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<title>GAP (MajoranaAlgebras) - Chapter 5: Functions for testing Majorana representations</title>
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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap4.html">[Previous Chapter]</a>    <a href="chap6.html">[Next Chapter]</a>   </div>

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<p><a id="X7CB84E157B2F902D" name="X7CB84E157B2F902D"></a></p>
<div class="ChapSects"><a href="chap5.html#X7CB84E157B2F902D">5 <span class="Heading">Functions for testing Majorana representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X80ABA2918548E108">5.1 <span class="Heading">The main function</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X867D4BF17C730633">5.1-1 MajoranaAlgebraTest</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X87C3D1B984960984">5.2 <span class="Heading">Other functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7AD4DBC27DB539EF">5.2-1 MAJORANA_TestFrobeniusForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E3499067808B4A0">5.2-2 MAJORANA_TestInnerProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7CB2F52386F07B68">5.2-3 MAJORANA_TestAxiomM2</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X85DFEF167A236627">5.2-4 MAJORANA_TestPrimitivity</a></span>
</div></div>
</div>

<h3>5 <span class="Heading">Functions for testing Majorana representations</span></h3>

<p>The output of the function <code class="func">MajoranaRepresentation</code> (<a href="chap3.html#X7F601CB47EBEAA6A"><span class="RefLink">3.1-1</span></a>) is guaranteed to be a commutative algebra generated by idempotents whose eigenspaces obey the Majorana fusion law. To check that the output is truly a Majorana algebra, one must also check that</p>


<ul>
<li><p>the inner product is a Frobenius form (see <code class="func">MAJORANA_TestFrobeniusForm</code> (<a href="chap5.html#X7AD4DBC27DB539EF"><span class="RefLink">5.2-1</span></a>));</p>

</li>
<li><p>the inner product is positive definite (see <code class="func">MAJORANA_TestInnerProduct</code> (<a href="chap5.html#X7E3499067808B4A0"><span class="RefLink">5.2-2</span></a>));</p>

</li>
<li><p>the inner product obeys axiom M2 (Norton's inequality) (see MAJORANA_TestAxiomM2 (5.2-3));



</li>
<li><p>the algebra is primitive (see <code class="func">MAJORANA_TestPrimitivity</code> (<a href="chap5.html#X85DFEF167A236627"><span class="RefLink">5.2-4</span></a>)).</p>

</li>
</ul>
<p><a id="X80ABA2918548E108" name="X80ABA2918548E108"></a></p>

<h4>5.1 <span class="Heading">The main function</span></h4>

<p><a id="X867D4BF17C730633" name="X867D4BF17C730633"></a></p>

<h5>5.1-1 MajoranaAlgebraTest</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MajoranaAlgebraTest</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the algebra given by <var class="Arg">rep</var> is indeed a Majorana algebra.</p>

<p>Note: does not check that the algebra obeys axiom M2 (Norton's inequality), this can be separately tested using MAJORANA_TestAxiomM2 (5.2-3).



<p><a id="X87C3D1B984960984" name="X87C3D1B984960984"></a></p>

<h4>5.2 <span class="Heading">Other functions</span></h4>

<p><a id="X7AD4DBC27DB539EF" name="X7AD4DBC27DB539EF"></a></p>

<h5>5.2-1 MAJORANA_TestFrobeniusForm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_TestFrobeniusForm</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the inner product given by <var class="Arg">rep.innerproducts</var> is a Frobenius form, otherwise returns false.</p>

<p><a id="X7E3499067808B4A0" name="X7E3499067808B4A0"></a></p>

<h5>5.2-2 MAJORANA_TestInnerProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_TestInnerProduct</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the inner product given by <var class="Arg">rep.innerproducts</var> is positive definite, otherwise returns false.</p>

<p><a id="X7CB2F52386F07B68" name="X7CB2F52386F07B68"></a></p>

<h5>5.2-3 MAJORANA_TestAxiomM2</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_TestAxiomM2</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the inner product given by <var class="Arg">rep.innerproducts</var> obeys axiom M2 (Norton's inequality), otherwise returns false.



<p><a id="X85DFEF167A236627" name="X85DFEF167A236627"></a></p>

<h5>5.2-4 MAJORANA_TestPrimitivity</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_TestPrimitivity</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <var class="Arg">true</var> if the 1-eigenspaces of all axes are 1-dimensional, otherwise returns false.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := AlternatingGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">T := AsList( ConjugacyClass(G, (1,2)(3,4)));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">input := ShapesOfMajoranaRepresentation(G,T);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rep := MajoranaRepresentation(input, 2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">NClosedMajoranaRepresentation(rep);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_IsComplete(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MajoranaAlgebraTest(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_TestFrobeniusForm(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_TestInnerProduct(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_TestAxiomM2(rep);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_TestPrimitivity(rep);</span>
true
</pre></div>


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