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<?xml version="1.0" encoding="UTF-8"?>


<Chapter Label="Chapter_Introduction">
<Heading>Introduction</Heading>

<Section Label="Chapter_Introduction_Section_A_quick_guide">
<Heading>A quick guide</Heading>


In order to construct the Majorana representation of a group <A>G</A> with respect
to a set of involutions <A>T</A>, you must first call
<Ref Func="ShapesOfMajoranaRepresentation"/>.

<Example><![CDATA[
gap> G := AlternatingGroup(5);;
gap> T := AsList( ConjugacyClass(G, (1,2)(3,4)));;
gap> input := ShapesOfMajoranaRepresentation(G,T);;
]]></Example>


This function outputs a record. One component of this record is labelled <A>shapes</A>
and contains the possible shapes of a Majorana representation of the form <Math>(G,T,V)</Math>.

<Example><![CDATA[
gap> input.shapes;
[ [ "1A", "2B", "5A", "3C", "5A" ], [ "1A", "2B", "5A", "3A", "5A" ],
 [ "1A", "2A", "5A", "3C", "5A" ], [ "1A", "2A", "5A", "3A", "5A" ] ]
]]></Example>


To construct the Majorana representation with shape at position <A>i</A> of this list,
call the function <Ref Func="MajoranaRepresentation"/> with <A>input</A> as its first argument
and <A>i</A> as its second.

<Example><![CDATA[
gap> rep := MajoranaRepresentation(input, 1);;
gap> rep.shape;
[ "1A", "2B", "5A", "3C", "5A" ]
]]></Example>


There are then a number of functions (see <Ref Chap="Chapter_funcs"/>) that one case use
on the (potentially incomplete) Majorana representation that this function has outputted.

<Example><![CDATA[
gap> MAJORANA_IsComplete(rep);
true
gap> MAJORANA_Dimension(rep);
21
]]></Example>


If an incomplete algebra is returned then the function
<Ref Func="NClosedMajoranaRepresentation"/> can be used to attempt to find the
3-closed part of the algebra.

<Example><![CDATA[
gap> G := AlternatingGroup(5);;
gap> T := AsList( ConjugacyClass(G, (1,2)(3,4)));;
gap> input := ShapesOfMajoranaRepresentation(G,T);;
gap> input.shapes;
[ [ "1A", "2B", "5A", "3C", "5A" ], [ "1A", "2B", "5A", "3A", "5A" ],
 [ "1A", "2A", "5A", "3C", "5A" ], [ "1A", "2A", "5A", "3A", "5A" ] ]
gap> rep := MajoranaRepresentation(input, 2);;
gap> MAJORANA_IsComplete(rep);
false
gap> NClosedMajoranaRepresentation(rep);;
gap> MAJORANA_IsComplete(rep);
true
gap> MAJORANA_Dimension(rep);
46
]]></Example>


</Section>

<Section Label="Chapter_Introduction_Section_Understanding_the_output">
<Heading>Understanding the output</Heading>


<Emph>Note that all vectors and matrices are given in sparse matrix format, as provided
by the GAP package <A>Gauss</A>. If <A>mat</A> is such a matrix then the integers in
<A>mat!.indices</A> refer to a spanning set of the algebra indexed by the list
<A>rep.setup.coords</A>. The list <A>mat!.entries</A> give their corresponding coefficients.</Emph>

The function <Ref Func="MajoranaRepresentation"/> outputs a record that encodes the
information required to perform calculations in the Majorana representation that
has been calculated. The record contains the following components.

<List>
 <C>group</C>
 <Item> The group <A>G</A>, as inputted by the user.</Item>
 <C>involutions</C>
 <Item> The set <A>T</A>, as inputted by the user.</Item>
 <C>shape</C>
 <Item> The shape of the representation, as chosen by the user in the input
 of <Ref Func="MajoranaRepresentation"/>.</Item>
 <C>eigenvalues</C>
 <Item> A list whose values give the eigenvalues of the adjoint action of the
 axes of the algebra. In this case, it must be equal to (or a subset of)
 <A>[0, 1/4, 1/32]</A>. Note that we omit the eigenvalue 1 as we assume the algebra
 to be primitive.</Item>
 <C>axioms</C>
 <Item> A string representing the axiomatic setting of the algebra's construction,
 potentially chosen by the user with the <A>options</A> record in the input of
 <Ref Func="MajoranaRepresentation"/>.</Item>
 <C>setup</C>
 <Item> Is itself a record, containing (among others) the following components.
 <List>
 <C>coords</C>
 <Item> A list whose elements index a spanning set of the algebra.</Item>
 <C>nullspace</C>
 <Item> Again a record such that <A>nullspace.vectors</A> gives a basis
 of the nullspace of the algebra (as the elements <A>rep.setup.coords</A>
 are not necessarily linearly independent).</Item>
 <C>orbitreps</C>
 <Item> A list of indices giving the representatives of the orbits of
 the action of the group <A>G</A> on <A>T</A>.</Item>
 <C>pairreps</C>
 <Item> A list of pairs of indices giving representatives of the orbitals
 of the action of the group <A>G</A> on <A>rep.setup.coords</A>.</Item>
 </List>
 </Item>
 <C>algebraproducts</C>
 <Item> A list where the vector at position <A>i</A> denotes the algebra product
 of the two spanning set vectors whose indices (in <A>rep.setup.coords</A>) are
 given by <A>rep.setup.pairreps[i]</A>. If the <A>i</A>th entry is set to <A>false</A>
 then this algebra product has not yet been found and the algebra is incomplete.</Item>
 <C>innerproducts</C>
 <Item> Performs the same role as <A>algebraproducts</A> except that, instead of vectors,
 the entries are rational numbers denoting the inner product between two spanning set
 vectors.</Item>
 <C>evecs</C>
 <Item> A list where if <A>i</A> is contained in <A>rep.setup.orbitreps</A> then
 <A>rep.evecs[i]</A> is bound to a record. This record has components <A>"ev"</A>
 where <A>ev</A> is an eigenvalue contained in <A>rep.eigenvalues</A>. This component gives a basis for the eigenspace of the axis corresponding to
 <A>rep.involutions[i]</A> with eigenvalue <A>ev</A>.</Item>
</List>

</Section>

<Section Label="Chapter_Introduction_Section_Info_levels">
<Heading>Info levels</Heading>


<ManSection>
 <InfoClass Name="InfoMajorana" />
<Description>
The default info level of <A>InfoMajorana</A> is 0. No information
is printed at this level. If the info level is at least 10 then <A>Success</A> is printed if the
algorithm has produced a complete Majorana algebra, otherwise <A>Fail</A>
is printed. If the info level is at least 20 then more information is printed
about the progress of the algorithm, up to a maximum info level of 100.
</Description>
</ManSection>

</Section>

</Chapter>

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