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<p><a id="X7A7065EB795C13B3" name="X7A7065EB795C13B3"></a></p>
<div class="ChapSects"><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>
</span>
<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>
</div></div>
<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>
</div></div>
<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>
</div></div>
</div>

<h3>4 <span class="Heading">Functions for calculating with Majorana representations</span></h3>

<p><a id="X7D4417837D9EA89A" name="X7D4417837D9EA89A"></a></p>

<h4>4.1 <span class="Heading">Calculating products</span></h4>

<p><a id="X8115D9E47F601DB2" name="X8115D9E47F601DB2"></a></p>

<h5>4.1-1 MAJORANA_AlgebraProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_AlgebraProduct</code>( <var class="Arg">u</var>, <var class="Arg">v</var>, <var class="Arg">algebraproducts</var>, <var class="Arg">setup</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the algebra product of vectors <var class="Arg">u</var> and <var class="Arg">v</var></p>

<p>The arguments <var class="Arg">u</var> and <var class="Arg">v</var> must be row vectors in sparse matrix format. The arguments <var class="Arg">algebraproducts</var> and <var class="Arg">setup</var> must be the components with these names of a representation as outputted by <code class="func">MajoranaRepresentation</code> (<a href="chap3_mj.html#X7F601CB47EBEAA6A"><span class="RefLink">3.1-1</span></a>). The output is the algebra product of <var class="Arg">u</var> and <var class="Arg">v</var>, also in sparse matrix format.</p>

<p><a id="X8545997F7E52FE1B" name="X8545997F7E52FE1B"></a></p>

<h5>4.1-2 MAJORANA_InnerProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_InnerProduct</code>( <var class="Arg">u</var>, <var class="Arg">v</var>, <var class="Arg">innerproducts</var>, <var class="Arg">setup</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the inner product of vectors <var class="Arg">u</var> and <var class="Arg">v</var></p>

<p>The arguments <var class="Arg">u</var> and <var class="Arg">v</var> must be row vectors in sparse matrix format. The arguments <var class="Arg">innerproducts</var> and <var class="Arg">setup</var> must be the components with these names of a representation as outputted by <code class="func">MajoranaRepresentation</code> (<a href="chap3_mj.html#X7F601CB47EBEAA6A"><span class="RefLink">3.1-1</span></a>). The output is the inner product of <var class="Arg">u</var> and <var class="Arg">v</var>.</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, 1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(rep.setup.coords);</span>
21
<span class="GAPprompt">gap></span> <span class="GAPinput">u := SparseMatrix( 1, 21, [ [ 1 ] ], [ [ 1 ] ], Rationals);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">v := SparseMatrix( 1, 21, [ [ 17 ] ], [ [ 1 ] ], Rationals);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_AlgebraProduct(u, v, rep.algebraproducts, rep.setup);</span>
<a 1 x 21 sparse matrix over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_InnerProduct(u, v, rep.innerproducts, rep.setup);</span>
-1/8192
</pre></div>

<p><a id="X85B9E746807414FF" name="X85B9E746807414FF"></a></p>

<h4>4.2 <span class="Heading">Basic functions</span></h4>

<p><a id="X7B229A8480CD11D3" name="X7B229A8480CD11D3"></a></p>

<h5>4.2-1 MAJORANA_IsComplete</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_IsComplete</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: true is all algebra products have been found, otherwise returns false</p>

<p>Takes a Majorana representation <var class="Arg">rep</var>, as outputted by <code class="func">MajoranaRepresentation</code> (<a href="chap3_mj.html#X7F601CB47EBEAA6A"><span class="RefLink">3.1-1</span></a>). If the representation is complete, that is to say, if the vector space spanned by the basis vectors indexed by the elements in <var class="Arg">rep.setup.coords</var> is closed under the algebra product given by <var class="Arg">rep.algebraproducts</var>, return true. Otherwise, if some products are not known then return false.</p>

<p><a id="X82D60F67810C6918" name="X82D60F67810C6918"></a></p>

<h5>4.2-2 MAJORANA_Dimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_Dimension</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the dimension of the representation <var class="Arg">rep</var> as an integer</p>

