<p>This chapter is a collection of tutorials that show how to work with wreath products in <strong class="pkg">GAP</strong> in conjunction with the package <strong class="pkg">WPE</strong>.</p>
<p>In this section we present an example session which demonstrates how we can create wreath products elements by specifying its components.</p>
<p>In the following we will work with the wreath product <span class=/
<div class="example"><pre>
<span <p>This chapter is a collectionof tutorials that show how with wreathproductsin <strongclass=""GAP/ conjunctionwiththe <strong class="pkg"><strong>/p
<span class="GAPprompt">gap></span> <span class="GAPinput">K := AlternatingGroup spanclass"GAPprompt">gapg;<span>< class=G"> : SymmetricGroup(4);/java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93 ""g;</pan <pan class="APinput> : WreathProductK );</span>
<permutation group of size 311040000 with 10 generators>
</pre></div>
<p>The resulting group <span class="SimpleMath">G</span> is embedded into a symmetric group on <span class="SimpleMath">5 ⋅ 4 = 20</span> points via the imprimitive action of the wreath product. The size of the group java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>Suppose we would like to input thelpermutation 311040000 with 10
<p class=> G A(5|4⋅(4| ^ 24=311040000.>
<p>as an element of <span class="SimpleMath">G</span>. The method <code class="code">WreathProductElementList</code> is the preferred way to create a wreath product element by specifying its components. Note that we first specify the four base components and at the end the top component as the last entry.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gList := [ (1,5,2,4,3), (1,3,5,2,4), (1,5,3,4,2), (1,4,5), (1,3)(2,4) ];;</span>
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(15,311,5,2,14)4,3)618,,20)(7,19,,17)(,16)
<span class="GAPprompt">gap></span> <span class="GAPinput">g in G;</span>
true
</pre></div>
<p>On the other hand, the method <code class="code">ListWreathProductElement</code> can be used to get a list containing the components of a wreath product element.</p>
<p>The package author has implemented the methods <code class="func">ListWreathProductElement</code> (<a href="../.span =""gap&t< class="GAPinput">g in G;</span>
<p>java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
<p>In this section we present an example session which demonstrates how we can display wreath product elements in an intuitive way. Wreathspan class=GAPprompt>gt;</span> <span class="GAPinput"> * (1,2,4) ^ Embedding(G, 5);</span>
<p>Suppose we are given some element <span class="SimpleMath">g<<span class="APprompt">gap></span> <span class="GAPinput">hList := ListWreathProductElement(G, h);</span>
<p>First we translate the element <span class="SimpleMath">g</span> into a generic wreath product element <span class="SimpleMath">w</span>. <strong class="pkg">GAP</strong> uses <code class="code">ViewObj</code> to print <span class="SimpleMath">w</span> in a compressed form.</p>
<div java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
<span class="GAPprompt"gap&;<span> <span class="GAPinput">w := g ^ iso;</span>
<span class="GAPprompt>gap&;</pan> <pan class="GAPinput">K := AlternatingGroup(5);;</span>
</pre>/div>
<p>If we want to print this element in a"machine-readable" way, we could use one of the span class="GAPprompt">gap></span> <span class="GAPinput">iso := IsomorphismWreathProduct(G);;
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print(w);</span>
[ ,3,5,2,4), 1,,3,,2), 14,),(1,,2,)]
class"APinput> =ListWreathProductElementjava.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<></span<span class"GAPinput"> =ListWreathProductElement(, g);/pan
true
</java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
<p=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 82
<=ejava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
<classGAPpromptgt<spanspan =GAPinput>(w;/java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80 1
,5,2,4) 15342,(,,) (, )
</pre></div>
<p>There are many display options [ (1,5,2,4,3), (1,3,5,2,4), (1,5,3,4,2), (12,)]
<div class="example"><pre>
<spanclass=GAPprompt>gap&t<span><span class=GAPinput>Display(,rec(horizontal )<span 1 (1,,,4,) 2: (1,3,5,2,4) 3: (1,5,3,4,2) 4: (1,4,5)
top: (,3,,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(w);</span> 1234 top
((,,,,), (3,,,4), (,5,,42) (,4,);(1,3,,4) java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
</pre></div>
<p>We can also change the global display options via the following command.</p>
<p>All changes to the global behaviour can be reverted to the default behaviour via the following command.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 16 1234 top
( (1,5,2,4,3), (1,3,5,2,4), (1,5,3,4,2), (1,4,5); (1,3,2,4) )
<pre/div
<p>But sometimes, it is sufficient to just look at some components of a wreath product element. We can directly use the list representation to access the components on a low-level or we can use high-level functions on wreath product elements instead.</p>
<div class="example"><pre>
<span p>We can change globaldisplay options followingcommand</p>
(1,5,3,4,2)
<span class="GAPprompt">gap></span> span"APprompt">apgt<spanspan class""SetDisplayOptionsForWreathProductElements :=false)<span
java.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4
=gt< span"java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 112
(1,3,2,4)
<span class="GAPprompt">gapjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
ue
</pre></div>
<><="8345CC25855C9406"nameX8345CC25855C9406a></>
