreathProductElement(w);</span>
(,)(,75java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
</pre></div>

<p><a id="X83C9D5E6846B299E" name="X83C9D5E6846B299E"></a></p        local matnewmatf;

<h5>1.2-1 <span class="Heading">The Power of Component-wise Representation</span></h5>

<p>This component-wise representation is often exactly the one that we encounter in research on wreath products. Thus having it available on the computer greatly sharpens our intuition. We can make very non-trivial statements by looking at the components of such an element, and for the case of the element <span class="SimpleMath">\(w\)</span> even without

<newmat := ]java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m := Order(w);</span>
30
<span class="GAPprompt">gap></span> <span class="GAPinput">First( [1 .. m], k -> IsOne(TopComponentOfWreathProductElement(w ^ k)) );</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(w ^ 6);</span>
       1            2            3       4        5       6        7       top
( (1,2,3,5,4), (1,4,5,2,3), (1,2,3,5,4), (),newmat2 :PluckerCoordinatesmat[],1]{[1.];
</pre></div>

<  newmat[]: (mat[],mat[])[1.5];


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(Territory(w));</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints(H);</span>
7
</pre></div>

<p>On the other hand, all such elements must have at least three non-trivial base components, since the wreath cycle decomposition of <span class="SimpleMath">\(w\)</span> (see <a href="chap2_mj.html#X83A8F3308644ACFA"><span class="RefLink">2.2</span></a


<div class="example"><pre>
<span class="GAPprompt">gap></span>        newmat[1]:PluckerCoordinates[mat[]+[],[1+[4]){1.] - newmat[] - newmat[;
3
</pre></div>

<p>Moreover, for each integer <span class="SimpleMath">\(k\)</span> with <span class="SimpleMath">\(3 \leq k \leq 6\)</span> there exists at least one conjugate element with exactly <span class="SimpleMath">\(k\)        newmat: perm5*newmat *perm5(frob^-1);

<p><a id="X7D96C25680814773" name="X7D96C25680814773"></a></p>

<h4>1.3 <span class="Heading">Efficient Computing</span></h4>

<p>One        newmat = [2.5}{[.5};

<p>In this section we present a highlight showcase of computational problems that benefit from the generic representation. As noted in the introduction, no additional setup is required if one wants to benefit from the optimizations for computationsreturn ProjElWithFrobcwinv *newmat * (frob-1,,)java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68

<p>We only give a summary of some computational problems that now become approachable on the computer, and include examples for such computations in Chapter <a href="chap3_mj.html#X81932F777898AD72"><span class="RefLink">3</span></a> containing extensive <strong java.lang.StringIndexOutOfBoundsException: Index 268 out of bounds for length 0

<p>In the following let <span class="SimpleMath">\(G = K \wr H\)</span> be a wreath product of finite groups, where <span class="SimpleMath">\(H \leq \textrm{Sym}(m)\)</span>. Further let <span class="SimpleMath">\(x, y \in P = K \wr \textrm{Sym}(m)\)</span> local newmat,at,ivec,,newvecz,zprime,


<dl>
<k>Conjugacy Problem<strong><dt
<dd><p>Solve the conjugacy problem for <span class="SimpleMath">\(x\)</span> and <span class="SimpleMath">\(y\)</span> over <span class="SimpleMath">\(G\)</span>, i.e. decide whether there exists <span class="SimpleMath">\(c \in G\)</span> with <java.lang.StringIndexOutOfBoundsException: Index 249 out of bounds for length 28

</dd>
<"Mark>Classes </</dt>
<dd><p>Enumerate representatives of all conjugacy classes of elements of <span class="SimpleMath">\(G\)</span>, i.e. return elements <span class="SimpleMath">\(g_1, \dots, g_\ell\)</span> such that <span class="SimpleMath">\(g_1^G, \dots, g_\ell^G\)</span> are the conjugacy classes of <span class="SimpleMath

</dd>
<dt><strong class=" Centralizer /strong<dt>
<dd><p>Compute the centralizer of <span class="SimpleMath">\(x\)</span> in <span class="SimpleMath">\(G\)</span>, i.e. compute a generating set of <span class="SimpleMath">\(C_G(x)\)</span>.</p>

</dd>
<dt><strong class="Mark"> Cycle Index Polynomial</strong></dt>
<dd><p>Compute the cycle index polynomial of <span class="SimpleMath">\(G\)</span> either for the imprimitive action or the product action.</p>

</dd>
</dl>

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