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<title>GAP (MajoranaAlgebras) - Chapter 7: Signed Permutations</title>
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<p><a id="X847F187C7DA356AA" name="X847F187C7DA356AA"></a></p>
<div class="ChapSects"><a href="chap7_mj.html#X847F187C7DA356AA">7 <span class="Heading">Signed Permutations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X828866AB7AE9C0CC">7.1 <span class="Heading">Different Representations</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7_mj.html#X8598044F844BF533">7.2 <span class="Heading">Low-Level Descriptions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7BFE3679813E5D08">7.2-1 IsSignedPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X83BC19B8812D211C">7.2-2 ListSignedPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7CAA502279CC9A8C">7.2-3 ListSignedPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X7C6AB5A27B2DA2F1">7.2-4 SignedPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X856041438008B08B">7.2-5 NewSignedPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X82C6D31583C213E1">7.2-6 NewSignedPerm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X87BCE2B280486669">7.2-7 IsSignedPermRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X8531225C7C224C62">7.2-8 IsSignedPermListRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X8033A20F7A7A60CB">7.2-9 OnPosPoints</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X84709BF07811FECB">7.2-10 LargestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X824068057D141DE0">7.2-11 RandomSignedPermList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7_mj.html#X85E05F9283E5B099">7.2-12 RandomSignedPerm</a></span>
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</div>

<h3>7 <span class="Heading">Signed Permutations</span></h3>

<p>We provide <em>signed permutations</em>, that is permutations that can additionally change the sign of their result.</p>

<p>Assume <span class="SimpleMath">\(n \in \mathbb{N}\)</span>, then a signed permutation on <span class="SimpleMath">\(n\)</span> points is a permutation <span class="SimpleMath">\(\pi\)</span> on <span class="SimpleMath">\(\{ 1 \ldots n \}\)</span> together with signs <span class="SimpleMath">\(sgn : \{ 1 .. n \} \rightarrow \{-1,1\}\)</span>. A signed permutation on <span class="SimpleMath">\(n\)</span> points acts on the set <span class="SimpleMath">\(\{ -n \ldots 1, 1 \ldots n \}\)</span> by <span class="SimpleMath">\( \omega ^ { (\pi, sgn) } = sgn(\omega)\cdot sgn(|\omega|^\pi) \cdot (|\omega|^\pi) \)</span>.</p>

<p>We provide two representations of signed permutations, one as a list of images <code class="func">IsSignedPermListRep</code> (<a href="chap7_mj.html#X8531225C7C224C62"><span class="RefLink">7.2-8</span></a>) and one formed as pair of a permutation and a sign map <code class="func">IsSignedPermRep</code> (<a href="chap7_mj.html#X87BCE2B280486669"><span class="RefLink">7.2-7</span></a>). Our benchmarks indicate that a list of images is the better representation, and hence this is the default.</p>

<p>To get started with signed permutations consider the following example</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s := SignedPerm([2,-1]);</span>
<signed permutation in list rep>
<span class="GAPprompt">gap></span> <span class="GAPinput">1 ^ s;</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">2 ^ s;</span>
-1
<span class="GAPprompt">gap></span> <span class="GAPinput">OnPoints(2, s);</span>
-1
</pre></div>

<p>One can form groups out of signed permutations</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r := SignedPerm([-1,3,-2,4]);; t := SignedPerm([3,1,4,2]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group(r,t);</span>
<group with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(G);</span>
32
<span class="GAPprompt">gap></span> <span class="GAPinput">Orbit(G, 1, OnPoints);</span>
[ 1, -1, 3, -3, -2, 4, 2, -4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(G, 1, OnPoints);</span>
<group of size 4 with 9 generators>
</pre></div>

<p>Note that currently the package does not make an effort to exploit the special structure of signed permutation groups as permutation groups.</p>