<p>Takes a Majorana representation <var class="Arg">rep</var>, as outputted by <code class="func">MajoranaRepresentation</code> (<a href="chap3_mj.html#X7F601CB47EBEAA6A"><span class="RefLink">3.1-1</span></a>) and returns its dimension as a vector space. If the representation is not complete (cf. <code class="func">MAJORANA_IsComplete</code> (<a href="chap4_mj.html#X7B229A8480CD11D3"><span class="RefLink">4.2-1</span></a>) ) then this value might not be the true dimension of the algebra.</p>

<p><a id="X79B8FEDC80748937" name="X79B8FEDC80748937"></a></p>

<h5>4.2-3 MAJORANA_Eigenvectors</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_Eigenvectors</code>( <var class="Arg">index</var>, <var class="Arg">eval</var>, <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a basis of the eigenspace of the axis as position <var class="Arg">index</var> with eigenvalue <var class="Arg">eval</var> as a sparse matrix</p>

<p><a id="X79FD73207DA6125F" name="X79FD73207DA6125F"></a></p>

<h5>4.2-4 MAJORANA_Basis</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_Basis</code>( <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a sparse matrix that gives a basis of the algebra</p>

<p><a id="X7CEDE1DE82115F38" name="X7CEDE1DE82115F38"></a></p>

<h5>4.2-5 MAJORANA_AdjointAction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_AdjointAction</code>( <var class="Arg">axis</var>, <var class="Arg">basis</var>, <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a sparse matrix representing the adjoint action of <var class="Arg">axis</var> on <var class="Arg">basis</var></p>

<p>Takes a Majorana representation <var class="Arg">rep</var>, as outputted by <code class="func">MajoranaRepresentation</code> (<a href="chap3_mj.html#X7F601CB47EBEAA6A"><span class="RefLink">3.1-1</span></a>), a row vector <var class="Arg">axis</var> in sparse matrix format and a set of basis vectors, also in sparse matrix format. Returns a matrix, also in sparse matrix format, that represents the adjoint action of <var class="Arg">axis</var> on <var class="Arg">basis</var>.</p>

<p><a id="X79716CB683A678A7" name="X79716CB683A678A7"></a></p>

<h4>4.3 <span class="Heading">The subalgebra structure</span></h4>

<p><a id="X85DC61987ED3A345" name="X85DC61987ED3A345"></a></p>

<h5>4.3-1 MAJORANA_Subalgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_Subalgebra</code>( <var class="Arg">vecs</var>, <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the subalgebra of the representation <var class="Arg">rep</var> that is generated by <var class="Arg">vecs</var></p>

<p>Takes a Majorana representation <var class="Arg">rep</var>, as outputted by <code class="func">MajoranaRepresentation</code> (<a href="chap3_mj.html#X7F601CB47EBEAA6A"><span class="RefLink">3.1-1</span></a>) and a set of vectors <var class="Arg">vecs</var> in sparse matrix format and returns the subalgebra generated by <var class="Arg">vecs</var>, also in sparse matrix format.</p>

<p><a id="X81AE43B17B7020A3" name="X81AE43B17B7020A3"></a></p>

<h5>4.3-2 MAJORANA_IsJordanAlgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MAJORANA_IsJordanAlgebra</code>( <var class="Arg">subalg</var>, <var class="Arg">rep</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: true if the subalgebra <var class="Arg">subalg</var> is a Jordan algebra, otherwise returns false</p>

<p>Takes a Majorana representation <var class="Arg">rep</var>, as outputted by <code class="func">MajoranaRepresentation</code> (<a href="chap3_mj.html#X7F601CB47EBEAA6A"><span class="RefLink">3.1-1</span></a>) and a subalgebra <var class="Arg">subalg</var> of rep. If this subalgebra is a Jordan algebra then function returns true, otherwise returns false.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := 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">MAJORANA_IsComplete(rep);</span>
false
<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">MAJORANA_Dimension(rep);</span>
46
<span class="GAPprompt">gap></span> <span class="GAPinput">basis := MAJORANA_Basis(rep);</span>
<a 46 x 61 sparse matrix over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">subalg := MAJORANA_Subalgebra(basis, rep);</span>
<a 46 x 61 sparse matrix over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">MAJORANA_IsJordanAlgebra(subalg, rep);</span>
false
</pre></div>


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