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<h1>MajoranaAlgebras</h1>


<h2>A package for constructing Majorana algebras and representations</h2>

<p>
    1.5.2</p>

<p>
    7 July 2024
  </p>

</div>
<p><b>
    Markus Pfeiffer



  </b>
<br />Email: <span class="URL"><a href="mailto:markus.pfeiffer@st-andrews.ac.uk">markus.pfeiffer@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://markusp.morphism.de/">https://markusp.morphism.de/</a></span>
</p><p><b>
    Madeleine Whybrow




  </b>
<br />Email: <span class="URL"><a href="mailto:mlw10@ic.ac.uk">mlw10@ic.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://madeleinewhybrow.wordpress.com">https://madeleinewhybrow.wordpress.com</a></span>
<br />Address: <br />Department of Mathematics, Imperial College, South Kensington, SW7 2AZ<br />
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2018 by Markus Pfeiffer and Madeleine Whybrow</p>

<p><strong class="pkg">MajoranaAlgebras</strong> package is free software; you can redistribute it and/or modify it under the terms of the <span class="URL"><a href="https://www.fsf.org/licenses/gpl.html">GNU General Public License</a></span> as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.</p>

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<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X840C49F287F4C385">1.1 <span class="Heading">A quick guide</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X81C7E9D57969211C">1.2 <span class="Heading">Understanding the output</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X79EF8C857EA0E22B">1.3 <span class="Heading">Info levels</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X824D39948644EB03">1.3-1 InfoMajorana</a></span>
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</div>
<div class="ContChap"><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>
</span>
<div class="ContSSBlock">
<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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</div>
<div class="ContChap"><a href="chap3_mj.html#X7B5CADA77CC27BB0">3 <span class="Heading">Majorana representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X80ABA2918548E108">3.1 <span class="Heading">The main function</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F601CB47EBEAA6A">3.1-1 MajoranaRepresentation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X85C2A4B97D4B47D0">3.2 <span class="Heading">The n-closed function</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8155D0F98405BD1E">3.2-1 NClosedMajoranaRepresentation</a></span>
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</div>
<div class="ContChap"><a href="chap4_mj.html#X7A7065EB795C13B3">4 <span class="Heading">Functions for calculating with Majorana representations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7D4417837D9EA89A">4.1 <span class="Heading">Calculating products</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8115D9E47F601DB2">4.1-1 MAJORANA_AlgebraProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8545997F7E52FE1B">4.1-2 MAJORANA_InnerProduct</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X85B9E746807414FF">4.2 <span class="Heading">Basic functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7B229A8480CD11D3">4.2-1 MAJORANA_IsComplete</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82D60F67810C6918">4.2-2 MAJORANA_Dimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79B8FEDC80748937">4.2-3 MAJORANA_Eigenvectors</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79FD73207DA6125F">4.2-4 MAJORANA_Basis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7CEDE1DE82115F38">4.2-5 MAJORANA_AdjointAction</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X79716CB683A678A7">4.3 <span class="Heading">The subalgebra structure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X85DC61987ED3A345">4.3-1 MAJORANA_Subalgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81AE43B17B7020A3">4.3-2 MAJORANA_IsJordanAlgebra</a></span>
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<div class="ContChap"><a href="chap5_mj.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_mj.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_mj.html#X867D4BF17C730633">5.1-1 MajoranaAlgebraTest</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.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_mj.html#X7AD4DBC27DB539EF">5.2-1 MAJORANA_TestFrobeniusForm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E3499067808B4A0">5.2-2 MAJORANA_TestInnerProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7CB2F52386F07B68">5.2-3 MAJORANA_TestAxiomM2</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X85DFEF167A236627">5.2-4 MAJORANA_TestPrimitivity</a></span>
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<div class="ContChap"><a href="chap6_mj.html#X7E210A267E85D052">6 <span class="Heading">Orbital Structures</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7A489A5D79DA9E5C">6.1 <span class="Heading">Examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7DE994FE7D595E0C">6.1-1 IsOrbitalStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X86F56EAD87C74596">6.1-2 OrbitalStructure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X87395C8C8592FF00">6.1-3 OS_OrbitRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X79BD0DBA7B72D4C9">6.1-4 OS_CanonisingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X815676077CFC8CF4">6.1-5 OS_CanonisingElementAndRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7CC301797B101011">6.1-6 OS_StabilizerOf</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7F2D94D9870E46B9">6.1-7 OrbitalRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X8595D3EA85AA992F">6.1-8 AllOrbitalRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7CDB09DF87DD88C5">6.1-9 OrbitalCanonizingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X79D2C53084F773B2">6.1-10 OrbitalCanonizingElementInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X81A62A257FAC1BC7">6.1-11 OrbitalTransversalIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7FDDCFB078F55048">6.1-12 UnorderedOrbitalRepresentative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X79E001B18591F1AE">6.1-13 AllUnorderedOrbitalRepresentatives</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X78ABE8E9844AC143">6.1-14 UnorderedOrbitalTransversalIterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7FA8E4BE7F8F47DD">6.1-15 UnorderedOrbitalCanonizingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X82CAF97C7AE60FA4">6.1-16 UnorderedOrbitalCanonizingElementInverse</a></span>
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<div class="ContChap"><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>
</span>
</div>
<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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