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<title>GAP (MajoranaAlgebras) - Chapter 2: Shapes of a Majorana representation</title>
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<div class="ChapSects"><a href="chap2_mj.html#X879CF99D7A9D77ED">2 <span class="Heading">Shapes of a Majorana representation</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X809C94E778B04047">2.1 <span class="Heading">The shapes functions</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7AEAA41E813BB13C">2.1-1 ShapesOfMajoranaRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7873676E79F8D08B">2.1-2 ShapesOfMajoranaRepresentationAxiomM8</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8740DD5585713A26">2.1-3 MAJORANA_IsSixTranspositionGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7922BE3C7961F33D">2.1-4 MAJORANA_RemoveDuplicateShapes</a></span>
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<h3>2 <span class="Heading">Shapes of a Majorana representation</span></h3>

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<h4>2.1 <span class="Heading">The shapes functions</span></h4>

<p><a id="X7AEAA41E813BB13C" name="X7AEAA41E813BB13C"></a></p>

<h5>2.1-1 ShapesOfMajoranaRepresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ShapesOfMajoranaRepresentation</code>( <var class="Arg">G</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record with a component <var class="Arg">shapes</var></p>

<p>Takes a group <var class="Arg">G</var> and a <var class="Arg">G</var>-invariant set of generating involutions <var class="Arg">T</var>. Returns a list of possible shapes of a Majorana Representation of the form <var class="Arg">(G,T,V)</var> that is stored in the <var class="Arg">shapes</var> component of the output.</p>

<p><a id="X7873676E79F8D08B" name="X7873676E79F8D08B"></a></p>

<h5>2.1-2 ShapesOfMajoranaRepresentationAxiomM8</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ShapesOfMajoranaRepresentationAxiomM8</code>( <var class="Arg">G</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record with a component <var class="Arg">shapes</var></p>

<p>Performs exactly the same function as <code class="func">ShapesOfMajoranaRepresentation</code> (<a href="chap2_mj.html#X7AEAA41E813BB13C"><span class="RefLink">2.1-1</span></a>) but gives only those shapes at obey axiom M8. That is to say, we additionally assume that if <span class="SimpleMath">\(t,s \in T\)</span> such that <span class="SimpleMath">\(|ts| = 2\)</span> then the dihedral subalgebra <span class="SimpleMath">\(\langle \langle a_t, a_s \rangle \rangle\)</span> is of type <span class="SimpleMath">\(2A\)</span> if and only if <span class="SimpleMath">\(ts \in T\)</span> (and otherwise is of type <span class="SimpleMath">\(2B\)</span>).</p>

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<h5>2.1-3 MAJORANA_IsSixTranspositionGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MAJORANA_IsSixTranspositionGroup</code>( <var class="Arg">G</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: true if <var class="Arg">(G,T)</var> is a 6-transposition group, otherwise returns false</p>

<p>For a group <var class="Arg">G</var> and a subset <var class="Arg">T</var> of <var class="Arg">G</var>, returns true if all of the following conditions are satisfied: *<var class="Arg">T</var> is a set of involutions that generate <var class="Arg">G</var>; *<var class="Arg">T</var> is closed under conjugation by <var class="Arg">G</var>; *the order of the product of two elements of <var class="Arg">T</var> is at most 6.</p>

<p><a id="X7922BE3C7961F33D" name="X7922BE3C7961F33D"></a></p>

<h5>2.1-4 MAJORANA_RemoveDuplicateShapes</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MAJORANA_RemoveDuplicateShapes</code>( <var class="Arg">input</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>If an automorphism of the group <var class="Arg">G</var> stabilises the set <var class="Arg">T</var> then it induces an action on the pairs of elements of <var class="Arg">T</var> and therefore on the shapes of a possible Majorana representation of the form <var class="Arg">(G,T,V)</var>. If one shape is mapped to another in this way then their corresponding algebras must be isomorphic.</p>

<p>This function takes the record <var class="Arg">input</var> as produced by the function <code class="func">ShapesOfMajoranaRepresentation</code> (<a href="chap2_mj.html#X7AEAA41E813BB13C"><span class="RefLink">2.1-1</span></a>) or <code class="func">ShapesOfMajoranaRepresentationAxiomM8</code> (<a href="chap2_mj.html#X7873676E79F8D08B"><span class="RefLink">2.1-2</span></a>) and replaces <var class="Arg">input.shapes</var> with a list of shapes such that no two can be mapped to each other by an automorphism of <var class="Arg">G</var>.</p>


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