<omputingjava.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 70
<p>As noted in the introduction, no additional setup is required if one wants to benefit from the java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 61
<p>We include in the following sections examples for each computational problem listed in <a href="chap1.html#X7D96C25680814773"><span class="RefLink">1.3</span></a>. For all such examples we fix the following wreath product.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K := "GAPprompt">gap></span> <span class="GAPinput">a := BaseComponentOfWreathProductElement
<java.lang.StringIndexOutOfBoundsException: Range [22, 5) out of bounds for length 120
<)</span>
<permutation group of size 768000000 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">P := WreathProduct
<permutation group of size 46080000000 with 4 generators>
<span class="GAPprompt">gap></<span class="GAPprompt">gap></span="APinput">b = L[5];</span>
true
< "gapg;/span><"> =IsomorphismWreathProduct();<span>
<
<p>java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<=""java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
<="GAPprompt">gap>spanspan" = (,23,12,6,424,3,,5,21,15,10,2,2514,7)3,2211,)(6,,,28(17,27,19,26)(18,29java.lang.StringIndexOutOfBoundsException: Range [162, 161) out of bounds for length 162
<span class="GAPprompt">gap></span> <span class="GAPinput">y := (1,12,26,8,3,14,28,7,2,13,27,10,5,11,30,6)(4,15,29,9)(16,23,20,22)(17,24,19,21)(18,25);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(x ^ iso);</span> 123456 top
( (1,3,2,5), (1,4,5,sclass=GAPprompt>gapgt<spanclass"GAPinput> :([ 1234,, 1243) ]); ()/>
<span""gapgt;/>spanclass"java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 86 123456G: java.lang.StringIndexOutOfBoundsException: Range [82, 80) out of bounds for length 94
( (1,2,3,4,5), (), (1,5,4,3,2),java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 63
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 4
true
<spanpre
true
<pre/>
<p><a id="X8017E4AF7EBB72D2" name="java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 0
<p>We continue the <strong class="pkg">GAP</strong> session from Section <a href="chap3.html#X8345CC25855C9406"><span class="RefLink">3.3</span135,(4,5,,(1,,4,),(1534,15,32), (1,,4,) 13)(,)java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
<p12,,,) ) 15,,,2,(,,,,) (,2(,5, 132;(,3,,2)(
<e"
<span class="java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 4
fail
</pre></div>
<p>The
<div
<""gtspanspan G" P, x,y;/java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
(2,5)(3,4)(6,8,9,7)(11,29,25)(12,26,21,13,28,22,15,27,24,14,30,23)
< "gt/><class=GAPinput"(^)<span 123456 top
(25)(,) (,,,) 1452,(,(13,,5) 2453; 36,5) )
<span class="GAPprompt">gap></span> <span class="GAPinput">x ^ c = y;</span>
true
</pre></div>
<p>We see, that indeed these elements are conjugate over the larger wreath
<p>Enumerate java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 0
<p2()6,897(1129,)(,,21,1327,30,23)
<p>To compute in <strong class="pkg">GAP</strong> the conjugacy span class="GAPprompt">gap></span> <span class="GAPinput">Display(c^iso);</span>
<"java.lang.StringIndexOutOfBoundsException: Range [22, 19) out of bounds for length 26
<class&;/ span =": (G);/span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(CC);</span> 1960
</pre></div>
<p>We see that there are <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>We can compute additional information about a conjugacy class on the go. For java.lang.StringIndexOutOfBoundsException: Index 87 out of bounds for length 0
<p>Compute the <p>To access the representof this , do following.</p>
<Wecontinue <class"kg"GAP<//strongfrom "hap3.html#X8345CC25855C9406"><span class="RefLink">3.3</span></a>. In particular recall the definition of the isomorphism <code class="code">iso</code>.</p>
classpkgGAP/> centralizer anina group < class">/>(< ="././.doc/ref/chap35.html#X7A2BF4527E08803C"><span class="RefLink">Reference: centraliser</span></a>).</p>
<p>To compute in <( (), (), (), (1,4,5,2), 132,);())
<div class="example"><pre>
<span class=" 4,2 ,52354),(), 1,,,) ) ( java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58 1/72*x_1^6+1/12*><a id="X87A59E867A83A798X87A59E867A83A798<ap
+1/12*x_2<h4>. span Heading Index java.lang.StringIndexOutOfBoundsException: Range [58, 52) out of bounds for length 64
</pre></div>
<p>For example, the java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
<pjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<p>If we want to considerp compute in strong""GAP< indexofjava.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 314
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G_prod := WreathProductProductAction(K, H);;</span>
<span class="GAPpromptjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 9
<span class="GAPprompt">gap></span> < =G> : (G_prod);/span> 1/72*x_1^9+1/6*x_1^3*x_2^3+1/8*x_1*x_2^4+1/4*x_1*x_4^2+1/9*x_3^3+1/3*x_3*x_6
</pre></div>
<p>However, we do not need to create the wreath product in order to java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 42
<div class="example"><pre>
<span class="<>For example, the second monomial <code class="code">1/12*x_1^4*x_2</code> tells us that there are exactly <span java.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 0
<;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">c2 := CycleIndexWreathProductProductAction(K, H);;</span>
<panclass="GAPprompt">gap></span> <span class="GAPinput">c_prod = c2;</span>
true
</pre></div>
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.