<p><a id="X828866AB7AE9C0CC" name="X828866AB7AE9C0CC"></a></p>

<h4>7.1 <span class="Heading">Different Representations</span></h4>

<p>To create signed permutations in the different representations, we provide a constructor.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">r := NewSignedPerm(IsSignedPermRep, [-1,3,-2,4]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t := SignedPerm(IsSignedPermRep, [3,1,4,2]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group(r,t);</span>
<group with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(G);</span>
32
<span class="GAPprompt">gap></span> <span class="GAPinput">r := NewSignedPerm(IsSignedPermListRep, [-1,3,-2,4]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t := SignedPerm(IsSignedPermListRep, [3,1,4,2]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := Group(r,t);</span>
<group with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(G);</span>
32
</pre></div>

<p><a id="X8598044F844BF533" name="X8598044F844BF533"></a></p>

<h4>7.2 <span class="Heading">Low-Level Descriptions</span></h4>

<p><a id="X7BFE3679813E5D08" name="X7BFE3679813E5D08"></a></p>

<h5>7.2-1 IsSignedPerm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSignedPerm</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>Category of signed permutations</p>

<p><a id="X83BC19B8812D211C" name="X83BC19B8812D211C"></a></p>

<h5>7.2-2 ListSignedPerm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ListSignedPerm</code>( <var class="Arg">perm</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Convert a signed permutation into a list of images, equivalent to List([1..LargestMovedPoint(s)], x -> x^s);</p>

<p><a id="X7CAA502279CC9A8C" name="X7CAA502279CC9A8C"></a></p>

<h5>7.2-3 ListSignedPerm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ListSignedPerm</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Convert a signed permutation to a list of images of length <var class="Arg">len</var>. Arguments perm, len</p>

<p><a id="X7C6AB5A27B2DA2F1" name="X7C6AB5A27B2DA2F1"></a></p>

<h5>7.2-4 SignedPerm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SignedPerm</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given a list of signed images create a signed permutation object in <code class="func">IsSignedPermListRep</code> (<a href="chap7_mj.html#X8531225C7C224C62"><span class="RefLink">7.2-8</span></a>).</p>

<p><a id="X856041438008B08B" name="X856041438008B08B"></a></p>

<h5>7.2-5 NewSignedPerm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NewSignedPerm</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var> )</td><td class="tdright">( constructor )</td></tr></table></div>
<p><a id="X82C6D31583C213E1" name="X82C6D31583C213E1"></a></p>

<h5>7.2-6 NewSignedPerm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NewSignedPerm</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var>, <var class="Arg">arg3</var> )</td><td class="tdright">( constructor )</td></tr></table></div>
<p><a id="X87BCE2B280486669" name="X87BCE2B280486669"></a></p>

<h5>7.2-7 IsSignedPermRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSignedPermRep</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>Representation of signed permutations as a permutation and a vector of signs.</p>

<p><a id="X8531225C7C224C62" name="X8531225C7C224C62"></a></p>

<h5>7.2-8 IsSignedPermListRep</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsSignedPermListRep</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>Representation of signed permutations as a list of signed images</p>

<p><a id="X8033A20F7A7A60CB" name="X8033A20F7A7A60CB"></a></p>

<h5>7.2-9 OnPosPoints</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; OnPosPoints</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Only act as a permutation on <span class="SimpleMath">\(\{ 1\ldots n\}\)</span></p>

<p><a id="X84709BF07811FECB" name="X84709BF07811FECB"></a></p>

<h5>7.2-10 LargestMovedPoint</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; LargestMovedPoint</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The largest point that is moved by the signed permutation, where moving includes changing the sign.</p>

<p><a id="X824068057D141DE0" name="X824068057D141DE0"></a></p>

<h5>7.2-11 RandomSignedPermList</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RandomSignedPermList</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Create a random list of images that can be used to create a signed permutation.</p>

<p><a id="X85E05F9283E5B099" name="X85E05F9283E5B099"></a></p>

<h5>7.2-12 RandomSignedPerm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RandomSignedPerm</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Create a random signed permutation</p>


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